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
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