The Role of Risk Metricsin

Insurer Financial ManagementGlenn Meyers

Insurance Services Office, Inc.

Joint CAS/SOS Symposium on Enterprise Risk Management

July 29, 2003

Determine Capital Needs for an Insurance Company

• The insurer's risk, as measured by its statistical distribution of outcomes, provides a meaningful yardstick that can be used to set capital needs.

• A statistical measure of capital needs can be used to evaluate insurer operating strategies.

Size

of L

oss

Random LossNeeded AssetsExpected Loss

Volatility Determines Capital NeedsLow Volatility

Volatility Determines Capital NeedsHigh Volatility

Size

of L

oss

Random LossNeeded AssetsExpected Loss

Define Risk

• A better question - How much money do you need to support an insurance operation?

• Look at total assets.• Some of the assets can come from

premium reserves, the rest must come from insurer capital.

Coherent Measures of Risk

• Axiomatic Approach• Use to determine needed insurer assets, A• X is random variable for insurer loss• An insurer has sufficient assets if:

(X) = A

Coherent Measures of Risk

• Subadditivity – For all random losses X and Y,(X+Y) (X)+(Y)

• Monotonicity – If X Y for each scenario, then(X) (Y)

• Positive Homogeneity – For all 0 and random losses X(X) = (X)

• Translation Invariance – For all random losses X and constants

(X+) = (X) +

Examples of Coherent Measures of Risk

• Simplest – Maximum loss

(X) = Max(X)

• Next simplest - Tail Value at Risk

(X) = Average of top (1-)% of losses

Examples of Risk that are Not Coherent

• Standard Deviation– Violates monotonicity– Possible for E[X] + T×Std[X] > Max(X)

• Value at Risk/Probability of Ruin– Not subadditive– Large X above threshold– Large Y above threshold– X+Y not above threshold

But – Assets Can Vary!

• If assets are fixed, we have sufficient assets if:

(X) = A• If assets can vary, we have sufficient

assets if:(X – A) = 0

• If assets are fixed, the new criteria reduce to the old because of translation invariance.

Illustrate Implications with a Model

• Losses, L, have lognormal distribution– Mean 10,000– Standard deviation will depend on example

• Asset Index, I, has lognormal distribution– Mean 10,000– Standard deviation will depend on example

• Assets are a multiple, , of the index.

Illustrate Implications with a Model

• Random effect, E, of economic conditions• Assets

A = I×(1+E)• Losses

X = L×(1+E)• Loss volatility multiplier – • E drives the correlation between assets

and liabilities

Illustrate Implications with a Model

• Calculate shares, , of the asset index so that:

TVaR(X–A) = 0• Also look at standard deviation risk metric

with T satisfying:E[X–A] + T×Std[X–A] = 0

• Normally T is fixed. Here I calculate the implied T as a way to compare risk metrics.

Illustrate Implications with a Model

• Select sample of 1000 L’s, I’s and E’s• Six cases varying:

– Standard deviation of L– Standard deviation of I– Standard deviation of E– Loss volatility multiplier,

• Fix: – TVaR level = 99%

Case 1Fixed Assets and Volatile Losses

• Required assets are larger than expected loss

Loss (L ) Asset (Lambda ×I ) Economic (E )Mean 10,000 18,158 0.000

Std Dev 2,500 0 0.000

Population SampleBeta 0.00 Std[X ] 2,500 2,417CV[I ] 0.000 Std[A ] 0 0

Shares 1.8158 Corr[X,A ] 0.000 (0.005)Alpha 99.0% Std[X –A ] 2,500 2,417

TVaR(X–A ) 0 Implied T 3.26 3.36

Case 2 Fixed Assets and Less Volatile Losses

• Value of assets smaller than Case 1.• Implied T smaller than that of Case 1.

– TVaR is more sensitive the large loss potential

Loss (L ) Asset (Lambda ×I ) Economic (E )Mean 10,000 12,823 0.000

Std Dev 1,000 0 0.000

Population SampleBeta 0.00 Std[X ] 1,000 965CV[I ] 0.000 Std[A ] 0 0

Shares 1.2823 Corr[X,A ] 0.000 (0.000)Alpha 99.0% Std[X –A ] 1,000 965

TVaR(X–A ) 0 Implied T 2.82 2.91

Case 3 Variable Assets

• Introducing asset variability increases expected value of assets – a bit.

Loss (L ) Asset (Lambda ×I ) Economic (E )Mean 10,000 12,918 0.000

Std Dev 1,000 258 0.000

Population SampleBeta 0.00 Std[X ] 1,000 965CV[I ] 0.020 Std[A ] 258 259

Shares 1.2918 Corr[X,A ] 0.000 0.005Alpha 99.0% Std[X –A ] 1,033 999

TVaR(X–A ) 0 Implied T 2.83 2.92

Asset Risk and Economic VariabilityModel with Std[E] = 2%

When economic inflation is high• Bond Index – Model with Std[I] = 0.02

– Interest rates are high and bond prices drop – Model loss inflation with = –2.00

• Stable Stock Index – Model with Std[I] = 0.02– Stock prices increase with inflation– Model loss inflation with = +2.00

• Volatile Stock Index – Model with Std[I] = 0.10– Stock prices increase with inflation– Model loss inflation with = +2.00

Case 4 Variable Assets – Bond Index

• When assets move in the opposite direction of losses, you need assets with higher expected value.

Loss (L ) Asset (Lambda ×I ) Economic (E )Mean 10,000 13,703 0.000

Std Dev 1,000 274 0.020

Population SampleBeta (2.00) Std[X ] 1,078 1,043CV[I ] 0.020 Std[A ] 388 376

Shares 1.3703 Corr[X,A ] (0.262) (0.251)Alpha 99.0% Std[X –A ] 1,192 1,152

TVaR(X–A ) 0 Implied T 3.11 3.21

Case 5 Variable Assets – Stable Stock Index

• You need assets with lower expected value than with Case 4 because stocks move in the same direction as losses .

Loss (L ) Asset (Lambda ×I ) Economic (E )Mean 10,000 12,966 0.000

Std Dev 1,000 259 0.020

Population SampleBeta 2.00 Std[X ] 1,078 1,035CV[I ] 0.020 Std[A ] 367 355

Shares 1.2966 Corr[X,A ] 0.262 0.254Alpha 99.0% Std[X –A ] 1,092 1,051

TVaR(X–A ) 0 Implied T 2.72 2.82

Case 6 Variable Assets – Volatile Stock Index

• Higher expected value with volatile stocks• Perhaps this explains why PC insurers stay out of

stocks despite the wrong correlation.

Loss (L ) Asset (Lambda ×I ) Economic (E )Mean 10,000 14,438 0.000

Std Dev 1,000 1,444 0.020

Population SampleBeta 2.00 Std[X ] 1,078 1,035CV[I ] 0.100 Std[A ] 1,473 1,482

Shares 1.4438 Corr[X,A ] 0.073 0.068Alpha 99.0% Std[X –A ] 1,793 1,779

TVaR(X–A ) 0 Implied T 2.48 2.52

Summary – Risk Metrics

• Introduced the latest and greatest (??) risk metric – TVaR

• Compared it to the current champion (??)• TVaR

– Has a strong axiomatic foundation– Does more to discourage risky business

Summary – Using Risk Metrics

• Use to determine the amount of assets needed to support insurance liabilities

• Takes into account– Insurance risk– Asset risk– Correlation between the two

References

• Artzner, Delbaen, Eber and Heath– Coherent Measures of Risk– Original paper– http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf

• Meyers– Setting Capital Requirements with Coherent Measures of

Risk – Part 1 and Part 2– http://www.casact.org/pubs/actrev/aug02/latest.htm– http://www.casact.org/pubs/actrev/nov02/latest.htm