Capital Allocation
2001 Casualty Loss Reserve Seminar
New Orleans, Louisiana
September 10-12, 2001
Stephen Philbrick
Swiss Re Investors
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• Introduction
Agenda
– Inspiration
– Recap of DFAIC
• Discussion of Risk Measures
– Tail Conditional Expectation as a Coherent Measure of Risk
• Introduction to Capital Allocation
– Introduction and Illustration of the Shapley Value allocation
methodology
– Consideration of Assets in Allocation
– Capital Allocation Results and Sensitivity Testing of TCE tolerance
• Reserving Application - Inflation Sensitivity
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Inspiration
• Already have decided level of capital to hold
• Based on DFA call for papers 2001
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Introduction - Recap of DFAIC
Distribution of Net Earned Premium
1999
1999 Underwriting Summary
• Loss & LAE Reserves $ 2,330 million
9%
• Direct Written Premium $ 2,565 million
• Net Written Premium
$ 2,350 million
31%
Workers Comp
Auto Liab
10%
Home/ CMP(Prop)
• Booked Accident Year Loss&LAE Ratio
Gross
86.3%
Net
82.0%
Auto Phys Dam
GL/ CMP(Liab)
22%
• Expense Ratio (including
policyholder dividends)
29.5%
28%
Asset Mix
Asset Summary
6%
Invested Assets
Book Value:
Market Value:
Fixed Income Analysis
Average Maturity:
Duration:
64%
$4,702 million
$4,746 million
Cash
Common Stock
7%
Preferred Stocks
Governments
5%
Corporates
9.2 years
5.3 years
17%
Other
1%
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Probability of Ruin:
Risk Measures
• In the banking industry also known
as Value at Risk (VaR)
A
• Specific Sense: Probability capital
falls below zero.
• General Sense: The corresponding
value, for a selected qth percentile
tolerance, for a given financial
variable.
q
1
Blue Line =
Inverse Cumulative Distribution Function for a given Risk Variable
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Risk Measures
Expected Policyholder Deficit (EPD):
• Unlike VaR, EPD takes into account
the magnitude of insolvency.
• Specific Sense: The amount or
percentage of total obligations that
will not be met.
YE
A
• General Sense: The corresponding
value, for a selected tolerance
percentage YE of all summed
potential outcomes, for a given
financial variable.
1
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Definition of Terms:
Coherent Risk Measures

Xi represent portfolios of risks. (Think of it as the liabilities of a particular insurance
company).

a be some constant

r(*) be a function that assigns a value of risk to a portfolio
Four Axioms of a Coherent Risk Measure:

r(X + a) = r(X) + a
Translation Invariance

r(X1+X2) r(X1) + r(X2)
Subadditivity

r(aX)=ar(X)
Positive Homogeneity

For X1<X2 , r(X1)< r(X2)
Monotonicity
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Risk Measures
Tail Conditional Expectation (TCE):
• Also known as Tail Value at Risk
• Unlike the standard deviation risk
measure, TCE concentrates on the
tail of the distribution.
YT
ZT
A
• Combination of VaR and EPD
• Unlike VaR and EPD it is a Coherent
Risk Measure
XT
WT
• Expected Value of the Largest (1-q)
of all possible outcomes
q
0
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1
Risk Variables
TCE Required Capital:
• Apply TCE(.99) to the
distribution of “required
assets” (Ã1).
• Takes into account the
volatility of entire balance
sheet.
~
• Ã1 = E[A1] -S1
~
• S1 is 1) stochastic and 2)
recognizes undiscounted
ultimate loss immediately.
• TCE Required Capital = TCE
Required Assets - E[L1]
YT
ZT
A1 (TCE)
TCE
Required
Capital
E[L1]
XT
WT
q=0.99 1
Blue Line =
Inverse Cumulative Distribution Function of Required Assets (A1)
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Capital Allocation - Introduction
• Identify different Allocation Methods
• Discuss the Methodology we selected to allocate DFAIC’s capital:
– Risk Metric: TCE Required Capital
– Allocation Method: Shapley Value
– Explicit allocation to both assets and liabilities
• Compare and contrast 3 different Allocation Methods:
– Shapley: an axiomatic approach
– Simple comparative example
• View DFAIC results
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Shapley Value Introduction

The Marginal “First-In” allocation method allocates based on
the stand-alone risk of each business segment.

The Marginal “Last-In” allocation method evaluates the
marginal risk addition to the business as a whole.

The Shapley Value allocation method expands on the
Marginal “Last-In” concept by considering all possible
permutations of entry.

Shapley Value evenly splits the mutual covariance between
the “With” and “Without” marginal scenarios.
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Shapley Value Axioms

Collective Rationality - It does not allow one line of business
to subsidize another. Removes incentive for groups of lines
of business to act independently of the company as a whole.

Monotonicity in Costs - Given an increase in the overall cost
of the company, each line must participate in the rise of the
total cost.

Additivity - The subdivision of a line of business into two
lines should not affect the total allocation to the combined.
line.
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Shapley Value Illustration
Line of
Business
A
B
C
D
Sum
Total Company
Ratio
Standalone Allocated
Capital
Capital
1,000
2,000
3,000
4,000
548
1,095
1,643
2,191
10,000
5,477
54.8%
5,477
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Shapley Value Illustration
Line of
Business
A
B
C
D
Sum
Total Company
Ratio
Marginal
Capital
Allocated
Capital
92
378
895
1,736
163
668
1,580
3,066
3,100
5,477
176.7%
5,477
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Shapley Value Illustration
Line of Standalone Allocated Allocated
Business
Capital
Capital
Capital
LOB
"First-in" "Last-in"
A
1,000
548
163
B
2,000
1,095
668
C
3,000
1,643
1,580
D
4,000
2,191
3,066
S um
10,000
5,477
5,477
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Shapley Value Illustration
Capital Allocation
3,500
3,000
2,500
2,000
1,500
1,000
500
-
"First-in"
"Last-in"
A
B
C
D
LOB
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Shapley Value Illustration
Line of Standalone Allocated Allocated Allocated
Business
Capital
Capital
Capital
Capital
LOB
"First-in" "Last-in"
Shapley
A
1,000
548
163
345
B
2,000
1,095
668
906
C
3,000
1,643
1,580
1,654
D
4,000
2,191
3,066
2,572
Sum
10,000
5,477
5,477
5,477
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Shapley Value Illustration
Capital Allocation
3,500
3,000
2,500
2,000
1,500
1,000
500
-
"First-in"
"Last-in"
Shapley
A
B
C
D
LOB
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Capital Allocation Methodology

We selected TCE Required Capital as the Risk Metric

All 3 identified Allocation Methods allocate capital based
on the Marginal Impact of a particular variable (e.g.
Difference between With and Without Workers Comp)

-
First-In Marginal Contribution (Standalone)
-
Last-In Marginal Contribution (Marginal)
-
Shapley Value
We selected the Shapley Value Allocation Method
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“Without” Line of Business Definition
On the liability side:
We selected to reinsure the new and existing business away.
BUT, what about assets?
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“Without” Assets Definition
Determine asset allocation that minimizes risk
measure. (ie minimum risk efficient frontier portfolio)
Advantages:

Eliminates possibility of negative capital
allocation
Disadvantages:

Much more complex

Need a solution for each marginal run

Very time consuming, especially for Shapley values
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Shapley Value Allocation Using TCE Required Capital
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Sensitivity Testing Capital Allocation Results for TCE Tolerances
Allocation of Cost of Capital to Line of
Business
40%
35%
WC
30%
Auto Liab
25%
Prop
20%
Auto Phys Dam
15%
GL
10%
Assets
5%
0%
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
TCE Tolerance
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Standard Deviation(000,000)
Decomposition of Economic Risk
Real
Underwriting
Risk
Inflation
300
250
200
150
Discount Rate
100
50
Assets
-
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One-Year
Five-Year
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Capital Allocation - Conclusion

Shapley is based on reasonable axioms.

Shapley can be time consuming and difficult to calculate.

Specific allocation of capital to assets (i.e. investment
department) can be considered.

Based on our selected tail sensitive risk measure (TCE
Required Capital), DFAIC allocated a relatively greater
portion of capital to its longer-tailed more inflation sensitive
lines of business.
Swiss Re Investors
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Capital Allocation
2001 Casualty Loss Reserve Seminar
New Orleans, Louisiana
September 10-12, 2001
Stephen Philbrick
Swiss Re Investors
Z