Through the Looking Glass:
CAPM for Guarantors and Fair Value of
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.