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Through the Looking Glass:
CAPM for Guarantors and Fair Value of Liabilities
By Philip E. Heckman

In the November 2004 issue of AR, I had the privilege of reviewing the GIRO paper on fair value insurance accounting published in British Actuarial Journal in 2003. One of the prominent loose ends cited by the working party was the issue of Market Value Margins (MVMs), the risk loading to be applied in valuing liabilities uncertain in amount or timing—in particular, whether MVMs should reflect the enterprise's own risk or the appropriate market risk. In this issue, we shall see that an application of portfolio theory to the economics of guarantorship yields a simple result for an MVM incorporating market risk. Because of what guarantors do, this is essentially the mirror image of CAPM for investors, hence the title. The present model is more for illustration than for practical use.

What About Guarantors?
The emphasis on guarantorship follows from the accounting definition, which equates fair valuation of a liability to pricing for transfer to a guarantor in an open market (or a model thereof). Rather than the usual conditional guarantee, payable on default, we shall consider an unconditional guarantee, wherein the guarantor, for a pro rata share of the recorded liability, agrees to pay the same share of the obligation on maturity. The guarantor is not excused from performing in case of default on the original obligation. In this simplified version of the model, we shall ignore acquisition and administration costs, focusing on the cost of bearing risk. The guarantor will invest the proceeds at the risk-free ratematched for duration at maturity. (To be assured of meeting an obligation certain in amount and timing, the guarantor must invest risk-free. This holds a fortiori for a risky obligation. See Butsic, 1988.(.pdf) ) The guarantor's yield is the risk-free rate less the growth rate (i.e., the discount rate for the liabilities) of the obligation to maturity. Thus, in constructing a portfolio of risks, a guarantor will prefer obligations expected to have lower growth—the opposite of an investor selecting assets. In fact, under certain conditions this growth may be negative.

CAPM and COPM Contrasted (.pdf)
I call the portfolio model for guarantors the Capital Obligation Pricing Model (COPM) to emphasize at once the difference from CAPM and the debt to CAPM. We have gotten this far without equations but can enjoy that luxury no longer. We shall list the main assumptions and results side by side in a table so that similarities and differences become clear. Please excuse omissions in the presentation. We do not pretend that a complete, efficient market for guarantees actually exists. Rather we attempt to derive some of the consequences if such were the case. In the process, we hope to illuminate the very important differences between asset valuation and liability valuation in the face of the widespread presumption that they are one and the same problem. The risk-free rate is rF throughout. We assume that the guarantor's initial wealth is invested in instruments that do not covary with the obligations.

It is seen that the developments are entirely parallel but that CAPM leads to a risk adjustment positive in return (negative in price) while the result for COPM is just the opposite. This is a consequence of the guarantor's short position in the obligations guaranteed. In both cases, the actor's return is maximized by throwing a tangent from the risk-free point to the efficient frontier, but the tangent points describe quite different portfolios: the investor is hoping for underpriced assets, the guarantor for overpriced obligations. An example of our guarantor is a casualty reinsurer, as considered in Butsic's 1988 Michelbacher Prize paper (.pdf) wherein he arrived at a very similar result and also used ingenious arguments to derive an average margin of about 3%. We have assumed that a mean/variance formulation is adequate to describe the returns in question. Russo and Van Slyke in the 1996 Spring Forum (.pdf) address the consequences of relaxing this assumption.

I should remark that a great many, if not most, liabilities are recorded at values that are not guarantable in the sense used above. Debt liabilities are recorded at a discount for default risk, and no guarantor would touch them without demanding a premium over the recorded value. Life companies like to tie funding to their own asset returns. We must consider the possibility that property/casualty is the only industry funded, by regulatory fiat, on a sound basis (assuming accurate reserve estimates).

Continuing in an insurance context, we can see that the MVM, defined this way, is a charge for true systematic risk affecting all insurers. As such, it belongs as a positive risk load in the insurance rates as well as in the recorded liability. Insurance rate warriors have often had to contend with the contrary opinion in some jurisdictions. This finding should provide ammunition for argument in such situations.

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