IV.B Economic capital for a multi-line life insurance …...IV.B Economic capital for a multi-line...

109
Stochastic Processes and Modeling in Financial Reporting and Capital Assessment IV. Case Studies Section IV 1 DRAFT FOR DISCUSSION PURPOSES ONLY. THIS DOCUMENT IS PROTECTED UNDER INTERNATIONAL COPYRIGHT LAWS. COPYING IS ILLEGAL. THIS IS A PRELIMINARY DRAFT AND IS STILL UNDER REVIEW BY MILLIMAN AND IAA. IV.B Economic capital for a multi-line life insurance company An economic-capital (EC) model is one component of an insurance company’s larger risk management framework. It is designed to identify and manage, through determination of an appropriate level of capital, the unique risks that threaten the solvency of an insurance company. In this sense, it is a powerful tool at the disposal of actuaries and risk managers against catastrophic risks (or “tail risks”) facing a company. This case study introduces the fundamental concepts of an EC framework, illustrates generation of stochastic scenarios for various risk factors, and presents EC results developed for a life and health insurance company using sophisticated stochastic models. It demonstrates how the stochastic techniques discussed in this monograph can be applied in a risk- management setting. IV.B.1 The case-Study company: Background on XYZ Life Insurance Company This case study presents results for an EC analysis of XYZ Life Insurance Company, a large multi-line life and health insurance company writing primarily par individual life, variable annuity, and group health business. The generic company could be headquartered anywhere in the world— the EC analysis discussed in this case study is not unique to any particular country, although certain references are made to U.S. stock market indices. IV.B.2 Fundamental concepts of an EC framework Economic capital is defined as the amount of capital required to cover losses at a given confidence level over a specified period of time. For example, a company may wish to be 99.5% confident that it will remain solvent over a 20-year time horizon. When implementing an EC framework, companies must first decide on: A risk metric Companies must define how they will measure risk. In other words, how will they quantify the variability, or risk, of the outcomes under a stochastic projection. Commonly used risk metrics are Value at Risk (VaR) or Conditional Tail Expectation (CTE or Tail-VaR), which are discussed in greater detail elsewhere in this monograph. A confidence level to target Users of EC generally examine high levels of confidence. After all, EC models are intended to reveal catastrophic events that could materially, adversely affect operating results. To understand these catastrophic events, it is necessary to examine the tails of the risk-factor distributions (e.g., extreme interest-rate, default, or mortality events). Common confidence levels for users of EC are 99.5%, 99.9% and 99.95%.

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Stochastic Processes and Modeling in Financial Reporting and Capital Assessment IV. Case Studies

Section IV 1 DRAFT FOR DISCUSSION PURPOSES ONLY. THIS DOCUMENT IS PROTECTED UNDER INTERNATIONAL COPYRIGHT LAWS. COPYING IS ILLEGAL. THIS IS A PRELIMINARY DRAFT AND IS STILL UNDER REVIEW BY MILLIMAN AND IAA.

IV.B Economic capital for a multi-line life insurance company

An economic-capital (EC) model is one component of an insurance company’s larger risk management framework. It is designed to identify and manage, through determination of an appropriate level of capital, the unique risks that threaten the solvency of an insurance company. In this sense, it is a powerful tool at the disposal of actuaries and risk managers against catastrophic risks (or “tail risks”) facing a company. This case study introduces the fundamental concepts of an EC framework, illustrates generation of stochastic scenarios for various risk factors, and presents EC results developed for a life and health insurance company using sophisticated stochastic models. It demonstrates how the stochastic techniques discussed in this monograph can be applied in a risk-management setting.

IV.B.1 The case-Study company: Background on XYZ Life Insurance Company

This case study presents results for an EC analysis of XYZ Life Insurance Company, a large multi-line life and health insurance company writing primarily par individual life, variable annuity, and group health business. The generic company could be headquartered anywhere in the world—the EC analysis discussed in this case study is not unique to any particular country, although certain references are made to U.S. stock market indices.

IV.B.2 Fundamental concepts of an EC framework

Economic capital is defined as the amount of capital required to cover losses at a given confidence level over a specified period of time. For example, a company may wish to be 99.5% confident that it will remain solvent over a 20-year time horizon. When implementing an EC framework, companies must first decide on:

A risk metric

Companies must define how they will measure risk. In other words, how will they quantify the variability, or risk, of the outcomes under a stochastic projection. Commonly used risk metrics are Value at Risk (VaR) or Conditional Tail Expectation (CTE or Tail-VaR), which are discussed in greater detail elsewhere in this monograph.

A confidence level to target

Users of EC generally examine high levels of confidence. After all, EC models are intended to reveal catastrophic events that could materially, adversely affect operating results. To understand these catastrophic events, it is necessary to examine the tails of the risk-factor distributions (e.g., extreme interest-rate, default, or mortality events). Common confidence levels for users of EC are 99.5%, 99.9% and 99.95%.

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A time horizon

For quick implementation, some companies will examine very short time horizons (e.g., one to five years). In this case, companies are looking for short-term shocks in risk factors that can have sudden, catastrophic consequences (e.g., what impact would a pandemic have on my term insurance block of business over the next year?). More commonly, companies perform longer-term projections, with time horizons ranging from 20 to 60 years.

Projection techniques

Given the objective of an EC analysis—to examine tail outcomes of company-specific risk factors—stochastic modeling is required. Alternative techniques, such as stress testing or scenario testing, are generally inadequate for EC work, although they may play a supporting role in helping to identify critical risk factors.

IV.B.3 Modeling risks

After addressing the issues above, a company must analyze which risks are most relevant to its business. EC models are intended to be company-specific, which means they must thoughtfully analyze which risks have the potential to materially impact the company’s profitability. Often, one can get a sense for which risks are material through the use of stress tests. If the present value of profits is sensitive to a particular assumption (or risk factor) then it is a candidate for inclusion in the EC model. Alternatively, insight into relevant risk factors can often be obtained by interviewing the actuary with responsibility for a particular line of business.

Commonly modeled risk factors for an EC framework include:

Risk-free interest rates and credit spreads

Asset defaults

Liquidity

Mortality

Morbidity

Lapses

Currency exchange rates

Mix of business (if future business is included in the model)

Strategic risk (often modeled as a load to EC, rather than stochastically)

Operational risk (often modeled as a load to EC, rather than stochastically)

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Risk analysis for XYZ Life revealed that the risks shown in Figure 17XX are relevant and will be included in the EC model:

Figure 17XX: Risks Modeled for XYZ Life Insurance Company Economic Capital

IV.B.4 General methodology

IV.B.4.a Risk Metrics

We examined two risk metrics:

Present value of future profits (PVFP)

PVFP is the discounted value of all future profits and losses over the entire projection period.

Greatest present value of accumulated loss (GPVL)

GPVL is the present value of all net losses from the projection start date to the end of the year, when the present value of the loss is greatest. If there are never any losses (and therefore never any accumulated losses) the GPVL is zero. GPVL does not allow future gains to mitigate the current need for capital.

IV.B.4.b Confidence level

We used a CTE-99 approach for the business risks, in which capital is set at the average of the worst 1% of losses over 30 years. For strategic and operational failure risks, we measured the 99.5% probability of solvency over five years.

Table 1 - Risks Modeled for XYZ life Insurance Company Economic Capital

IndividualIndividual Variable Fixed Asset Disability Group Capital/ Consolidated

Life Annuity Annuity Management Income Health Surplus CorporateInterest Rate X X X X X

Mortality X

Lapse X X

Market/Equity Returns X X X

Morbidity X X

Currency Exchange Rates X

Credit X

Strategic X

Operational X

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IV.B.4.c Time horizon

We used a long-term runoff approach, with a 30-year projection horizon, in determining required EC, consistent with developments in C3-Phase II, variable-annuity Commissioners Annuity Reserve Valuation Method (VACARVM), and a principles-based approach to reserves and capital in the United States and with European standards such as Embedded Value (EV) and Solvency II. The long-term runoff approach we used for liabilities is different from the one-year Economic Balance Sheet approach used by many European companies, which looks at the fair value of assets and liabilities at the end of one year. To develop a fair value of liabilities, long-term modeling must be performed as in the long-term approach.

IV.B.4.d Projection techniques

We started with the company’s deterministic Cash Flow Testing (CFT) or EV models (when such models existed) or developed new models (when CFT or EV models did not already exist). We then performed the following:

1. Using best-estimate assumptions, we developed a deterministic “baseline” 30-year profit projection for each line of business (LOB) and each risk factor.

2. For each risk to be modeled, we generated 1,000 stochastic scenarios (the generation of these scenarios is discussed in greater detail later in this case study).

3. The 1,000 scenarios for each risk were projected independently. For example, the 1,000 mortality scenarios were run separately from the 1,000 interest-rate scenarios. This tells us the amount of EC required for each risk factor, but does not tell us the aggregate amount of EC because it ignores the correlation of risk factors. The total EC is not the sum of the indicated EC from each risk factor; total EC must reflect the interaction of all risk factors simultaneously. However, running risk factors independently does provide the risk manager with insight into which individual risks pose the greatest threat to company profitability

4. A decision was next needed as to how to generate a large number of scenarios, within the limits of projection run times, to produce a meaningful number and reasonable result. Two approaches could have been taken. First, we could have run every scenario with every risk. Since we had six generators and risks, this would have equated to 1,0006 or 1,000,000,000,000,000,000 (one quintillion) runs. Since this was not plausible, we took a second approach, which we will call “synthetic” scenarios. Under this approach, we utilized a PVFP measure (the difference between the present value of future profits for a scenario and the baseline) from each of the 1,000 scenarios for each independent risk. Each one of these was assigned a number of one to 1,000. We then generated 1,500,000 random integers

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between one and 1,000 (inclusive) using Microsoft Excel. This would generate 250,000 choices of scenarios from each risk (a grid of 250,000 by six). To verify the distribution, we reviewed each random number to ensure that it occurred roughly 1,500 times. Then, using the grid of scenarios, we performed a look-up of the present value and summed across all the risks. The sums were then ranked in ascending order so that the worst combined independent scenarios appeared first. It was felt that this produced a reasonable number of scenarios that showed poor results and would reflect the essence of EC, that is, setting aside enough capital to cover a very high level of possible outcomes.

5. To account for the interaction/correlation of various risk factors, we developed the “worst 2,500” scenarios from Step 4, and combined all risk factors (except for credit, operational, and strategic risks) into a single set of scenarios. We then ran a deterministic projection using these scenarios, which were originally generated stochastically. The “worst 2,500” scenario reflect the interaction of all risk factors in order to more accurately reflect how risk affects the company in the real world (i.e., the real-world company faces all risks simultaneously; it does not face ten separate risks independently).

Scenario reduction techniques were used to aid in selecting the adverse scenarios, which drive the tail of the distribution of results and hence drive EC.

The Excel file ____.xls illustrates how individual risk scenarios were combined to

create the 2,500 worst scenarios [still being developed].

Credit defaults were projected at the aggregate company level. Gains and losses on assets, backing capital, and surplus were projected separately from LOB profits. As discussed above, strategic risk and operational risk were projected separately at the 99.5% confidence interval over five years.

IV.B.5 Scenario generation

This section discusses how to generate 1,000 stochastic scenarios for the following risk factors included in the EC model:

Economic scenarios, including:

● Interest rates

● Equity returns

● Spot currency exchange rates

Credit/default

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Mortality

Morbidity

Lapses

Operational risk

Strategic risk

In several cases, Microsoft Excel workbooks are provided, giving the reader an opportunity to practice generating stochastic scenarios for use in an EC setting.

IV.B.5.a Economic-scenario generator

This section summarizes the economic-scenario generator. The generator produces robust and internally consistent economic scenarios of equity index returns, yield curves, and spot exchange rates. An Excel file is provided that illustrates the concepts for interest rates and equity returns [still

under construction].

Equity model

The underlying model for the equity index is a stochastic volatility model, namely the model of Heston (1993). The dynamics follow the equations below:

(4.B.1)

where

S(t) is the index level at time t

σ2t is the underlying variance of the equity index process at time t

f(t) is the expected return for the current period

q is the continuously compounded dividend yield over the same time interval

ρ is the correlation between return innovations and variance innovations

κ is strength of mean reversion of volatility

β is equilibrium variance

ν is volatility of volatility

dz1 and dz2 are Brownian motion processes

⎥⎦

⎤⎢⎣

⎥⎥⎦

⎢⎢⎣

⎡ −+⎥⎥⎥

⎢⎢⎢

−−=⎥⎦

⎤⎢⎣

2

12

2

2

20

1

)(2

)()ln(dzdz

vdtqtfS

d t

t

t

t

t ρρσσβκ

σ

σ

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The volatility of the equity index is itself a stochastic process with its own volatility and mean reversion.

The advantages of this form relative to the commonly used lognormal model is that stochastic volatility models allow generated scenarios to be consistent with the following stylized facts of the historical time series of the S&P 500 index:

● The underlying volatility of the S&P 500 changes over time, but reverts to some long-term mean

● Changes in the underlying volatility are negatively related to index movements over time

● Distributions of equity price movements exhibit fatter tails and skew relative to lognormal distributions

Interest-rate model

The model for interest rates posits the yield curve as having three stochastic components, namely the level, the slope, and the curvature of the forward-rate curve. This framework allows for generated scenarios to reflect the following features of historical yield curve movements:

● The points on the yield curve generally tend to move together

● Volatility of the short end of the yield curve is generally higher than the relative volatility of the long end of the yield curve

● Yield curves tend to flatten as they rise and steepen as rates drop

● Inversions may occur at a frequency of 5% to 20%

Spot-exchange-rate model

The model for spot exchange rates is a lognormal model that maintains the principle of interest-rate parity by using the difference between the domestic and foreign interest-rate curves as the drift component.

Model parameterization

All model parameters and coefficients were estimated using historical data. We used 20 years of monthly data of S&P 500 index levels, risk-free yield curves, and the VIX index to estimate parameters.

We parameterized the scenario generator assuming a start date of September 30, 2006, and generated 1,000 30-year scenarios projecting forward from that date.

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The following sections present the parameters used in the model to produce the scenarios.

Starting interest rates (foreign and domestic)

The risk-free yield curves as of September 30, 2006, were the following:

Maturity Constant Maturity Risk-Free Rates (Years) U.S. Eurozone Great Britain Switzerland Japan

1 4.91% 3.60% 5.02% 2.17% 0.33% 2 4.86% 3.59% 4.92% 2.17% 0.66% 3 4.80% 3.59% 4.84% 2.18% 0.76% 5 4.70% 3.63% 4.73% 2.28% 1.14% 7 4.60% 3.65% 4.63% 2.37% 1.37% 10 4.57% 3.71% 4.52% 2.38% 1.68% 20 4.84% 3.91% 4.34% 2.50% 2.18% 30 5.03% 3.90% 4.10% 2.53% 2.44%

This translates to starting forward rates in our three-factor model of:

Maturity Starting Rate (Years) U.S. Eurozone* Great Britain* Switzerland* Japan*

1 4.85% 3.53% 4.96% 2.16% 0.33% 2 4.21% 3.66% 4.23% 2.65% 2.27% 30 5.80% 3.95% 3.41% 2.69% 3.16%

*Only the domestic rate was simulated stochastically. The foreign interest-rate curves were determined by adding (or subtracting) the

starting spread to the domestic curves.

For the domestic curve, assuming a target slope and curvature of 1.52% and -2.30%, to produce a target yield-curve shape consistent with the historically observed average shape, and assuming our target one-year rate stays at 4.91%, our target long-term par semiannual yield curve is the following:

Maturity Constant Maturity Risk-Free Rates(Years) U.S.

1 4.91% 2 5.04% 3 5.16% 5 5.40% 7 5.63% 10 5.80% 20 5.87% 30 5.83%

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Interest-rate parameters

Maturity Starting

Rate Volatility (Monthly)

Mean Reversion

Target

Mean Reversion Strength

Short Rate 1 4.85% 5.77% 3.97% 0.80%Medium Rate (Slope Parameters) 7 4.21% 0.35% 1.52% 1.22%Long Rate (Curvature Parameters) 30 5.80% 0.78% -2.30% 2.75%

Duration parameters (for bond-fund calculations)

The domestic-bond duration parameter used in the bond-fund calculations was based on the duration of the SBBIG bond index. For the foreign bond funds, we used market-value weighted averages of XYZ’s foreign bond funds as of September 30, 2006.

U.S. Eurozone Great Britain Switzerland Japan

Duration 4.0 yrs 4.6 yrs 14.5 yrs 6.9 yrs 9.6 yrs

Correlation matrix

Short Rate

Yield Curve Slope

Yield Curve Curvature S&P 500

Short Rate 1.000 -0.730 0.537 -0.178 Yield Curve Slope -0.730 1.000 -0.929 0.077 Yield Curve Curvature 0.537 -0.929 1.000 -0.035 S&P 500 -0.178 0.077 -0.035 1.000

Equity returns

S&P 500 parameters:

Initial Total

Return

Mean Reversion Strength

Target Total

Return Dividend Yield Drift

Process 8.50% 100.00% 8.50% 0.00%

Initial

Variance

Mean Reversion Strength

Target Variance

Volatility of Variance

Correlation (between underlying

and variance) Equity

Volatility 2.96% 260.00% 2.96% 25.00% -43.75%

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We set the dividend-yield assumption to zero to generate our equity index as a total return index.

Other correlated indices: S&P 500 Russell 2000 NASDAQ 100 EAFE S&P 500 1.000 0.796 0.826 0.782 Russell 2000 0.796 1.000 0.873 0.784 NASDAQ 100 0.826 0.873 1.000 0.764 EAFE 0.782 0.784 0.764 1.000 Volatilities 25.0 % 21.3 % 21.3 %

Currency returns

Currency-exchange-rate correlations and volatilities were developed based on the historical correlations and volatilities observed in the foreign exchange rates. The following matrix shows the correlations and volatilities of historical weekly data from January 1999 to September 2006.

To reflect currency hedging, the volatility of the currency return is reduced by a factor based on the proportion of currency hedged and the historical hedge effectiveness.

FX

EUR/USDFX

GBP/USDFX

CHF/USD FX

JPY/USD FX EUR/USD 1.000 0.708 0.940 0.379 FX GBP/USD 0.708 1.000 0.679 0.361 FX CHF/USD 0.940 0.679 1.000 0.401 FX JPY/USD 0.379 0.361 0.401 1.000 Volatilities 10.0 % 8.4 % 10.6 % 10.2 %

In addition to generating the four correlated equity-index returns, we used the interest rates and currency returns to generate domestic money-market returns and domestic and foreign bond returns.

Money market

We assumed the money-market account earned the one-year forward rate at the beginning of the period.

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Domestic bond

The domestic bond fund takes an assumed duration as a parameter. It is simulated by purchasing a semiannual coupon bond with the given duration and holding it for one period, then rolling the position over into a new bond with the given duration parameter.

Foreign bond

The foreign bond fund also takes an assumed duration as a parameter. It is simulated by purchasing a semiannual coupon bond with the given duration and holding it for one period, then rolling the position over into a new bond with the given duration parameter. However, the par coupon at issue and the revaluation of the bond a week later are based on a set of foreign-interest-rate curves. The foreign-interest-rate curves are the stochastically generated domestic-interest-rate curves plus a given spread. The spread is the difference between the foreign and domestic term structures provided at time 0 and stays constant throughout the projection. This gives the value of the foreign bond in the local currency. To calculate the value of the foreign bond in the domestic currency, we multiply the value of the foreign bond in the local currency by the spot exchange rate.

The foreign bond fund value in the local currency for currency f and time t, Ff,t , is therefore

Ff,t = Ff,t-1 * a) (4.B.2)where a) is the end-of-period bond value in local currency based on change in foreign-

interest-rate curve

The spot exchange rate is calculated as

])())(2/exp[( 2/12,,,, ttrree fffTfTdTftTf Δ+Δ−−×=Δ+ σφσ (4.B.3)

The foreign-bond-fund value in the domestic currency is

Fd,t = Ff,t * ef,t (4.B.4)

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IV.B.5.b Credit-risk model

Credit risk is a large driver of corporate bond prices. Valuation and risk management of these instruments, and derivatives written on these instruments, require models that reflect the risk of the counterparty defaulting on a payment. There are many different ways to model credit risk. In this section, we describe a model for simulating default events to generate scenarios that represent the number of defaults in a certain year for a portfolio of credit-risky bonds. The model analyzed in this section is similar to the model used for analyzing the EC requirement for Basel II. The first subsection will describe the dynamics of the model, and the second subsection presents key simplifications that can be made to make the model more tractable. The final subsection summarizes and presents further improvements.

Description of the model

We selected the Asymptotic Single Risk Factor (ASRF; Vasicek, 1987) model for generating heavy-tailed credit loss distributions to estimate the risk associated with a portfolio of bonds. The model assumes that the borrower’s end-of-period state (default or no default) is driven by a normally distributed variable that describes the hypothetical firm’s asset return. A default event, upgrade event, or downgrade event occurs if this variable tiR , falls below or rises above a specific level. Therefore, firm i defaults if and only if:

Diti CR ,, <

downgrades if and only if:

DGiti CR ,, <

and upgrades if and only if:

UGiti CR ,, >

where Ci is the point on the standard normal distribution associated with the desired probability of a particular event. For example, if we want the probability of default to be 50%, we set Ci,DG equal to zero, so that N(Ci,DG) = 50%.

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The probability of upgrade, downgrade, or default for firm i can be given by

)()Pr( ,,, EiEiiEi CNCRW =<= (4.B.4)

where N(.) is the cumulative standard normal distribution function.

We approximate this probability by taking it from the agency-rating transition matrix for a given ratings class. To determine the value of C that results in a default, upgrade, or downgrade, we can invert the normal cumulative density function.

)( ,1

, EiEi WNC −= (4.B.5)

To simulate the return for each period, we use the following equation

tiititi XR ,, 1 εββ −+= (4.B.6)

This equation shows that the asset return on each firm can be described as a systematic factor X, and a firm-specific factor epsilon.

Potential simplifications to the model

In order to calculate the return for each firm in the above equations, we need to estimate the betas and also the correlations for each firm with the market parameter. However, if the portfolio is large enough and contains bonds from different ratings classes, the return and correlations can be approximated by the ratings-class correlation with the market parameter. This information can be generated from the transition matrix, which describes the probability of a particular ratings class to default. Also, due to the law of large numbers, we can conclude that every bond in a particular ratings class has the same correlation with the market parameter tX . Therefore, the corresponding beta for each firm will be the same if the bonds are in the same ratings class. The formula for calculating the correlation coefficient for each ratings class is given by the Basel II risk-weight function below

default ofy probabilit,1

1124.01

112.0 50

*50

50

*50

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−−

= −

PDe

ee

e PDPD

β (4.B.7)

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The probabilities of default for the ratings classes are taken from the transition matrices provided by Moody’s below:

Public Transition matrix (exhibit 24) taken from Moody’s public bond default study:

Private Transition matrix (table 15) taken from the SOA private placement study:

Therefore, we can calculate the return thresholds and the rating-wise asset correlations to simulate the return for each bond in the portfolio and then compare the return at the end of the period to the return threshold and record a default event, upgrade event, or downgrade event if the simulated return is more or less than the threshold for the corresponding event.

Calculating cost of a credit event

When calculating the cost of a particular upgrade or downgrade event we can use the target duration of the portfolio and the change in the credit spread from the event. For example, if a bond rated AAA is downgraded to AA, the credit spread will widen by x basis points. The loss generated by this event is the portfolio duration times the change in the credit spread. For this analysis, the duration of XYZ Life’s portfolio is assumed to be seven years.

Aaa Aa A Baa Ba B Caa-C DAaa 89.90% 6.72% 0.54% 0.19% 0.01% 0.00% 0.00% 0.00%Aa 1.04% 87.89% 6.92% 0.27% 0.05% 0.02% 0.00% 0.01%A 0.06% 2.57% 88.12% 4.95% 0.52% 0.10% 0.02% 0.02%Baa 0.05% 0.21% 4.92% 84.72% 4.44% 0.79% 0.25% 0.18%Ba 0.01% 0.06% 0.48% 5.65% 76.68% 7.61% 0.62% 1.18%B 0.01% 0.05% 0.17% 0.41% 5.55% 74.54% 5.44% 5.37%Caa-C 0.00% 0.04% 0.04% 0.20% 0.74% 7.17% 60.65% 19.52%

Aaa Aa A Baa Ba B Caa-C DAaa 93.81% 2.99% 2.16% 0.84% 0.08% 0.07% 0.03% 0.01%Aa 2.04% 88.78% 7.47% 1.25% 0.18% 0.18% 0.04% 0.08%A 1.09% 1.74% 88.78% 7.37% 0.53% 0.25% 0.12% 0.12%Baa 0.17% 0.75% 4.72% 89.12% 3.28% 0.85% 0.48% 0.63%Ba 0.20% 0.34% 0.80% 9.22% 81.11% 3.81% 0.93% 3.59%B 0.14% 0.31% 0.99% 4.87% 6.44% 79.19% 2.11% 5.96%Caa-C 0.62% 0.16% 1.87% 2.18% 3.35% 10.12% 75.49% 6.23%

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To calculate the cost of a default event we simply pay the inverse of the recovery rate. The following tables give an example set of credit spreads and recovery rates.

Public Spreads (market spreads as of 1/05/07) and Recovery Rates (exhibit 14) taken from Moody’s public bond default study:

Private Spreads (pg 138) and Recovery Rates (pg 182) taken from the SOA Private Placement study:

Rating Recovery Rate Credit SpreadAAA 100.00% 0.55%AA 95.40% 0.62%A 46.40% 0.69%BBB 48.10% 0.98%BB 40.10% 1.70%B 36.60% 2.67%CCC 30.40% 6.00%

Rating Recovery Rate Credit SpreadAAA 99.98% 2.10%AA 99.55% 2.10%A 99.78% 2.10%BBB 99.43% 2.65%BB 98.27% 4.85%B 98.31% 7.70%CCC 87.18% 7.70%

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Results

The table in Figure XX reports statistics when the model is applied to the company’s portfolio of credit risky bonds.

Figure XX: Default Costs

Where Public Total is all publicly traded bonds, Private Pl is private placements and Public IG is investment grade bonds. Investment grade bonds are typically bonds rated BBB or higher.

The ASRF model can be used as a robust tool to simulate credit events for a portfolio of credit-risky bonds. It has a simple form and its assumptions hold well for large bond portfolios. The inputs can be determined from published data by ratings agencies and include the transition matrix that these agencies publish specifically for credit analysis.

Default Costs Total Porfolio Public Total Private Pl Public IG

Asset Book Value 23,134,742,000$ 14,713,695,912$ 4,372,466,238$ 13,742,036,748$

Mean (337,037,030) (348,273,805) 11,236,774 (245,032,621) Std Dev 160,597,325 148,995,351 57,576,405 111,491,777

Pct Rank0% (992,296,673) (993,856,730) (160,794,768) (878,724,306) 1% (813,838,956) (798,796,529) (124,871,684) (535,887,498) 5% (630,155,806) (621,287,441) (83,171,480) (440,604,228)

10% (540,453,752) (541,293,231) (62,906,371) (381,568,692) 15% (490,512,886) (497,860,302) (47,820,458) (353,159,947) 20% (455,573,306) (455,733,777) (37,780,423) (329,906,181) 25% (431,157,732) (429,045,515) (28,268,226) (311,554,238) 50% (318,695,254) (329,016,849) 10,192,706 (232,370,643) 75% (233,001,008) (246,116,664) 50,529,800 (162,232,191) 80% (214,108,654) (222,387,267) 59,346,554 (149,306,072) 85% (181,353,285) (200,371,434) 71,374,346 (133,437,155) 90% (140,938,398) (173,549,935) 83,661,549 (113,831,379) 95% (98,727,220) (140,797,629) 103,146,259 (86,993,860) 99% (14,785,000) (81,036,909) 142,276,413 (42,650,643)

100% 72,361,748 35,515,008 183,600,163 39,347,181

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IV.B.5.c Mortality

The mortality scenarios were developed using the mortality model described earlier in this monograph. Interested readers can find the full details in Chapter ?/Section? of this monograph.

We generated 30,000 random numbers using this process. This allows us to create 1,000 scenarios, each with a 30-year projection horizon. The results of this analysis generated mortality scaling factors to apply to actual mortality. This was used for all lines of business where stochastic mortality was reflected.

Summary statistics for the first 10 years of the projection are summarized in the tables attached. Factors were generated for three risks: base mortality, pandemic mortality, and terrorism risk.

The Excel file ____.xls demonstrates how stochastic mortality scenarios can be generated.

IV.B.5.d Morbidity

This section illustrates how claim costs can be developed for a stochastic projection model. This section was developed for the individual disability income LOB. However, the discussion is general enough that it applies equally well to any health product.

Probability distributions for new claim costs and the runoff of the September 30, 2006, claim liability were created using the company’s September 30, 2006, policy in-force data and its September 30, 2006, disabled life data.

Probability distributions for new claim costs

A probability distribution for the September 30, 2006, policy in-force data was produced using Milliman’s Risk Simulator™ software, which is a Monte-Carlo analytical tool. Following is an overview of the process:

Expected claim incidence and termination rates were based on the company’s cash flow testing assumptions

A table was created from the expected claim incidence and termination rates to reflect the probability of going on claim in a year (based on age, occupation class, sex, etc.) and staying on claim for one month, two months, etc.

The probability distribution used in the stochastic liability model for 2006 claim costs was derived by performing 10,000 risk-simulation iterations over the full policy file. For each policy, within each iteration, a random number was chosen that determined whether the policy incurred a claim and, if so, how long it would remain on claim. The present value of

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paid benefits if the policy incurred a claim was calculated and accumulated over all in-force policies. Thus, one complete iteration through the policy in-force file represented one aggregate random claims-cost scenario; 10,000 of these aggregate random claims-cost scenarios were created, on which the probability distribution used in the stochastic liability model was based.

Because the projected in-force business will be shrinking due to policy lapses and expiries, we aged the in-force file every five years and created new probability distributions.

The table in Figure XX compares the probability distributions for years 2006, 2011, 2016, 2021, 2026, and 2031 for certain percentiles.

Figure XX: Probability Distribution for Annual Claim Costs

Probability Distribution for Annual Claim Costs Variances from Expected Annual Claim Costs by Year Incurred

Percentile 2006 2011 2016 2021 2026 20310.010 -12.01% -12.28% -12.78% -13.06% -14.77% -19.69%0.050 -8.69% -8.99% -8.92% -9.24% -10.27% -14.30%0.100 -7.07% -6.96% -7.22% -7.49% -8.03% -10.97%0.150 -5.82% -5.76% -5.91% -6.19% -6.94% -8.92%0.200 -4.58% -4.52% -4.62% -4.88% -5.84% -7.56%0.250 -3.76% -3.69% -3.75% -4.01% -4.72% -5.56%0.300 -2.93% -2.90% -2.91% -3.15% -3.60% -4.22%0.350 -2.11% -2.07% -2.05% -2.28% -2.49% -3.55%0.400 -1.29% -1.66% -1.63% -1.40% -1.92% -2.21%0.450 -0.87% -0.86% -0.78% -0.97% -0.82% -0.88%0.500 -0.04% -0.03% 0.07% -0.09% -0.25% -0.21%0.550 0.38% 0.78% 0.53% 0.78% 0.84% 1.15%0.600 1.21% 1.19% 1.37% 1.21% 1.41% 1.81%0.650 2.02% 2.01% 2.22% 2.08% 2.51% 3.16%0.700 2.85% 2.82% 2.66% 2.95% 3.08% 4.50%0.750 3.68% 3.63% 3.51% 3.83% 4.18% 5.85%0.800 4.48% 4.85% 4.80% 4.69% 5.29% 7.16%0.850 5.73% 5.67% 5.66% 6.00% 6.96% 9.20%0.900 6.98% 6.91% 7.38% 7.30% 8.07% 11.21%0.950 9.01% 9.34% 9.52% 9.47% 10.87% 14.55%0.990 12.73% 13.05% 13.35% 13.83% 15.91% 21.34%

The results in Figure XX show a general widening of the distributions as the policyholders age and their number shrinks.

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As an example of how we use the table in the Stochastic Liability model to determine the random annual claim cost for the year 2006, we choose a random number. If the random number were 0.850, then the random annual claim cost would be 5.73% higher of the baseline claim cost in 2006.

Because the annual claim costs represent the present value of future paid claims, the 5.73% variance factor is applied to a vector of future paid claims (for which the present value is equal to the baseline annual claim cost) to produce variances in future aggregate paid claims arising from the variances in claim costs incurred in 2006. The 5.73% is also applied to the vectors of the various claim reserves arising from claims incurred in 2006 to produce variances in future aggregate claim reserves.

Probability distribution of claim runoff

Risk Simulator was also used to produce a probability distribution of the September 30, 2006, claim runoff. The table in Figure XX shows the resulting probability distribution at selected percentiles.

Figure XX: Probability Distribution for Claim Runoff

Probability Distribution for Claim Runoff

Percentile Variance from Average Payout 0.010 -2.41% 0.050 -1.67% 0.100 -1.29% 0.150 -1.07% 0.200 -0.84% 0.250 -0.69% 0.300 -0.55% 0.350 -0.40% 0.400 -0.25% 0.450 -0.10% 0.500 -0.02% 0.550 0.13% 0.600 0.27% 0.650 0.42% 0.700 0.50% 0.750 0.72% 0.800 0.88% 0.850 1.10% 0.900 1.32% 0.950 1.62% 0.990 2.36%

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Because there is no claim incidence risk, the probability distribution in Figure XX is much tighter around the mean than those in the table in Figure XX for the annual claim costs, which reflect both claim incidence and termination risks.

Pricing risk

The probability distributions for annual claim costs and runoff assume that the expected morbidity bases as defined by company’s cash-flow-testing assumptions are correct. However, these morbidity bases represent estimates, and there is a risk that the current “true” expected basis is different. Consequently, we incorporate a pricing-risk factor to these baseline claim projections to reflect this pricing risk.

We used the probability distribution in the table in Figure XX to represent the pricing risk for both annual claim costs and claim runoff:

Figure XX: Probability Distribution for the Pricing Risk

Probability Distribution for the Pricing Risk Percentile Variance

0.00 -5.0% 0.10 -2.5% 0.35 0.0% 0.65 2.5% 0.90 5.0%

IV.B.5.e Lapses

Lapse scenarios were developed based on historical company experience and an assumption that deviations from the expected rates will follow a normal distribution. The steps needed to develop the lapse scenarios are as follows:

1. Develop the expected lapse rates based on historical company experience. These rates constitute our best-estimate assumptions and were used in the baseline projection. Any amount of company experience can be used in developing the lapse assumptions. In order to ensure the results are credible, it is preferable to use at least five years (and usually 10 years or more) in the experience study.

2. Calculate the standard deviation of experience lapse rates based on the same historical company data that was used in Step 1.

3. Develop a string of random numbers using Excel’s Random Number Generator. The random numbers represent multiplicative scalars that will be applied to the best-estimate lapse rates to generate the 1,000 scenarios.

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For Step 3, we used the normal distribution with mean = 1 and standard deviation = 0.16. The mean should always be set equal to 1 because, on average, we expect the lapses to equal the best-estimate assumption, and therefore the average multiplicative scalar should be 1. The standard deviation of 0.16 was based on XYZ Life company experience. The random-number seed can be any value preferred by the user (we chose 1).

We generated 30,000 random numbers using the process described above. This allows us to create 1,000 scenarios, each with a 30-year projection horizon.

Summary statistics for the first 10 years of the projection are summarized in the table in Figure XX.

Figure XX: Illustration of Stochastic Lapse Scenarios

The Excel file Lapse Scenarios.xls demonstrates how stochastic lapse scenarios can be generated.

Operational and strategic risks

In order to estimate the appropriate level of capital that the company should hold for the mitigation of risks arising from operational or strategic sources, a series of scenarios have been developed to describe the risk exposure it faces in these areas. Two pieces of analysis have been used to ensure that the scenarios are relevant to the company’s business.

First, management facilitated a data-gathering exercise from the profit centers relating to potential operational losses that were both high-impact and low-frequency. The business experts contributing to this exercise were able to draw on their extensive knowledge of the organization’s

Table X - Illustration of Stochastic Lapse Scenarios

1 2 3 4 5 6 7 8 9 10

Mean 1.0045 1.0040 1.0032 0.9936 1.0018 0.9976 1.0018 0.9872 1.0005 0.9997Std Dev 0.1606 0.1615 0.1552 0.1561 0.1580 0.1574 0.1567 0.1626 0.1617 0.1597

Percentile0.0% 0.4725 0.3585 0.5557 0.5334 0.4869 0.5298 0.4848 0.4664 0.5580 0.45580.5% 0.5879 0.5837 0.5987 0.5925 0.5970 0.6009 0.6134 0.5888 0.5892 0.60291.0% 0.6094 0.6165 0.6372 0.6250 0.6433 0.6219 0.6384 0.6129 0.6176 0.63495.0% 0.7498 0.7318 0.7408 0.7348 0.7483 0.7278 0.7532 0.7164 0.7364 0.738710.0% 0.8036 0.7926 0.8001 0.7981 0.8030 0.8010 0.8056 0.7759 0.8008 0.801525.0% 0.8998 0.8974 0.8973 0.8837 0.8885 0.8947 0.8892 0.8799 0.8874 0.886150.0% 1.0077 1.0087 1.0044 0.9891 1.0027 0.9962 1.0017 0.9890 0.9946 1.000175.0% 1.1101 1.1087 1.1120 1.1025 1.1047 1.1006 1.1112 1.0929 1.1060 1.106990.0% 1.2113 1.2079 1.1945 1.2045 1.2053 1.2024 1.2123 1.1973 1.2111 1.204695.0% 1.2701 1.2750 1.2440 1.2549 1.2598 1.2577 1.2584 1.2524 1.2665 1.261699.0% 1.3677 1.3483 1.3580 1.3460 1.3505 1.3608 1.3502 1.3627 1.3832 1.369299.5% 1.3793 1.4248 1.3824 1.3707 1.4022 1.3930 1.3625 1.4073 1.4218 1.4085

100.0% 1.4740 1.5275 1.5484 1.5131 1.8583 1.5484 1.4244 1.4693 1.5275 1.5198

Projection Year

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historical experience and also to include knowledge of similar events occurring in other organizations.

Second, in parallel, the senior company team was engaged using the STRATrisk methodology to determine the key strategic risks that the company faces. STRATrisk is a cutting-edge methodology that has been developed to determine the key sources of risk that can profoundly impact an organization and ultimately prevent it from achieving its strategic goals. The methodology is the result of a research program led by Neil Allan of Bath University in the UK and funded by the Department of Trade and Industry.

The two pieces of analysis are combined to identify scenarios that, in aggregate, suitably describe the company’s exposure in these areas. These risks were not modeled stochastically, but were added as a load to the otherwise indicated EC.

IV.B.6 Presentation of results

This section presents the results of an EC analysis conducted based on the assumptions and methods discussed above. We present results using two risk metrics:

The 99% Conditional Tail Expectation (CTE-99) on PVFP

The CTE-99 on GPVL

PVFP risk metric

The table in Figure XX shows the PVFP for:

● The baseline scenario (using best-estimate assumptions)

● The “2,500 worst” scenarios by LOB

● The worst of the 1,000 scenarios run for each LOB and each risk factor

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Figure XX: Indicated EC Using CTE-99 PVFP

The table in Figure XX summarizes the indicated required capital when using this risk metric.

Figure XX: Summary of Indicated EC

The present value of future profits is greater than zero in all three scenarios. Under this risk metric, EC is only required if the PVFP is less than zero. Therefore, we conclude that there is no indicated EC and the entire initial capital is redundant.

Indicated Economic Capital Using CTE-99 Present Value of Future Profits

Difference Worst 1,000from for each

Baseline 2,500 Worst Baseline risk factorIndividual Life 2,640 2,368 (273) 1,858 Individual Annuity 43 43 - 37 Disability Income 523 217 (306) (482) Group Health 773 393 (380) 48 Pensions 30 28 (2) 31 Variable Annuities 105 58 (47) 113 Subtotal 4,114 3,106 (1,007) 1,605

Credit Defaults (380) (668) (289) (1,170)

Total 3,734 2,438 (1,296) 436

Summary of Indicated Economic Capital

Worst 1,000for each

Baseline 2,500 Worst risk factorInsurance Risk inLines of Business 4,114 3,106 1,605

Credit Defaults (380) (668) (1,170)

Strategic and Operational Risk (306) (306) (306)

Total Present Value of Profits 3,428 2,132 130

Indicated Economic Capital - - -

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GPVL risk metric

The table in Figure XX shows the Present Value of Future Profits for:

● The baseline scenario (using best-estimate assumptions)

● The “2,500 worst” scenarios by LOB

Figure XX: Indicated EC Using CTE-99 GPVL

The table in Figure XX summarizes the indicated required capital when using this risk metric.

Figure XX: Summary of Indicated EC (GPVL risk metric)

Indicated Economic Capital Using CTE-99 Greatest Present Value of Accumulated Losses

Baseline 2,500 WorstIndividual Life 2,640 - Individual Annuity 43 (3) Disability Income 523 (60) Group Health 773 (10) Pensions 30 - Variable Annuities 105 (39) Subtotal 4,114 (113)

Credit Defaults (380) (960)

Total 3,734 (1,073)

Summary of Indicated Economic Capital (GPVL Risk Metric)

Baseline 2,500 WorstInsurance Risk inLines of Business 4,114 (113)

Credit Defaults (380) (960)

Strategic and Operational Risk (306) (306)

Total Present Value of Profits 3,428 (1,379)

Indicated Economic Capital - 1,379

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The results presented in the tables in Figures XX and XX assume that each LOB is operated as a stand-alone company and, therefore, gains in one LOB cannot be used to offset losses in other LOBs. On this stand-alone basis, risk exists. Assuming each LOB were a separate company, the total indicated EC is $1.4 billion. We also ran the stochastic projection on an aggregate company basis and found that risks are completely mitigated by aggregation. That is, gains in some LOBs fully offset losses in other LOBs.

IV.B.6.a Calibration, validation, and review

As discussed in Section 3 of this monograph, a lengthy checking process was utilized. Existing projection models for each LOB had already been set up and validated. The purpose of this analysis was to add stochastic generators into the model to perform the EC analysis. For each LOB, a base case or best-estimate projection was first completed. This projection was deterministic, meaning that all assumptions were fixed at the start of the projection and not allowed to vary. The best-estimate projection was then utilized in each case to test and validate the stochastic generator.

Calibration

The stochastic models were calibrated based on historical experience. Much of this discussion is contained earlier in the case study. Since this was a projection and the stochastic generators were only applied to the forecast, the initial projection and generators tied to this experience. One additional test was performed to review the “dynamic” validation; that is to ensure that the projection trended forward from the valuation date as expected.

Validation

Validation of the scenarios generated was performed using expanded versions of the tables attached. In all cases, the means were compared to projected expected values based on the deterministic projections. For equity and interest-rate scenarios, a review was performed of the percentile distributions and individual scenarios by investment personnel at the company. Other scenarios were graphed, similar to those shown in Section 3, to provide a visual of the projected scenarios and the projected variation from the mean over time.

For a specific generator, the component of the projection affected by the generator was compared to a base case. For example, on the Life LOB, projected best-estimate death claims were compared to specific mortality scenarios, which were projected using the stochastic mortality generator. A comparison of death claims ensured that the projected death claims for the mortality scenario moved in a similar fashion to the base-case death claims adjusted by the mortality for the specific scenario. Each component was tested for this.

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As an additional test, the mean projection across all of the 1,000 scenarios generated was compared to the base case. This was done for each independent risk assessment. Some variation was expected but the deviation from the base was not material.

Additional tests were performed, particularly on extreme scenarios, to ensure results were plausible and appropriate for the scenario.

Peer review and checking

Two forms of review occurred on this project. First, a “bottom-up” approach where individual inputs to the models and generators were reviewed and confirmed with the documentation prepared. The second approach, “top-down,” looked at projected financial results and variations in aggregate, at various CTE levels, and for specific scenarios. This review occurred both on the individual risk projections, i.e., interest rate only, mortality only, morbidity only, …, as well as for aggregated risk projections, that is, scenarios where the risks were combined.

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Table 4.C.1 Summary of Stochastic Mortality Scenarios

Total Mortality Risk Percentage Gross-Up Applied to Mortality Input by Projection Year

Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Year 10Mean 0.49% 0.63% 0.77% 0.41% 0.98% 0.47% 0.46% 0.52% 0.63% 0.46%Std Dev 2.65% 3.51% 9.49% 3.91% 11.88% 4.37% 4.67% 5.02% 8.13% 5.58%Percentile Distribution

0.0% -4.39% -7.10% -8.47% -10.82% -11.21% -12.81% -14.49% -15.49% -16.48% -17.95%0.1% -4.34% -5.88% -7.39% -8.86% -10.66% -10.91% -12.71% -12.54% -14.22% -14.84%0.5% -3.94% -5.48% -6.60% -7.95% -7.93% -9.61% -10.48% -10.85% -12.50% -12.91%1.0% -3.45% -4.63% -6.29% -6.85% -7.37% -8.85% -9.83% -10.02% -11.17% -11.97%5.0% -2.26% -3.24% -4.12% -4.61% -5.36% -6.01% -6.54% -7.11% -7.79% -8.22%

25.0% -0.88% -1.26% -1.72% -1.90% -2.08% -2.32% -2.47% -2.99% -2.96% -3.23%50.0% 0.24% 0.27% 0.34% 0.10% 0.28% 0.30% 0.21% 0.48% 0.30% 0.42%75.0% 1.38% 1.75% 2.14% 2.38% 2.57% 2.82% 3.07% 3.59% 3.85% 3.70%90.0% 2.40% 3.36% 3.99% 4.45% 5.50% 5.71% 5.86% 6.35% 6.45% 7.32%95.0% 3.47% 5.18% 5.27% 5.74% 7.48% 7.41% 8.03% 8.25% 8.66% 9.27%99.0% 12.32% 14.00% 14.76% 9.35% 14.42% 13.50% 11.67% 12.12% 13.53% 13.61%99.5% 18.71% 20.88% 19.67% 19.17% 28.45% 16.58% 18.68% 19.78% 20.19% 15.81%99.9% 22.95% 32.64% 39.56% 35.10% 57.10% 22.00% 27.85% 30.95% 27.50% 30.51%

100.0% 25.04% 37.25% 277.67% 35.72% 345.37% 34.10% 29.12% 31.21% 197.77% 31.48%

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Table 4.C.2 Summary of Stochastic Mortality Scenarios

Base Mortality Risk Percentage Gross-Up Applied to Mortality Input by Projection Year

Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Year 10Mean 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Std Dev 1.53% 2.09% 2.62% 3.03% 3.41% 3.77% 4.12% 4.47% 4.80% 5.14%Percentile Distribution

0.0% -4.42% -7.11% -8.48% -10.85% -11.24% -12.82% -14.51% -15.49% -16.49% -17.96%0.1% -4.40% -6.02% -7.40% -8.86% -10.88% -10.93% -12.76% -12.67% -14.27% -14.86%0.5% -3.97% -5.76% -6.62% -7.96% -8.07% -9.79% -10.50% -11.09% -12.68% -12.91%1.0% -3.49% -5.07% -6.31% -7.13% -7.46% -8.86% -9.86% -10.23% -11.64% -11.99%5.0% -2.53% -3.34% -4.33% -4.79% -5.44% -6.25% -6.73% -7.51% -7.89% -8.36%

25.0% -1.03% -1.38% -1.86% -2.02% -2.34% -2.43% -2.68% -3.15% -3.12% -3.46%50.0% 0.04% 0.02% 0.09% -0.06% 0.02% -0.03% -0.02% 0.13% 0.03% 0.17%75.0% 0.99% 1.36% 1.76% 2.01% 2.11% 2.48% 2.79% 3.05% 3.32% 3.36%90.0% 1.89% 2.74% 3.46% 4.01% 4.52% 4.94% 5.46% 5.52% 6.00% 6.40%95.0% 2.47% 3.34% 4.33% 5.06% 6.03% 6.47% 6.94% 7.26% 7.62% 8.57%99.0% 3.61% 4.81% 6.05% 6.88% 8.13% 8.91% 9.41% 10.38% 11.81% 11.79%99.5% 4.13% 5.53% 6.52% 7.26% 8.82% 9.42% 10.37% 11.20% 12.38% 12.91%99.9% 4.99% 6.59% 7.27% 8.17% 9.68% 10.59% 11.00% 12.09% 13.51% 13.77%

100.0% 5.77% 6.89% 8.37% 9.32% 9.80% 10.72% 11.47% 12.99% 14.21% 13.82%

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Table 4.C.3 Summary of Stochastic Mortality Scenarios

Pandemic Mortality Risk Percentage Gross-Up Applied to Mortality Input by Projection Year

Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Year 10Mean 0.36% 0.47% 0.65% 0.35% 0.82% 0.37% 0.38% 0.38% 0.50% 0.33%Std Dev 2.14% 2.33% 9.06% 2.39% 11.30% 1.84% 2.24% 1.99% 6.62% 2.00%Percentile Distribution

0.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.1% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.5% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%1.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%5.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

25.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%50.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%75.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%90.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%95.0% 1.31% 2.07% 1.67% 1.16% 1.78% 1.76% 1.71% 1.94% 1.40% 1.25%99.0% 8.72% 11.13% 11.15% 7.90% 10.76% 9.24% 10.37% 8.30% 6.80% 9.50%99.5% 19.12% 14.46% 17.30% 13.00% 18.92% 13.32% 16.84% 12.55% 13.63% 15.44%99.9% 24.89% 23.47% 36.23% 33.08% 55.07% 20.99% 32.17% 25.27% 30.79% 21.10%

100.0% 25.74% 38.08% 278.91% 33.15% 346.47% 21.11% 34.33% 27.30% 202.79% 33.40%

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Table 4.C.4 Summary of Stochastic Mortality Scenarios

Terrorism Mortality Risk Percentage Gross-Up Applied to Mortality Input by Projection Year

Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Year 10Mean 0.13% 0.15% 0.12% 0.07% 0.16% 0.10% 0.09% 0.13% 0.12% 0.13%Std Dev 0.92% 1.36% 1.35% 0.23% 1.55% 1.04% 0.45% 1.09% 0.87% 1.09%Percentile Distribution

0.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.1% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%0.5% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%1.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%5.0% 0.01% 0.00% 0.01% 0.00% 0.01% 0.01% 0.00% 0.01% 0.01% 0.01%

25.0% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01%50.0% 0.02% 0.02% 0.02% 0.02% 0.02% 0.02% 0.02% 0.02% 0.02% 0.02%75.0% 0.04% 0.04% 0.03% 0.04% 0.04% 0.04% 0.04% 0.04% 0.03% 0.04%90.0% 0.11% 0.11% 0.09% 0.10% 0.10% 0.11% 0.09% 0.10% 0.09% 0.10%95.0% 0.30% 0.26% 0.21% 0.21% 0.21% 0.25% 0.18% 0.25% 0.21% 0.24%99.0% 2.72% 2.01% 1.02% 1.11% 2.40% 1.03% 1.60% 2.27% 2.66% 1.96%99.5% 3.86% 3.63% 2.70% 1.72% 5.22% 1.36% 3.25% 4.40% 3.03% 4.85%99.9% 13.89% 19.93% 18.42% 2.73% 29.09% 10.92% 4.97% 9.81% 11.17% 11.49%

100.0% 19.76% 32.74% 37.47% 3.17% 35.17% 30.54% 8.74% 29.00% 21.46% 27.31%

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IV.D Unpaid claim variability for a multi-line non-life insurance company

IV.D.1 INTRODUCTION

Building an effective Economic Capital (EC) model for an insurance company is a complex process, with many important features discussed in this monograph. Because risk transfer is at the core of all insurance policies, understanding the financial impact of risk transfer is at the core of building an EC model. Quite often the impact of risk transfer is the largest single risk element of the EC model.

For all non-life insurance companies, the best estimate of the remaining financial impact of the insurance policies it wrote in the past is shown in the liability section of the balance sheet. In many countries this financial impact is split between the estimated cost of fulfilling the unexpired portion of the insurance contracts (the unearned premium), the estimated value of the claims, and the estimated costs associated with settling those claims on the balance sheet, as well as related exhibits that show how the estimates relate to historical time periods, i.e., loss development. The terminology used to describe the claim cost liabilities in the financial statements varies by country—e.g., loss reserves, claim provisions, etc.—but for purposes of this case study we will use the more generic term of "unpaid claim estimates," which will include both the value of the claims and the associated settlement costs unless noted otherwise.

While the best estimate of the historical liability can be a very large amount, it is generally only shown as a single-point, or deterministic, estimate in financial statements. In order to use this in an EC model, we will also need to estimate the distribution of possible outcomes, which we will refer to as claim variability. In the process of estimating the variability of the historical claims, we will also be able to estimate the variability of the future claims and cash flows. In so doing, we will quantify the three major components of insurance risk. Namely, unpaid claim risk (for claims that have already occurred), pricing risk (for claims that will occur in upcoming financial periods) and cash-flow risk (for the loss of investment income when actual cash flows exceed expectations). The variations in the projected cash flows can also be used to analyze one of the major financial risks—asset/liability matching risk, or more precisely, the effects of mismatching cash inflows and outflows.

With the distributions of possible outcomes estimated, we can use them to quantify insurance- and financial-risk components directly. In addition, we can use the simulated outputs as inputs to a larger EC model or to parameterize an EC model that will simulate possible outcomes within the model. As part of this case study we will illustrate the calculation of some insurance-risk

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components. The next case study will illustrate the use of the distributions as part of a larger EC model.

IV.D.1.a Model selection

As discussed in Section II.E., there are a wide variety of casualty models that can be used to estimate the claim variability, ranging from individual-claim to aggregate-claim models. The types of claim variability models used as building blocks of an EC model will depend on the types of data available, user familiarity with the models, available software, and other factors.1 While these criteria are important for model selection, the more important criteria are the uses of the model output in the EC model.

For example, if the EC model will be used to analyze facultative reinsurance then models based on individual-claim data will be required. The level of detail required in the EC model will determine the types of models to consider for analyzing claim variability. In turn, the types of models being considered may also cause the modeler to seek data sources other than published financial data in order to build the type of claim-variability model needed.2

Once the types of models to be used have been selected, the next step will be to analyze the data and parameterize them to make sure the distributions of possible outcomes are reasonable. There are a variety of criteria that can be used to both test underlying model assumptions against the data and to test the model output to ensure both the model assumptions and results are reasonable.3 One of the overriding goals in this process is to make sure that each iteration of the simulation model is a plausible outcome—stated differently, are there any outcomes that are implausible and should thus be removed, or constrained, from the model? However, care must be exercised in this process to not remove improbable outcomes, because these tail events are a critical part of the distribution of possible outcomes.4

The model parameterization step actually involves multiple steps because the modeler will generally want to use more than one model. Actuaries have long recognized that the procedures and assumptions underlying each model are approximations to reality. This is not a bad thing, just recognition that reality is extremely complex and that mathematical models require simplifications. Since no one model is perfect, statistical theory recognizes that blending the best features of multiple 1 See [1], Section 3.1.1., pp. 45-47 for a more complete discussion of model selection criteria. 2 The other data sources are likely to be the raw data underlying the financial statements or subgroups of the

financial data aligned with the EC modeling goals rather than the financial reporting goals. No matter the source, the data should still tie to the financial statements.

3 See [1], Sections 3.1.2. and 3.1.3., pp. 47-51 for a more complete discussion of model results and model assumption testing criteria, respectively.

4 For some models the most likely implausible outcomes are negative claims costs when only positive amounts are possible (e.g., no recoveries).

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models will generally result in a better overall model.

Once the results from multiple models have been combined, or credibility weighted, into a single distribution of possible outcomes for each segment of business, the next step will be to correlate the results in an aggregate distribution for the entire business unit. This step may also involve sub-units based on management or legal organizations within the entire business unit. The calculation of the correlation coefficients my vary depending on which models are used, but there are three general areas where correlation can be applied.

First, there may be correlation between variables of different business segments. For example, there can be correlation between inflation parameters in different segments or contagion effects where some claims affect multiple business segments.

Second, there can be correlation between the random incremental payments between business segments. This is correlation of the payments for claims that have already occurred.

Finally, there is the correlation between the ultimate loss ratios between business segments. This can also be thought of as price-adequacy correlation related to underwriting cycles. Just like the estimation of model parameters, the estimation of correlation coefficients is not an exact science, so testing different correlation-assumption scenarios is prudent in order to gain insight into the sensitivity of these assumptions.

IV.D.1.b Bootstrap modeling

While there are many models to choose from, we will illustrate the concepts in this case study using the bootstrap model. Like all models, the bootstrap model has advantages and disadvantages, but for our purposes it has three key advantages.

First, setting up the model involves aggregate loss triangles of both paid and incurred losses and volume-weighted age-to-age factors (i.e., the chain ladder method), which should be familiar to most people. The use of aggregate triangle data means that the required data should be readily available in most cases and perhaps even readily tie to published financial statements.

Second, while the bootstrap model has components that are commonly known and understood, it also has components that are sophisticated enough to produce robust estimates of distributions of possible outcomes. Even for users with a modest mathematical background, the sophisticated components are straightforward and easy to learn. Understanding the theory behind the model would require a more sophisticated background in statistics, but that is beyond the scope

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of this case study.5

Third, because the framework of the bootstrap model is based on the chain ladder method, other methods can also be used with a bootstrap model, thus giving us multiple models we can credibility-weight into a more robust distribution than we could derive with only one model. For this case study we will be using both paid and incurred data models based on the chain ladder, Bornhuetter-Ferguson, and Cape Cod methods, for a total of six different models. The basic mechanics of these models are illustrated in Appendix A.

By using a model that is both familiar and sophisticated, we hope to illustrate the concepts in sufficient detail so that the interested reader can learn from the example and then apply the concepts to other types of models as needed.

IV.D.2 BUILDING A MODEL

Sample Insurance Company (SIC) is a local property/casualty insurer that writes coverage in three primary segments—short-tail property lines, medium-tail casualty lines, and long-tail casualty lines.6 SIC has been operating with a stable underwriting and claims-handling staff for many years and has had only modest growth in exposures in the past 10 years. While the property exposures are subject to weather-related claims, the exposures are geographically diverse and not subject to major catastrophes.

The financial data includes paid and incurred loss-development triangles by accident year, earned premiums, and earned exposures by calendar year. For simplicity, the loss data contains all claims-handling expenses in addition to the value of the claims. The data for all three segments is contained in Exhibit IV-1, but the remainder of this section will focus only on the medium-tail casualty business.

IV.D.2.a Diagnostic testing

Like all models and methods, the quality of a bootstrap model depends on the quality of the assumptions. This point cannot be overemphasized because the results of any simulation model are only as good as the model used in the simulation process. If the model does not “fit” the data then the results of the simulation may not be a very good estimate of the distribution of possible

5 For the interested reader, the references contain several papers that one can read to gain a deeper understanding

of the theory. 6 The fundamental process of modeling a business segment only requires one segment to illustrate, but three are

used to illustrate the correlation concepts later in the case study. We will only focus on the medium-tail casualty segment for most of the model-building discussion, but these processes can be extended to any number of segments in practice.

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outcomes.

Another advantage of the bootstrap model is that it can be “tailored” to some of the statistical features found in the data under analysis. Thus, we can use a variety of diagnostic tools to help us judge the quality of those assumptions and to change or adjust the parameters of the model depending on what statistical features we find in the data. In essence, we use the diagnostic tools to find the model that provides the best fit to the data.

The diagnostic tests are designed to either test various assumptions in the model, to gauge the quality of the model fit, or to help guide the adjustment of model parameters. Most, if not all, of these tests are relative in nature, which means that we can use them to “improve” the fit of the model by comparing the tests from one set of model parameters to another. The model will generally run properly with any set of parameters, so the question we are trying to answer is, “Which set of model parameters will give us simulations that are most realistic and most consistent with the statistical features found in the data (or that we believe are in the data)?”

Also included with some of the diagnostics are statistical tests, which can be viewed as a type of pass/fail test for some aspects of the model assumptions. However, for these statistical tests it is important to note that a “fail” result for a given test does not generally invalidate the entire model. A “fail” signal should only be interpreted as the possibility that a better model (or model parameterization) might be possible.

Residual graphs

Since the simulations in the bootstrap model use the residual values to create sample triangles of historical data, an important diagnostic test is to review graphs of the residuals to make sure that they are indeed random. Moreover, rather than simply looking at the residuals compared to the predicted (or actual) values, it can also be very useful to look at them by development period, accident period, and calendar period.7 Consider the residual plots in the graphs in Figure 1XX.

At first glance, the residuals in these graphs appear reasonably random as this model is providing a reasonably good fit of the data. In addition to the residual plots (blue dots), the development, accident, and calendar graphs also include a trend line (pink line) that links the averages for each period. These graphs are important as they may reveal potential features in the data that we can account for to provide an improved model fit.

7 Note that for each of these four graphs the same residuals are being plotted. In other words, this is four different

views of the same data.

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Figure 1XX: Residual Graphs Prior to Heteroscedasticity Adjustment

In this example there do not appear to be any issues with the trends, but the development-period graph does illustrate a common issue with insurance triangle data. In this graph (upper left), you can see that the range of the residuals in the first three periods is not the same as the middle four or the last two. What this means is that the residuals in these three different sub-groups may be representative of three separate standard deviations.

While the bootstrap model does not require a specific type of distribution for these residuals, it is nevertheless important that all of the residuals are independent and identically distributed. This means that, at a minimum, they must share a common mean and standard deviation. This is important since we will be sampling with replacement from all residuals during the simulations. If the residuals do have different standard deviations (as shown in Figure 1XX) then this is referred to in statistical terms as heteroscedasticity.8

In order to adjust for this heteroscedasticity, the bootstrap model allows us to identify groups of development periods and then adjust the residuals to a common standard deviation value.

8 While some papers on bootstrap modeling have discussed whether the mean of the residuals should be zero or

not, this is usually not as important a consideration as the standard deviations and adjusting for heteroscedasticity. Even after adjusting for heteroscedasticity, the mean of the residuals will usually not equal zero.

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A set of tools for identifying these groups is included below. In the table in Figure 2XX, we have identified the heteroscedasticity adjustment groups by group number and then sets of relativities have been calculated for all development periods within each group based on the relative standard deviations of each group.

Figure 2XX: Heteroscedasticity Relativities and Groups

Graphs of the standard deviation and range relativities can be used to help visualize the proper groupings of periods. These are illustrated in the graph in Figure 3XX for the residuals shown in Figure 1XX and Figure 2XX.

Figure 3XX: Residual Relativities Prior to Heteroscedasticity Adjustment

The relativities illustrated in Figure 3XX confirm that the residuals in the first three periods are not the same as the middle four or the last two, but a little testing is required to determine the optimal groups using the other diagnostic tests noted below. Consider the residual plots in Figure 3XX, which are from the same data model after the first three groups (plus the seventh) and the last two development periods, respectively, have been adjusted (separately). This grouping resulted in better improvement in the other tests described below compared to other groupings that were tested.

Comparing the residual plots in Figures 1XX and 4XX, you can see that the general “shape” of the residuals has not changed and the “randomness” is still consistent, but now the residuals do

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appear to exhibit the same standard deviation (or homoscedasticity).9 Comparing the residual relativities in Figures 3XX and 5XX, you can also see that the relativities are also more consistent.

Figure 4XX: Residual Graphs After Heteroscedasticity Adjustment

Figure 5XX: Residual Relativities After Heteroscedasticity Adjustment

Normality test

Another test of the residuals is whether they come from a normal distribution or not. While

9 During the simulation process, after the residuals are randomly selected from this homoscedastic group they are

adjusted back to their original heteroscedastic group sizes to insure that the simulation process produces sample triangles that exhibit the same statistical properties as the original data. In addition, the application of the process variance (illustrated in Step 8 in the Appendix) is similarly adjusted.

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the bootstrap model does not depend on the residuals being normally distributed, this is still a useful test for comparing parameter sets and for gauging the skewness of the residuals. For this test, we can use both graphs and calculated test values. Consider the following graphs for the same heteroscedasticity groups used earlier.

Figure 6XX: Normality Plots Prior to and After Heteroscedasticity Adjustment

Even before the heteroscedasticity (hereafter “hetero”) adjustment, the normality plot looks quite good. In addition to the graph, the p-value is a statistical test for normality that can be thought of as a pass/fail test with a value greater than 5.0% generally considered a “passing” score (of the normality test, not whether the bootstrap model passes or fails).10 In addition to the pass/fail test, the p-value (prior to the hetero adjustment) can be thought of as a gauge for the process-variance distribution since a low value would be more indicative of the gamma or lognormal distribution and a higher value more indicative of the normal distribution.11 Also shown with these graphs is N, which is the number of data points, and the well known R2 test. After the hetero adjustment, the plot looks even better and the values improve, indicating that the hetero adjustment did improve the model fit.

Outliers

Another useful test is to check for outliers in the data, or values outside of the “typical” range. A very useful graphical test for this purpose is a box-whisker plot. Consider the graphs in Figure 7XX, again from the same hetero groups.

10 For the interested reader, this is known as the Shapiro-Francia test. The null hypothesis is typically considered to

be invalid if the value is less than 5.0%, but as noted earlier failing this test does not generally invalidate the bootstrap model.

11 There are no definitive values here for changing from one distribution to another and sticking with the underlying theory (and using the gamma distribution) is advisable. But for those looking for guidance on when a different distribution may be beneficial in improving the model fit to the underlying data, this test can be useful.

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Before the hetero adjustment, you can see that there were four outliers in the data model, corresponding to the two highest and two lowest values in all the previous graphs. The basic design of the box-whisker plot is to use the box to show the inter-quartile range (the 25th to 75th percentiles) and the median (50th percentile) of the residuals. The whiskers then extend to the largest values that are less than three times the inter-quartile range. Any values beyond the whiskers are generally considered outliers and are identified individually with a point.

Figure 7XX: Box-Whisker Plots Prior to and After Heteroscedasticity Adjustment

After the hetero adjustment, the residuals do not contain any outliers. If the data continued to contain outliers, specific examples of them (e.g., cells or observations) could be removed from the model (i.e., be given zero weight in the calculations and not used in the sampling with replacement). However, if there are still a lot of outliers then this typically means that the model is not sufficiently “optimized” yet—i.e., some other aspect of the model should be adjusted first—or there is significant skewness in the data that will be replicated in the bootstrap simulations. Finally, removing only one or a few outliers should be done with caution as they may still represent realistic “extreme” or “skewed” values in the simulations.

Indeed, the outlier issue also highlights a potential weakness of the bootstrap model that needs to be considered when assessing both the model assumptions and results. Specifically, since the number of data points used to parameterize the model are limited (to 55 in this example), it is hard to determine whether the most extreme observation is a one-in-100 or a one-in-1,000 event (or simply, in this example, a one-in-55 event).12 Of course, the nature of the extreme observations in the data will also affect the level of the extreme simulations in the results. For purposes of this case study, we will assume that the modeler is satisfied with the level of the extreme simulations in the

12 For the bootstrap model described in Appendix A, a way to overcome a lack of extreme residuals would be to

parameterize a distribution for the residuals and resample using the distribution (e.g., use a normal distribution if the residuals are normally distributed).

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results that follow.

While the three diagnostic tests shown above are illustrative of tests common to most types of models, they are not the only tests available. In addition to the paid data, the same tests are equally applicable to the incurred data. Rather than complete an exhaustive set of tests, we refer the interested reader to [1]. For the examples that follow, these tests were applied to both the paid and incurred data for all three business segments.

IV.D.2.b Model results

After the diagnostics have been reviewed, the next step in the evaluation of each model is to run the simulations for the segment being analyzed. In many ways, the simulation results are an additional diagnostic tool for evaluating the quality of the model. To illustrate the diagnostic elements of the simulation output, we will review the results for the medium-tail casualty data. The estimated-unpaid results shown below in Figure 8XX were simulated using the hetero adjustments noted above.

Figure 8XX: Estimated-unpaid Model Results

Estimated-unpaid results

The first diagnostic element of the estimated-unpaid table13 in Figure 8XX can be seen by reviewing the columns labeled Standard Error and Coefficient of Variation. As general rules, the standard error should go up when moving from the oldest years to the most recent years and the standard error for the total of all years should be larger than any individual year. In the table above, the standard errors follow these general rules. For the coefficients of variation, they should go down 13 The output tables from a bootstrap model could include any number of percentiles, but we are only showing

three percentiles to keep the table more manageable.

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when moving from the oldest years to the more recent years and the coefficient of variation for all years combined should be less than for any individual year.14 Except for the 2007 accident year, the coefficients of variation in the table above seem to follow the general rules, although there are also some random fluctuations.

The reason the standard errors (value scale) tend to go up is that they tend to follow the magnitude of the mean or expected-value estimates. The reason the coefficients of variation (percent scale) tend to go down has more to do with the independence in the incremental claims-payment stream. For the oldest accident year, there is typically only one (or a few) incremental payment(s) left, so the variability is (almost) fully reflected in the coefficient. For the most current accident year, the “up and down” variations in the future incremental payment stream can offset each other thus causing the total variation to be a function of the correlation between each incremental payment for that accident year (i.e., the incremental payments are assumed independent).

The coefficient of variation rules noted above are a reflection of the Step 8s that are described in the Appendix, in the sense that they describe the process variance in the model. While the coefficients of variation should go down, if they do start going back up in the most recent year or years, as illustrated above for 2007, then this could be the result of the following issues:

The parameter uncertainty tends to increase when moving from the oldest years to the more recent years as more and more parameters are used in the model. In the most recent years, the parameter uncertainty could be “overpowering” the process uncertainty, causing the coefficient of variation to start going back up. At the very least, the increasing parameter uncertainty will cause the rate of decrease in the coefficient of variation to slow down.

If the increase in the most recent years is significant, then this could indicate that the model is overestimating the uncertainty in those years. If this is the case, then an adjustment to the model parameters may be needed (e.g., limit incrementals to zero) or you may need to use a Bornhuetter-Ferguson or Cape Cod model instead of a chain-ladder model.15

In addition to the correlation within each accident year, the rules for the standard error and coefficient of variation for the total of all years also include the impact of the correlation between

14 These standard-error and coefficient-of-variation rules are based on the independence of the incremental process

risk and assume that the underlying exposures are relatively stable from year to year—i.e., no radical changes. In practice, random changes do occur from one year to the next that could cause the actual standard errors to deviate from these rules somewhat. In other words, these rules should generally hold true, but are not considered hard and fast in every case. Strictly speaking, the total-of-all-years rules assume that the individual years are not positively correlated.

15 Caution should be exercised in the interpretation and adjustments for increases in the coefficient of variation in recent years. While keeping the theory in mind is appropriate, this must be balanced with the need to keep from underestimating the uncertainty of the more recent years.

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accident years. In essence, when one or more accident years are “bad” we do not expect all accident years to be “bad.” To see this impact, you can add the accident year standard errors and note that they will not sum to the standard error for all years combined.

The next diagnostic element in the estimated-unpaid table in Figure 8XX are the columns labeled Minimum and Maximum. In these columns, the smallest and largest values from among all iterations of the simulation are displayed. These values can be reviewed and judged to make sure that they do not fall outside the “realm of possibility.” If they do seem a bit unrealistic then they could indicate the need to review the model assumptions. For example, the presence of negative numbers might lead to limiting incremental values to zero. Sometimes “extreme” outliers in the results will show up in these columns and may also distort the histogram (discussed later in this case study).

As a final observation regarding the table in Figure 8XX, notice that there are three rows of percentile numbers and then four rows of TVaR numbers at the bottom of these tables under the totals row. The three percentile rows—the normal, lognormal and gamma distributions, respectively—have been fit to the total unpaid-claim distribution. The fitted mean, standard deviation, and selected percentiles are shown under the columns labeled Mean, Standard Error, and Percentile, respectively, so that the smoothed results can be used to judge the quality of fit or for other purposes and so that the smoothed results can be used for estimates of the extreme values.16

The Tail Value at Risk (TVaR) is the average of all of the simulated values equal to or greater than the percentile value. For example, in the table in Figure 8XX the 90th percentile value for the total unpaid for all accident years combined is 465,418 and the average of all simulated values that are greater than or equal to 465,418 is 502,553. The “Normal TVAR,” “Lognormal TVaR,” and “Gamma TVaR” rows are calculated the same way, except that instead of using the actual simulated values from the model the respective fitted distributions are used in the calculations.

To interpret the TVaR numbers, the question we are trying to answer is “if the actual outcome does exceed the X percentile value, by how much might it exceed that value on average?” This is an important question related to risk-based capital calculations and other technical aspects of enterprise risk management. It is worth noting, however, that the purpose of the normal, lognormal, and gamma TVaR numbers is to provide “smoothed” values in the sense that some of the random noise is kept from distorting the calculations.

16 Of course the use of the extreme values assumes that the moseare reliable.

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Estimated cash flow results

In addition to the results by accident year, we can also review the model output by calendar year (or by future diagonal) in the Estimated Cash Flow table shown below in Figure 9XX. Comparing the tables in Figures 9XX and 8XX, notice that the total row is identical; the total is the same whether you add the parts horizontally or diagonally. Similar diagnostic issues can be reviewed in the table in Figure 9XX, except that the relative values of the standard errors and coefficients of variation move in the opposite direction for calendar years compared to accident years. This should make intuitive sense as the “final” payments projected the furthest out into the future should be the smallest yet relatively most uncertain.

Figure 9XX: Estimated-cash-flow Model Results

Estimated-ultimate-loss-ratio results

Another output table shows the estimated ultimate-loss ratios by accident year as shown in the table in Figure 10XX. Unlike the estimated-unpaid and estimated-cash-flow tables, the values in the estimated-ultimate-loss-ratios table are calculated from all simulated values, not just the values beyond the end of the triangles. In other words, since the simulated sample triangles represent other possibilities of what could have happened in the past, and the “squaring of the triangle” and process variance represent what could happen as those other possible past values play out into the future, we have enough information to estimate the complete variability in the loss ratio from day one in each accident year until all claims are completely paid and settled.17

17 If we are only interested in the “remaining” volatility in the loss ratio, then the values in the estimated-unpaid

table can be added to the cumulative values in the data-input table and divided by the premiums.

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Figure 10XX: Estimated-loss-ratio Model Results

Because we are using all simulated values, the standard errors in the table in Figure 10XX should be proportionate to the means while the coefficients of variation should be relatively constant by accident year. Diagnostically, the increases in standard error and coefficient of variation for the latest few years are consistent with the reasons cited earlier for the estimated-unpaid tables. In addition, the drop-off in the mean loss ratio in the last two years could indicate the need to switch to another model like the Bornhuetter-Ferguson.18

Estimated incremental results

The next two output-result tables are designed to help us take a deeper look at the simulations and to understand the reasons for increases in the coefficients of variation illustrated in the tables in Figures 8XX and 9XX. The next two tables show the mean and standard errors, respectively, by accident year by incremental period. As illustrated in the table in Figure 11XX, the mean values by incremental period can be reviewed down each column or across each row to look for any irregularities in the expected patterns. The standard errors can be similarly reviewed as illustrated in the table in Figure 12XX.

As you can see by looking down the columns for 24 and 36 months in the tables in Figures 11XX and 12XX, it appears as though the means have dropped off a bit in 2006 and 2007 compared to the exposures and that there might be “too much” variability in future incremental values for 2006 and 2007—i.e., the values below the diagonal line do not appear consistent with the values above the line. This does not imply that the values above the diagonal line are correct and that the values below are overstated, just that they are not always consistent. These inconsistencies appear to be affecting both the unpaid and loss-ratio results for 2006 and 2007.

18 The decrease in mean loss ratios (and the increase in coefficient of variation) is consistent with what one might

find in a similar deterministic analysis—i.e., a lower-than-average paid to date could project to an ultimate that is too low.

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Figure 11XX: Estimated Means by Accident Year by Incremental Period

Figure 12XX: Estimated Standard Errors by Accident Year by Incremental Period

Tying these tables together, we can note that the mean incremental values in Figure 11XX will add to the mean unpaid values in Figures 8XX and 9XX. A couple of individual-standard-deviation incremental values in Figure 12XX will match the corresponding value in Figures 8XX or 9XX, respectively, but summing a row or diagonal can only be done by noting the independence of the values.19

Distribution graph

The final model output we will illustrate is a histogram of the estimated-unpaid amounts for the total of all accident years combined, as shown in the graph in Figure 13XX below. The total-unpaid-distribution graph, or histogram, is created by dividing the range of all values from the simulation (using the maximum and minimum values) into 100 “buckets” of equal size and counting the number of simulations that fall within each “bucket.” Dividing by the total number of simulations (10,000 in this case) results in the frequency or probability for each “bucket” in the graph.

Since the simulation results often look “jagged,” as they do in Figure 13XX, a Kernel density function is also used to calculate a “smoothed” line fit to the histogram values. The Kernel density distribution is represented by the blue line in Figure 13XX. In simple terms, a Kernel density function can be thought of as a weighted average of values “close” to each point in the “jagged” distribution with progressively less weight being given to values the further they are from the point

19 The sum of a row or diagonal is equal to the square root of the sum of the squares of the individual values.

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being evaluated.20

Figure 13XX: Total Unpaid Claims Distribution

IV.D.2.c Combining model results

Once the results for each model have been reviewed and “finalized” in the iterative process involving the diagnostics and model output, they can be “combined” by assigning a weight to the results of each model. Similar to the process of weighting the results of different deterministic methods to arrive at an actuarial “best estimate,” the process of weighting the results of different stochastic models will result in an actuarial “best estimate of the distribution.”

Figure 14XX: Model Weights by Accident Year

By comparing the results for all six models (or fewer if fewer are used) a qualitative

20 For a more detailed discussion of Kernel density functions, see Wand & Jones, Kernel Smoothing, Chapman &

Hall, 1995.

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assessment of the relative merits of each model can be determined. Bayesian methods can be used to inform the selection of weights based on how well each model forecasts. The weights can be determined separately for each year so that different weights can be used for each year. For example, the table in Figure 14XX shows an example of weights for the medium-tail casualty data.21 The weighted results are then displayed in the “Best Estimate” column of Figure 15XX.

Figure 15XX: Summary of Results by Model

Since we are concerned with the entire distribution and not just a single point estimate, the weights by year are used to randomly sample the specified percentage of the iterations from each model. An example of the “weighted” results is shown in the table in Figure 16XX.

Figure 16XX: Estimated-unpaid Model Results (Best Estimate)

21 For simplicity, the weights are judgmental and not derived using Bayesian methods.

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Of course, for the weighted results we should also be able to populate tables of estimated cash flow, estimated ultimate-loss ratios, mean incrementals, and standard-deviation incremental, along with a graph of the total unpaid distribution, similar to what are shown in Figures 9XX, 10XX, 11XX, 12XX, and 13XX, respectively. In addition to these tables and graph, for the weighted results an additional graph could be created to summarize the distribution graphs for all models. An example this graph is shown in Figure 17XX.

Figure 17XX: Summary of Model Distributions

The estimated unpaid, estimated cash flow, estimated ultimate-loss ratios, mean incrementals, and standard-deviation incrementals, along with graphs of the total unpaid distribution and of a summary of model distributions for, respectively, the short-tail property, medium-tail casualty, and long-tail casualty segments are shown in Exhibits 2, 3, and 4.

IV.D.3 AGGREGATION OF RESULTS

Once the modeling is complete for each segment, we need to aggregate the results for the entire business unit using the correlation between the segments. However, rather than simply “add up” the distributions for each segment, in order to estimate the distribution of possible outcomes for the company as a whole we need to take correlation into account.22

22 The remainder of the case study assumes the reader is familiar with correlation. For the interested reader,

Appendix B provides some background on correlation.

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Figure 18XX: Impact of Correlation on Aggregate Distributions

For most insurance coverages, insurance claims tend to happen independently of each other. For example, personal-property claims are usually not related to automobile claims. However, one can usually find examples of positive correlation—e.g., catastrophes would tend to cause claims for multiple coverages at the same time—as well as examples of negative correlation, e.g., workers’ compensation and unemployment claims tend to move in opposite directions, such as when an increase in unemployment causes more unemployment claims to be filed while there are correspondingly fewer employed workers that can file workers’ compensation claims.

As such, these “correlations” between multiple segments will have an effect on the distribution of possible outcomes for all segments combined. If we are only concerned with the expected value of the aggregate distribution, then we can calculate the expected value for each segment separately and add all the expectations together. However, if we are concerned about the distribution or trying to quantify a value other than the mean, such as the 75th percentile, we cannot simply sum the segments. The only time the sum of the distributions would be appropriate for the aggregate is when all segments are 100% correlated with each other—a highly unlikely situation!

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The impact of correlation on the aggregate distribution can be illustrated graphically. In the graph in Figure 18XX we have kept the example simple by starting with three identical distributions. The sum of the three segments (which is equivalent to assuming 100% correlation) results in the same identical distribution, except that all numbers along the value axis (estimated liability) are three times as large.

Alternatively, if we assume 0% correlation between the three segments (i.e., independence), then the expected value (or mean) would be the same as the sum, but the resulting aggregate distribution is narrower since some positive outcomes would be offset by some negative outcomes, and vice versa, which means that other parts of the distribution will not be the same as the sum. Thus, the values for every part of the aggregate distribution, except the mean, would not equal three times the corresponding value for one of the individual distributions.

The degree to which the segments are correlated will influence the shape of the aggregate distribution. How significant will this impact be? That primarily depends upon three factors—how volatile (i.e., wide) the distributions are for the individual segments, the relative values of the amounts. and how strongly correlated the segments are with each other. All else being equal, if there is not much volatility then the strength of the correlation will not matter that much. If, however, there is considerable volatility, the strength of correlations (or lack thereof) will produce differences that could be significant.

It is important to note that the correlation between individual segments does not affect the distribution of any individual segment. It only influences the aggregate distribution of all segments combined.

IV.D.3.a Calculating correlation

There are several ways to measure correlation, both parametric (for example, Pearson’s R) and non-parametric (Spearman’s Rank Order, or Kendall’s Tau). Pearson's correlation coefficient may be less helpful if the underlying values are not “normally” distributed, whereas Spearman's and Kendall's formulas may be more useful when distributions are not normal.

For the bootstrap model, Spearman’s Rank Order calculation is often used to assess the correlation between each pair of segments. Since we want to measure the correlation of the incremental payments between pairs of segments, we can use the residuals as they reflect those movements directly. However, rather than calculating the correlation of the residuals themselves, Spearman’s formula calculates the correlation of the ranks of those residuals.23

23 See Appendix B for an example of the calculation.

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IV.D.3.b The correlation process

A common method of simulating correlated variables is to simulate them from a multivariate distribution, after specifying the parameters and the correlations for each variable in the distribution. This type of simulation is most easily applied when the distributions are all the same and they are known in advance (e.g., if they are all from a multivariate normal distribution) and are quite common in EC models. However, since we are estimating the distributions with bootstrapping, we don’t know them in advance and they might have different shapes so a different approach is needed.

Two useful correlation processes for the bootstrap model are location mapping and re-sorting.24 The basic idea for location mapping is that residuals for each pair of segments represent the correlation between the segments when comparing cell by cell the location of each residual, so we can use this to our advantage in the simulation process. For each iteration in the simulation process, the residuals for the first segment are sampled noting the location in the residual triangle of each sampled residual. Each of the other segments is sampled using the residuals at the same locations for their respective residual triangles. Thus, for each iteration, the correlation of the residuals is preserved in the sampling process.

The location-mapping process is easy to implement in Excel and actually does not require the need to calculate a correlation matrix. However, there are two drawbacks to the location-mapping process. First, it requires all of the business segments to have data triangles that are the exact same size and no missing values or outliers when comparing each location of the residuals.25 Second, the correlation of the residuals is used in the model and no other correlation assumptions can be used for stress testing of the aggregate results.

The re-sorting process can be accomplished using algorithms such as Iman-Conover, Copulas, etc. The primary advantages of the re-sorting option include: a) the triangles for each segment can be different shapes and sizes, b) different correlation assumptions can be used, and c) the different algorithms can have other beneficial impacts on the aggregate distribution. For example, using a t-distribution Copula with low degrees of freedom instead of a normal-distribution Copula will effectively “strengthen” the focus of the correlation in the tail of the distribution. This type of consideration is important for risk-based capital and other EC modeling issues.

In order to induce correlation between different segments in the bootstrap model, we will calculate the correlation matrix using Spearman's Rank Order and use re-sorting based on the ranks of the total unpaid for all accident years combined and the normal distribution. The calculated 24 For a useful reference see [5]. 25 It is possible to fill in “missing” residuals in another segment using a randomly selected residual from elsewhere

in the triangle, but in order to maintain the same amount of correlation the selection of the other residual would need to account for the correlation between the residuals, which complicates the process considerably.

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correlations based on the paid residuals after hetero adjustments are displayed in the table in Figure 19XX.

Figure 19XX: Estimated Correlation and P-values

Using these correlation coefficients and the simulation data used for Exhibits 2, 3, and 4, the aggregate results for the three business segments combined is summarized in the table in Figure 20XX. The complete set of tables for the aggregate results is shown in Exhibit 5.

Figure 20XX: Aggregate Estimated Unpaid

IV.D.4 COMMUNICATION

The communication of results should start with an internal peer review to ensure quality of work and consistency of analysis. All of the tables and graphs in Section 3 of this case study, including the assumptions, diagnostics, and results, will need to be reviewed for each segment. In addition, comparisons between segments could also reveal inconsistencies or key issues that might need further review.

Once the analysis has passed peer-review inspection, communication to management (as well as to any outside audiences) can continue. This communication can take various forms such as visual slides, printed exhibits, and written reports. The form of the communication will likely depend

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on the purpose and goals of the communication. The tables and graphs of Section 3 of this case study should be useful for most communication. The discussion in Section 3 will provide a useful road map for the peer-review and communication process. One of the key points to keep in mind is that we are trying to convey the depth and breadth of the risk profile we have calculated.

IV.D.5 COMPONENTS OF ECONOMIC CAPITAL

The results from a stochastic unpaid-claims analysis can be leveraged into various applications such as the insurance-risk component of economic capital, the allocation of required capital, the related underwriting profit targets by business segment and return-on-equity targets, various reinsurance options, and mergers and acquisitions, including capital management, strategic planning, performance management, and rating-agency discussions, to name some of the more common. As noted earlier, we can use the simulated outputs as inputs to a larger EC model or to parameterize an EC model that will simulate possible outcomes within the model. The next case study will illustrate the use of the distributions as part of a larger EC model, so we will illustrate the calculation of some insurance-risk components here.

IV.D.5.a Required capital

The required capital can be viewed as a “shared asset” in the sense that each segment can “tap into” and in fact “use all” the capital, as needed. Thus, the calculation of required capital is based on the aggregate distribution of all segments combined. Because one of the primary purposes of capital is to guard against the potential for adverse outcomes, the difference between a relatively unlikely but possible outcome, and the booked liability, is a quantifiable measure of required capital. For example, the difference between the 99th percentile and the mean estimate would signify the amount of capital required to cover 99.0% of the possible outcomes. This situation is illustrated in the graph in Figure 21XX.

While each business segment can “tap into” the total capital for a firm, we would not expect every segment to “go bad” (or need capital) at the same time. Quite the contrary—in general, we would expect “worse than expected” results from some segments to be offset by “better than expected” results from other segments. Of course, in practice we would also expect some events, such as catastrophes, unanticipated major changes in inflation, or major changes in interest rates and industry underwriting cycles, to affect multiple segments at the same time so the “better” and “worse” results by segment will generally not balance themselves out. In total, however, the effects of these correlation issues will generally result in a significant reduction in total required capital (or diversification benefit) for the firm, as illustrated in Figure 21XX.

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Figure 21XX: Required Capital

In addition to correlation, two other key issues of quantifying required capital are discounting and risk measure. Discounting of liabilities is generally an accounting issue by country, but it can also be thought of as a statutory (non-discounting, risk-based capital, conservative) issue versus an economic (discounting, economic capital, going concern) issue. While we have illustrated Value at Risk (VaR) as the risk measure, there are other choices (e.g., Tail Value at Risk, Expected Policyholder Deficit, etc.) that could be used.26 Finally, note that the required capital illustrated in Figure 21XX assumes that the mean of the distribution is used as the booked liability held on the balance sheet. Should the management of our case-study company, SIC, decide to book a liability different than the mean estimate, then the required capital would change by the difference between the calculated mean and the selected liability.27

Based on the unpaid claim estimates in Exhibits 2, 3, 4, and 5 we calculated the unpaid-claims-risk portion of required capital using the 99th percentile and summarized the results in the table in Figure 22XX. As illustrated in Figure 21XX, the unpaid-claims-risk capital calculated by summing the required capital for the segments is 465,000 in Figure 22XX, but after the impact of correlation the required capital becomes 261,000. The “savings” of 204,000 in required capital is the benefit from diversification. The total required capital is then allocated to the segments in proportion to the required capital by segment.28

26 See Section II.E for more discussion of risk measures. 27 Rather than illustrate various combinations of these options, we will only use undiscounted results and VaR in

the quantification of required capital. 28 Other methods of allocating capital by segment are more complex. For a more complete discussion of the

different methods of allocating capital see [9] and [12].

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Figure 22XX: Required Capital from Unpaid-claims Risk

Indicated Unpaid-claims-risk Portion of Required Capital as of December 31, 2007

(No Discounting, Modeled Correlation, in 000s)

(1)

Mean Unpaid

(2)

99.0% Unpaid

(3) (2) – (1)

Indicated VaR Risk Capital

(4) Allocated 29

Unpaid Claim Risk

Capital

(5) (4) / (1)

Ratio

Short-tail Property 137 205 67 37 27.5% Medium-tail Casualty 542 784 242 136 25.1%

Long-tail Casualty 546 701 155 87 16.0% Sum 1,225 1,690 465 SIC Aggregate 30 1,225 1,486 261 261 21.3%

In addition to unpaid-claims distributions, we can use the distributions of ultimate-loss ratios and future cash flows to quantify other insurance-risk components of required capital in a similar fashion to the unpaid-claims risk. Using the ultimate-loss ratio distributions for 2006 by segment, we calculated the pricing-risk portion of required capital using the 99th percentile and summarized the results in the table in Figure 23XX.31 Rather than using the calculated-correlation matrix, we assumed that the future pricing correlation between segments will be stronger than indicated by the calculations because of contagion risk and underwriting cycles. Thus, we assumed all of the correlation coefficients are 50% for pricing risk.

29 The allocated unpaid-claims-risk capital amounts, shown in Column (4) in Figure 22XX, are based on shares for

each segment, which sum to the aggregate unpaid-claims-risk capital and the proportionate share of indicated capital, shown in Column (3) by segment.

30 The SIC aggregate VaR is less than the sum of the segment VaRs due to the impact of correlation between the three segments.

31 We simply used 2006 as a proxy for 2008. In practice, consideration of actual rate-level changes and underwriting-cycle information could lead to a better estimate of the expected ultimate loss ratio for 2008 and beyond.

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Figure 23XX: Required Capital from Pricing Risk

Indicated Pricing-risk Portion of Required Capital as of December 31, 2007

(No Discounting, 50% Correlation, in 000s)

(1)

2008 Premium

(2)

Mean Unpaid

(3)

99.0% Unpaid

(4) (3) – (2)

Indicated VaR Risk Capital

(5)

Allocated Pricing

Risk Capital

(6) (5) / (1)

Ratio

Short-tail Property 1,700 1,132 1,270 138 79 4.6% Medium-tail Casualty 300 201 385 183 105 34.8%

Long-tail Casualty 350 279 386 105 60 17.1% Sum 2,350 1,612 2,039 427 SIC Aggregate 2,350 1,612 1,855 242 242 10.3%

The future premiums shown in Figure 23XX are the estimated premiums for 2008, so that the ratios in Column (6) are equivalent to the familiar premium-to-surplus ratio. In practice, the pricing-risk analysis can be extended for more future years and would add future values to the cash-flow analysis. Finally, note that the complete required-capital calculation will require additional inputs for other risks such as credit risk, market risk, and operational risk.

IV.D REFERENCES

[1] CAS Working Party on Quantifying Variability in Reserve Estimates. “The Analysis and Estimation of Loss & ALAE Variability: A Summary Report.” CAS Fall Forum (2005): 29-146.

[2] England, Peter D. and Richard J.Verrall. “Analytic and Bootstrap Estimates of Prediction Errors in Claims Reserving,” Insurance: Mathematics and Economics 25 (1999): 281-293.

[3] England, Peter D. and Richard J.Verrall. “A Flexible Framework for Stochastic Claims Reserving” PCAS Vol. LXXXVIII (2001): 1-38.

[4] England, Peter D. and Richard J.Verrall. “Stochastic Claims Reserving in General Insurance,” British Actuarial Journal 8 (2002): 443-544.

[5] Kirschner, Gerald S., Colin Kerley, and Belinda Isaacs. “Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business,” Variance Vol. 2-1 (2008): 15-38.

[6] Mack, Thomas. “Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates,” ASTIN Bulletin 23:2 (1993): , 213-25.

[7] Milliman’s P&C Insurance Software Team. “Using the Milliman Reserve Variability Model: An Overview of the Milliman Bootstrap Modeling System,” Version 1.2, 2008.

[8] Pinheiro, Paulo J. R., João Manuel Andrade e Silva and Maria de Lourdes Centeno. “Bootstrap Methodology in Claim Reserving,” ASTIN Colloquium, 2001.

[9] Ruhm, David L. and Donald F.Mango. “A Method of Implementing Myers-Read Capital Allocation in Simulation,” CAS Fall Forum (2003): 451-458.

[10] Shapland, Mark R. “Loss Reserve Estimates: A Statistical Approach for Determining ‘Reasonableness’,” Variance Vol.1-1 (2007): 121-148.

[11] Venter, Gary G. “Testing the Assumptions of Age-to-Age Factors,” PCAS Vol. LXXXV (1998): , 807-47. [12] Venter, Gary G. “A Survey of Capital Allocation Methods with Commentary Topic 3: Risk Control,” ASTIN

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Colloquium, 2003. [13] Verrall, Richard J. “A Bayesian Generalized Linear Model for the Bornhuetter-Ferguson Method of Claims

Reserving,” North American Actuarial Journal Vol. 8, No. 3 (2004): 67-89. Abbreviations and notations Collect here in alphabetical order all abbreviations and notations used in the paper EC, economic capital

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Exhibit IV.D-1: Raw Data by Segment

Short-tail Property Business Cumulative Paid-loss Triangle

Accident Year 12 24 36 48 60 72 84 96 108 120 1998 633,138 702,081 702,776 702,597 702,582 702,574 702,600 702,603 702,603 702,6061999 690,044 753,278 753,974 753,728 753,699 753,755 753,759 753,776 753,7402000 715,149 788,987 789,751 789,417 789,483 789,454 789,458 789,4592001 776,683 869,154 869,418 869,503 869,526 869,443 869,4532002 835,881 969,913 970,431 970,570 970,667 970,6872003 852,298 962,409 962,621 962,724 962,7912004 925,163 1,043,937 1,044,530 1,044,5802005 996,162 1,111,148 1,111,6662006 1,037,082 1,146,0982007 1,097,779

Cumulative Incurred-loss Triangle Accident

Year 12 24 36 48 60 72 84 96 108 120 1998 676,785 702,528 702,863 702,637 702,602 702,578 702,600 702,603 702,603 702,6061999 735,945 753,820 754,069 753,769 753,719 753,762 753,766 753,776 753,7402000 771,742 789,568 789,908 789,482 789,515 789,462 789,458 789,4592001 843,292 869,893 869,524 869,548 869,544 869,450 869,4532002 934,776 970,853 970,647 970,630 970,685 970,6912003 928,419 963,823 962,801 962,755 962,7942004 1,008,315 1,044,810 1,044,644 1,044,6122005 1,073,005 1,111,893 1,111,7872006 1,100,650 1,146,6412007 1,166,903

Calendar

Year Earned

Premium Earned

Exposures 1998 691,456 4,526 1999 810,079 4,905 2000 952,501 5,278 2001 1,087,494 5,582 2002 1,189,652 5,846 2003 1,291,574 6,019 2004 1,458,613 6,420 2005 1,646,366 6,907 2006 1,721,347 7,265 2007 1,715,950 7,504

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Medium-tail Casualty Business

Cumulative Paid-loss Triangle Accident

Year 12 24 36 48 60 72 84 96 108 120 1998 9,517 29,915 46,152 58,997 73,021 88,298 92,190 93,679 94,269 94,2691999 9,365 32,876 57,714 89,184 101,039 109,576 123,613 126,445 127,5392000 7,726 34,370 59,003 86,042 106,011 109,918 123,109 126,0922001 8,261 37,736 58,379 99,932 107,179 116,542 122,0282002 11,786 30,222 56,605 77,070 90,497 103,0142003 10,535 35,458 57,998 79,408 98,1842004 11,724 34,268 64,358 92,6362005 9,561 37,796 76,1832006 10,018 36,2582007 9,149

Cumulative Incurred-loss Triangle Accident

Year 12 24 36 48 60 72 84 96 108 120 1998 61,031 62,004 82,200 74,078 79,713 103,520 100,403 97,249 94,269 94,2691999 47,864 61,938 101,627 121,076 147,448 144,723 135,012 130,060 127,5392000 27,940 69,617 108,167 104,884 133,231 125,720 125,973 127,5632001 40,334 76,821 95,402 129,831 147,298 159,224 136,5152002 61,777 57,617 93,842 102,773 115,469 140,6892003 74,724 68,238 105,243 97,981 136,6172004 60,430 69,929 111,746 126,2292005 48,068 76,242 135,3322006 48,261 67,0122007 48,873

Calendar

Year Earned

Premium Earned

Exposures 1998 159,259 1,665 1999 171,133 1,782 2000 184,964 1,903 2001 190,587 1,999 2002 192,215 2,078 2003 197,982 2,127 2004 216,206 2,267 2005 245,144 2,446 2006 278,137 2,583 2007 297,604 2,667

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Long-tail Casualty Business

Cumulative Paid-loss Triangle Accident

Year 12 24 36 48 60 72 84 96 108 120 1998 26,453 50,795 60,576 64,988 68,047 69,557 71,158 71,603 73,120 73,3211999 24,968 47,880 56,324 62,265 65,595 68,776 70,277 71,230 71,8432000 26,426 47,491 58,548 63,709 67,454 71,041 72,546 73,3272001 26,922 55,991 68,865 76,951 85,351 90,432 94,5512002 27,727 57,487 67,796 77,489 81,584 83,9462003 32,901 68,559 84,330 94,676 102,4702004 49,268 109,278 140,617 163,0742005 66,695 137,801 171,6842006 65,820 148,0362007 68,147

Cumulative Incurred-loss Triangle Accident

Year 12 24 36 48 60 72 84 96 108 120 1998 50,947 65,810 69,642 70,593 70,926 71,267 73,263 75,084 74,068 73,4341999 46,488 60,277 65,964 69,713 71,788 72,780 73,396 73,619 75,3242000 49,839 63,077 68,256 69,440 71,718 74,568 74,448 75,6462001 61,884 80,397 86,551 94,218 102,053 101,615 113,5132002 53,356 71,295 77,395 86,863 89,656 91,3562003 66,344 88,119 101,862 105,915 111,3282004 101,272 154,280 176,998 192,5952005 149,776 190,519 213,0072006 153,855 236,1132007 151,973

Calendar

Year Earned

Premium Earned

Exposures 1998 182,934 530 1999 162,791 536 2000 150,917 541 2001 145,400 563 2002 160,840 597 2003 181,833 629 2004 264,025 700 2005 326,343 747 2006 360,235 752 2007 373,839 758

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Exhibit IV.D-2: Short-tail Property Results

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Exhibit IV.D-3: Medium-tail Casualty Results

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Exhibit IV.D-4: Long-tail Casualty Results

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Exhibit IV.D-5: Aggregate Results

(Using Model Correlations)

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(Using 50% Correlation)

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IV.D.6 Appendix A – THE BOOTSTRAP MODEL

The term “bootstrapping” originates in German literature, from legends about Baron von Münchhausen, who was known for the often unbelievable tales he told of his fantastic adventures. Supposedly he could lift himself out of a swamp or quicksand by pulling himself up by his own hair. In later versions of similar tales, he pulled himself out of the sea by pulling up on his bootstraps, thus forming the basis for the term bootstrapping.32

The term has taken on broad application in many fields of study, including physics, biology and medical research, computer science, and statistics. Though ultimately having grown to mean different things in each of the above applications, they generally start from the premise of using simple information and building it bit by bit into more complex systems, often using only one’s own data to estimate or project more complex information about that data (like the Baron, who needed nothing more than what he already had to pull himself out of the sea).

Bradley Efron, chairman of the Department of Statistics at Stanford University, is credited with expanding the concept of the bootstrap estimate into the realm of statistics. In his work, "bootstrap" means that one available sample gives rise to many others by resampling the existing data. He suggests duplicating the original sample as many times as computing resources will allow, and then treating this expanded sample as a virtual population. Then samples are drawn with replacements from this population to verify the estimators.

Several writers in actuarial literature have applied this concept to the process of loss reserving. The most commonly cited examples are from England and Verrall ([2] and [4]), Pinheiro, et al. [8], and Kirschner, et al. [5]. In its simplest form, they suggested using a basic chain-ladder technique to square a triangle of paid losses, repeating that randomly and stochastically a large number of times, and then evaluating the distribution of the outcomes.

IV.D.6.a A simple paid-loss chain-ladder simulation

For purposes of providing a simple review of the algorithm’s steps, we will follow the most basic form, a simulation of possible future outcomes based on the paid-loss triangle and the basic chain-ladder approach. With random sampling from a triangle of residuals, the model simulates a large number of “sample” triangles, uses the chain-ladder model to estimate the future payment triangles (lower right), and then calculates a distribution from the many random estimates of future payments.

32 This Appendix borrows heavily from [7].

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In very simple terms, the model performs the following steps (of course, the reality is a bit more complex):

1. Use a triangle of cumulative paid losses as input. Calculate a triangle of age-to-age development factors. (Initially we’re using the chain-ladder model with volume-weighted averages.)

2. Calculate a new triangle of “fitted values”—i.e., use the age-to-age factors to “undevelop” each value in the latest diagonal to form a new triangle of values predicted by the model assumptions.

3. Working from the incremental versions of these triangles, calculate a triangle of residuals using the fitted triangle and the original data. These are called “unscaled Pearson residuals” in the model.

4. Standardize the residuals so that they are independent and identically distributed (i.i.d.).

5. Adjust the standardized residuals for degrees of freedom, resulting in “scaled Pearson residuals.”

6. Create a new incremental sample triangle by selecting randomly with replacement from among the triangle of scaled Pearson residuals.

7. Develop and square that sample triangle, adding tail factors and estimating ultimate losses.

8. Add process variance to the future incremental values from Step 7 (which will change the estimated ultimate).

9. Calculate the total future payments (estimated unpaid amounts) for each year and in total for this iteration of the model.

10. Repeat the random selection, new triangle creation, and resulting unpaid calculations in Steps 6 through 9, X times.

11. The result from the X simulations is an estimate of the distribution of possible outcomes. From this we can calculate the mean, standard deviation, percentiles, etc.

IV.D.6.b A simple incurred-loss chain-ladder simulation

We can also use incurred-loss data and follow the same steps as outlined above for paid data. When paid data is used, the resulting distribution is for total unpaid amounts since we are estimating the difference between the ultimate values and the payments to date. However, when incurred data is used (with the same algorithm) the resulting distribution is for the total incurred but not reported (IBNR) amount because we are estimating the difference between the ultimate values and the incurred-to-date.

In order to make an apples-to-apples comparison of the distribution of unpaid claims, as opposed to IBNR, the incurred model needs to include an additional step to convert the squared triangle of incremental incurred amounts to a squared triangle of incremental paid amounts. (Note that the paid and incurred calculations are identical through Step 8.)

So why not just add the case reserves to the distribution of IBNR to get a distribution of

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total reserves that will match the paid method? This would indeed result in a distribution of total reserves, but it would not be a consistent comparison since there would be no variation in the case-reserve portion of the distribution. By using the steps outlined below for the incurred data, the simulated sample triangles will include variations in the case reserves, which results in a complete apples-to-apples comparison of the unpaid claims (note, however, that the loss-ratio distribution will still be based on the incurred data).

In very simple terms, the model performs the following steps:

1. Use a triangle of cumulative incurred losses as input. Calculate a triangle of age-to-age development factors. (Initially we’re using the chain-ladder model with volume-weighted averages.)

2. Calculate a new triangle of “fitted values”—i.e., use the age-to-age factors to “undevelop” each value in the latest diagonal to form a new triangle of values predicted by the model assumptions.

3. Working from the incremental versions of these triangles, calculate a triangle of residuals using the fitted triangle and the original data. These are the unscaled Pearson residuals.

4. Standardize the residuals so that they are i.i.d.

5. Adjust the standardized residuals for degrees of freedom, resulting in scaled Pearson residuals.

6. Create a new incremental sample triangle by selecting randomly with replacement from among the triangle of scaled Pearson residuals.

7. Develop and square that sample triangle, adding tail factors and estimating ultimate losses.

8. Add process variance to the future incremental values from Step 7 (which will change the estimated ultimate) and calculate the ultimate value for each year estimated by this iteration of the model.

9. Simulate in parallel using paid losses (Steps 1 to 8). Adjust the simulated incremental paid losses so that the total ultimate value for each accident year matches the total ultimate value for the incurred losses in Step 8.

10. Calculate the total future payments (estimated unpaid amounts) for each year and in total for this iteration of the model, using the paid losses after they are adjusted to match the ultimate incurred amounts.

11. Repeat the random selection, new triangle creation, and resulting adjusted unpaid calculations in Steps 6 through 10 X times.

12. The result from the X simulations is an estimate of the distribution of possible outcomes. From this we can calculate the mean, standard deviation, percentiles, etc.

IV.D.6.c A simple Bornhuetter-Ferguson simulation

We can extend both the paid and incurred chain-ladder models by incorporating the Bornhuetter-Ferguson method into the model steps. The Bornhuetter-Ferguson bootstrap model requires an additional set of parameters for the ultimate losses (i.e., the mean or a priori) per year.

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Replacing Step 7 of the basic chain-ladder model, the ultimate loss will be calculated using the Bornhuetter-Ferguson method. Using the paid Bornhuetter-Ferguson with premiums and an a priori loss-ratio assumption is typical:

Ultimate Loss = Paid to date + (1 – %Paid) x Premium x Loss Ratio (4.D.1)

The a priori loss ratio can be simulated from a distribution to include uncertainty in the a priori ultimate. The paid-to-date has been simulated in Step 6. The Bornhuetter-Ferguson method gives the total remaining unpaid amount for that year. The total unpaid needs to be divided into incremental losses. This can be done by using the age-to-age factors to determine the proportion that should fall in each incremental period. After this step, we can proceed to Step 8 of the normal chain-ladder method, where we add process variance to the future incremental values.

The incurred Bornhuetter-Ferguson model uses the same extra step to convert the total IBNR to total unpaid as the incurred chain-ladder model, except that it will use a parallel paid Bornhuetter-Ferguson model instead of a paid chain-ladder model in Step 9. For simplicity, we will only illustrate the paid Bornhuetter-Ferguson model later in this appendix.

IV.D.6.d A simple Cape Cod simulation

Similarly, we can extend both the paid and incurred chain-ladder models by incorporating the Cape Cod method into the model steps. The Cape Cod bootstrap model requires these additional parameters:

• Premium index factors (to adjust premiums to current rate level)

• Loss trends (to adjust for loss cost inflation)

• Weights (to mark which periods to include in the loss-ratio weighted average)

• Decay rates (to systematically reduce the weight given to each year the further that each year is from the year being calculated)

Step 7 of the basic chain-ladder model will use the Cape Cod method to calculate the ultimate loss. The advantage of the Cape Cod over the Bornhuetter-Ferguson is that the uncertainty of the ultimate values can be based on the stochastic iterations instead of an input variable.

The calculation of the incremental values for the next step can be done using the age-to-age factors to determine the proportion that should fall in each incremental period of the Bornhuetter-Ferguson model. After this step, we can proceed to Step 8 of the normal chain-ladder method, where we add process variance to the future incremental values.

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The incurred Cape Cod model uses the same extra step to convert the total IBNR to total unpaid as the incurred chain-ladder model, except that it will use a parallel paid-Cape-Cod model instead of a paid chain-ladder model in Step 9. Keeping with the simplicity theme, we will illustrate only the paid-Cape-Cod model, later in this Appendix.

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IV.D.6.e A walkthrough of the basic calculation, based on paid-loss data

Step 1. Build a basic development model.

Use the standard chain-ladder method and the all-period volume-weighted average (VWA) to calculate age-to-age factors.

Cumulative paid-loss data

12 24 36 48 60 2003 352 783 1,045 1,183 1,295 2004 255 572 710 750 2005 279 638 767 2006 311 717 2007 308

12-24 24-36 36-48 48-60 60+

VWAs 2.264 1.265 1.101 1.095 1.000

Step 2. Create a “fitted” triangle.

Start with the most recent cumulative diagonal and “un-develop” backwards using the appropriate VWAs.

Triangle fitted backwards from latest diagonal

12 24 36 48 60 2003 375 849 1,074 1,183 1,295 2004 238 538 681 750 2005 268 606 767 2006 317 717 2007 308

Step 3. Calculate Pearson residuals.

Working from the incremental forms of both triangles, subtract the fitted from the actual amount for each cell. Divide each result by the square root of the fitted amount. This results in unscaled Pearson residuals.

)ˆ(

ˆ

mabsmCrUP

−=

C = actual incremental amount

m̂ = fitted incremental for matching location in each triangle

Unscaled Pearson residuals

12 24 36 48 60

2003 -1.18 -1.97 2.45 2.78 2004 1.12 0.96 -0.40 -3.50 2005 0.69 1.12 -2.51 2006 -0.32 0.28 2007

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Step 4. Standardize the residuals.

Multiply each unscaled residual by the hat matrix adjustment factor.33

fij = 1

1 ijh−

hij = the diagonal of the hat matrix H H = X(XTWX)-1XTW X = the design matrix from the GLM model W = the weight matrix from the GLM model.

Hat matrix adjustment factors

12 24 36 48 60 2003 1.4922 1.6088 1.4725 1.6849 1.0000 2004 1.3675 1.4675 1.3033 1.3415 2005 1.4172 1.5350 1.3485 2006 1.5606 1.7546 2007 1.0000

Standardized Pearson residuals

12 24 36 48 60 2003 -1.76 -3.17 3.60 4.69 2004 1.54 1.40 -0.53 -4.69 2005 0.98 1.72 -3.39 2006 -0.50 0.50 2007

Step 5. Adjust the residuals for degrees of freedom.

Multiply each standardized residual by the degrees of freedom (D. of F.) adjustment factor (dfa below).

dfa = )( pN

N−

N = number of non-zero residuals (less outliers)

p = number of parameters in the model (typically the number of columns in the residual triangle + the number of rows in the residual triangle + the number of additional hetero groups – 1)

Scaled Pearson residuals (adjusted for degrees of freedom)

12 24 36 48 60 2003 -2.60 -4.66 5.30 6.91 2004 2.26 2.07 -0.78 -6.91 2005 1.44 2.53 -4.99 2006 -0.74 0.74 2007

Additional statistics calculated

Pearson chi-squared statistic = sum of squares of standardized Pearson residuals = 90.65 Degrees of freedom = # of non-zero residuals in the model minus

# of parameters ( # columns + # rows – 1) = 13 – 7 = 6 Scale parameter = chi-squared statistic ÷ degrees of freedom

= 15.11 D. of F. adjustment factor = square root of [N ÷ (N – p)] = 1.472

Simulation steps:

33 See [8] for more details.

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Step 6. Randomly create a new triangle of “sample” data.

6a. Build a new triangle by randomly selecting (with replacement) from among the non-zero scaled Pearson residuals in Step 5.

6b. Create a triangle of sample data based on the randomly selected residuals. For each cell:

mmrC SP ˆˆ' +=

Randomly selected residuals

12 24 36 48 60 2003 2.26 -0.74 -4.99 -0.74 2.26 2004 -4.66 -4.66 2.26 -6.91 2005 -0.78 6.91 -0.74 2006 2.07 2.26 2007 6.91

Sample incremental triangle calculated based on the random residuals

12 24 36 48 60

2003 419 458 150 101 136 2004 166 220 170 12 2005 255 465 152 2006 353 446 2007 429

Step 7. Complete the new randomly generated triangle.

Calculate new VWAs and use them to complete the bottom right of the triangle.

Sample cumulative triangle

12 24 36 48 60

2003 419 877 1,027 1,128 1,264 2004 166 385 555 567 2005 255 720 872 2006 353 799 2007 429

12-24 24-36 36-48 48-60 60+

VWAs 2.332 1.238 1.071 1.120 1.000 Completed cumulative triangle with future payments

12 24 36 48 60 2003 419 877 1,027 1,128 1,264 2004 166 385 555 567 635 2005 255 720 872 934 1,047 2006 353 799 989 1,060 1,187 2007 429 1,001 1,239 1,327 1,487

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Step 8. Introduce process variance.

Calculate the incremental payments in the future payment stream from the cumulative completed triangle.

To add process variance in the simulation, replace every future incremental paid amount with a randomly selected point from a gamma distribution34 where:

Mean = the incremental paid loss amount

Variance = Mean x Scale Parameter (see Step 5)

Completed incremental triangle

12 24 36 48 60

2003 419 458 150 101 136 2004 166 220 170 12 68 2005 255 465 152 62 113 2006 353 446 190 71 128 2007 429 571 238 88 160

Randomly generated future incremental payments based on above

12 24 36 48 60

2003 2004 64 2005 47 79 2006 194 76 143 2007 505 256 69 198

Step 9. Calculate total unpaid amounts.

Sum the future incremental values to estimate the total unpaid loss by period and in total (in this example, 1,632).

This provides one estimated possible outcome.

Total estimated future incremental payments

12 24 36 48 60 Total

2003 2004 64 64 2005 47 79 125 2006 194 76 143 414 2007 505 256 69 198 1,028

1,632

Step 10. Repeat and summarize.

Repeat Steps 6 through 9 the specified number of times (e.g., 10,000 or more), capturing the resulting cash flows and the unpaid amounts by period and in total for each iteration. The resulting distribution of unpaid amounts from all the iterations can be used to calculate means, percentiles, etc.

Total simulated results

Mean

Unpaid Standard

Error Coefficient of

Variation

2003 0 0 0.0% 2004 72 44 61.2% 2005 161 68 42.5% 2006 382 115 30.0% 2007 767 233 30.4%

1,382 311 22.5%

34 As a technical note, the gamma distribution is used as an approximation to the over-dispersed Poisson. The use

of volume-weighted average LDFs corresponds to the GLM assumption of an over-dispersed Poisson distribution, but the gamma is a very close approximation and runs much faster in simulation software.

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IV.D.6.f A walkthrough of the basic calculation, based on incurred-loss data

Step 1. Build a basic development model.

Use the standard chain-ladder method and the all-period volume-weighted average (VWA) to calculate age-to-age factors.

Cumulative incurred loss data

12 24 36 48 60 2003 715 1,077 1,184 1,285 1,295 2004 654 794 804 835 2005 655 886 910 2006 837 937 2007 747

12-24 24-36 36-48 48-60 60+

VWAs 1.291 1.051 1.066 1.008 1.000

Step 2. Create a “fitted” triangle.

Start with the most recent cumulative diagonal and “un-develop” backwards using the appropriate VWAs.

Triangle fitted backwards from latest diagonal

12 24 36 48 60 2003 888 1,146 1,205 1,285 1,295 2004 577 745 783 835 2005 671 866 910 2006 726 937 2007 747

Step 3. Calculate Pearson residuals.

Working from the incremental forms of both triangles, subtract the fitted from the actual amount for each cell. Divide each result by the square root of the fitted amount. This results in unscaled Pearson residuals.

)ˆ(

ˆ

mabsmCrUP

−=

C = actual incremental amount

m̂ = fitted incremental for matching location in each triangle

Unscaled Pearson residuals

12 24 36 48 60

2003 -5.80 6.44 6.32 2.35 2004 3.21 -2.16 -4.55 -2.91 2005 -0.60 2.56 -3.05 2006 4.13 -7.66 2007

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Step 4. Standardize the residuals.

Multiply each unscaled residual by the hat matrix adjustment factor.

fij = 1

1 ijh−

hij = the diagonal of the hat matrix H H = X(XTWX)-1XTW X = the design matrix from the GLM model W = the weight matrix from the GLM model

Hat matrix adjustment factors

12 24 36 48 60 2003 2.2611 1.3536 1.3399 1.6454 1.0000 2004 2.0614 1.2555 1.1985 1.3264 2005 2.2445 1.2904 1.2379 2006 2.4376 1.3153 2007 1.0000

Standardized Pearson residuals

12 24 36 48 60 2003 -13.12 8.71 8.47 3.86 2004 6.61 -2.71 -5.46 -3.86 2005 -1.34 3.30 -3.77 2006 10.07 -10.07 2007

Step 5. Adjust the residuals for degrees of freedom.

Multiply each standardized residual by the degrees of freedom (D. of F.) adjustment factor (dfa below).

dfa = )( pN

N−

N = number of non-zero residuals (less outliers)

p = number of parameters in the model (typically the number of columns in the residual triangle + the number of rows in the residual triangle + the number of additional hetero groups – 1)

Scaled Pearson residuals (adjusted for degrees of freedom)

12 24 36 48 60 2003 -19.31 12.82 12.46 5.68 2004 9.74 -3.99 -8.03 -5.68 2005 -1.98 4.86 -5.55 2006 14.82 -14.82 2007

Additional statistics calculated

Pearson chi-squared statistic = sum of squares of standardized Pearson residuals = 660.07 Degrees of freedom = # of non-zero residuals in the model minus

# of parameters ( # columns + # rows – 1) = 13 – 7 = 6 Scale parameter = chi-squared statistic ÷ degrees of freedom

= 110.01 D. of F. adjustment factor = square root of ( N ÷ (N – p) ) = 1.472

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Simulation steps:

Step 6. Randomly create a new triangle of “sample” data.

6a. Build a new triangle by randomly selecting (with replacement) from among the non-zero scaled Pearson residuals in Step 5.

6b. Create a triangle of sample data based on the randomly selected residuals. For each cell:

mmrC SP ˆˆ' +=

Randomly selected residuals

12 24 36 48 60

2003 -1.98 9.74 -5.68 -3.99 12.82 2004 4.86 -3.99 12.46 -5.55 2005 -19.31 -3.99 -19.31 2006 4.86 9.74 2007 14.82

Sample incremental triangle calculated based on the random residuals 12 24 36 48 60

2003 829 415 15 44 51 2004 694 116 115 12 2005 171 139 -84 2006 857 353 2007 1,152

Step 7. Complete the new randomly generated triangle.

Calculate new VWAs and use them to complete the bottom right of the triangle.

Sample cumulative triangle

12 24 36 48 60

2003 829 1,244 1,259 1,303 1,354 2004 694 810 925 937 2005 171 310 226 2006 857 1,210 2007 1,152

12-24 24-36 36-48 48-60 60+

VWAs 1.401 1.019 1.026 1.039 1.000 Completed cumulative triangle with future incurred amounts

12 24 36 48 60 2003 829 1,244 1,259 1,303 1,354 2004 694 810 925 937 973 2005 171 310 226 232 241 2006 857 1,210 1,233 1,265 1,314 2007 1,152 1,615 1,646 1,688 1,754

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Step 8. Introduce process variance and calculate totals.

Calculate the incremental incurred amounts from the cumulative completed triangle.

To add process variance in the simulation, replace every future incremental incurred amount with a randomly selected point from a gamma distribution where:

Mean = the incremental incurred loss amount

Variance = Mean x Scale Parameter (see Step 5)

Completed incremental triangle

12 24 36 48 60

2003 829 415 15 44 51 2004 694 116 115 12 36 2005 171 139 -84 6 9 2006 857 353 23 32 49 2007 1,152 462 31 42 65

Randomly generated process variance

12 24 36 48 60 Total

2003 829 415 15 44 51 1,354 2004 694 116 115 12 115 1,052 2005 171 139 -84 7 7 240 2006 857 353 9 55 28 1,301 2007 1,152 541 11 71 143 1,918

5,865

Note that up to this point, the calculations for the incurred model have been identical to those in the paid simulation.

Step 9. Convert to paid-loss development pattern.

Starting with the ultimate incurred values, convert to incremental paid losses using the paid-loss development pattern from Step 8 of the paid simulation. This is necessary to provide a distribution of unpaid loss as opposed to IBNR.

Note: We are using the values from Step 8 in the paid example for illustration purposes only. The simulation process does not store the results from the paid model to use with the incurred model, but will in effect run a “new” paid model (using the paid parameters) in order to generate results for the incurred model.

Incremental paid with process variance (from paid method, Step 8)

12 24 36 48 60 Total 2003 419 458 150 101 136 1,264 2004 166 220 170 12 64 631 2005 255 465 152 47 79 997 2006 353 446 194 76 143 1,213 2007 429 505 256 69 198 1,458

5,563

Incremental triangle converted to paid pattern

12 24 36 48 60 Total 2003 448 490 161 108 146 1,354 2004 276 366 283 19 107 1,052 2005 61 112 36 11 19 240 2006 379 478 209 81 154 1,301 2007 565 665 336 91 261 1,918

5,865

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Step 10. Calculate total unpaid amounts.

Sum the future incremental values to estimate the total unpaid loss by period and in total (in the example, 1,934).

This provides one estimated possible outcome.

Incremental unpaid with process variance

12 24 36 48 60 Total 2003 2004 107 107 2005 11 19 30 2006 209 81 154 444 2007 665 336 91 261 1,353

1,934

Step 11. Repeat and summarize.

Repeat Steps 6 through 10 the specified number of times (e.g., 10,000 or more), capturing the resulting cash flows and the unpaid amounts by period and in total for each iteration. The resulting distribution of unpaid amounts from all the iterations can be used to calculate means, percentiles, etc.

Total simulated results

Mean

Unpaid Standard

Error Coefficient of

Variation

2003 0 0 0.0% 2004 72 49 68.8% 2005 166 86 51.4% 2006 365 152 41.7% 2007 785 357 45.5%

1,388 416 30.0%

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IV.D.6.g A walkthrough of the basic Bornhuetter-Ferguson calculation (paid data)

Note that, for simplicity, we are using the data from Steps 1 through 6 of the paid chain-ladder example. In the actual simulations the Bornhuetter-Ferguson (BF) model is simulated independently of the chain-ladder model.

Step 7. Complete the new randomly generated triangle.

Calculate new VWAs and BF unpaid ratios. A priori loss ratios are simulated from the selected distribution with the selected mean and CoV. Use these to complete the bottom right of the triangle.

The differences in the BF unpaid ratios are used to allocate the total unpaid to incremental periods in Step 7. The results of the allocation are shown in Step 8.

Sample cumulative triangle

12 24 36 48 60 2003 419 877 1,027 1,128 1,264 2004 166 385 555 567 2005 255 720 872 2006 353 799 2007 429

12-24 24-36 36-48 48-60 60+

VWAs 2.332 1.238 1.071 1.120 1.000 CDF 3.465 1.486 1.200 1.120 1.000 BFUnpd 0.711 0.327 0.167 0.108 0.000

Premium Mean L/R CoV

Simulated L/R

A Priori Ultimate

Total Unpaid

(1)

(2)

(3)

(4)

(5)

(4) x (1) (6)

(5) x Unpd 2003 2,000 55.0% 20.0% 60.2% 1,204 0 2004 2,000 55.0% 20.0% 50.9% 1,018 109 2005 2,000 55.0% 20.0% 49.9% 999 167 2006 2,000 55.0% 20.0% 39.7% 794 260 2007 2,000 55.0% 20.0% 87.5% 1,750 1,245

Step 8. Introduce process variance.

To add process variance in the simulation, replace every future incremental paid amount with a randomly selected point from a gamma distribution where:

Mean = the incremental paid loss amount

Variance = Mean x Scale Parameter (see Step 5)

Completed incremental triangle

12 24 36 48 60

2003 419 458 150 101 136 2004 166 220 170 12 109 2005 255 465 152 59 107 2006 353 446 127 47 85 2007 429 672 280 104 188

Randomly generated future incremental payments based on above

12 24 36 48 60 2003 2004 106 2005 44 74 2006 130 51 97 2007 601 300 83 230

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Step 9. Calculate total unpaid amounts.

Sum the future incremental values to estimate the total unpaid loss by period and in total (in this example, 1,716).

This provides one estimated possible outcome.

Total estimated future payments

12 24 36 48 60 Total 2003 2004 106 106 2005 44 74 118 2006 130 51 97 278 2007 601 300 83 230 1,214

1,716

Step 10. Repeat and summarize.

Repeat steps 6 through 9 the specified number of times (e.g., 10,000 or more), capturing the resulting cash flows and the unpaid amounts by period and in total for each iteration. The resulting distribution of unpaid amounts from all the iterations can be used to calculate means, percentiles, etc.

Total simulated results

Mean

Unpaid Standard

Error Coefficient of

Variation 2003 0 0 0.0% 2004 95 65 68.2% 2005 187 85 45.6% 2006 381 125 32.8% 2007 782 198 25.4%

1,446 278 19.2%

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IV.D.6.h A walkthrough of the basic Cape Cod calculation (paid data)

Note that, for simplicity, we are using the data from Steps 1 through 6 of the paid chain-ladder example. In the actual simulations the Cape Cod model is simulated independently of the chain-ladder model.

Step 7. Complete the new randomly generated triangle.

Calculate new VWAs. Use the Cape Cod method to calculate the a priori ultimate-loss ratios and use them to complete the bottom right of the triangle.

For simplicity, we are assuming that the trend is 5.0% for each period, the weight is 100.0% for all periods, and the decay rate is 75.0%.

The differences in the paid ratios are used to allocate the total unpaid to incremental periods in Step 7. The results of the allocation are shown in Step 8.

Sample cumulative triangle

12 24 36 48 60 2003 419 877 1,027 1,128 1,264 2004 166 385 555 567 2005 255 720 872 2006 353 799 2007 429

12-24 24-36 36-48 48-60 60+

VWAs 2.332 1.238 1.071 1.120 1.000 CDF 3.465 1.486 1.200 1.120 1.000 Paid 0.289 0.673 0.833 0.892 1.000

Premium Premium

Index On-Level Premium Paid

Trend Index

Trended Paid

(1)

(2)

(3)

(1) x (2) (4)

(5)

(6)

(4) x (5) 2003 2,000 1.420 2,840 1,264 1.252 1,582 2004 2,000 0.970 1,940 567 1.191 675 2005 2,000 1.050 2,100 872 1.133 988 2006 2,000 1.100 2,200 799 1.078 861 2007 2,000 1.000 2,000 429 1.025 440

Percent

Paid On-Level

Ratio Weighted

Ratio De-Trended

Ultimate

Ultimate

(7)

(8)

(6) ÷ (7) ÷ (3) (9)

(10)

(9) x (3) ÷ (5) (11)

(10) x [1–(7)] + (4)2003 1.000 55.7% 53.1% 1,204 1,264 2004 0.892 39.0% 52.4% 853 659 2005 0.833 56.5% 53.9% 999 1,039 2006 0.673 58.2% 55.2% 1,127 1,168 2007 0.289 76.2% 56.4% 1,101 1,212

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Step 8. Introduce process variance.

To add process variance in the simulation, replace every future incremental paid amount with a randomly selected point from a gamma distribution where:

Mean = the incremental paid loss amount

Variance = Mean x Scale Parameter (see Step 5)

Completed incremental triangle

12 24 36 48 60

2003 419 458 150 101 136 2004 166 220 170 12 92 2005 255 465 152 59 107 2006 353 446 180 67 121 2007 429 423 176 65 118

Randomly generated future incremental payments based on above

12 24 36 48 60 2003 2004 88 2005 44 74 2006 185 72 136 2007 366 191 48 151

Step 9. Calculate total unpaid amounts.

Sum the future incremental values to estimate the total unpaid loss by period and in total (in this example, 1,355).

This provides one estimated possible outcome.

Total estimated future payments

12 24 36 48 60 Total 2003 2004 88 88 2005 44 74 118 2006 185 72 136 393 2007 366 191 48 151 756

1,355

Step 10. Repeat and summarize.

Repeat Steps 6 through 9 the specified number of times (e.g., 10,000 or more), capturing the resulting cash flows and the unpaid amounts by period and in total for each iteration. The resulting distribution of unpaid amounts from all the iterations can be used to calculate means, percentiles, etc.

Total simulated results

Mean

Unpaid Standard

Error Coefficient of

Variation

2003 0 0 0.0% 2004 75 53 70.0% 2005 169 74 43.9% 2006 378 106 27.9% 2007 740 137 18.5%

1,362 244 17.9%

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IV.D.7 Appendix B – CORRELATION

Correlation is a way of measuring the strength and direction of a relationship between two or more sets of numbers. Essentially, this is a way to measure the tendency of two variables to “move” in the same or opposite directions.35

The correlation coefficient between two variables is indicated (and calculated) using a range of values from -100% to 100%. With positive correlation, the two variables tend to “change” in the same direction—i.e., when the X variable is high, the Y variable tends to be high; if X is low, Y tends to be low. The closer the measured correlation is to 100%, the stronger the tendency is to “move” in the same direction.

Conversely, with negative correlation, the two variables tend to “move” in opposite directions—i.e., as the X variable increases, the Y variable tends to decrease, and vice versa. The closer the measured correlation is to -100%, the stronger the tendency is to “move” in the opposite direction.

A correlation of zero indicates that no relationship is anticipated between the two variables—i.e., you would not expect to use the values or movements in one variable to tell you anything about the value or movements in the second variable. In statistical terms, this is referred to as being independent.

IV.D.7.a The correlation matrix

Since we are usually concerned about multiple segments, the best way to quickly see and understand the relationships between the various segments is to use a correlation matrix. This is a symmetric matrix of the various correlation coefficients between each pair of segments.36 It lists each segment down the first column and across the top row. At each intersection of two segments you will find the correlation coefficient describing the expected relationship between those two segments.

35 Movement in this context is relative and does not necessarily imply that the average values of the observations

are moving (i.e., the averages may or may not be increasing or decreasing). More simply stated, we are describing the movements between one observation and the next and whether the corresponding movements from one observation to the next for another variable are similar or not. Finally, these movements are also relative in size and shape as, for example, one variable could be very small and stable and the other could be very large and volatile.

36 In other words, the top right and bottom left triangles of the matrix are mirror images. In addition, the center diagonal, where each segment intersects with itself, is always filled with 1s because any variable is always perfectly correlated with itself.

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Figure 1XX: Sample Correlation Matrix

1 2 3 4 5 1 1.00 0.25 0.11 0.34 0.42 2 0.25 1.00 0.15 0.15 0.27 3 0.11 0.15 1.00 -0.19 0.02 4 0.34 0.15 -0.19 1.00 -0.36 5 0.42 0.27 0.02 -0.36 1.00

In the table in Figure 1XX, Segments 1 and 5 are expected to exhibit the strongest positive

correlation with each other, while 4 and 5 are expected to exhibit the strongest negative correlation with each other. In addition to correlation coefficients, an additional output is a matrix of the p-values that correspond to each correlation coefficient in the correlation matrix. For each correlation coefficient, the p-value is a measure of the significance of the correlation coefficient, with a low value (less than 5.0%) indicating the coefficient is significant (i.e., likely to be correct or very close to correct) and a high value (greater than 5.0%) indicating the coefficient is not significantly different from zero.

Figure 2XX: P-values for Sample Correlation Matrix

1 2 3 4 5 1 0.00 0.03 0.81 0.04 0.02 2 0.03 0.00 0.45 0.63 0.07 3 0.81 0.45 0.00 0.29 0.92 4 0.04 0.63 0.29 0.00 0.06 5 0.02 0.07 0.92 0.06 0.00

In the table in Figure 2XX, Segments 1 and 5 exhibit the strongest p-value, while 3 and 5 are

not likely to exhibit correlation significantly different from zero.

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IV.D.7.b Measuring correlation

There are several ways to measure correlation, both parametric (for example, Pearson’s R) and non-parametric (Spearman’s Rank Order, or Kendall’s Tau). Pearson's correlation coefficient may be more helpful if the underlying values are “normally” distributed, whereas Spearman's and Kendall's formulas may be more useful when distributions are not normal.

For this case study we will use Spearman’s Rank Order calculation to assess the correlation between each pair of segments in the model. Rather than calculating the correlation of the residuals themselves, Spearman’s formula calculates the correlation of the ranks of those residuals.37 The residuals to be correlated are converted to a rank order, and the differencesbetween the ranks of each observation of the two variables, D, are calculated. The correlation coefficient (ρ) is then calculated as:

)1(6

1 2

2

−−= ∑

NND

ρ (4.D.2)

Two examples should prove useful at this point. Consider the residuals for the following two segments:

Figure 3XX: Sample Residuals for Two Segments Segment A Segment B

12 24 36 48 60 12 24 36 48 60

2001 -1.90 -3.07 3.85 4.38 0.00 2001 -1.23 2.57 -0.98 -1.64 0.00 2002 1.77 1.51 -0.65 -5.50 2002 -1.14 3.19 -4.74 1.56 2003 1.10 1.74 -3.95 2003 0.16 -4.28 6.94 2004 -0.48 0.43 2004 2.78 -2.74 2005 0.00 2005 0.00

From these residuals (resid.), we can calculate the ranks and differences (diff.) as follows:

37 In point of fact, the model calculates the correlation coefficients for the unadjusted scaled Pearson residuals and

the heteroscedasticity-adjusted residuals. This allows the user to see the impact of the hetero adjustment on the correlation of the residuals.

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Figure 4XX: Calculation of Residual Ranks Segment A Segment B Ranks

Obs. Resid. Rank Obs. Resid. Rank Diff. Diff. 2 1 -1.90 4 1 -1.23 5 -1 1 2 1.77 11 2 -1.14 6 5 25 3 1.10 8 3 0.16 8 0 0 4 -0.48 6 4 2.78 11 -5 25 5 -3.07 3 5 2.57 10 -7 49 6 1.51 9 6 3.19 12 -3 9 7 1.74 10 7 -4.28 2 8 64 8 0.43 7 8 -2.74 3 4 16 9 3.85 12 9 -0.98 7 5 25 10 -0.65 5 10 -4.74 1 4 16 11 -3.95 2 11 6.94 13 -11 121 12 4.38 13 12 -1.64 4 9 81 13 -5.50 1 13 1.56 9 -8 64

496

The Spearman Rank Order correlation coefficient is calculated as:

363.0)113(13

49661)1(

61 22

2

−=−×

×−=

−−= ∑

NND

ρ

For the two segments represented above, there is negative correlation between them.

As a second example, consider the following residuals:

Figure 5XX: Sample Residuals for Two Segments Segment C Segment D

12 24 36 48 60 12 24 36 48 60

2001 -1.90 -3.07 3.85 4.38 0.00 2001 -1.64 -2.74 3.19 6.94 0.00 2002 1.77 1.51 -0.65 -5.50 2002 2.78 1.56 -1.23 -4.74 2003 1.10 1.74 -3.95 2003 0.16 2.57 -4.28 2004 -0.48 0.43 2004 -1.14 -0.98 2005 0.00 2005 0.00

From these residuals, we can calculate the ranks and differences as follows:

Figure 6XX: Calculation of Residual Ranks

Segment C Segment D Ranks

Obs. Resid. Rank Obs. Resid. Rank Diff. Diff. 2 1 -1.90 4 1 -1.64 4 0 0 2 1.77 11 2 2.78 11 0 0 3 1.10 8 3 0.16 8 0 0 4 -0.48 6 4 -1.14 6 0 0 5 -3.07 3 5 -2.74 3 0 0

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6 1.51 9 6 1.56 9 0 0 7 1.74 10 7 2.57 10 0 0 8 0.43 7 8 -0.98 7 0 0 9 3.85 12 9 3.19 12 0 0 10 -0.65 5 10 -1.23 5 0 0 11 -3.95 2 11 -4.28 2 0 0 12 4.38 13 12 6.94 13 0 0 13 -5.50 1 13 -4.74 1 0 0

0

The Spearman Rank Order correlation coefficient is calculated as:

000.1)113(13

061)1(

61 22

2

=−×

×−=

−−= ∑

NND

ρ

Thus, for this second pair the business is perfectly positively correlated.

IV.D.7.c Modeling correlation

Looking deeper at the previous examples, the residuals for Segment C and Segment D are identical to the residuals for Segment A and Segment B, respectively. This was done to illustrate the concept of “inducing” correlation. Even though the values are the same, they are not in the same order, but comparing the relative orders to each other is how we measure correlation—i.e., how likely is one variable to move in the same direction as the other. Thus, the correlation between two simulated variables can be changed by re-sorting one compared to the other—i.e., the desired level of correlation can be induced by re-sorting.

In order to induce correlation between different segments in the bootstrap model, we first calculate the correlation matrix using the Spearman Rank Order correlation as illustrated above for each pair of segments. The model then simulates the distributions for each segment separately. Using the estimated correlation matrix, or another correlation assumption, we can calculate the correlated aggregate distribution by re-sorting the simulations for each segment based on the ranks of the total unpaid for all accident years combined.38

Another example will help illustrate this process. As noted above, the first step is to run the bootstrap model separately for each segment. The sample output for three segments is shown below (based on 250 iterations). For this example, we have included results by accident year as well as by future calendar year, which sum to the same total. Note that other parts of the simulation output (e.g., loss ratios) could also be included as part of the correlation process.

38 Note that the re-sorting is based on total values and the coefficients are based on residuals. Correlating the

residuals reflects a far more complex algorithm, but our research has indicated that these different approaches are reasonably consistent. Thus, re-sorting is an easier, although quite robust, algorithm.

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Step 1: Simulate individual segment distributions.

Segment A

Total Unpaid Total Cash Flow

Accident Year Calendar Year

Iteration 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 4 18 219 366 818 … 14,513 3,669 3,268 2,160 2,393 1,469 … 14,513

2 1 75 413 667 1,100 … 16,200 4,280 4,173 2,271 1,887 1,363 … 16,200 3 11 -6 861 773 1,379 … 16,826 4,769 4,120 2,400 1,670 1,347 … 16,826

4 0 126 335 1,299 543 … 17,504 5,233 3,707 3,237 1,505 1,486 … 17,504

: : : 250 38 122 470 575 1,191 … 20,210 4,370 3,767 2,807 2,145 1,450 … 20,210

AVG 19 118 512 782 1,015 … 19,256 5,330 4,260 3,227 2,180 1,613 … 19,256

Segment B

Total Unpaid Total Cash Flow

Accident Year Calendar Year

Iteration 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 420 35 446 592 1,212 … 45,151 11,058 12,762 8,898 5,921 3,024 … 45,151

2 233 802 302 1,484 1,621 … 23,077 10,107 6,151 2,458 2,107 941 … 23,077

3 330 177 737 344 2,548 … 37,989 10,990 12,038 7,029 3,847 2,144 … 37,989 4 738 68 589 540 803 … 18,430 5,291 4,377 4,267 1,776 1,833 … 18,430

: : :

250 0 15 440 1,113 2,453 … 30,816 12,148 8,186 7,066 2,008 1,178 … 30,816

AVG 207 500 658 954 2,213 … 31,930 10,072 8,363 5,681 3,395 2,114 … 31,930

Segment C

Total Unpaid Total Cash Flow

Accident Year Calendar Year

Iteration 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 0 0 3 3 184 … 4,045 1,445 941 815 239 288 … 4,045 2 0 0 8 2 15 … 3,022 1,432 925 305 235 29 … 3,022

3 8 0 0 39 82 … 3,233 1,181 817 561 324 175 … 3,233

4 0 0 5 40 86 … 3,972 1,475 1,017 748 342 327 … 3,972 : : :

250 0 0 0 121 156 … 2,599 1,113 767 365 142 22 … 2,599

AVG 3 3 5 45 74 … 3,495 1,228 985 584 335 201 … 3,495

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Step 2: Rank the simulation results.

The second step is to sort the simulation results based on the values of the total unpaid for all years combined. Sorting in ascending order gives us the rank of each simulation.39

Segment A

Total Unpaid Total Cash Flow

Accident Year Calendar Year

Rank 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 27 84 545 425 581 … 12,123 3,787 3,056 1,868 1,312 1,118 … 12,123

2 27 73 250 750 491 … 12,792 3,800 3,166 2,267 936 1,081 … 12,792 3 1 -209 23 377 692 … 12,883 3,240 3,266 2,337 1,841 1,009 … 12,883

4 -25 200 470 488 812 … 13,240 4,100 2,611 2,392 1,440 969 … 13,240

: : : 250 2 307 653 1,366 1,015 … 30,200 4,522 3,472 2,506 1,874 1,519 … 30,200

AVG 19 118 512 782 1,015 … 19,256 5,330 4,260 3,227 2,180 1,613 … 19,256

Segment B Total Unpaid Total Cash Flow

Accident Year Calendar Year

Rank 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 136 216 263 584 697 … 15,042 5,271 4,843 2,268 1,697 334 … 15,042

2 30 434 222 142 1,822 … 15,658 6,157 4,088 2,122 1,274 1,623 … 15,658

3 463 101 597 104 1,031 … 17,012 6,930 6,794 1,244 954 283 … 17,012 4 182 148 618 835 1,411 … 17,236 6,957 5,052 2,522 1,942 265 … 17,236

: : :

250 382 1,167 1,333 2,264 3,256 … 64,376 14,018 16,007 12,046 9,279 3,238 … 64,376

AVG 207 500 658 954 2,213 … 31,930 10,072 8,363 5,681 3,395 2,114 … 31,930

Segment C Total Unpaid Total Cash Flow

Accident Year Calendar Year

Rank 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 0 27 18 200 59 … 2,285 789 747 349 211 95 … 2,285 2 92 5 1 27 15 … 2,367 807 496 652 143 116 … 2,367

3 0 0 0 4 2 … 2,451 673 818 374 160 351 … 2,451

4 1 0 0 5 90 … 2,524 1,068 658 145 343 213 … 2,524 : : :

250 0 0 0 215 211 … 5,351 1,318 1,159 1,009 629 442 … 5,351

AVG 3 3 5 45 74 … 3,495 1,228 985 584 335 201 … 3,495

39 This step is actually for illustration purposes only. During actual simulations, only the ranks are needed for each

iteration (separately by segment), which are then used in Step 4.

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Step 3: Generate correlation matrix.

The third step is to calculate the rank orders that will give us the desired correlation between each segment pair. The method illustrated here is to use a multivariate normal distribution with the desired correlation matrix and simulate random values from that distribution. Using the simulated values, the ranks of those simulated values will give us the desired rank orders for the desired correlation.40

Multivariate Normal Ranks

Iteration A B C A B C 1 -2.652 -0.978 0.030 1 41 128 2 -0.070 0.445 1.136 118 168 218

3 0.030 0.643 -0.536 128 185 74

4 0.915 1.491 0.274 205 233 152 : : :

250 -0.885 1.080 0.412 47 215 165

Step 4: Re-sort the simulation results based on correlation ranks.

The fourth step is to re-sort the individual simulation results for each segment based on the rank orders from Step 3. After this re-sorting, the correlation coefficients for each pair should match the desired correlations specified in the correlation matrix from Step 3.

Segment A Total Unpaid Total Cash Flow

Accident Year Calendar Year

Rank 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 27 84 545 425 581 … 12,123 3,787 3,056 1,868 1,312 1,118 … 12,123

118 0 293 525 805 661 … 18,819 5,095 3,899 2,812 1,826 1,648 … 18,819

128 1 439 388 1,038 1,215 … 19,079 4,838 3,637 3,255 1,995 2,609 … 19,079

205 10 386 362 1,110 1,080 … 21,851 4,398 4,341 4,008 2,644 2,665 … 21,851 : : :

47 1 78 516 697 740 … 16,779 5,551 4,405 2,638 1,708 1,129 … 16,779

AVG 19 118 512 782 1,015 … 19,256 5,330 4,260 3,227 2,180 1,613 … 19,256

Segment B Total Unpaid Total Cash Flow

Accident Year Calendar Year

Rank 1 2 3 4 5 … Total 1 2 3 4 5 … Total 41 37 240 135 462 1,170 … 23,945 6,994 6,500 5,222 2,546 1,982 … 23,945

168 134 705 212 1,604 3,135 … 33,764 9,755 9,307 5,790 3,849 1,819 … 33,764 185 329 18 149 1,163 1,579 … 37,006 12,598 9,414 7,085 3,852 2,311 … 37,006

233 181 227 962 897 1,505 … 46,286 14,619 11,230 7,839 4,036 4,029 … 46,286

: : : 215 55 997 763 2,100 2,944 … 41,587 13,804 8,545 8,269 4,718 2,923 … 41,587

AVG 207 500 658 954 2,213 … 31,930 10,072 8,363 5,681 3,395 2,114 … 31,930

40 As a technical note, the multivariate T random number generator used in our model uses the Cholesky

decomposition to induce the desired correlation between independent normal random values.

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Segment C Total Unpaid Total Cash Flow

Accident Year Calendar Year

Rank 1 2 3 4 5 … Total 1 2 3 4 5 … Total 128 0 0 0 3 70 … 3,480 1,456 788 672 167 65 … 3,480

218 0 12 0 55 159 … 4,036 1,565 1,461 314 274 195 … 4,036 74 0 0 2 24 29 … 3,163 1,373 824 388 410 157 … 3,163

152 0 0 0 99 112 … 3,595 1,027 1,061 560 377 452 … 3,595

: : : 165 0 0 0 8 17 … 3,644 1,205 853 786 237 357 … 3,644

AVG 3 3 5 45 74 … 3,495 1,228 985 584 335 201 … 3,495

Step 5: Sum the correlated values.

The fifth step is to sum the values across the segments to get aggregate results for each iteration.

Total All Segments Combined Total Unpaid Total Cash Flow

Accident Year Calendar Year

Iteration 1 2 3 4 5 … Total 1 2 3 4 5 … Total 1 64 324 680 890 1,822 … 39,548 12,237 10,344 7,762 4,026 3,165 … 39,548

2 134 1,010 737 2,464 3,956 … 56,619 16,415 14,667 8,916 5,950 3,661 … 56,619

3 330 457 540 2,225 2,824 … 59,249 18,809 13,875 10,728 6,256 5,078 … 59,249 4 191 612 1,324 2,106 2,698 … 71,731 20,044 16,632 12,406 7,057 7,146 … 71,731

: : :

250 56 1,075 1,279 2,805 3,701 … 62,010 20,560 13,803 11,693 6,662 4,408 … 62,010

AVG 230 621 1,175 1,781 3,301 … 54,681 16,629 13,609 9,493 5,910 3,928 … 54,681

Step 6: Summarize.

The final step is to use the aggregate results for all simulations to describe the distribution of unpaid claims from all the results, including means, percentiles, etc. All of the same summaries that are created for each individual segment can also be created for the aggregate results (e.g., cash flows, loss ratios, or graphs).

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IV.E Economic capital for a multi-line non-life insurance company

Non-Life Economic Capital Case Study

This case study presents an Economic Capital analysis of DFA Insurance Company (DFAIC), a

regional non-life multi-line insurance company writing personal lines and main street commercial

business. Background information and financial statement data for DFAIC were obtained from a

call paper request41. This case study will briefly describe the simulation model, but will mainly focus

on parameterizing the simulation model. More detailed descriptions of the model and the economic

capital framework have been provided in other sections of this monograph.

Risk Metric and Time Horizon

The risk metric used is the 99% Conditional Tail Expectation (CTE-99), also known as the Tail

Value at Risk (TVAR99) of economic value capital over a one year horizon. The CTE-99 is the

average of the worst 1% of losses in surplus over the next year. Surplus is calculated as the excess of

the market value of assets over the market value of liabilities. Because insurance company reserves

are not commonly traded we will estimate their market value by using a risk adjusted discount rate42.

This case study will use a forecast horizon of one year. This assumes that shortfalls in surplus are

able to be replenished in one year. Companies commonly budget one year ahead, so this horizon is

consistent with that practice. An advantage of a short horizon is that it decreases the difficulty in

modeling reasonable outcomes for economic scenarios, operational changes and underwriting cycles.

A disadvantage of the short horizon is that it does not allow sufficient time for strategy changes to

be modeled. It also does not allow dynamic management response to be modeled. If significant

strategy changes are anticipated, it may be more appropriate to use a longer forecast horizon.

Description of the Model

The simulation model is composed of four modules: an economic scenario module, an underwriting

module, an asset module, and an accounting and taxation module. The economic scenario generator

41 “2001 Call for Papers- Dynamic Financial Analysis, A Case Study” Casualty Actuarial Society Forum Casualty

Actuarial Society - Arlington, Virginia 2001: Spring 1-24 <http://www.casact.org/pubs/forum/01spforum/01spf001.pdf>.

42 Butsic, Robert, “Determining the Proper Discount Rate for Loss Reserve Discounting: An Economic Approach,” 1988 Casualty Actuarial Society Discussion Paper Program - Evaluating Insurance Company Liabilities, pp. 147-188. http://www.casact.org/pubs/dpp/dpp88/88dpp147.pdf

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is used as the simultaneous input to both the asset module and the underwriting module. The

results of the asset and underwriting modules are then combined and fed through the accounting

and tax calculations. With this type of model we can properly pick up the correlation between the

asset returns and underwriting returns caused by changes in the underlying economic values such as

interest rates, inflation rates, and other economic variables.

Economic Scenario Module

The economic scenario module generates stochastic scenarios of the economic variables that are

used as inputs to the underwriting and assets models. The economic variables generated by this

module include short term interest rates for bank deposits or government notes, bond yields for

government and corporate bonds of various maturities, equity appreciation and dividend yields,

economic growth rates, and inflation rates. There is a detailed description of an economic scenario

generator in the Life Insurer Economic Capital case study.

Underwriting Module

The underwriting module models insurance company operations and calculates the resulting cash

flows relating to issuing policies, claim payments and company expenses.

Within the underwriting module, lines of business with similar characteristics are typically grouped

for modeling purposes. The level of grouping depends on the volume of business and similarity of

the lines, the purpose of the analysis, and the availability of data. For this case study we will model

using the following grouping of lines

• Private Passenger Auto Liability

• Commercial Auto Liability

• Homeowners

• Workers Compensation

• Commercial Multi peril

• Property

• Other Direct

• Assumed

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Expenses are modeled in four groups.

• Commissions (as a percentage of Written Premium)

• General Underwriting Expense (as a percentage of Earned Premium)

• Unallocated Loss Adjustment Expenses (one half as a percentage of Paid Loss, one half as a

percentage of Earned Premium)

• Allocated Loss Adjustment Expenses (modeled with losses)

Asset Module

The asset module processes the investment activity of the company. Asset market values are

updated based on the economic scenario generators. Cash flows from investments (coupons,

dividends and maturities) and net cash flows from insurance operations are invested. Investments

are rebalanced to a desired mix by asset type.

Assets are modeled using the following groupings

• Tax Free Bonds

• Taxable Bonds

• Equities

• Cash

• Other

Accounting and Taxation Module

The accounting and taxation module combines the results of the underwriting and investment

modules to produce financial statements and calculate income taxes. For this case study we will

create undiscounted statements for comparison with the historical financials and as the basis of the

income tax calculation. We will also create economic value (market value) statements, to determine

our desired metric, economic capital.

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Inputs and Parameterizing

The base scenario for this case study will be a simple model. We will assume that the company will

write business at a steady state for one additional year. No planned rate changes or exposure

changes are modeled.

The calculation of the parameters for the Private Passenger Auto Liability line will be shown in this

section and in the accompanying Excel workbook “PC EC Case Study.xls”. The parameters for the

other lines of business are not discussed or shown in the workbook, but were calculated in a similar

manner. In many cases, the supplied data were not sufficient to calculate the needed parameters. In

these cases, assumptions or approximations were used to derive the parameters. The additional

assumptions or approximations used will be described in this section.

Premium

The premium calculations are shown in the premium tab of the Excel workbook.

Direct Written Premium is modeled by multiplying the prior year written premium by a random

error term.

WrittenPrem1 = WrittenPrem0 * (1 + error term),

where error term is a random draw from a distribution with a mean of 0.

The error term represents the difference between the expected or budgeted amount and the amount

actually realized. For premium this would include differences from expected in the number of

policies issued, differences in the average policy size, and differences in the final charged rates.

We would like to calculate the error term based on a history of expected versus actual premiums.

However, this information is not available so we will approximate by fitting an exponential

regression to ten years direct earned premium. The error term is selected to be a normal

distribution, with a mean of 0, and a standard deviation equal to the standard error of the y estimate

from the regression.

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For PPA liability

WrittenPrem1 = 793,144 * ( 1 +N(0, 0.0929) )

Earned Premium is calculated by multiplying written premium by an earning pattern. We will

assume that premium earning and collection occur over two years. The earning pattern is calculated

from the ratio of unearned premium to written premium. Collected Premium and the collection

pattern are calculated in a similar manner.

EarningPattern2 = Unearned Premium / Written Premium

EarningPattern1 = 1 – Earningpattern2

EarningPattern2 = 211,134 / 593,000 = 0.356

EarningPattern1 = 1.0 - 0.356 = 0.644

The reinsurance rate, the ratio of ceded premium to direct premium, was selected at 26% for PPA

Liability after observing the historical ratios over the previous ten years. If we had more details

about historical reinsurance, rates, and retentions we could refine our estimate, but the simple

approach used here should produce an adequate estimate.

Expenses

The expense calculations are shown in the “Expenses” tab of the Excel workbook. Expenses are

calculated as

Commission Expense = Written Premium * Rate * (1 + error term)

Other Underwriting Expense = Earned Premium * Rate * (1 + error term)

ULAE = (Earned Premium + Paid Loss) / 2 * Rate * (1 + error term)

The commission rate is calculated from the ratio of direct commission to direct written premium.

The other underwriting expense rate is calculated from the ratio of the remaining underwriting

expenses to earned premium. The unallocated loss adjustment expense rate is selected based on a

review of the ratios for the prior ten years.

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Only the ULAE data has the historic information needed to estimate the standard deviation of the

error terms. We used the coefficient of variation (the ratio of the standard deviation to the mean)

for ULAE to calculate the standard deviations for the commission rate error and the other

underwriting expense error.

Losses

Losses are modeled by line of business in four groups

• Runoff of Starting Loss Reserve

• Individual Large Losses

• Aggregate Small Losses

• Catastrophe Losses

Runoff of Starting Reserve

The runoff of the starting reserve, also known as reserving risk, projects the possible outcomes of

future payments made on claims that have already occurred as of the projection date.

Often there is less information about the losses in the starting reserve than we will have about the

future simulated losses. We often do not know the size of individual claims or historical reinsurance

retentions so we cannot calculate the ceded portion of each claim. For this reason, we modeled the

runoff separately from the future claims. The runoff was modeled on a net of reinsurance basis.

Runoff Losses = Starting Net Loss Reserve * (1 + error term)

The error term for the starting reserve and the payout pattern were estimated using the procedures

described in Case Study D.

Future Claims

Accurate modeling of reinsurance is important for an economic capital analysis. Modeling excess of

loss reinsurance requires knowledge of the size of each individual claim size to properly apply the

per claim retention and limit. However, it is usually not feasible from a computing power and

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runtime standpoint to generate every individual claim. For this reason, the simulation will generate

large claims individually using a frequency/severity model, and generate the aggregate amount of

small claims using a loss ratio model. For these purposes, large means any claim in excess of the

XOL reinsurance retention.

Catastrophe losses are modeled separately because they may have a different payment pattern than

other losses, and can also affect multiple lines of business.

The range of possible outcomes of the total of these three types of future claims is a part of pricing

risk.

Individual Large Losses

Large losses are modeled by generating individual claims which exceed the reinsurance retention of

500,000. We need to simulate the number of claims and the size of each claim.

The data provided with the call paper did not include any individual large claim data so we need to

supplement the company data with industry data which we will assume is representative of the

company’s large claim experience.43 The data does not include private passenger auto liability so we

have used commercial auto liability as a proxy.

The maximum likelihood method described earlier in the monograph was used to fit lognormal and

pareto distributions to the data. Before doing the fits, the claims must be trended to the starting

date of the simulation, developed to ultimate value, adjusted for any differences in policy limits, and

grouped into intervals by claim size. For this case study, we will assume that these steps produce the

grouped data shown in the “Data” sheet of the excel spreadsheet.

The lognormal and pareto parameters were obtained by using Solver to minimize the sum of

negative log-likelihood for claim size intervals exceeding 500,000. Both of these distributions have

two parameters so we are able to determine which distribution has a better fit to the data based on

which has a smaller sum of negative log-likelihood. If distributions have different numbers of

43 Texas Department of Insurance, “The 2000 Texas Liability Insurance Closed Claim Annual Report”,

http://www.tdi.state.tx.us/reports/pc/documents/taccar2000.pdf

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parameters the Akaike (AIC) and Bayesian (BIC) information criteria can be used to select between

alternative distributions.

The large loss frequency is the number of large claims per million of premium. This can be

calculated by dividing the ceded losses by the average ceded loss per large claim. For this case study

we will use a Poisson distribution for the claim size, with the parameter lambda equal to the large

loss frequency.

Aggregate Small Losses

Small losses are the remaining non-catastrophic losses. They are modeled by

Small Losses = Direct Earned Premium * Small Loss Ratio * (1 + error term)

The small loss ratio is calculated by first selecting a net loss ratio based on the historic net data.

This is converted to a direct premium loss ratio by grossing up by the ceded premium percentage.

Next, the retained portion of large losses is subtracted. This leaves the portion of losses that needs

to be modeled by the small loss ratio.

Using a simplifying assumption, the standard deviation of the small loss ratio error term is calculated

by assuming that the coefficient of variation of small losses is the same as the coefficient of variation

of net losses. If additional data was available, we could determine the standard deviation of the

small loss ratio error using its own historical experience.

The loss payment pattern is calculated using a paid loss development triangle. The ratio of paid to

ultimate loss is estimated for each development age, and the incremental differences between ages

are used as the payment pattern. The same payment pattern is also used for large losses, cat losses,

and the starting reserves. A refinement to the parameterization would be to derive a specific

payment pattern for each type of losses, small, large and catastrophe, but the data required was not

available for this case study.

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Catastrophe Losses

The call paper does not give very much data for catastrophe exposure. A realistic analysis with

significant cat exposure would typically require inputs from a cat model. For this case study we

generate catastrophic losses in a similar manner as the large loss model, by simulating from a

frequency distribution and a severity distribution. Based on the limited information provided in the

Call for Papers, we assume a catastrophic loss annual frequency of 0.2, and for the severity

distribution we use a pareto distribution with a minimum severity of 1 million, and a 1% chance of a

loss exceeding 160 million.

Investment Model

For this case study assets are grouped into broad categories, taxable bond, nontaxable bond, stock,

cash, and other. Investment returns are obtained from the economic scenario generator. We will

assume that the asset mix will remain the same as the starting assets from the financial statements.

Correlation

Many insurance company financial results are correlated between lines of business. Different lines

of business are often affected by common inflation, and changes in company operations are often

changed across the entire company.

Some correlation is induced by the use of a common economic scenario for all lines of business.

Additional correlation is induced through the error terms for the small loss ratio, the large claim

frequency.

The correlation matrix calculation is shown on the Correlation tab of the excel workbook. The

correlation between the small loss ratio error was calculated by calculating the correlation of the net

loss ratios between lines of business. The net loss ratios were first adjusted by backing out the

impact of changes in the inflation rate. Loss ratio changes caused by changes in inflation were

calculated assuming a duration of 2 for the short tailed Homeowners and Property lines, and a

duration of 3 for the remaining lines.

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The correlations for large claim frequency were assumed to be the same as the correlations for the

small loss ratios. If individual large claim data was available the frequency correlations could have

been calculated directly.

Validation and Peer Review

Similar to the life and health company case study, two forms of review were used for this case study.

Calculation of the input parameters to the model were reviewed and confirmed with the historical

data. Input assumptions were reviewed for reasonableness in comparison to the historical.

Projected financial results outputs were also reviewed.

The correlations calculated based on the data were accepted as is for this case study. Normally, one

would expect to see small positive correlations between lines of business. However, the data for

this case study produced a mixture of positive and negative correlations, with strong correlations

between seemingly unrelated lines of business. This is probably due to the use of fictitious data, and

would not normally be seen using real world data.

Communication of Results

The results of the simulation can be presented in graphical or tabular form. A histogram of the

change in economic value surplus is graphed below. The smooth line is the kernel density estimate,

which can be calculated using a statistical package such as R.

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The simulation results are shown below several confidence levels. In 1% of the simulation trials,

DFAIC suffered a loss of more than 226 million. The average loss for these 1% of trials was 308

million. This is our economic capital metric, the 99% Tail Value at Risk of the change in economic

surplus, so the required economic capital for DFAIC, for a one year horizon, is 308 million.

DFA Insurance Company

Economic Capital

Confidence

Level

Value at Risk

(VAR)

Tail Value at Risk

(TVAR or CTE)

.999 -346,641,338 -648,271,933

.995 -271,909,060 -368,965,600.99 -226,240,694 -308,324,433.95 -96,751,912 -179,315,382.90 -40,003,833 -122,235,015

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The historical financial statements were provided on a undiscounted basis. For comparison

purposes, the simulation results are provided here on a undiscounted basis at various percentiles.

DFA Insurance Company

Summary of Financial Model Results

Undiscounted Basis (000's)

5th 25th 50th 75th 95th Prior Percentile Percentile Percentile Percentile Percentile Earned Premium 2,353,625 2,181,235 2,212,566 2,235,188 2,257,952 2,289,674 Incurred Losses 1,778,434 1,829,770 1,907,566 1,961,451 2,017,351 2,095,763 Total UW Expenses 693,794 782,484 798,549 810,680 822,475 838,980 UW Income -118,604 -667,568 -590,283 -536,871 -485,799 -408,758 Investment Income 351,388 204,400 223,155 238,902 254,335 288,376 Total Income 226,648 -427,245 -350,524 -297,400 -244,580 -167,304 Invested Assets 4,792,399 4,238,132 4,350,847 4,428,306 4,524,588 4,093,531 Loss Reserve 2,328,543 2,067,837 2,110,921 2,142,285 2,173,587 2,217,587 Surplus 1,204,304 1,169,260 1,275,288 1,351,693 1,425,495 1,531,029

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Stochastic Processes and Modeling in Financial Reporting and Capital Assessment V. References

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V. REFERENCES

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