Investment and Reinsurance Risk

1998 CAS DFA Seminar

Managing Risk in a Portfolio Context

Reinsurance and Reserves

Gary G. Venter

Sedgwick Re Insurance Strategy, Inc.

(INSTRAT)Investment and Reinsurance Risk

Goal

Manage investment and reinsurance risk simultaneously

Test strategies by their impact on bottom line income probability distributions

Create risk/return efficient frontiers from strategies tested

Risk/Return Efficient Frontier @ probability=p

Any strategy above or to the right of the efficient frontier provides less benefit than points on or below the efficient frontier.

Inefficient options can be quickly recognized.

By establishing an efficient frontier of options, one can discover and create new or hybrid solutions that provide greater benefit

Measuring Risk

Problems with standard deviation

Inconsistent meaning among distributions

Treats upside and downside risk the same

Look at key individual percentiles

Preference might be to increase mean return, reduce downside risk by giving up chance of big gain

Risks to Income

Investment performance

Market results

Need to liquidate to pay losses

Underwriting result

Current year

Reserve development

Reinsurance

Modeling Requirements

Model the risk elements

Investment market, current u/w, develop-ment, reinsurance, cash flow, taxes, GAAP, statutory

Get probability distribution of income for each strategy to find efficient frontier of strategies at each probability level

Need to generate scenarios by probability, not just wide variety

Modeling Investment Risk

Yield curve

Diffusion model

Other assets

Regression models

Why a Diffusion Model?

Rates are moving continuously

Process generating movement of short-term rates also generates yield curve

Can guarantee arbitrage-free yield curve movements in model

Can calibrate to market - e.g., to bond options

The CIR Model

dr = a(b - r)dt + sr1/2dz.

r is short-term rate

b is reverting mean

a is speed of reversion

s is volatility measure

What is z?

z is standard Brownian motion

Starts at zero

After time t, z is normally distributed with mean 0 and variance t

Notation sdz means that after time t, variance is s2t

Thus for CIR sr1/2dzmeans that after time t, variance is s2rt

Use this to simulate short intervals t

Yield Curve under CIR

Y(T) = A(T) + rB(T) for term T

A(T) = -2(ab/s2T)lnC(T) – 2aby/s2

B(T) = [1 – C(T)]/yT

C(T) = (1 + xyeT/x – xy)-1

x = [(a - )2 + 2s2]-1/2 y = (a - x)/2

This has a free parameter  called the market price of risk

Yield is a linear function of r for fixed T

Testing - Yield Curve Model Should:

Closely approximate current yield curve

Produce patterns of changes in the short-term rate that match history

Over longer simulations, produce distribu-tions of yield curves that, for any given short-term rate, match historical conditional distribution of yield curves

Historical Yield Curve Distribution

Measure shape by 1st and 2nd differences

E.g., 1 year rate minus 3 month rate

Look at historical shape measures as a function of r

Yield Spread 3-Month to 1-Year

1-Year to 3-Year Yield Spread

CIR Compared to Historical

AndersenLund Model Comparison

AndersenLund Comparison

AL + Variable Market Price of Risk

Long spread matches historical

Modeling Loss Development

Need paid development for cash flow

Need reserve development for liabilities

One strategy: simulate paid diagonal and develop triangle to get new incurred

Simulating whole diagonal - as opposed to just simulating sum - requires model of development process

Diagonal Development Issues

Is development proportional to emerged?

Is development proportional to ultimate?

Is it inflation sensitive?

What is variance proportional to?

Can you tell from data?

Classifying Development Models

Six yes-no questions

Can test the questions using triangles

Provides 64 classes of development models

May be several models in each class

Other class schemes possible

Six Questions

Development depends on emerged?

Purely multiplicative development?

Independent of diagonal effects?

Stable parameters - e.g. factors?

Normally distributed disturbances?

Constant variance of disturbances?

For each age - not proportional to anything

E.g., All Yes

new incrmnt = f*[prev emerged] + e

e normal in 0 and s2

MLE of f is chain ladder (i.e., link ratio) estimate

Alternatives

Development proportional to ultimate - BF

new incrmnt = f*ultimate + e

Additive not multiplicative - Cape Cod

new incrmnt = a + e

Diagonals inflation sensitive - separation

new incrmnt = f*[prev emerged]*(1+i) + e

More Alternatives

Factors change over time - smoothing

Last 3 diagonals, exponential smoothing . . .

Lognormal disturbances - take logs

Disturbance proportional - weighted MLE

Inverse weights to variance

Proportional to emerged to date: weighted least squares takes ratios of column means

Still More

Combined models - e.g.:

new incrmnt = a + f*ultimate*(1+i) + e

Reduced parameters - e.g.:

new incrmnt = (1+i)*ultimate/(1+j)k + e

Which model does the data like?

Fit models to the triangle

Measure goodness of fit

Statistical significance of parameters

Sum of squared errors

Penalize for extra parameters

Divide sse by (obs - params)2

Testing a Triangle

0 to 1 is a constant

Adjusted SSE’s for Incrementals

SSE ModelPrmsSimulation Formula

157,902 CL 9qw,d = fdcw,d + e

81,167 BF 18qw,d = fdhw + e

75,409 CC 9qw,d = fdh + e

52,360 BF-CC 9qw,d = fdhw + e

h5 - h9 same, h3=(h2+h4)/2, f1=f2, f6=f7, f8=f9

44,701 BF-CC+ 7qw,d = fdhwgw+d + e

4 unique age factors, 2 diagonal, 1 acc. yr.

Stability of Factors

Normal Residuals

Is Disturbance Proportional?

Reserve Model Choice Affects Risk

No Inflation on Reserves

Mean / 1% / 10% / 90% / 99%
Short / 3422 / 3085 / 3220 / 3626 / 3821
Medium / 3443 / 2600 / 3024 / 3786 / 4096
Long / 3470 / 2182 / 2801 / 4117 / 4784
Stocks+ / 3540 / 1762 / 2287 / 4661 / 6120

With Post-Event Inflation

Mean / 1% / 10% / 90% / 99%
Short / 3429 / 3021 / 3205 / 3635 / 3859
Medium / 3438 / 2589 / 2972 / 3816 / 4289
Long / 3538 / 1899 / 2848 / 4242 / 4879
Stocks+ / 3569 / 1358 / 2294 / 4613 / 6197

Reinsurance & Investment Scenarios

Higher Retention

Mean / 1% / 10% / 90% / 99%
Short / 3422 / 3085 / 3220 / 3626 / 3821
Medium / 3443 / 2600 / 3024 / 3786 / 4096
Long / 3470 / 2182 / 2801 / 4117 / 4784
Stocks+ / 3540 / 1762 / 2287 / 4661 / 6120

Lower Retention

Mean / 1% / 10% / 90% / 99%
Short / 3227 / 3271 / 3351 / 3500 / 3618
Medium / 3255 / 2884 / 3156 / 3749 / 3951
Long / 3345 / 2197 / 2865 / 4202 / 4630
Stocks+ / 3473 / 1773 / 2564 / 4909 / 5642

Conclusions

DFA can be used to jointly manage reinsurance and investment strategy

Requires probabilistic generation of asset and underwriting scenarios

Modeling issues

Getting reserve model right necessary for relevant analysis

Interest rates follow complex processes that require careful attention