The Economics of Capital
Allocation
Glenn Meyers
Insurance Services Office
Presented at the Bowles Symposium
April 10, 2003
Capital Allocation
Controversial in theory
– All the capital supports all the business.
Useful in practice
– CXO’s want line managers to share corporate
goals.
How do CXO’s use capital allocation?
Expected Profit for Line
Total Expected Profit
=
Allocated Capital for Line
Total Capital
Economics First
First devise an economically sound
strategy to assemble an insurance portfolio.
Next devise a capital allocation system that
matches the economic strategy.
Economically sound?
Strategy leads to decisions that increase
insurer’s return on capital.
Capital Allocation Second
Once you have an economically sound
portfolio strategy - - Devise an equivalent capital allocation
method.
First Task – Quantify Needed Capital
Capital = Needed Assets – Expected Payout
Let X = Random Loss
r(X) = Needed Assets
Capital = r(X) – E(X)
Conditions on r(X)
Subadditivity – r(X+Y) ≤ r(X)+r(Y)
– Reflects the diversification effect of insurance,
Positive homogeneity – r(k·X) = k·r(X)
These are two of the four axioms that
define the so-called coherent measures of
risk.
We don’t need the other axioms for what
follows.
Economically Sound Decision
Let P = Profit and C = Capital. Then the
insurer is better off by adding a line/policy if:
P  P P

C  C C
 P  C  C  P  C  P  P  C
P P


C C
 Marginal return on new business 
return on existing business.
Increase Return on Capital
If you can find business where the return
on marginal capital is greater than current
return on capital - - Adding the business and adding capital
increases return on capital.
Numerical examples are in the paper.
Capital is a Scarce Resource
Economics is about allocating scarce
resources.
But here, we are allocating capital to the
insurer, not the individual insurance contract.
If capital is limited, you do not pick just any
business that increases return on marginal
capital.
You go for the business that gets you the
best return on marginal capital.
Optimization Problem
Assume capital is limited.
Assume we can continuously adjust
exposures.
Increase exposure in lines where return on
marginal capital is greatest.
Decrease exposure in lines where the
return on marginal capital is least.
Optimal result occurs when the return on
marginal capital is equal for all lines!
Comment on the
Optimization Problem
Let l* be the return on marginal capital
with the optimal exposure mix.
– In the paper l* is a Lagrange multiplier
Let r be insurer’s return on capital.
If the insurer continues to attract capital,
and - - The insurer is operating efficiently then
l* ≥ r
Question: When does l* = r ?
Now – Allocate Capital
Assume insurer is making efficient use of
capital – as defined above.
Total expected profit is equal to:
r × Total Capital
l* × Total Marginal Capital
Define:
Allocated Capital 
l*
r
 Marginal Capital
Allocated Capital 
l
r
*
 Marginal Capital
Consistent with how allocated capital is used.
Expected Profit for Line
Total Expected Profit
=
Allocated Capital for Line
Total Capital
Satisfies the optimality condition that the
return on marginal capital is equal for all
lines of insurance.
When does l* = r ?
If the random loss X = U·e for exposure
measure e and random number U, then
the distribution of X is homogeneous
with respect to e.
Proposition: If for each line of insurance i,
Xi is homogeneous with respect to ei, then
l* = r.
Related results due to:
– Myers/Read
– Mildenhall
Heterogeneity Multiplier = l*/r
Allocated Capital 
l*
 Marginal Capital
r
The multiplier is equal to one, and is
unnecessary when all losses are homogeneous
with respect to some exposure measure.
Hence the name – “Heterogeneity Multiplier”
An example is given in the paper where the
multiplier is greater than one.
The example has a diversifiable, and a nondiversifiable component. The multiplier
approaches one as the exposure gets larger.
Implementing the Strategy
Establish a target rate of return, r, for the
insurer.
Observe the premium for each line of
insurance that the marketplace allows.
If market allows a profit provision P with
P
l
*
 Marginal Capital
r
then increase exposure in the line.
Otherwise decrease exposure in the line.
Key Assumption in the Above
Ability to continuously adjust exposure
At best, an approximation
The approximation is pretty good if individual
insurance contracts are a small part of the line of
business.
Paper provides an example of what can happen
if this assumption is not good.
Essence of the example – 2 contracts (Zanjani)
–
–
–
–
Marginal capital equal
Expected profit unequal
Significant reward for pooling the two contracts
If you reject the contract with the lowest return on
allocated capital, you lose the pooling reward.
Summary
Capital Allocation as a Risk Charge
Input: Risk-based capital.
– I like to think I made a strong case for allocating in
proportion to marginal capital.
Input: Expected rate of return
– Determined by investors’ expectations and other
investments, i.e. Wall Street.
Right now, no consensus on risk-based capital
Our challenge as actuaries is to come up with a
workable risk-based capital formula.
– I think the problem is more one of getting realistic insurer
aggregate loss distributions.