Best Estimates for Reserves

Glen Barnett

and

Ben Zehnwirth

email: GlenBarnett@insureware.com, BenZehnwirth@insureware.com,

or find us on http://www.insureware.com

Summary

I. Ratio techniques and extensions

• Ratios are regressions

• Regressions have assumptions       (know what you assume when using ratios)

• Assumptions need to be checked        (When do ratios work?)

• Assumptions often don't hold

• What does this suggest?

Summary

II. Statistical modeling framework            (Probabilistic Trend Family of models)

• Model the logarithms of the incrementals

• Parameters to pick up trends in the three directions

• Probability Distribution to every cell

• Assumptions generally met

• Assessing stability of trends        (confidence about the future)

III. Reserve Figure

Ratio techniques and extensions

j-1 j

x{ }y

(Mack 1993)

Chain Ladder Ratio

E( y | x ) = bx and Var( y | x ) = 2 x

- weighted least squares with w = 1/ x

- weighted average with w = x

b = = = ^ x y ·1/ x

x2 ·1/ x

y/ x · x x

y

x

Ratio techniques are regression estimatorsRatio techniques are regression estimators

y

xtrend

y/x

Equivalently, y = bx +

where E( ) = 0 and Var ( ) = 2x

· Chain ladder is weighted regression through origin

· Derives standard errors of link ratios, forecasts for chain ladder

· Introduces weighting (: Var()2x) parameter:Ave. Dev. Factor Chain LadderOrdinary regression through origin

Regression methodology advantages· Standard Errors of ratios and forecasts· Testing of assumptions

Incurred losses analysed by Mack(1993)

Residuals of chain ladder ratios

W td . S td . R es id u al sv s . F i t ted V alu es

0 5000 10000 15000 20000 25000 30000

-1

0

1

2

Is E(y|x) = bx satisfied by the data?

1982 is underfitted

1982: low incurred development

U n ad ju s ted D a tav s . D ev . Y ear

0 1 2 3 4 5 6 7 8 9

0

5e3

1e4

1.5e4

2e4

2.5e4

W td . S td . R es id u al sv s . A cc. Y ea r

81 82 83 84 85 86 87 88 89

-1

0

1

2

1984: high incurred developmentU n ad ju s ted D a ta

v s . D ev . Y ear

0 1 2 3 4 5 6 7 8 9

0

5e3

1e4

1.5e4

2e4

2.5e4

W td . S td . R es id u al sv s . A cc. Y ea r

81 82 83 84 85 86 87 88 89

-1

0

1

2

1984 is overfitted

The best line has an intercept - typical with real, exposure adjusted data

Why is E(y|x) = bx not satisfied by the data?

y

x

(Murphy, 1995)- Including an intercept will give a better fit.

- Unbiased “Development Factors”.

y = a + bx + where Var ( ) = 2x

• Works wholly within a regression framework

• Advocates use of intercept

• Derives standard errors of forecasts for

Ratios with Intercepts

Devel. Intercept Ratio(Slope)Period Est. S.E. P-value Est. Ratio-1 S.E. P-value00-01 4,329.21 516.31 0.00007 1.21445 0.21445 0.42131 0.626401-02 4,159.69 2,531.37 0.15144 1.06962 0.06962 0.35842 0.852402-03 4,235.92 2,814.52 0.19266 0.91968 -0.08032 0.24743 0.758603-04 2,188.79 1,133.11 0.12557 1.03341 0.03341 0.07443 0.6767904-05 3,562.27 2,031.41 0.17778 0.92675 -0.07325 0.11023 0.5538905-06 743.46 2,387.33 0.78494 1.00608 0.00608 0.12204 0.9647706-07 792.81 153.13 0.12147 0.99109 -0.00891 0.00824 0.4748307-08 **** **** **** 1.01689 0.01689 0.01495 0.4614208-09 **** **** **** 1.00922 0.00922 **** ****

ELRF Parameters

Delta () = 1 AIC = 760.5 In order for the test to be conducted at an overall 5% level, a parameter is regarded as insignificant if the corresponding P-Value is greater than 0.00320.

(Venter, 1996)

Incremental at dev. period j

Cumulative at dev. period j-1

y – x = a + (b – 1) x +

j-1 j

x{ }y

Case (ii) b = 1, a 0

a = Ave (incrementals)

Use link-ratios for projection

Abandon Ratios - No predictive power

Case (i) b > 1, a = 0

^

Plot of Development Year 1vs Development Year 0

Cumulative (1) vs.

Cumulative (0)

0

4000

8000

12000

0 2000 5000

1000

3000

5000

7000

9000

0 2000 5000

Corr=-0.117 P-value=0.764

Incremental (1) vs.

Cumulative (0)

Weighted Standardized Residuals of Cape Cod model

W td . S td . Residu alsv s. D ev . Y ear

1 2 3 4 5 6 7 8 9

-2

-1

0

1

2

W td . S td . Residu alsv s. A cc. Y ear

81 82 83 84 85 86 87 88 89

-2

-1

0

1

2

W td . S td . Residu alsv s. Pay . Y ear

82 83 84 85 86 87 88 89 90

-2

-1

0

1

2

W td . S td . Residu alsv s. F itted V alu es

50 00 100 00 150 00 200 00 250 00

-2

-1

0

1

2

y – x = a + a = Ave (y–x)^

Forecasts

Cape Cod ModelChain Ladder Ratios Model

Accident Mean Standard Coeff. OfYear Forecast Error Variation

1981 0 0 ***1982 155 148 0.961983 616 586 0.951984 1633 702 0.431985 2779 1404 0.511986 3671 1976 0.541987 5455 2190 0.41988 10934 5351 0.491989 10668 6335 0.591990 16360 24606 1.5

Total 52272 26883 0.51

Accident Mean Standard Coeff. OfYear Forecast Error Variation

1981 0 0 ***1982 172 41 0.241983 483 464 0.961984 1112 498 0.451985 1971 1167 0.591986 4230 1526 0.361987 6908 1651 0.241988 10283 3231 0.311989 14905 3644 0.241990 19367 4521 0.23

Total 59430 8425 0.14

Trend Parameter For IncrementalsTrend Parameter For Incrementals

y – x = a0 + a1w + (b – 1) x +

Intercept Acc Yrtrend

“Ratio”

j-1 j

x{ }y12

n

w

An increasing trend down the accident years will ‘induce’ a correlation between (y-x) and x.

where Var()2x

Extended Link Ratio Family of Models

Wtd. Std. Residuals vs. Payment Year

78 79 80 81 82 83 84 85 86 87

-1.5

-1

-0.5

0

0.5

1

Wtd. Std. Residuals vs. Fitted Values

0 2500000 5000000 7500000 10000000

-1

0

1

2

Commonly have changing payment year trends

Often have non-normality

Even this extended family of models is generally inadequate:

(ABC)

(Pan6)

Part II

1122

100d

t = w+d

Development year

Payment year

Accident yearw

Trends occur in three directions:

Trend properties of loss development arrays

• Trends in payment year direction project onto the    other two directions and vice versa

• Changing trends can be hard to pick up in the    presence of noise, unless main trends are removed    first (regression as a form of adjustment)

• Modeling a changing trend as a single trend will result in    pattern in the residual plots

Underlying Trends in the DataUnderlying Trends in the Data

Projection of trends onto other directions

Changing trends hard to pick without removing main development and payment year trend.

Distribution of data about those trends

d10 2 3 4 5

y(d)

1

2

Development trend for single accident year, data on log scale:

(Data = Trends + Random Fluctuations)

log(p(d)) = y(d) = + i + d

y(0) = + 0

y(1) = + 1 + 1

y(2) = + 1 + 2 + 2

d

i=1

d10 2 3 4 5

p(d)

On the original (dollar) scale, each payment has a lognormal distribution, related by the trends.

All years - trends in 3 directions

log(p(w,d)) = y(w,d) = w+ i + j + w,d

d

i=1

w+d

j=1

Different levels for accident years

Payment year trends

You would never use all these parameters at the same time - parsimony is as important as flexibility.

A model with too many parameters will give poor forecasts.

Wtd. Std. Residuals vs. Payment Year

78 79 80 81 82 83 84 85 86 87 88 89 90 91

-2

-1

0

1

2

Fitting a single trend to changing trends

- estimates as an average trend

- changes show up in the residuals

• if the modeling framework “works”, it should be hard to differentiate between real data and data simulated from an identified model

Checking the modeling framework

• if you create (simulate) data, you should be able to identify the (known) changing trends in the data; mean forecasts should usually be within about 2 standard errors of the true mean

Smooth data can conceal changing trends

Individual Link Ratios by Delay

0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Very smooth data

(real array - ABC - values in paper)

Very smooth ratios

Smooth data can conceal changing trends

W td . S td . Residualsv s. D ev . Y ear

0 1 2 3 4 5 6 7 8 9 10

-1

0

1

2

W td . S td . Residualsv s. A cc. Y ear

77 78 79 80 81 82 83 84 85 86 87

-1

0

1

2

W td . S td . Residualsv s. P ay . Y ear

77 78 79 80 81 82 83 84 85 86 87

-1

0

1

2

W td . S td . Residualsv s. F i tted V alues

9.00 10.00 11.00 12.00

-1

0

1

2

Residuals after removing all accident year and development year trends.

• Volatile (noisy) data can be predictable (within model uncertainty) if trends are stable

The data in the following example is very volatile (noisy) -see the paper for this real data array (Pan6).

Noisy data is not necessarily hard to predict

Wtd. Std. Residualsvs. Pay. Year

86 87 88 89 90 91 92 93 94 95 96

-2

-1

0

1

Wtd. Std. Residualsvs. Acc. Year

86 87 88 89 90 91 92 93 94 95 96

-2

-1

0

1

Wtd. Std. Residualsvs. Dev. Year

0 1 2 3 4 5

-2

-1

0

1

Wtd. Std. Residualsvs. Fitted Values

10.00 11.00 12.00 13.00 14.00 15.00

-2

-1

0

1

Residual plots after removing a single development year and payment year

trend.

• change in development year trend (the trend between 0-1 is different from the later years)

• no obvious trend changes in other directions

• wider spread for first two development years

• single superimposed inflation parameter is not significantly different from 0

one accident year level, two development year trends, no payment year trend, weighted regression

Noisy data is not necessarily hard to predict

• residual plots and other diagnostics for that model are good

• forecast of this model yields an outstanding mean forecast of $20.6 million and a standard deviation of $9.3 million, so the standard deviation is high (volatile data).

• It is important to see how the forecasts compare as we remove the most recent years (validation):

Noisy data is not necessarily hard to predict

Years inEstimation

N Trend(dev period 1+)

standarderror

MeanFcst

Standard Error

86-96 44 0.6250 0.1432 20,352,011 9,136,87086-95 41 0.6102 0.1479 21,410,781 9,839,12786-94 35 0.6149 0.1681 21,037,520 10,654,17386-93 29 0.5024 0.1977 19,755,944 11,647,27486-92 25 0.5631 0.2143 18,567,664 11,529,359

Part III

• any single figure will be wrong, but we can find the probability of lying in a range.

• include both process risk and parameter risk. Ignoring parameter risk leads to underreserving.

• forecast distributions are accurate if assumptions about the future remain true.

Prediction intervals and uncertainty

• Distribution of sum of payment year totals important for dynamic financial analysis. Distributions for future underwriting years important for pricing.

• For a fixed security level on all the lines combined, the risk margin per line decreases as the number of lines increases.

• future uncertainty in loss reserves should be based on a probabilistic model, which might not be related to reserves carried by the in the past.

• uncertainty for each line should be based on a probabilistic model that describes the particular line

• experience may be unrelated to the industry as a whole.

• Security margins should be selected formally. Implicit risk margins may be much less or much more than required.

Risk Based Capital

Booking the Reserve

• extract information, in terms of trends, stability of trends and distributions about trends, for the loss development array. Validation analysis.

• formulate assumptions about future. If recent trends unstable, try to identify the cause, and use any relevant business knowledge.

• select percentile (use distribution of reserves, combined security margin, and available risk capital). Increased uncertainty about future trends may require a higher security margin.

Other Benefits of the Statistical Paradigm

• Credibility - if a trend parameter estimate for an individual company is not credible, it can be formally shrunk towards an industry estimate.

• Segmentation and layers - often the statistical model (parameter structure) identified for a combined array applies to some of its segments. These ideas can also be applied to territories etc. and to layers.