Best Estimates for Reserves Glen Barnett and Ben Zehnwirth

Best Estimates for Reserves

Glen Barnett and

Ben Zehnwirth

Overview

• Assessing ratios (loss development) (old paradigm)– the mathematical framework underlying ratio techniques– gives a tool for assessing whether ratio techniques ‘work’– Find: often powerless to provide meaningful forecasts

• The solution? (new paradigm)– a probabilistic modelling framework – powerful and flexible enough to capture the information in

the data– benefits unimaginable under the old paradigm –

Segmentation, Pricing Excess Layers, Value-at-risk

- 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

Introduction

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

Introduction

5,012 8,269 10,907 11,805 13,539 16,181 18,009 18,608 18,662 18,834106 4,285 5,396 10,666 13,782 15,599 15,599 16,272 16,807

3,410 8,992 13,873 16,141 18,735 22,214 22,863 23,4665,655 11,555 15,766 21,266 23,425 26,083 27,0671,092 9,565 15,836 22,169 25,955 26,1801,513 6,445 11,702 12,935 15,852

557 4,020 10,946 12,3141,351 6,947 13,1123,133 5,3952,063

8,269 / 5,012= 1.65

8,992 / 3,410= 2.64

0 1 2 3 4 5 6 7 8 9

Ratio (loss development) techniques start with a (cumulative) loss development array:

Ratio techniques

- analysed by Mack (1993)

Incurred Losses, Historical Loss Development Study, 1991 Ed. (RAA)

We can graph the points in a cumulative array:

Ratio techniques

5,012 8,269 10,907 11,805 13,539 16,181 18,009 18,608 18,662 18,834106 4,285 5,396 10,666 13,782 15,599 15,599 16,272 16,807

3,410 8,992 13,873 16,141 18,735 22,214 22,863 23,4665,655 11,555 15,766 21,266 23,425 26,083 27,0671,092 9,565 15,836 22,169 25,955 26,1801,513 6,445 11,702 12,935 15,852

557 4,020 10,946 12,3141,351 6,947 13,1123,133 5,3952,063

Cum.(1) vs Cum.(0)

4,0002,0000

12,000

11,000

10,000

9,000

8,000

7,000

6,0005,000

4,000

3,000

2,000

1,000

0

8,269

5,012

Any ratio is the slope of a line through the origin.

Ratio techniques

Slope = rise/run (the ratio)

= 8,269/5,012

= 1.65

What is the SLOPE of this line?

Cum.(1) vs Cum.(0)

4,0002,0000

12,000

11,000

10,000

9,000

8,000

7,000

6,0005,000

4,000

3,000

2,000

1,000

0

2.641.65

8,269

5,012

Ratio techniques

Cum.(1) vs Cum.(0)

4,0002,0000

12,000

10,000

8,000

6,000

4,000

2,000

0

Average ratios are “average” trends through the origin

Select ratios to be typical (“average”) ratios

e.g. arithmetic average, chain ladder, average of last three years,...

Cum.(1) vs Cum.(0)

4,0002,0000

12,000

10,000

8,000

6,000

4,000

2,000

0

Arithmetic Average ratio

1/n (y/x)

Chain Ladder Ratio

= y/ x

(Volume weighted average)

Regression Line through

origin

Ratio techniques

BUT an average trend line through the origin is a regression line through the origin!

Average ratio techniques are forms of regression.

j-1 j

x{ }y

(Chain Ladder - Mack 1993)

Regression equation:

y = bx + ,

where E( ) = 0

and Var( ) = 2x

Ratio techniques

If ratios are regressions, we can describe ratio techniques using formal statistical language.

Equivalently

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

which is the Chain Ladder Ratio (or Weighted Average by Volume, or Volume Weighted Ratio)

Ratio techniques

To determine an average ratio (slope) b we use weighted least squares. Method aims to estimate b by minimising

w(y-bx)2,

where weight w = 1/ x 1/ Var( )

We obtain

b = = = ^ x y ·1/ x

x2 ·1/ x

y/ x · x

x

y

x

Ratio techniques

We have now seen that the Chain Ladder Ratio is a regression estimator through the origin

Mack (‘93) ctd:• Can generalise by Var()2x

• By changing obtain other ‘averages’:– = 2: Arithmetic Average– = 1: Chain Ladder (Average Weighted by Volume)– = 0: Ordinary regression through the origin (Wtd Volume2

Ratio)

Any average ratio is a regression estimator through the origin

We know that when we apply a ratio technique, we are actually fitting a regression model through the origin.

Any regression model is based on a set of assumptions.

If the assumptions of a model are not supported by the data then any subsequent calculations made using the model (eg forecasts) are meaningless – they are based on the model, not the data.

Regression methodology allows us to test the assumptions made by a model.

Assessing ratio techniques

How does this formal regression methodology help us to assess whether ratio techniques ‘work’?

1 E( y | x ) = bx – i.e. to obtain the mean cumulative at development period

j, take the cumulative at the previous period and multiply by the ratio.

2 No trend in the payment period (diagonal) direction– ratio techniques do not allow for changes in inflation.

3 Normality– the models assume that observations are values from a

normal distribution.


What are the major assumptions made by the models based on ratio techniques?

Fitted values

- value given by the model (the value on the line)

called the fitted (or predicted) value, y


^

Fitted line

x

y

xi

yi

yi^


Residual = Observed value - Fitted value

Fitted line

x

y

xi

residual

yi

yi

yi^

Residual Analysis Residual = Data – Fit

Raw residuals have different standard deviations - need to adjust to make them comparable

Many model checks use standardized residuals

Standardized residual = Residual

Std. dev.(residual)

______________


What can we do with the residuals?

What features of the data does this model not capture?

1990

6

1996

x

e.g.:

Plot vs dev. yr

Plot vs pmt. yr

Plot vs acci. yr

Plot vs fitted

Residual plots should appear random about 0, without pattern.



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

Std. Residuals vs. Fitted Values

0 5000 10000 15000 20000 25000 30000

-1

0

1

2Underfit small values

Overfit large values

1982 is underfitted

1982: low incurred development

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

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 ear

81 82 83 84 85 86 87 88 89

-1

0

1

2

1984: high incurred development

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

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 ear

81 82 83 84 85 86 87 88 89

-1

0

1

2

1984 is overfitted

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



Cum.(1) vs Cum.(0)

4,0002,0000

12,000

11,000

10,000

9,000

8,000

7,000

6,0005,000

4,000

3,000

2,000

1,000

0

Regression line through origin causes underfit/overfit

‘Best’ line

Overfit large

values

Underfit small

values

Typical with real, exposure-adjusted data.

’Best’ line not through origin - has an intercept.


Including an intercept term :

y = a + bx + ,

or equivalently

y–x = a + (b – 1) x + ,

Incremental Cumulative at j at j -1

yx j-1 j

j-1 j

}y-x

Cumulative

Incremental

(Murphy 1995)


y - x = a + (b - 1)x +

Is b - 1 significantly different from zero? (Venter 1998)

Case 1: b = 1 a ≠ 0 - ratio has no power to predict next incremental

- abandon ratios and predict next incremental

by:

a = Average(incrementals) (=0)

Case 2: b > 1

- could use link-ratio techniques for projection with possibly an intercept

^

Cum.(1) vs Cum.(0)

4,0002,0000

12,000

10,000

8,000

6,000

4,000

2,000

0

Incr.(1) vs Cum.(0)

Corr. = -0.117, P-value = 0.764

4,0002,000

8,000

7,000

6,000

5,000

4,000

3,000

“Ratio not a predictor of future emergence”

Assessing ratio techniquesMack data:

Cum.(1) vs Cum.(0)

150,000100,00050,0000

400,000

350,000

300,000

250,000

200,000

150,000

100,000

50,000

0

Incr.(1) vs Cum.(0)

Corr. = 0.985, P-value = 0.000

150,000100,000

220,000

200,000

180,000

160,000

140,000

120,000

100,000

Ratios do seem to work for some arrays...


• What about Case 2: b > 1, a = 0?

ABC

Incr.(1) vs Acc. Yr

Corr. = 0.985, P-value = 0.000

77 78 79 80 81 82 83 84 85 86

220,000

200,000

180,000

160,000

140,000

120,000

100,000

Incr.(1) vs Cum.(0)

Corr. = 0.985, P-value = 0.000

150,000100,000

220,000

200,000

180,000

160,000

140,000

120,000

100,000

... most often arrays with trends down the accident years

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



Condition 1:

Condition 2:

y-x

w

w

j-1 j

}y-x

Incremental

x{ j-1 j

Cumulative

}yx{

y-x

w

Constant Trend

y - x = a + (b - 1) x +

usually ratios not helpful

ratios work: - see acc.yr. trend


y – x = a0 + a1w + (b – 1) x +

Intercept Acc Yrtrend

‘Ratio’

This gives the Extended Link Ratio Family of models

Include a trend parameter in the model:

j-1 j

x{ }y12

n

w


Case 3: b = 1, a1 0

• abandon ratios – model ‘correlation’ using trend parameter

Trend Adj. Incr.(1) vs Cum.(0)

Corr. = 0.110, P-value = 0.762

160,000140,000120,000100,00080,000

10,000

5,000

0

-5,000

-10,000

-15,000

After adjusting for accident trends, the relationship with the previous cumulative often disappears.


Summary of Assumption 1: E( y | x ) = bx

• Very often not satisfied by the data– residuals suggest intercept is needed

• Include intercept term:– Case 1: ratios abandoned in favour of intercepts– Case 2: projection using link ratios

• A relationship between y and x is often better modeled using a trend parameter– Case 3: ratios abandoned in favour of trend terms

• ‘Optimal’ model in the ELRF is not likely to include ratios!


Assumption 2: Are there trends in the payment period direction?

• Commonly have changing payment period trends

• Indicated here by Chain Ladder Ratio model residuals

• Assumption 2 is not supported by this data

Wtd Std Res vs Pay. Yr

78 79 80 81 82 83 84 85 86 87

2

1.5

1

0.5

0

-0.5

-1


Why are changing payment period trends important?

0 1

1990

1991

1992

1993

Change in inflation between 1992 and 1993

- Relationship between development years differs over accident years

- Must account for payment trends, or can’t understand development patterns

Average ratio incorporates effect of change in inflation

10% 10% 0%

Incremental

1600 2480 2964 3206 1760 2728 3212 1936 2904 1936

Cumulative

Ave ratio: 1.53 1.19 1.08

968 484 242

1033 553 288Forecast Incremental:

Actual Incremental:

% increase over actual:

Implied inflation:

6.7% 14.3% 18.8%

6.7% 7.1% 4.0%

Final acci. year:


What are you projecting?

1600 800 400 200

1600 880 484 242

1760 968 484

1936 968

1936

Very unlikely you’d want to assume this!

- Cumulatives “smear” over the breaks

- With randomness and cumulation means no hope of seeing the trend changes

- Lack control over future inflation assumption

- Even ‘best’ model in ELRF can’t capture such trend changes



Pan6Assumption 3: Often have non-normality:

Considerable right-skewness - even if model for mean was correct, estimates may be very poor

Wtd Std Res vs Fitted

8,000,0006,000,0004,000,0002,000,0000

2.5

2

1.5

1

0.5

0

-0.5

-1


Combining answers from several techniques

- It is a misguided exercise to combine information from several models without assessing their appropriateness (fidelity to data, assumptions, validation) — why mix good with bad?

- Answers should not be selected merely on the basis of their similarity to each other — not borrowing strength - Projections from the best models unlikely to come from the centre of the range of answers.


- Misguided to try to fit a continuous probability distribution to the results from a collection of methods. Different methods do NOT simply yield random values from some underlying process centered on the correct answer - many methods will share similar biases. e.g. may miss the same features.

- More methods do NOT mean more information about the process.

- the range of answers from a variety of methods does NOT reflect the uncertainty in the process generating the losses.


Summary of ratio techniques

• Average ratios are regression estimators through the origin.

• Regression methods allow us to test the assumptions of a model.

• Often find that major assumptions of ratios model are not satisfied by the data– lack predictive power– cannot capture payment period trend changes– non-normality.

• Need to work from a different paradigm.

Where to from here?

- very important to check if the technique is appropriate!

- lack of predictive power of cumulative to predict next incremental suggests abandoning ratios

- changing trends (e.g. against payment years) suggests modeling trend changes

- non-normality and nonlinear trends suggests transformation - particularly log transform


100

d

t = w+d

Development year

Payment year

Accident yearw

Trends occur in three directions:

19861986

19871987

19981998Futu

re

Past

Probabilistic modelling

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 changing 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


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


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)


The modeling framework (containing many possible models) consists of four components:

Model the mean in each cell by:

- the level of its own accident year

- plus the development trends to that development

- plus the payment trends to that payment year

Model the distribution about the mean


0 1 2 3 4 5 6 7 8

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 0

500

1000

1500

2000

2500

3000

Real data show the same features as the model

Log scale

Original scale

- spread related to mean


100000 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

Probabilistic modellingProperties (axioms) of trends

Underlying Trends in the DataUnderlying Trends in the Data


0.1

0.3

0.15

10.5

11

11.5

12

12.5

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

Accident year 1983

Accident year 1978 Accident year 1979

Development year trends

Projection of trends onto other directions


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

10.5

11

11.5

12

12.5

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

Trends + randomness

Development Year


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


log data

Payment Period Trends

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

15.63% trend 10% trend 78-82, 30% trend 82-83, and 15% trend 83+

13.25

12.75

12.25

11.75

11.25

13.75

Summary of trend properties of loss development arrays


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

• You cannot determine payment period trend changes without including parameters in that direction

• 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

Checking the modeling framework


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

– trends are the ‘same’ and change in the same periods

– amount of randomness is the same

• 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 (ABC)

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

Noisy data is not necessarily hard to predict


Very noisy data

2 orders of magnitude

• residual plots and other diagnostics for a simple model are good

• forecast 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).

• compare forecasts as most recent years removed (validation):

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


We can simulate from the distribution of the sums of log-normals (accident year totals, payment year totals, total reserve).


• can obtain means (and s.d.’s) for reinsurance layers.• can calculate margins at given reserve/premium

HistogramKernelLognormalGamma

Density Comparisons

1 Unit = $1,000,000

10080604020

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

• distributions can be computed for every cell and sums of cells

• include both process risk and parameter risk. Ignoring parameter risk leads to underreserving and underpricing (especially in reinsurance).

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

Prediction intervals and uncertainty

Prediction and Reserving

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

• identify trends, stability of trends and distributions about trends. 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.


• 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.

Other Benefits of the Statistical Paradigm

Value at Risk

- PALD results useful for Value-at-Risk calculations

- for any given quantile, the Value at Risk is simply that quantile minus the provision.

- maximum dollar loss at the 100(1–)% confidence level is the V-a-R for the (1) quantile.


Value at Risk

Selected Statistics (Kernel Density Estimate) Mean Median Std Dev. Sel.Value % #Std.Devs V-a-R 479.942 474.406 73.610 550.000 83.084 0.958 70.490

/


Fitted vs Predictive Distributions

- predictive distributions incorporate parameter uncertainty

- fitted distributions underestimate the uncertainty (variance)

- since mean and variance are related for losses, fitted distributions lead to under-reserving

- predictive/fitted distinction a critical consideration when performing DFA


Predictive Aggregate Loss Distributions (PALD)

- can’t calculate distribution of totals directly (acci year, pmt year, total outstanding)

- generally can’t use CLT: skewness, dependence, different distributions larger sample size

- can simulate from the predictive distribution of aggregates

e.g. predictive distribution of payment or calendar year totals important for liability-side of DFA


Predictive Aggregate Loss Distributions (PALD)

Simulated distribution of total outstanding + various approximations to predictive distribution

HistogramKernelLognormalGamma

Density Comparisons

1 Unit = $1,000

800600400

0.006

0.005

0.004

0.003

0.002

0.001

0


Pricing

- Can simulate future year for pricing purposes

(assumptions about level, etc)

- Obtain entire distribution of predicted losses, not just

mean and variance


Several Reinsurance Layers

Layer 1 Ground-Up

Data

Trends change together - “same” model works for each triangle

Layer 2


Example

Same parameter structure - payment year trends change together

Same thing in other directions

W t d . S t d . R e s i d u a l sv s . P a y . Y e a r

8 8 8 9 9 0 9 1 9 2 9 3 9 4 9 5

- 2

- 1

0

1

2

W t d . S t d . R e s i d u a l sv s . P a y . Y e a r

8 8 8 9 9 0 9 1 9 2 9 3 9 4 9 5

- 2

- 1

0

1

2

XS10 (used here like GU data) 40XS10 (a layer of XS10)


Correlation between layers

Layer1 Ground-Up

DataLayer2

Correlation: matters when simulating or forecasting layer sums/differences

Corresponding wtd residuals

-2.5

0

2.5

-1-3 -2 0 1 2

40XS10

XS10

DY 0-4

DY 5+(wt = 0.219)


Equality of parameters across layers

- Often development and payment year trend parameters nearly identical (same percentage changes).

- Can set equal in a combined regression model, taking the correlations into account.

- More parsimonious model.


Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

2

0

-2

-4

Acc. Yr Trends

85 86 87 88 89 90 91 92 93 94 95

12

10

8

6

Pay. Yr Trends

88 89 90 91 92 93 94 95

4

2

0

-2

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10

1.2

0.8

0.4

0


XS10

40XS10

No significant differences between layers



reduced model

As we eliminate parameters, the estimates of the remaining parameters generally become more precise

XS10 40X10-XS10Years Estimate S.E. t-ratio Diff. S.E. t-ratio

1985-89 10.3745 0.2006 51.71 0.0000 0.0000 Alpha

1990-95 8.8064 0.2239 39.33 0.0000 0.0000

0:1 2.4386 0.1891 12.90 0.0000 0.0000 Gamma

3:10 -1.0841 0.0773 -14.02 0.0000 0.0000

Iota 1991-93 0.5212 0.0796 6.55 -0.0546 0.0242 -2.26


Segments often related too

ALL2M (combined data) Segment of ALL2M (Hospital S)

Payment year trends

Trends change in same years


89 90 91 92 93 94 95 96 97 98

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

-3

-3.5

-4


89 90 91 92 93 94 95 96 97 98

3

2.5

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

-3


Excess layers as differences of limited layers

Limited to $2M

- Still same parameter structure as the limited layers

- Excess layer has larger relative uncertainty

- Larger uncertainty on parameter estimates

Limited to $1M

$1M XS $1M— =


Iota (inflation) Parameter Estimates

Payment 2M 1M 1Mxs1M

Years Iota S.E. Iota S.E. Iota S.E.

1989-98 0.11 0.03 0.08 0.03 0.06 0.05

- Superimposed inflation estimate not credibleon 1Mxs1M alone


- Fit a combined model for the two limited layers, or the base layer and the excess layer

- If some parameters are equal (particularly the inflation parameter in this case), may yield a better inflation estimate for the excess layer


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

Summary

RATIO TECHNIQUES

RATIO TECHNIQUES

EXTENDED LINK RATIO FAMILY

EXTENDED LINK RATIO FAMILY

PREDICTIVE AGGREGATE LOSS DISTRIBUTION

PREDICTIVE AGGREGATE LOSS DISTRIBUTION

OPTIMAL ASSET ALLOCATIONOPTIMAL ASSET ALLOCATION

REINSURANCEREINSURANCE

RISK BASED CAPITALRISK BASED CAPITAL

SIMULATIONSIMULATIONPROBABILISTIC TREND MODELSPROBABILISTIC TREND MODELS

Summary

Contact: [email protected]

or see some of the papers and examples on our website: www.insureware.com

Further Information

Documents

Best Estimates for Reserves Glen Barnett and Ben Zehnwirth