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?

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  (X) = Needed Assets Capital =  (X) – E(X)

Conditions on  (X) Subadditivity –  (X+Y) ≤  (X)+  (Y) –Reflects the diversification effect of insurance, Positive homogeneity –  (k·X) = k·  (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:  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 * be the return on marginal capital with the optimal exposure mix. –In the paper * 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 * ≥ r * ≥ r Question: When does * = r ?

Now – Allocate Capital Assume insurer is making efficient use of capital – as defined above. Total expected profit is equal to: r × Total Capital * × Total Marginal Capital * × Total Marginal CapitalDefine:

Consistent with how allocated capital is used. Satisfies the optimality condition that the return on marginal capital is equal for all lines of insurance.

When does * = 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, X i is homogeneous with respect to e i, then * = r. Related results due to: –Myers/Read –Mildenhall

Heterogeneity Multiplier = */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 non- diversifiable 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 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.