The Common Shock Modelfor

Correlations BetweenLines of Insurance

Glenn MeyersInsurance Services Office

CAS Annual MeetingNovember 16, 2004

The Common Shock IdeaMultiple Line Parameter Uncertainty• Select from a distribution with E[] = 1

and Var[] = b.

• For each line h, multiply each loss by .

• Can calculate if desired.

| |

, | , | | , |

,

Var X E Var X Var E X

Cov X Y E Cov X Y Cov E X E Y

Cov X Y

Std X Std Y

Multiple Line Parameter Uncertainty

A simple, but nontrivial example

1 2 31 3 , 1, 1 3b b

1 3 2Pr Pr 1/ 6 Pr 2 / 3and

E = 1 and Var[] = b

X1 and X2 – Low Volatility b = 0.01 r= 0.50

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

0 1,000 2,000 3,000 4,000

Y 1 = X 1

Y2=

X2

X1 and X2 – Low Volatility b = 0.03 r= 0.75

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

0 1,000 2,000 3,000 4,000

Y 1 = X 1

Y2=

X2

X1 and X2 – High Volatility b = 0.03 r= 0.45

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

0 1,000 2,000 3,000 4,000

Y 1 = X 1

Y2=

X2

X1 and X2 – High Volatility b = 0.01 r= 0.25

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

0 1,000 2,000 3,000 4,000

Y 1 = X 1

Y2=

X2

Correlation Depends Upon

• The volatility of the common shocks

• The volatility of the lines of insurance

• Which in turn depends upon the– Claim severity distribution– Claim count distribution and the size of the

insurer

Line of Insurance VolatilityThe Collective Risk Model

• Select from a distribution with mean 1 and variance c.

• Select claim count N from a Poisson distribution with mean ·.

• For each claim

• Select a claim size from a distribution with mean and standard deviation .

• Total loss = Sum of all N claims

Additional Assumptions of the Collective Risk Model

• The size of the risk is proportional to the expected claim count, .

• The parameters of the claim severity distribution, and , are the same for all risk sizes.

• The contagion parameter, c, is the same for all risk sizes.

• We will test assumptions below.

Implications of Assumptions

• Behavior of loss ratio as risk size increases

2 2 2 2

Standard Deviation R

cc

Implications of AssumptionsLoss Ratios for the Collective Risk Model

= 15,000 = 60,000 c = 0.01

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

1,000 10,000 100,000 1,000,000

Expected Loss (000)

Stan

dard

Dev

iati

on

Introduce Common ShocksAcross Lines of Insurance

• For claim count , where E[] = 1 and Var[] = g

• For claim severity , where E[] = 1 and Var[] = b

• Assume:– b and g are the same for all risk sizes– b and g are the same for all lines of insurance

• I back off on this one in general, but I think it is good for the examples that follow.

Effect of b and g on loss ratioLoss Ratios for the Collective Risk Model

= 15,000 = 60,000 c = 0.01

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

1,000 10,000 100,000 1,000,000

Expected Loss (000)

Stan

dard

Dev

iati

on

b=g=0.001

b=g=0

Effect of b and g on correlationLoss Ratios for the Collective Risk Model

= 15,000 = 60,000 c = 0.01 b = g = 0.001

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

1,000 10,000 100,000 1,000,000

Expected Loss (000)

Coe

ffic

ient

of

Cor

rela

tion

Does Correlation Matter?• Consider total loss over n line of insurance• Look at covariance matrix

• Variance is the sum of n diagonal elements plus n2-n off diagonal elements

• For large n, even small correlations matter.

1 1 2 1

2 1 2 2

2 2

, ... ,

, ... ,

... ... ... ...

, , ...

n

n

n n n

Var X Cov X X Cov X X

Cov X X Var X Cov X X

Cov X X Cov X X Var X

Effect of correlation with two lines

Loss Ratios for the Collective Risk Model for the Sum of Two Risks = 15,000 = 60,000 c = 0.01

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1,000 10,000 100,000 1,000,000

Expected Loss (000) per Risk

Stan

dard

Dev

iati

on

b=g=0.001

b=g=0

Effect of correlation with ten lines

Loss Ratios for the Collective Risk Model for the Sum of Ten Risks = 15,000 = 60,000 c = 0.01

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

1,000 10,000 100,000 1,000,000

Expected Loss (000) per Risk

Stan

dard

Dev

iati

on

b=g=0.001

b=g=0

Empirical Tests of Model• Schedule P loss ratios for 55 insurers

– Remove “predictable” changes over time

• Difficulties– Catastrophes– Loss development understates ultimate variability– Reinsurance is probably more prevalent for smaller

insurers.

• Used auto (commercial and personal) lines– Not as affected as other lines by these difficulties

Model predicts how the standard deviation of the loss ratio decreases with insurer size

Commercial Auto Loss RatiosActual - Schedule P, HiLim and LowLim from Collective Risk Model

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

1,000 10,000 100,000 1,000,000

Expected Loss (000)

Stan

dard

Dev

iati

on

Actual

HiLim

LowLim

Use model to simulate loss ratios based on actual insurer sizes

Commerical Auto Loss RatiosSimulated from the Collective Risk Model with LowLim Parameters

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

1,000 10,000 100,000 1,000,000

Expected Loss (000)

Stan

dard

Dev

iati

on

Simulated

HiLim

LowLim

Model predicts that the coefficient of correlation will increase with insurer size

Commercial Auto and Personal Auto Loss RatiosActual – Schedule P, Predicted – Collective Risk Model

(1.00)

(0.80)

(0.60)

(0.40)

(0.20)

0.00

0.20

0.40

0.60

0.80

1.00

10,000 100,000 1,000,000 10,000,000

Expected Loss (000)

Cor

rela

tion

Corr Actual

Corr Predicted

Model predicts that the coefficient of correlation will increase with insurer size

Commercial Auto and Personal Auto Loss RatiosSimulated and Predicted from the Collective Risk Model

(1.00)

(0.80)

(0.60)

(0.40)

(0.20)

0.00

0.20

0.40

0.60

0.80

1.00

10,000 100,000 1,000,000 10,000,000

Expected Loss (000)

Cor

rela

tion

Corr Sim

Corr Predicted

Model predicts that the coefficient of correlation will increase with insurer size

Discussion of the above slides

• The plot of actual vs predicted is at best a weak confirmation.

• The plot of simulated vs predicted shows we should not expect anything more than a weak confirmation.

• Further analysis is necessary.• Add the assumption that common shocks affect

all insurers simultaneously.

Further Analysis

• Let R=X/E[X]

• A consequence of the model is that:

1 21 2

1 1 2 2

,1 1

Cov X XE R R b g b g

• The average of empirical (R1 – 1)·(R2 – 1) products is independent of insurer size.

Further Analysis

• We have already demonstrated that the standard deviation of loss ratios decrease with size.

• We can then conclude that increases with size if the numerator does not also decrease with size.

1 2

1 21 2

1 1,

E R RR R

Std R Std R

Further Analysis

• Data gathered– r1 and r2 and associated expected losses were

taken for same year and different insurers– 15,790 pairs

• Did a regression of (r1 – 1)·(r2 – 1) against average expected loss.

• Slope = 1.95×10-10 (not negative)– Increasing coefficient of correlation is

consistent with Schedule P data.

Estimating b + g + b·g

• To estimate b + g + b·g, take the average of the 15,790 (r1 – 1)·(r2 – 1) products.

• Average = 0.00054

1 21 2

1 1 2 2

,1 1

Cov X XE R R b g b g

Statistical Significance

• Standard statistical tests of significance do not apply.– Non uniform variance of loss ratios– Non normality of loss ratios

• Test hypotheses by simulating null hypothesis.– Use insurer size (as measured by expected

loss) as input– Loss ratios have lognormal distribution– Did 200 simulations

Results of Hypothesis Test

• H0: Slope of regression = 0

• H1: Slope of regression < 0

• Did 200 simulations under the null hypothesis

• Slope of 1.95×10-10 is out of any critical

region for H1.

• The actual slope was below 49 of the 200 simulated slopes – not that unusual for H0.

Results of Hypothesis Test

• H0: b + g + b·g = 0

• H1: b + g + b·g > 0

• Estimate = 0.000540

• Top estimate from a simulation derived from H0 = 0.000318

• Positive correlation consistent with Schedule P data.

Summary of Results

This version of the collective risk model– c is the same for all risk sizes– b and g generate correlation

is consistent with Schedule P data.• I think of this as a first order approximation. I

would not be surprised to see more refined versions.

• But – It takes a lot of data to test statistically.• And – The results do have consequences!

Comments on Copulas

• One can calculate the coefficients of correlation with this model and use them as input for a normal copula.

• Good approximation!

• Question – Why use the copula?

• This talk addressed a larger question, what is a good model for a multivariate loss distribution.

Relevance

• IAA Solvency Subcommittee and “Blue Book”– Standard conservative formula for capital requirements.– Optional use of internal model for capital. This model will

be subject to regulator standards and audit. Standing of judgmental correlations?

• Progress on new capitalization standards– Swiss are well on their way– EU is also moving in this direction– US is uncertain