An Economic Perspective on Capital Allocation George Zanjani Federal Reserve Bank of New York The views expressed in this presentation are those of the author and do not necessarily represent the position of the Federal Reserve Bank of New York or the Federal Reserve System.

Capital Cost Allocation: An Economic View Basic Questions How much capital “should” an insurer hold? How “should” capital be allocated to line of business?

Outline of Presentation Why allocate capital? Capital allocation: What doesn’t work (Merton & Perold (1993)) What works: Marginal capital allocation A market-based approach to capital cost allocation (Zanjani (2002)) Allocating capital at the margin (Myers & Read (2001); Zanjani (2002))

Do capital management and allocation matter? Components of price: - loss cost - production costs - capital costs Correct pricing in a multi-line company requires an allocation. Some lines of business may need more capital than others. Sometimes, capital costs are the largest component of price.

Merton & Perold (1993) In general, allocation of capital to business lines is NOT feasible. Example - The Power of Diversification Line 1 alone Line 2 aloneLines 1 & 2 Capital $75 $75 $100 Premiums $75 $75 $150 Expected Profit $8 $5 $13 ROE 10.7% 6.7% 13% Target ROE 12% 12% 12% Assessment BAD BAD GOOD

The Merton/Perold finding is disturbing. What is a “right” price for a contract in a multi-line setting? How can one evaluate business line performance? How much capital should an insurer hold? What can we expect market forces to deliver?

So what can be done? Capital can be allocated at the margin. The marginal allocation can be connected to basic economic principles. Example Organization Capital$100 Line 1 Premiums $75 Line 2 Premiums $75 Line 1 Marginal Capital X Line 2 Marginal Capital Y Depending on the allocation rule chosen and the underlying economics, X and Y may “add up” (sum to 100).

But what is “marginal capital”? Marginal capital is the amount that “supports” expansion or contraction in a given line at the margin. Loosely, if we were to expand Line 1 premiums from $75 to $76, we would need approximately (75 / X ) in additional capital. Thus, it is useful in assessing the underlying marginal cost of risk and determining the cost of adding a (small) contract to an existing book. It is NOT the Holy Grail of capital allocation rules. The depressing logic of Merton and Perold (1993) still applies for infra- and supra-marginal decisions. For example, marginal allocations may still lead you astray when deciding whether to exit or enter a line of business.

Consumer i demand price: p i quality: q Economics of Capital Cost Allocation (Simplified) y i ( p i, q ) Production realized loss: y i x i = L i ( L = Σ i L i ) expected loss per unit of demand: μ i capital: R cost of holding capital: δ surplus: S = R + Σ i ( p i - μ i ) y i realized default savings: D = max[ 0, L - E[L] - S ]

Firm objective: Choose prices and capital to solve max { Σ i ( p i - μ i ) y i + E[D] - δR } First order conditions: (price) (capital) But quality depends on surplus and the risk assumed in each of the markets: q( S, y 1, …, y N ). Thus, dq/dp j and dq/dR are messy.

Application: Normal Risk Under competition (no market power), the pricing condition becomes: The markup over expected loss varies according to the line’s “beta.” Fair price components: expected loss, default savings, capital costs The beta pricing rule allocates the sum of default savings and capital costs to line based on the line’s risk:

How does this relate to capital allocation? If quality is the probability of default, marginal default savings and capital costs are allocated individually to line based on a beta rule: Marginal cost pricing implies Consider the amount of capital you need to maintain a certain level of quality This adds up and is easy to remember. But other rules are possible.

Capital Allocation in General In general, a variety of marginal allocation rules are possible. The crucial determinants are: the definition of quality (e.g., probability of ruin, default-value-to-liability ratio) the nature of the risk (e.g., the loss distributions assumed, the composition of lines) Use these to generate an implicit capital (or surplus) function. Myers and Read (2001) can be analyzed in this context (with minor adjustments) by defining quality as the default-value-to- liability ratio and using lognormal or normal risk.

Conclusions Marginal capital allocations are essential inputs in assessing the marginal cost of production. They have a sound basis in financial theory and a plausible connection with market-based economics. The trick is determining the relevant “quality” measure. From an engineering standpoint, a variety of approaches are possible. In practice, appropriate allocations will be company- specific, even when similar approaches are used. Marginal capital allocations should NOT be used for all types of business decisions. There is no “one size fits all” allocation rule.

References Merton, R.C., and Perold, A.F. (1993), “Theory of Risk Capital in Financial Firms,” Journal of Applied Corporate Finance 6, Myers, S.C., and Read, J.A. (2001), “Capital Allocation for Insurance Companies,” Journal of Risk and Insurance 68, Zanjani, G. (2002), “Pricing and Capital Allocation in Catastrophe Insurance,” Journal of Financial Economics, forthcoming.