More information will be available at the St. James Room

4th floor 7 – 11 pm.

ICRFS-ELRF will be available for FREE!

Much of current discussion included in the software

ELRFELRF

PTFPTF

BF/ELRBF/ELR

MPTFMPTF LRTLRT

Models – Coin Versus Roulette WheelModels – Coin Versus Roulette Wheel

““Best Estimates for Reserves” is now included in 2005 CAS Best Estimates for Reserves” is now included in 2005 CAS Syllabus of Examinations. ReviewSyllabus of Examinations. Review http://casact.org/pubs/actrev/may01/latest.htm

Unique Benefits afforded by Paradigm Shift

SEE NEXTSEE NEXTSLIDESLIDE

MPTF discussed in Session 3 (3:15 Arlington) and 6 (10:30am White Hill)

Summary- Examples

• Many myths grounded in a flawed paradigm Ranges? Confidence Intervals? Myths Loss Reserve Upgrades. Myths• Link Ratios cannot capture trends and volatility• Link Ratios can give very false indications• Must model Paids and CREs separately Cannot determine volatility in paids from

incurreds!

Some real life examples taken from “Best Estimates..”

RELATIONSHIPS/CORRELATIONSRELATIONSHIPS/CORRELATIONS

CREDIBILITY MODELLINGCREDIBILITY MODELLING

CAPITAL ALLOCATION

CAPITAL ALLOCATION

ModelingModeling

MULTIPLE LINES/ SEGMENTS/LAYERS

MULTIPLE LINES/ SEGMENTS/LAYERS

V@RV@R

ADVERSE DEVELOPMENT

COVER

ADVERSE DEVELOPMENT

COVER

EXCESS OF LOSS

EXCESS OF LOSS

PAD PAD PTFPTFMPTFMPTF

REINSURANCEREINSURANCE

The Pleasure of Finding Things The Pleasure of Finding Things Out !Out !

The Pleasure of Finding Things The Pleasure of Finding Things Out !Out !

Unique Benefits afforded by Paradigm Shift

0 1 2 3 4 5 6 7 8 9 10 11 12

If we graph the data for an accident year against development year, we can see two trends.

e.g. trends in the development year direction

x xxxxx x xx x xx x

x

x

xx

xxx

x

x xx

x

x

ProbabilisticProbabilistic ModellingModelling

0 1 2 3 4 5 6 7 8 9 10 11 12

x

x

xx

xxx

x

x xx

x

xCould put a line through the points, using a ruler.

Or could do something formally, using regression.

x xxxxx x xx x xx x

Probabilistic ModellingProbabilistic Modelling

yy ˆ

Variance =313

)ˆ( 2

yy

The model is not just the trends in the mean, but the distribution about the mean

0 1 2 3 4 5 6 7 8 9 10 11 12

x

x

xx

xx

xx

x xx

x

x

(Data = Trends + Random Fluctuations)

Models Include More Than The TrendsIntroduction to Probabilistic ModellingIntroduction to Probabilistic Modelling

(y – ŷ)

)313()ˆ( 2

2

yy

oo o

oo

o oo o

o o o

o

Simulate “new” observations based in the trends and standard errors

0 1 2 3 4 5 6 7 8 9 10 11 12

x

x

xx

xx

xx

x xx

x

x

Simulating the Same “Features” in the Data

Introduction to Probabilistic ModellingIntroduction to Probabilistic Modelling

Simulated data should be indistinguishable from the real data

- Real Sample: x1,…,xn

- Fitted Distribution fitted lognormal

- Random Sample from fitted distribution: y1,…,yn

What does it mean to say a model gives a good fit? e.g. lognormal fit to claim size distribution

Model has probabilistic mechanisms that can reproduce the data

Does not mean we think the model generated the data

y’s look like x’s: —

PROBABILISTIC MODEL

RealData

S1 S2

Simulated triangles cannot be distinguished from real data – similar trends, trend changes in same periods, same amount of random variation about trends

S3

Based on Ratios

Models project past volatility into the future

yxX = Cum. @ j-1Y = Cum. @ j

Link Ratios are a comparison of columns

j-1 j

y

x

We can graph the ratios of Y to X

y/x y

x

y

x

y/x

ELRFELRF (Extended Link Ratio Family)(Extended Link Ratio Family)x is cumu. at dev. j-1 and y is cum. at dev. jx is cumu. at dev. j-1 and y is cum. at dev. j

Mack (1993)Mack (1993)

Chain Ladder Ratio ( Volume Weighted Average )

xVbxy 2 :

xy

xxy

xb̂ ,1 .1

x

y

nb, . δ

1ˆ22

x

w

bxyw

1 where

2 Minimize

Arithmetic Average

Intercept (Murphy (1994))Intercept (Murphy (1994))

Since y already includes x: y = x + p

Incremental Cumulative

at j at j -1

Is b -1 significant ? Venter (1996)

y a bx V x : 2

xVxbap 2 : 1

Use link-ratios for projection

Case (ii) b a 1 0

xVxbap 2 : 1 Case (i) b a 1 0

a Ave Incrementals

Abandon Ratios - No predictive power

yx j-1 j j-1 j

} p

x xx x

x

x

x

pCumulative Incremental

Is assumption E(p | x ) = a + (b-1) x tenable?

Note: If corr(x, p) = 0, then corr((b-1)x, p) = 0

If x, p uncorrelated, no ratio has predictive power

Ratio selection by actuarial judgement can’t overcome zero correlation.

p

x

xVxbap 2 : 1

Condition 1:

Condition 2:

j-1 j

} p yx j-1 j

Cumulative Incremental

wp

w

x

xx

x

x

x

x

x

909192

p

Now Introduce Trend Now Introduce Trend Parameter For IncrementalsParameter For Incrementals12

n

w

x y

xbwaap 110

a0Intercept

a1Trend

b Ratio

p

Condition 3:

Incremental

Review 3 conditions:

Condition 1: Zero trend

Condition 2: Constant trend, positive or negative

Condition 3: Non-constant trend

FORECASTING AND FORECASTING AND STATISTICAL MODELSSTATISTICAL MODELS

FUTURE PAST

(i) RECOGNIZE POTENTIAL ERRORS

(ii) ON STRAIGHT STRETCHES NAVIGATE

QUITE WELL

100

d

t = w+d

Development year

Calendar year

Accident yearw

Trends occur in three directions:

19861986

19871987

19981998Futu

re

Past

Probabilistic ModellingProbabilistic Modelling

M3IR5 DataM3IR5 Data100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072 7427

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530

100000 81873 67032 54881 44933 36788 30119 24660 20190

100000 81873 67032 54881 44933 36788 30119 24660

100000 81873 67032 54881 44933 36788 30119

100000 81873 67032 54881 44933 36788

100000 81873 67032 54881 44933

100000 81873 67032 54881

100000 81873 67032

100000 81873

100000

- 0.2d

d

0 1 2 3 4 5 6 7 8 9 10 11 12 13

PAID LOSS = EXP(alpha - 0.2d)

-0.2

alpha = 11.513

Axiomatic Properties of TrendsAxiomatic Properties of Trends

Probabilistic ModellingProbabilistic Modelling

0.1

0.3

0.15

Sales FiguresSales Figures

0.15

0.3

0.1

Resultant development year trendsResultant development year trends

WHEN CAN ACCIDENT YEARS BE REGARDED AS DEVELOPMENT YEARS?

GLEN BARNETT, BEN ZEHNWIRTH AND EUGENE DUBOSSARSKYAbstract

The chain ladder (volume-weighted average development factor) is perhaps the most widely used of the link ratio (age-to-age development factor) techniques,

being popular among actuaries in many countries. The chain ladder technique has a number of interesting properties. We present one such property, which indicates that the chain ladder doesn’t distinguish between accident years and development

years. While we have not seen a proof of this property in English language journals, it appears in Dannenberg, Kaas and Usman [1]. The result is also

discussed in Kaas et al [2]. We give a simple proof that the chain ladder possesses this property and discuss its other implications for the chain ladder technique. It

becomes clear that the chain ladder does not capture the structure of real triangles.

The Chain Ladder ( Volume Weighted Average )

Transpose Invariance property

Use Volume Weighted Average to project incremental data: Take incremental array, cumulate across, find

ratios, project, and difference back to incremental data.

Now: transpose incremental*, do Volume Weighted Average , transpose back same forecasts!

(equivalently, perform chain ladder ‘down’ not ‘across’: cumulate down, take ratios down, project down, difference back)

The Chain Ladder ( Volume Weighted Average )

1p

1p

The Chain ladder (Volume weighted aveage) - Transpose Invariance property

Chain ladder does not distinguish between accident and development directions. But they are not alike:

Log paid

3.54

4.55

5.56

6.57

7.58

8.5

0 1 2 3 4 5 6 7 8

Dev. year

Log paid

3.54

4.55

5.56

6.57

7.58

8.5

1968 1970 1972 1974 1976 1978 1980

Acc. year

Adj Log paid

5

5.5

6

6.5

7

7.5

8

1968 1970 1972 1974 1976 1978 1980

Adj log paid

-4

-3

-2

-1

0

1

2

0 2 4 6 8

raw data

adjusted for trend in other direction

The Chain Ladder ( Volume Weighted Average )

Additionally, chain ladder (and ratio methods in general) ignore abundant information in nearby data.

Log paid

3.54

4.55

5.56

6.57

7.58

8.5

0 1 2 3 4 5 6 7 8

Dev. year

* If you left out a point, how would you guess what it was?

- observations at same delay very informative.

-1.5

-1

-0.5

0

0.5

1

1.5

2 3 4 5 6

The Chain Ladder ( Volume Weighted Average )

Additionally, chain ladder (and ratio methods in general) ignore information in nearby data.

Log paid

3.54

4.55

5.56

6.57

7.58

8.5

0 1 2 3 4 5 6 7 8

Dev. year

* If you left out a point, how would you guess what it was?

- observations at same delay very informative.

- nearby delays also informative (smooth trends)

(could leave out whole development)

-1.5

-1

-0.5

0

0.5

1

1.5

2 3 4 5 6

Chain ladder ignores both

Unstable Trends, Low Process VariabilityData (ABC)

Major Calendar Year shifts satisfying Condition 3

The plots indicate a shift from calendar periods 84-85-86. However, we cannot adjust for accident period trends to diagnostically view what is left over along the calendar periods as we can with PTF models.This example is in “Best Estimates”

Do U assign zero weight to all years save last two or three?

The link-ratio type models cannot capture changes along the calendar periods (diagonals). Determine the optimal model and note that several of the ratios are set to 1. The residuals of the optimal model are displayed below.

Model DisplayDev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

-3

-2.5

-2

-1.5

-1

-0.5

00.1661

+-0.0134

-0.3994+-0.0131

-0.4692+-0.0054

-0.3944+-0.0090

-0.3362+-0.0100

Acc. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

9.5

10

10.5

11

11.5

12

12.5

13

0.1610+-0.0131

0.0473+-0.0105

0.0691+-0.0149

-0.0691+-0.0149

-0.0473+-0.0105

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0.0652+-0.0035

0.1083+-0.0123

0.1691+-0.0074

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10

0

1e-4

2e-4

3e-4

4e-4

5e-4

6e-4

7e-4

8e-4

Note that as you move down the accident years the 16%+_ trend kicks in at earlier development periods. If variance was not so small, we would not be able to see this on the graphs of the data themselves – the trend change would be ‘obscured’ by random fluctuation.

1977 Run-off

1978 Run-off 1979 Run-off

Link Ratios can give answers that are much too high (LR high) - Case Study 5.

This case study illustrates how the residuals in ELRF can be very powerful in demonstrating that methods based on link ratios can sometimes give answers that are much too high.

The ELRF module also allows us to assess the predictive power of link ratio methods compared to trends in the incremental data. For the data studied, trends in the incremental data have much more predictive power. Moreover, link ratio methods do not capture many of the features of the data.

Overview

Bring up the Weighted Residual Plot using the button.

Residuals represent the data minus what has been fitted to the data. Observe that the residuals vs calendar years (Wtd Std Res vs Cal Year) trend downwards (negative trend).

This means that the trends fitted to the data are much higher than the actual trends in the data.

Accordingly any forecast produced by this method will assume trends that are much higher than the trends in the data. Therefore, the forecast will be much too high.

Table 5.1 - Summary of Forecasts

Model

Forecast Outstandi

ng($000's)

Forecast SD ($000's)

CV

Volume Weighted Average(Chain Ladder)

896,133 104,117 11.6%

Arithmetic Average 1,167,464 234,466 20.1%

Below are the forecasts based on volume weighted average ratio and the arithmetic average ratio.

Do link ratio method have any predictive power for this data?

The Best Model in ELRF rarely uses link ratios and treats development periods as separate problems- Show them!

A good model for this data has the following trends and volatility about trends

With the calendar year trend of 8.71% ± 0.97%, we obtain a distribution with a mean $593,506,000, and a st. deviation of $42,191,000. Scenario 1

If we revert to the trend of 18.3% ± 2.6% experienced from 1981-1984, we obtain a reserve distribution with the mean of $751,912,000 with a standard deviation of $79,509,000. Scenario 2.

Returning to the calendar year trend changes, it is important to try and identify what caused those changes. We cannot just assume that the most recent trend (8.71% ± 0.97%) will continue for the next 17 years. Modelling other data types such as Case Reserve Estimates (CRE) and Number of Claims Closed (NCC) can assist in formulating assumptions about future calendar year trends. See below.

This would make it easier to decide on a future trend scenario along the calendar years.

The estimated trend between 1974-1978 is 47.5% ± 3.2%. If we assume that trend only for the next calendar year (1991-1992) and revert to a trend of 18.3% ± 2.6% thereafter, we obtain a reserve distribution with a mean of $1,007,496,000 and a standard deviation of $112,234,000. Scenario 3

Conclusions

There is some evidence that even the mean of $593,506,000 based on scenario 1 is too high. Accordingly, scenarios 2 and 3 appear to be even more unlikely given the features in these three triangles in the past.

Comparing PL vs CRE

OverviewThis case study illustrates how comparing a model for the Paid Losses with a model for the Case Reserve Estimates (CRE) gives additional critical information that cannot be extracted from the Incurred Losses triangle. For most portfolios, we find that CREs lag Paid Losses in respect of calendar year trends.

Bring up the Weighted Residual Plot using the button.

The residual display is quite informative. Early accident years, and late accident years and calendar years are under-fitted.

The link ratio regression model does not describe (nor capture) the structure in the data.

Most importantly, we cannot extract any information about the CREs and Paid Losses, and their relationship. This is done in Section 7.3 below using the PTF modelling framework.

Do link ratio methods have any predictive power for this data?

All the trends are not captured even with this model that is the best we can do in ELRF.More importantly, we cannot extract any information about the trends in the data and the volatility about the trends.

Paids vs CREs

In which year was this company purchased?

Loss Reserving Myths

1. Reserve Upgrades2. Ranges and Confidence Intervals

More information will be available at the St. James Room

4th floor 7 – 11 pm.

ICRFS-ELRF will be available for FREE!