The Role of Risk Metrics
in
Insurer Financial Management
Glenn 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.
Volatility Determines Capital Needs
Low Volatility
Size of Loss
Chart 3.1
Random Loss
Needed Assets
Expected Loss
Volatility Determines Capital Needs
High Volatility
Size of Loss
Chart 3.1
Random Loss
Needed Assets
Expected 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:
r(X) = A
Coherent Measures of Risk
• Subadditivity – For all random losses X and Y,
r(X+Y)  r(X)+r(Y)
• Monotonicity – If X  Y for each scenario, then
r(X)  r(Y)
• Positive Homogeneity – For all l  0 and random
losses X
r(lX) = lr(X)
• Translation Invariance – For all random losses X
and constants a
r(X+a) = r(X) + a
Examples of Coherent
Measures of Risk
• Simplest – Maximum loss
r(X) = Max(X)
• Next simplest - Tail Value at Risk
r(X) = Average of top (1-a)% 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:
r(X) = A
• If assets can vary, we have sufficient
assets if:
r(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, l, of the index.
Illustrate Implications with a Model
• Random effect, E, of economic conditions
• Assets
A = lI×(1+E)
• Losses
X = L×(1+bE)
• Loss volatility multiplier – b
• E drives the correlation between assets
and liabilities
Illustrate Implications with a Model
• Calculate shares, l, of the asset index so
that:
TVaRa(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, b
• Fix:
– TVaR level a = 99%
Case 1
Fixed Assets and Volatile Losses
Mean
Std Dev
Beta
CV[I ]
Shares
Alpha
TVaR(X–A )
Loss (L )
10,000
2,500
0.00
0.000
1.8158
99.0%
0
Asset (Lambda ×I ) Economic (E )
18,158
0.000
0
0.000
Std[X ]
Std[A ]
Corr[X,A ]
Std[X –A ]
Implied T
Population
2,500
0
0.000
2,500
3.26
Sample
2,417
0
(0.005)
2,417
3.36
• Required assets are larger than expected loss
Case 2
Fixed Assets and Less Volatile Losses
Mean
Std Dev
Beta
CV[I ]
Shares
Alpha
TVaR(X–A )
Loss (L )
10,000
1,000
0.00
0.000
1.2823
99.0%
0
Asset (Lambda ×I ) Economic (E )
12,823
0.000
0
0.000
Std[X ]
Std[A ]
Corr[X,A ]
Std[X –A ]
Implied T
Population
1,000
0
0.000
1,000
2.82
Sample
965
0
(0.000)
965
2.91
• Value of assets smaller than Case 1.
• Implied T smaller than that of Case 1.
– TVaR is more sensitive the large loss potential
Case 3
Variable Assets
Mean
Std Dev
Beta
CV[I ]
Shares
Alpha
TVaR(X–A )
Loss (L )
10,000
1,000
0.00
0.020
1.2918
99.0%
0
Asset (Lambda ×I ) Economic (E )
12,918
0.000
258
0.000
Std[X ]
Std[A ]
Corr[X,A ]
Std[X –A ]
Implied T
Population
1,000
258
0.000
1,033
2.83
Sample
965
259
0.005
999
2.92
• Introducing asset variability increases expected
value of assets – a bit.
Asset Risk and Economic Variability
Model 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 b = –2.00
• Stable Stock Index – Model with Std[I] = 0.02
– Stock prices increase with inflation
– Model loss inflation with b = +2.00
• Volatile Stock Index – Model with Std[I] = 0.10
– Stock prices increase with inflation
– Model loss inflation with b = +2.00
Case 4
Variable Assets – Bond Index
Mean
Std Dev
Beta
CV[I ]
Shares
Alpha
TVaR(X–A )
Loss (L )
10,000
1,000
(2.00)
0.020
1.3703
99.0%
0
Asset (Lambda ×I ) Economic (E )
13,703
0.000
274
0.020
Std[X ]
Std[A ]
Corr[X,A ]
Std[X –A ]
Implied T
Population
1,078
388
(0.262)
1,192
3.11
Sample
1,043
376
(0.251)
1,152
3.21
• When assets move in the opposite direction of
losses, you need assets with higher expected
value.
Case 5
Variable Assets – Stable Stock Index
Mean
Std Dev
Beta
CV[I ]
Shares
Alpha
TVaR(X–A )
Loss (L )
10,000
1,000
2.00
0.020
1.2966
99.0%
0
Asset (Lambda ×I ) Economic (E )
12,966
0.000
259
0.020
Std[X ]
Std[A ]
Corr[X,A ]
Std[X –A ]
Implied T
Population
1,078
367
0.262
1,092
2.72
Sample
1,035
355
0.254
1,051
2.82
• You need assets with lower expected value
than with Case 4 because stocks move in the
same direction as losses .
Case 6
Variable Assets – Volatile Stock Index
Mean
Std Dev
Beta
CV[I ]
Shares
Alpha
TVaR(X–A )
Loss (L )
10,000
1,000
2.00
0.100
1.4438
99.0%
0
Asset (Lambda ×I ) Economic (E )
14,438
0.000
1,444
0.020
Std[X ]
Std[A ]
Corr[X,A ]
Std[X –A ]
Implied T
Population
1,078
1,473
0.073
1,793
2.48
Sample
1,035
1,482
0.068
1,779
2.52
• Higher expected value with volatile stocks
• Perhaps this explains why PC insurers stay out
of stocks despite the wrong correlation.
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