Risk and Return - Part 1Introduction to VaR and RAROC

• Glenn Meyers - Insurance Services Office• Tim Freestone/Wei-Keung Tang

– Seabury Insurance Capital LLC• Peter Nakada - eRisk, Inc.

Risk and Return - Part 1Introduction to VaR and RAROC

• The purpose of Part 1 is to provide an overview of the issues involved in determining the cost of capital for an insurer.

• We don’t all agree on how to deal with these issues.

• Go to Part 2 to see some different points of view on this issue.

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.

Chart 3.1

Size

of L

oss

Random LossNeeded AssetsExpected Loss

Volatility Determines Capital NeedsLow Volatility

Volatility Determines Capital NeedsHigh Volatility

Chart 3.1

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

unearned premium reserves and loss reserves, the rest must come from insurer capital.

Coherent Measures of Risk

• Axiomatic Approach• Use to determine insurer assets• X is random variable for insurer loss

(X) = Total Assets

Capital = (X) – Reserves(X)

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

Representation Theorems

X sup E X P P• Artzner, Delbaen, Eber and Heath

Maximum of a bunch of generalized scenarios

• Wang, Young and Panjer

Expected value of X with probabilities distorted by g, where g(0)=0, g(1)=1 and g is concave down.

0

1X g F x dx

CorrelationMultiple Line Parameter Uncertainty

• Select from a distribution with E[] = 1 and Var[] = b.

• For each line h, multiply each loss by .• Generates correlation between lines.

Multiple Line Parameter Uncertainty

A simple, but nontrivial example

1 2 31 3 , 1, 1 3b b

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

E[] = 1 and Var[] = b

Correlation and Capital b = 0.00

Chart 3.4Correlated Losses

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Random Multiplier

Sum

of R

ando

m L

osse

s

Correlation and Capital b = 0.03

Chart 3.4Correlated Losses

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

0.7 1.3 1.3 1.0 1.0 0.7 1.0 0.7 1.3 1.3 0.7 1.3 1.3 1.0 0.7 0.7 1.0 1.3 0.7 1.0 1.3 1.0 0.7 0.7 1.0

Random Multiplier

Sum

of R

ando

m L

osse

s

Positive Correlation Means More Capital

• A good insurer strategy will try to reduce correlation between its insureds.– Unless the price is right

• Example – Avoid geographic concentration in catastrophe-prone areas.

Long-Tailed Lines of Insurance

• Uncertainty in loss reserve must be supported by capital.

• Release capital over time as uncertainty is reduced.

Reinsurance

• Reduces capital needs• Reduces the cost of capital• Adds reinsurance transaction costs• Insurer strategy - Minimize the

combined capital and reinsurance transaction costs.

Allocating Capital

• Actually – Allocate the cost of capital• In total, the cost of capital must come

from the profit provisions of individual insurance policies.

• Allocate capital implicitly, or explicitly.• See session C-3.

Measure Risk/Determine Capital• Build insurer’s aggregate loss

distribution.– Claim count distribution– Claim severity distribution– Dependencies/Correlation– Catastrophes– Reinsurance

• Hard part is to get the information.• Should be fast as to evaluate various

line/reinsurance strategies.

Measure Risk/Determine Capital• For various line/reinsurance strategies

– Calculate your favorite measure of risk/needed assets/capital.

– Allocate cost of capital to business segments.

– Compare resulting costs with market driven premiums.

• Select the most desirable strategy

Measure Risk/Determine Capital• Links to a comprehensive example• “The Cost of Financing Insurance”

– CAS Ratemaking Seminarhttp://www.casact.org/coneduc/ratesem/2002/handouts/meyers1.ppt

– Papershttp://www.casact.org/pubs/forum/01spforum/meyers/index.htm