THE IMPACT OF CREDIT RISK ON THE FAIR VALUATION OF A LIABILITY
Trent R. Vaughn
Abstract
This paper discusses the impact of credit risk on the determination of the fair value of a liability. In the first section, we explore the problem of determining the fair value of a (generic) liability from first principles, utilizing standard financial valuation theory. Section 2 discusses the approach to fair value described in the recent AAA monograph, including a critique of the credit risk comments in that monograph. Next, Section 3 provides a link between the “cost of capital method” for fair valuation (as described in the AAA monograph) and the more-common property/casualty actuarial approach of “risk loads.” Finally, Section 4 emphasizes the need for sound actuarial judgment in determining the fair value of an insurance liability.
ACKNOWLEDGEMENT
The author would like to thank Philip Heckman and Paul Brehm for several helpful comments regarding the ideas in this paper. Any errors, of course, are solely the responsibility of the author.
- THE FAIR VALUE OF A LIABILITY VERSUS THE MARKET VALUE
OF THE COUNTERVAILING ASSET
Setting the Stage: Four Key Questions Regarding a Generic Liability
In order to clarify the issues, let's consider a corporation that has issued a bond with a maturity of one year and a face value of $75. Assume that this corporation owns one asset that will produce a single cash flow at the end of the year; there is an 80% probability that this cash flow will be $125, and a 20% probability that it will be $60. The current value of the asset is $100 (that is, it offers a 12% expected annual return). In the event that the cash flow on the asset turns out to be $60, shareholders will default, and the bondholder will not receive the full promised payment of $75. According to financial theory, the risky bond in this example can be valued according to the following formula.[1]
Present Value (PV) of Bond = PV of $75 assuming no chance of default (i.e. discounted at risk-free rate)
- PV of Default Option (1.1)
The "default option" is a put option on the corporation's assets with an exercise price of $75. The simple put option in this example can be valued by the binomial pricing model.[2] Assuming a risk-free rate of 5% per annum, the current value of the bond in this example will be $67.03 (that is, $75 / 1.05 minus $4.40).[3] In this example, consider the following four questions:
1. What did the bondholder originally pay for the bond? We don't know the answer to this question. Fortunately, however, this question is not relevant to the problem of determining the current value of the bond. In economic theory, historical costs are irrelevant to determining current values.
2. What would another third party pay for the bond? Assuming that this bond is actively traded in a perfectly competitive, secondary market, financial theory tells us that the bondholder could sell this bond for $67.03.
3. How much would the company have to pay the bondholder to extinguish its liability? The bondholder would require at least $67.03, since that is the amount that he would receive by selling this bond in the open market. Thus, $67.03 represents the "floor", or the minimum amount that the company would have to pay the bondholder to extinguish this liability. In fact, the exact amount that the company would have to pay depends on the characteristics of the individual bondholder (e.g. the bondholder’s personal tax situation, liquidity needs, etc.). As such, the answer to this question is difficult to determine in practice, and is therefore not generally used as a guide to the value of the bond.
4. How much would the company have to pay a third party (in an arm's length transaction) to take over the liability? Note that the original bond indenture is between the company and the bondholder. Thus, there is no (easy) way for the company to transfer the liability fully to a third party. However, the company can establish a contract for a third party to reimburse the company for the amount of the liability, which will then be transferred to the bondholder. How much would the company have to pay for this type of contract? The price depends not on the credit risk of the company (the third party will need to honor the contract whether or not the company is solvent), but on the credit risk of the third party. For instance, let's assume that the third party will be financially able to make the $75 payment with absolute certainty. In that case, the company will need to pay $71.43 for this third party to take over the liability (ignoring transaction costs and federal income taxes). If there is some chance that the third party will not be able to meet its obligation, then the company will pay less than $71.43.
As discussed, the first and third questions are not generally considered in determining the current value of the bond. The answer to the second question is generally regarded as a relevant indicator of the market value of the bond, when we are considering the bond as a financial asset. In the example above, we utilized standard financial theory to estimate the market value of this particular bond. However, there is actually no need to utilize the financial formula to determine the market value of an actively-traded bond; instead, we can directly observe the price at which the bond is trading in the marketplace. In fact, this is an application of a fundamental principle of finance: look first to market values.[4] That is, when you are analyzing the value of an asset that trades in a competitive market, it is unwise to use methods that value the asset "from scratch". Instead, look first to the market value of the asset.
On the other hand, the answer to the fourth question is generally regarded as the relevant factor for determining the fair value of the corporate liability that is created by the bond. For instance, the Insurance Steering Committee of the IASB defines the fair value of a corporate liability as follows: “In particular, the fair value of a liability is the amount that the enterprise would have to pay a third party at the balance sheet date to take over the liability.”
As demonstrated in the example above, the answers to the second and fourth questions are not necessarily identical. That is, the fair value of the liability is not necessarily equivalent to the market value of the countervailing asset. In addition, the market value of the bond, when considered as a financial asset, depends on the credit risk of the company that has issued the bond; the fair value of the liability created by the bond does not depend on the credit risk of the issuer, but does depend on the credit risk of the relevant third party. In this sense, the IASB definition of the fair value of a liability is incomplete; in order to pin down the answer to the fourth question, we need to specify the credit standing of the third party that is taking over the liability.
Determining the Fair Value of an Insurance Liability
When we are working with financial models for pricing insurance policies in the primary marketplace, two important factors must be incorporated. First, it is very likely that an insurance company's credit risk is reflected in the market price. Assuming that policyholders have adequate information to evaluate the credit risk differences between various insurance companies, a rationale policyholder will pay less for a "promise" from a shaky company than from a solid company, all other things equal. Second, in order to compensate the insurance shareholders for their opportunity cost of capital, the insurance premium must include a provision for federal income taxes on the insurance company’s investment income.[5] In light of these considerations, the financial formula for pricing P/L insurance policies in the primary market is as follows:[6]
Premium = PV of Expected Losses (assuming no risk of default) + Insurance Underwriting Expenses – PV of Default Option + PV of Federal Income Taxes on Investment Income (1.2)
Regarding the impact of credit risk in the primary market, we must include an important caveat regarding guarantee fund protection. If guaranty funds reimburse policyholders of insolvent companies in a perfectly certain, full and timely manner, then the financial condition of each particular company is irrelevant to the policyholder. Nonetheless, a recent empirical study in the Journal of Risk & Insurance indicated that competitive P/L premiums are negatively related to default probabilities, even in the presence of guaranty funds [8].
According to financial theory, the above formula provides an estimate of the market price of an insurance policy.[7] In other words, the formula provides an answer to the first question; that is, it tells us how much the buyer initially payed for the liability. As in the case of the publicly-traded bond, we do not necessarily need to apply the formula to answer this question; instead, we can directly observe the market prices (or premiums) that emerge in the primary market. In any event, we have already established that historical costs are irrelevant to current valuations.
Next, let’s assume that this insurance liability is a financial asset, and estimate the market value that investors would pay for such an asset. If this insurance liability really was traded in an actual security market, then we could directly observe the market’s price for the liability. Yet, unlike the bond in our previous example, insurance liabilities are (generally) not traded in a competitive financial marketplace. Nonetheless, we can still utilize financial theory to estimate the market price that would emerge for this insurance liability, in a hypothetical, perfectly competitive, secondary security market.
Specifically, according to financial theory, the present (or market) value of the insurance liability – when considered as a hypothetical financial asset -- is given by the following formula:[8]
Present Value = PV of Expected Losses (assuming no risk of default) – PV of Default Option(1.3)
In deriving formula (1.3), we simply eliminated the following two terms from formula (1.2): Insurance Underwriting Expenses and PV of Federal Income Taxes on Investment Income. The policyholder in the primary market is required to pay these two items (insurance underwriting expenses and federal income taxes on investment income) in order for the shareholders to earn a fair return on the insurance transaction. But these items only need to be paid once. In other words, if the corresponding insurance liability were actively traded in the secondary market, buyers would be unwilling to pay these additional amounts; in fact, if buyers were required to pay these additional amounts, they would not earn their opportunity cost of capital. To see this clearly, consider a “fixed” insurance liability of (say) $100. Assume that the insurer is financially-solid, and that there is no probability of default. Investors can purchase a risk-free government bond with a face amount of $100 for a price of $100 discounted at the risk-free rate. Thus, if this insurance liability were actively traded in the secondary market, investors would be unwilling to pay more than $100 discounted at the risk-free rate.[9]
Notice the close correspondence between formula (1.3) for the present value of an insurance liability and formula (1.1) for the present value of our hypothetical bond. The hypothetical bond offers a fixed face value, which is discounted at the risk-free rate. In the case of the insurance liability, the “face amount” (or promised payment) is a random variable; thus, we discount the expected loss payment at a rate that reflects the systematic risk of this random variable, and assuming that there is no chance of default. Moreover, the “PV of Default Option” is also trickier to model in the case of the insurance liability. For the bond, the PV of default is a put option on the firm’s assets with an exercise price that equals the face amount of the bond. For the insurance liability, the “exercise price” of this put option is a random variable; we can still value put options with a stochastic exercise price, but the methods are more complex.
In summary, formula (1.3) provides the market value of the insurance liability, when considered as a financial asset. This formula provides a hypothetical answer to the second question in the previous subsection. In this formula, the credit risk of the insurance company is reflected in the market value of the financial asset via the “PV of Default Option.” The greater the credit risk of the insurance company, the less investors will be willing to pay for this hypothetical financial asset.
By comparison, let’s consider the fourth question from the previous subsection in an insurance context. Specifically, let's assume that an insurance company has an existing liability to policyholders. According to the IASB, the fair value of this liability is the amount that the insurance company would need to pay for a third party (a reinsurance company) to reimburse it for these losses (i.e. a loss portfolio transfer). Importantly, the logic underlying Formula (1.2) is equally valid in both the primary insurance market and the reinsurance market. Accordingly, financial formula for the reinsurance premium would be given as follows:
Reinsurance premium = PV of expected losses assuming no chance of default - PV of Reinsurer's Default Option + Reinsurer's Underwriting Expenses + PV of Reinsurer’s federal income taxes on investment income (1.4)
As in the primary market, the amount that the insurance company will pay for this contract depends on the financial strength, or credit risk, of the reinsurer. In fact, this is even more true in the reinsurance market, because guaranty funds do not apply to reinsurers and insurance companies are generally well-equipped to evaluate the credit risk of various reinsurance companies. Provided the selected reinsurer is fairly solid, however, the PV of the Reinsurer's Default Option will be close to zero. Even so, due to the existence of the third and fourth terms on the right-hand-side of this equation, the reinsurance premium will be larger than the PV of expected losses assuming no chance of default. For instance, if the proper discount rate in the absence of default risk is equal to the risk-free rate, then the corresponding adjusted discount rate for determining the fair value of the liability is implicitly less than the risk-free rate.
As in the case of the primary market, we don’t necessarily need a formula to determine the premium that would be required for such a transaction; in theory, we can observe the prices that actually emerge in the reinsurance market. If the financial theory of pricing insurance is correct, however, we will observe reinsurance premiums that exceed the PV of expected losses assuming no chance of default (since, as noted in the previous paragraph, the reinsurer will need to be compensated for both underwriting expenses and FIT’s on investment income). Moreover, since reinsurance company’s may vary in degree of financial strength, the PV of the Reinsurer’s Default Option will vary from firm to firm; thus, financially-solid reinsurers will be able to charge more than shaky reinsurers.
Fair Value of an Insurance Liability: A Specific Example
In order to clarify the insurance issues, let's consider a simple, single-policy insurance company. At the end of one year, there is a 90% chance that the insurer will pay losses of $95 on the policy, but there is a 10% chance that losses will be $195. Hence, the expected loss amount is $105. Also, we will assume that the variability in actual losses is unrelated to the return on the “market porfolio”; in terms of the CAPM, the insurance loss amount has a “beta” of zero, and thus expected losses can be discounted at the risk-free rate. The total premium (net of expenses) is $100, and is invested in a one-period risk-free bond with a 5% return. The insurer's shareholders have contributed $100 of surplus, which will be invested in a common stock portfolio with the following return distribution: probability of +40% return = 0.60 and probability of -30% return = 0.40. The insurer’s underwriting expenses on this policy are $20.
The end-of-period assets and loss amounts for each of the four possible states-of-the-world are as follows:
Scenario End-of-Period AssetsEnd-of-Period Loss
High Return, Low Loss$245$95
High Return, High Loss$245$195
Low Return, Low Loss$170$95
Low Return, High Loss$170$195
Under three of these scenarios, the insurer will be financially able to meet its obligation to the policyholders. In low return/high loss scenario, the insurer's end of period assets ($170) will fall short of the actual loss amount ($195), and the company will be insolvent. In this hypothetical example, the PV of Default Option is difficult to model. For our purposes, however, the actual amount is not relevant to the general conclusions; as such, we will assume that PV of Default Option is equal to $1. Lastly, in order to isolate the impact of credit risk on fair valuation, we will ignore federal income taxes.
According to the stated assumptions and formula (1.2), the market price, or premium, that will emerge in the insurance marketplace will be as follows:
Premium = $105 / 1.05 - $1 + $20 = $119
Note that the model produces an answer for premium that differs from the actual premium collected of $100. In this case, the actuary should re-examine the assumptions regarding expected losses and underwriting expenses. Or, it could just be that the policy has been mispriced by the company. Such an example would illustrate the general principle that historical costs are irrelevant to current valuations.