2011 Chicago Ebig Mtg Phillips

Equity-Based Insurance Guarantees Conference November 14-15, 2011

Chicago, IL

Managing Basis Risk

Peter Phillips

1Managing Basis Riskg g

Annuity Solutions GroupAon Benfield Securities IncAon Benfield Securities, Inc.Peter M. Phillips

November 15, 2011 11:15 am to 12:00 pm, pEquity-Based Insurance Guarantees ConferenceChicago, IL

2Legal Disclaimer

This was prepared for informational purposes only and is intended only for the designated recipient. It is neither intended, nor should be considered, as (1) an offer to sell, (2) a solicitation or basis for any contract for purchase of any security, loan or other financial product, (3) an official confirmation, or (4) a statement of Aon Benfield Securities, Inc. or any of their affiliates. With respect to indicative values, no representation is made that any transaction can be effected at the values provided and the values provided are not necessarily the

l i d A B fi ld S iti I b k d d Th i i t f thi d t i d i d tvalue carried on Aon Benfield Securities, Inc. books and records. The recipient of this document is advised to undertake an independent review of the legal, tax, regulatory, actuarial and accounting implications of any transaction described herein Aon Benfield Securities, Inc. does not provide legal, tax, regulatory, actuarial or accounting opinions. Any offer will be made only through definitive agreements and such other offering materials as provided by Aon Benfield Securities, Inc. or their appropriately licensed affiliate(s) prior to closing which contain important information regarding among other things certain risks associated with anywhich contain important information regarding, among other things, certain risks associated with any transaction described in this document and should be read carefully before determining to enter into such a transaction.

Annuity Solutions Group | Aon Benfield Securities, Inc. | November 15, 2011 1

3Agenda

Section 1 Introduction

Section 2 Fund Mapping

S ti 3 M d li B d F d Ri kSection 3 Modeling Bond Fund Risk

Section 4 Basis Risk Slippages

2Annuity Solutions Group | Aon Benfield Securities, Inc. | November 15, 2011

4Section 1: Introduction


5Basis Risk is Important

In the second half of 2008 hedge breakage losses at four companies exceeded $1.5 Billion according to J.P. Morgan*

For some of these companies a great deal of the hedge breakage in 2008 came from basis risk

It can play a key role in hedge program performance

For some of these companies a great deal of the hedge breakage in 2008 came from basis risk between actual account value movements and the hedge instrument index movements

This basis risk issue hit large direct writers and small reinsurers Interesting facts

One company lost close to 1B USD on this problem in less than a single year One company lost close to 1B USD on this problem in less than a single year Another company produced a graph showing their hedge program performance excluding

basis risk Actively managed fund exposure was dialed down after 2008

There is a lot of good literature in Finance on the topic of basis risk There is a lot of good literature in Finance on the topic of basis risk Roll, A Mean/Variance analysis of tracking error Sharpe, Asset Allocation: Management Style and Performance Measurement Famma & French, ICAPM Grinold and Kahn, Active Portfolio Management

* Variable Annuity Market Trends, J.P.Morgan May 29 2009


6Section 2: Fund Mappingpp g


7Fund Mapping

VA guarantees are written on a basket of underlying assets The underlying assets are often actively managed mutual funds Mutual fund NAV (Net Asset Value) are reported daily at the end-of-day, and are not available in real-time

Definition

Mutual fund NAV (Net Asset Value) are reported daily at the end of day, and are not available in real time NAV returns must be modeled in terms of returns in observable/investible market indices and there are two

distinct issues to consider:1. Approximating NAV in real-time (required for intra-day liability Greeks)

What is the current value of the underlying?

2. Hedging liability Greeks using tradable instruments (e.g. index futures, total return swaps, etc.) What are the optimal hedge ratios?


8Fund Mapping

Model


9Fund Mapping

Global funds Bond funds

Additional Modeling Considerations

Dividends (fund dividends and index dividends) Fund fees (MER) Currency exposure and hedging Data sampling frequency (high frequency, daily, monthly) Sample Window Mechanics Sample Window Mechanics

Moving or fixed duration Hold out sample size Number of observations

Time-varying weightsy g g Index selection process Multicollinearity Confidence intervals for estimates Trading instrument basis risk (cash index versus futures index return) Ex-ante versus Ex-post tracking error


10

Fund Mapping

Qualitative Approach

stylemappingS&P500 x%Russell2000 x%EAFE x%

x%

AggressiveFund1

indexmapping

x%100%

S&P500 x%Russell2000 x%EAFE x%

x%

Fund2 Balanced

x%100%


11

Fund Mapping

Quantitative Approach

Example Multivariate Fund Mapping Data


12

Fund MappingMulticollinearity

Multicollinearity refers to the case when the predictor variables are highly correlated

This is an acute problem with financial index time-series data Multicollinearity can be seen in these plots of NAV, S&P 500 and

Russell 2000 returns. All three are highly correlated and form a 1-g ydimensional line in 3-dimensional space this is equivalent to trying to fit a plane through a line, which would lead to an unstable estimate

Regression on such data can lead to unstable and unintuitiveweight estimates. For example, it can lead to negative weights.g p g g

Solutions require reducing dimensionality of the predictor set: Feature reduction (eliminate one or more predictors) Feature extraction (Principal Component Regression, Partial

L t S R i )-0.5

0

0.5

1

1.5

2

N

A

V

Least Squares Regression) Regularization (Ridge regression, LASSO)

0

1

2

-2

-1.5

-1

-2-1.5

-1-0.5

00.5

11.5

2

-3

-2

-1

SPX INDEX

RTY INDEX


13

Fund MappingMulticollinearity Bond Fund Example

Vanguard Bond Fund NAV return modeled using AA Corporate Zero Curve (using 15 different tenors)

Daily NAV return versus change in yields:


14


If we simply apply least-squares regression to the data without PCA (nave approach), the regression coefficients on the 15 tenors are sporadic and counter-intuitive, even though the model has reasonably good fit (in-sample R2 = 84%)


15


If we apply Principal Component Analysis to the yield curve data, we find that 98% of variance in the curve is accounted for by the first three principal components. The 15-dimensional yield curve may be reduce to 2 or 3 dimensions. These three principal components may interpreted as the level, slope, and curvature of the yield curve*

Applying linear regression on the 3 principal components results in regression coefficients that all Applying linear regression on the 3 principal components results in regression coefficients that all have negative sign (intuitive) and are stable

* LITTERMAN, R. AND J. SCHEINKMAN (1991). Common factors affecting bond returns, Journal of Fixed Income, vol. 1, no. 1, pp. 54-6


16

Fund MappingMulticollinearity Conclusion

Deciding which market indices to use for modeling a given fund is a complex statistical problem. This problem is called feature selection and in determines the number of type of hedging instruments that are selected

Correlation between market indices must be handled carefully and requires identifying underlying, uncorrelated risk factor drivers. This problem is called feature extraction and it is necessary for obtaining stable and sensible hedge ratios

Feature selection and extraction must be done jointly and should be guided by qualitative analysis of each funds investment mandate


17

Fund MappingConfidence Intervals

Confidence intervals for regression parameters have several applications, including selection of hedging instruments and basis risk monitoring

If a Beta confidence interval for an index includes zero then it makes sense to remove the index from the model

Confidence intervals for Betas are also useful for studying and detecting time-varying Beta (change in Beta over time due to market structure changes)

Classical linear regression theory assumes that the tracking errorsClassical linear regression theory assumes that the tracking errors (residuals) are normally distributed

However in practice tracking errors are not normally distributed. Tracking errors often exhibit fat tails and negative skew (can be seen through formal statistical tests for normality or through Normal Quantile-Quantile plots)Quantile Quantile plots)

This means that the classical theory, which assumes normal distribution of errors, may paint a different picture of the actual tracking error risk one could face in practice.


18

Fund MappingConfidence Intervals - Bootstrap

Statistical bootstrapping can be used to obtain confidence intervals of regression parameters without making assumptions about the distribution of data*

Bootstrap is a resampling method that is simple and relatively effective but can require significant computational infrastructure

The method essentially requires randomly sampling points from the time-series and repeatedly re-running the regression, in order to approximate the probability distribution of Beta (regression coefficient)

95% confidence interval

* Efron, B.; Tibshirani, R. (1993). An Introduction to the Bootstrap.


19

Fund MappingModel Testing

In order to asses model performance correctly, the data must be split into a training set (used to estimate the model parameters) and a test set (used to measure the models performance)

Various cross-validation techniques exist for splitting the data (such as random q p g (sampling), however for time-series data the training set should be data occurring chronologically prior to the test set (i.e. use past data to predict the future)

Note that the training/test windows can be rolled through the data, to obtain a better estimate of the models performance


20


Failure to use out-of-sample datasets will lead to overly optimistic tracking error predictions and can lead to completely opposite conclusions when comparing different models

Increasing error

Decreasing error

Model Complexity


21



22


Basic model testing Data sampling frequency (high-frequency, daily, weekly, monthly, quarterly) Training window size (1 year, 5 years, etc.)

In the example below, we find that out-of-sample tracking error is persistently higher than in-p , p g p y gsample

In-sample tracking error suggests that longer training windows are better, while out-of-sample tracking error points to the opposite conclusion (shorter training windows are better)

Out-of-sample tracking errors based on 300-day forward predictionsforward predictions

Annuity Solutions Group | Aon Benfield Securities, Inc. | November 15, 201121

23

Fund MappingTime-Varying Beta

Critical questions when estimating Beta: What amount of historical data should be used? How often should the estimate be updated?

Like market volatility correlation is a stochastic process and is related to the Like market volatility, correlation is a stochastic process and is related to the underlying market returns

Market correlation structure can change rapidly and may cause systematic hedge program losses if fund mapping weights are not updated


24

Fund MappingTime-Varying Beta

Possible methods for handling time-varying Beta: Rolling window update

Revise fund weights every X days using T years of historical data Trigger based updateTrigger based update

Monitor regression parameters / correlation, update if estimates move outside of confidence interval

Sequential update modelsK l Filt P ti l Filt Kalman Filter, Particle Filter

Market Implied Beta Betas can be also be obtained from option-implied correlation

Betas derived from Kernel regressionsg

Time-varying Beta has been studied extensively in the CAPM literature (see Campbell Harvey 1991). Several researchers find the Kalman Filter outperforms other methods of estimating Beta, in terms of tracking error (Root Mean Squared Error)*g , g ( q )

* Mergner, S. & Bulla, J. (2005), Time-varying beta risk of paneuropean industry portfolios: A comparison of alternative modeling techniques.


25

Fund MappingTime-Varying Beta Kalman Filter


26

Fund MappingTime-Varying Beta Example 1

Kalman 150day 300day OLSRMSE 7.40% 7.50% 7.60% 7.60%#ofPoints 2,360 2,241 2,096 2,395

TrackingError(outofsample,annualized)


27

Fund MappingTime-Varying Beta Example 2

TrackingError(outofsample,annualized)Kalman 150day 300day OLS

RMSE 0.84% 9.90% 9.00% 3.60%#ofPoints 2,365 2,246 2,096 2,395


28

Fund MappingModel Selection

Which type of model to use is a complex decision that has a material economic impact on a hedging program

Model selection criteria to consider include Model performancep

Out-of-sample tracking error should be stable, in-sample error should be low Different models should be tested (different combinations of indices, training

windows sizes, dynamical models, etc.)Business process manageability Business process manageability Dynamic models may be difficult to manage without sophisticated daily operations

controls, reporting and fail-safes Static models may lose predictive power in the event of sudden structural

changes in the marketchanges in the market Hedging instruments

Choice of models and indices depends on availability of liquid hedging instruments

Real-time data Selected indices should be quoted in real-time


29

Section 3: Modeling Bond Fundsg


30

Consequences of using GBM as process for bond fund returns Bond fund returns are highly dependent on long term interest rate levelsBond fund returns are highly dependent on long term interest rate levels

when interest rates increase bond prices decline Experiment: explicitly link interest rate movements to bond fund price changes using a stochastic equity and

interest rate generator and price a European put Option when T= 30, _S=0.2, _r=0.01, moneyness = 140%

correlation=1 correlation=0 correlation=1

Stochasticrates BS %Diff Stochasticrate BS %Diff Stochasticrate BS %Diff

PutPrice 1.7164 1.7164 0.00% 5.0269 5.0269 0.00% 8.1371 8.1371 0.00%

Delta 0.0376 0.0360 4.13% 0.0602 0.0574 4.67% 0.0668 0.0632 5.30%

Naively using GBM to model bond fund returns may have the following unwelcomed consequences

Rho 36.0957 35.0724 2.83% 72.8377 70.9825 2.55% 97.6541 95.3216 2.39%

Vega 44.5833 43.3457 2.78% 63.1159 63.0164 0.16% 70.7413 67.9621 3.93%

Gamma 0.0010 0.0009 6.82% 0.0011 0.0010 7.09% 0.0009 0.0009 7.59%

Naively using GBM to model bond fund returns may have the following unwelcomed consequences Biased hedge ratios and larger hedge breakage numbers

Dealing with the issue in practice Use a stochastic equity and interest rate models to model your risk Consider using multi-factor interest rate models Make careful adjustments to how you calculate your hedge program Greeks and to how you measure

hedge program performance


31

Modeling Bond Funds

The common procedure includes the following steps: Simulate forward curves for each projection step based on a calibrated stochastic interest rates model,

such as HW1/2 LMM or String model

Additional Modeling Considerations

such as HW1/2, LMM or String model Estimate the average bond duration based on liability cash flows Compute the zero coupon bond yield rate with duration and duration+1 for each projection step t The difference of yield rates between step t-1 and step t is the bond fund return

Other considerations: Some companies layer in other stochastic processes like credit spreads and inflation to better model

bond fund return process Many companies adjust how they calculate their Hedge program Greeks to capture the dynamics

b t h i i t t t d b d f d tbetween changes in interest rates and bond fund returns Many companies also adjust their performance attribution model to reflect such assumptions and to

properly partition hedge program performance


32

Modeling Bond Funds

The Hull-White Two-Factor Model defines the short interest rate as:HW2F Approach

0)0(),()()()( rrtYtXttr

Where the two factors and are stochastic processes defined by the following linear SDEs under the risk neutral measure:

)(tX )( tY

0)0(),(

0)0(),(

2

1

YtdWbYdtdY

XtdWaXdtdX

With a two dimensional wiener process with correlation

From this short rate model, forward curves are able to generated at each time step t, where

dttdWtdW )()( 21)1( sstfFrom this short rate model, forward curves are able to generated at each time step t, where

s=1,2,T years.)1,,( sstf


33

Modeling Bond Funds

Assuming the average cash flow duration is D, define two bond yield rates at time t

Compute Zero Coupon Bond Yield Rates

1 )1,,()( sstftBond

Then the bond fund return can be defined as the difference of two yields

12

1

)1,,()(

),,()(

Ds

Ds

sstftBond

f

Then the bond fund return can be defined as the difference of two yields

Here is the maintenance fee for this bond fund and this logic need to be adjusted if using stub year.

MfeetBondtBondtbond )()1()( 12Mfee


34

Section 4: Quantifying Slippages due to Basis Risky g pp g


35

Simulating Basis Risk in Hedge OverlayA Simple Model

21122122

22

21

21 ..2 YTE2211 ~~ rrY

1. R Squared4. Tracking Error

2.Correlation3. Volatility Estimates

dttdZtdZ )()( 21

2. Simulated Correlation Estimate = Sqrt(R Squared)=Sqrt( 92.67%) = .9627

3. Account Value equity Vol assumption is 20% and the Cash Index Vol used for hedging simulation is 20%4 P j t d T ki E S t[ *(1 C l ti 2)] S t[ 202 *(1 96272)] 5 41%**2

1. R Squared assumption is 92.67%

4. Projected Tracking Error = Sqrt[ *(1-.Correlation 2)] = Sqrt[.202 *(1-.96272)] = 5.41%**2AV

**Weighted average tracking error based on daily data from the top 5 US equity sub account funds according to VARDS


36

Quantifying the Impact of Basis RiskA Simple Model

The table below presents results on repeated hedging simulations of a 5 year put option with a strike 1000, AV=1100, T=5, Q=0, sig=20%, r = 5% with a weekly rebalancing interval, and a correlation between the account value and index returns of .9627, a tracking error of 4.5%, and a real world drift of 8%

Across 10,000 paths

Optn Optn OptionCost Cost Cost

w/o Tracking Error w 5.4% Tracking Error NakedMax 154 365 681Min 55 -119 0Min 55 -119 0Mean 102 102 60Std 8 43 117Std/Option Cost 7.90% 41.86% 194.15%

Theoretical Option Cost 102

Note how the standard deviation of the cost of hedging with basis risk is 5 times higher than without basis risk In the real world you have volatility of the fund, the volatility of index, and the correlation changing, whereas in

our simulation these three factors were fixedIt is also worth noting the underlying has a return volatility of 20% as previously mentioned while the option It is also worth noting, the underlying has a return volatility of 20% as previously mentioned, while the option position itself at time zero has a return volatility of 56%*

* Option return volatility=delta*(S/put value)*underlying return volatilityAnnuity Solutions Group | Aon Benfield Securities, Inc. | November 15, 2011 35

37

Simulated Impact of Basis Risk on CostA Simple Model


38

Simulated Impact of Basis Risk on SurplusA Simple Model

4000

6000Simulated Surplus with no basis risk

300 200 100 0 100 200 3000

2000

C

o

u

n

t

4000

6000Simulated Surplus with 5.4% Tracking Eror

-300 -200 -100 0 100 200 300Surplus

0

2000

4000

C

o

u

n

t

With tracking error, even if you hedge properly in a laboratory setting, you could face very significant gains or losses even AFTER hedging

-300 -200 -100 0 100 200 3000

Surplus


39

Basis Risk

Basis Risk can have a dramatic impact on hedge program performance Even with highly efficient hedging programs considerable residual risk remains due to the following

Conclusion

Gearing of the Liability position Ex ante basis risk between hedged items and hedging instruments in the laboratory Changing levels of correlation and absolute volatility in practice

Suggestions: Suggestions: Monitor tracking error closely Use index funds that can be hedged Simulate hedging new product designs to better appreciated the impact basis risk has on hedge program

performance, and on capital and reserve levels Implement more advanced fund mapping techniques Develop, test and implement more complex models of basis risk inside of hedge program simulations Move your hedge program Greeks calculations to after the NAVs have been updated from the market

close


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2011 Chicago Ebig Mtg Phillips

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