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.
1.
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
For a discussion and derivation of this formula, see Chapters 20 and 23 of Brealey and Myers [2].
See chapters 20 and 21 of Brealey and Myers [2] for a detailed discussion of the binomial method for option valuation.
3
In the context of the binomial pricing model, the value of this put option can be determined by either the replicating
portfolio method or the risk-neutral method. In the case of the risk-neutral method, we “pretend” that investors are risk
neutral. Thus, the expected return on the stock must equal the risk-free rate of 5%. This results in the following
equation: (+25%) x p + (-40%) x (1-p) = 5%, where “p” is the probability of a stock price rise in a hypothetical, riskneutral world. Solving this equation gives us p = 69.2%. Now, if there is a rise in the value of the underlying asset, the
put option is worthless; but if there is a fall in the value of the underlying asset, the put option is worth $15.
2
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.
Accordingly, the expected value of the option (in a risk-neutral world) is given by $0 x 69.2% + $15 x 30.2% = $4.62.
We then discount this amount at the risk-free rate to get the current option value of $4.40.
4
For a discussion of this financial principle, see chapter 11 of Brealey and Myers [2].
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)
5
Insurance shareholders always have the option of investing directly in financial assets, instead of investing in financial
assets indirectly via the common stock of an insurance company. Thus, the policyholders must “pay” the federal income
taxes on the insurance company’s asset portfolio. For a detailed discussion, see Myers/Cohn [7].
6
Note: this formula assumes that underwriting expenses are paid immediately and are thereby not discounted.
7
Note that this formula, unlike most traditional P/L insurance pricing methods, does not include a specific profit
provision or “risk load”. Instead, the “risk load” is implicitly included by discounting the expected loss amount at a riskadjusted discount rate. Section 3 of this paper discusses the relationship between methods that utilize risk-adjusted
discount rates and methods that utilize explicit risk-loads.
8
Technical note: in order to utilize this formula, we must assume that security markets are complete in an Arrow-Debreu
sense. Pricing of insurance liabilities in incomplete security markets is more complicated, and is beyond the scope of
this paper.
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-handside 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
Note, however, that future federal tax payments still serve to reduce the market value of the insurance company’s
equity. In effect, there are three claimants on the right-hand-side of the balance sheet; that is, future payments will be
made to insurance liability holders, the federal government, and the shareholders.
9
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
High Return, Low Loss
High Return, High Loss
Low Return, Low Loss
Low Return, High Loss
End-of-Period Assets
$245
$245
$170
$170
End-of-Period Loss
$95
$195
$95
$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.
According to formula (1.3), the market value of the insurance liability, when considered as a financial asset in a
hypothetical, perfectly competitive, secondary market, is given as follows:
Fair Value = $105 / 1.05 - $1 = $99
In order to determine the fair value of the liability (formula (1.4)), we need to specify the underwriting expenses for
an appropriate reinsurance loss portfolio transfer. Since reinsurance underwriting expenses are generally less than
primary underwriting expenses, let’s assume that this amount is $10. Also, let’s assume that we are dealing with a
financially-solid reinsurer with a PV of Reinsurer’s Default Option that is equal to $0. Thus, the second method of
valuation will produce the following answer:
Fair Value = $105 / 1.05 - $0 + $10 = $110
As an aside, what should we do if the quoted price for a loss portfolio transfer is significantly different from the
financial model’s predicted price of $110? As noted earlier, a fundamental principle of finance is to look first to
market prices, at least as a starting point. However, unlike financial assets, insurance and reinsurance policies may
not always trade in a perfectly competitive market. In this case, we may observe prices that diverge from long-run
equilibrium values. In practice, it is probably difficult to simulate an “arm’s length” transaction; in which case, one
would rely on the financial formula.
In the first subsection, we used a generic bond to demonstrate that the fair value of a liability is not necessarily
identical to the market value of the countervailing asset. The previous example demonstrates this important point in
an insurance setting.
In closing, let’s summarize the factors that influence the fair value of a liability and the market value of the
countervailing asset:
The Fair Value of an Insurance Liability depends on the following three items: (1) The PV of the insurance liability
assuming no risk of default10, (2) the underwriting expenses of the reinsurance company, and (3) the financial
strength of the reinsurance company.
The Market Value of the Countervailing Asset depends on the following two items: (1) The PV of the insurance
liability assuming no risk of default, and (2) the financial strength of the insurance company.
2. THE AAA MONOGRAPH AND THE IMPACT OF CREDIT RISK
The American Academy of Actuaries [1] recently released a public policy monograph entitled “The Fair Value of
Insurance Liabilities: Principles and Methods.” This monograph provides an introduction to the theoretical and
practical issues involved in determining the fair value of an insurance liability. Unfortunately, the monograph
presents a confusing and misleading discussion of the impact of credit risk on the fair value of an insurance liability.
In the first section of this paper, we have discussed the problem of fair value from first principles, utilizing financial
theory. If you take this approach, it is clear that the credit risk of the enterprise is irrelevant to the fair value of a
liability (under the ISC definition of fair value, at least). The credit risk of the third party, however, is relevant.
But instead of approaching the issue from first principles, let's take a look at the AAA monograph. Specifically, in
Section 2 of the monograph, on pages 7-9, the authors provide an example of how the "cost of capital method" can
be utilized to determine the risk-adjusted discount rate for the liabilities. As a result of this discussion, the authors
derive the following formula for the risk adjusted discount rate for the insurance liabilities (on the bottom of p. 8):
rL = rA - (e x (rE - rA)).
In this formula, the variables are defined as follows: rL is the risk-adjusted discount rate for the liabilities, rA is the
expected return on the insurer’s investment portfolio, rE is the required return on equity, and “e” is the insurer’s
“capital ratio”, which is defined as the ratio of marginal surplus to the expected loss amount. 11 (As an aside, the
formula on the bottom of page 8 of the monograph contains an error -- namely, it's missing a minus sign. The
formula above is the corrected version.)
The "proof" of this formula is not difficult -- it's basically the same argument that Miller/Modigliani [6] used to
derive their "Proposition II" from their "Proposition I". However, the AAA monograph is unclear as to whether the
variables in this formula apply to the ENTERPRISE or the THIRD PARTY. Judging from much of the later
discussion and examples, it appears that they intend for the variables to apply to the enterprise. Unfortunately, this
In the CAPM, this present value depends on the “beta” of the loss amount.
Property/Liability actuaries may also recognize this formula from Butsic’s article loss reserve discounting [3], which
has been included on the CAS exam syllabus for many years. Specifically, see formula (3) on p. 157.
10
11
is inconsistent with the ISC definition, and it leads to a risk-adjusted discount rate (and corresponding fair value)
that depends on the credit risk of the enterprise.
For instance, consider the following statement from pp. 21-22 of the monograph:
Actuaries should be aware that some liability valuation methods may reflect the credit
standing of the liability holder implicitly, even when there is no explicit assumption
regarding credit standing or the likelihood of default. For example, one can argue that the
cost of capital method outlined in this paper reflects credit standing implicitly. To
understand how credit standing may be reflected, consider the effect of either increasing
investment risk or decreasing the level of capital.
-- An increase in investment risk leads to an increase in the investment return rA. This in
turn increases the liability valuation rate rL and decreases the value of the liability.
-- A decrease in the capital ratio "e" also increases the liability valuation rate rL and
decreases the value of the liability.
Therefore assumption changes that are consistent with increased credit risk lead to decreases
in liability value when using the cost of capital method, even though there is no explicit
assumption concerning default."
The primary message of this quotation is as follows: if we use the cost of capital method (as described on pp. 7-9 of
the monograph) to determine the risk-adjusted discount rate (and corresponding fair value of the liability), then an
increase in the riskiness of the enterprise will result in a decrease in the fair value of the liability. But that message
is inconsistent with the ISC definition of fair value. And the reason that it is inconsistent is because it's a different
method! When you use the "cost of capital method" with the variables defined according to the enterprise (not the
third party), then you are effectively answering the second question posed in the first section of this paper: in other
words, you are using financial theory to estimate the market value of the insurance liability, assuming that the
insurance liability was an asset that was traded in a hypothetical, perfectly competitive, secondary security market.
According to financial theory, this market value depends on the credit risk of the enterprise. But this market value is
not necessary identical to the "amount that the enterprise would have to pay a third party to take over the liability”,
as we demonstrated in the previous section.
In fact, the amount that the enterprise would have to pay a third party to take over the liability can be determined by
the formula on the bottom of page 8, but only if we interpret the variables as pertaining to the third party; in this
manner, we are in effect determining the amount that the third party would charge to take over the liability and still
earn its required return on capital rE. That is, "e" is the leverage ratio of the third party, rA is the expected return on
the asset portfolio of the third party, and rE is the required return on capital of the shareholders of the
third party. Once you've done this, order is restored. For instance, let's assume that we decrease the capital ratio of
the third party. According to the formula on the bottom of p. 8, a decrease in the capital ratio will increase the
liability valuation rate rL and, hence, decrease the value of the liability. That makes sense -- by lowering "e" (all
other things equal) you have increased the credit risk of the third party, and this will logically lead to a decrease in
the amount that the enterprise would pay for the third party to take over the liability. Importantly, however, the
values of "e", rA, etc., for the enterprise have no impact on the risk-adjusted discount rate -- as they shouldn't, since
the third party will have to pay the liability regardless of the future status of the enterprise.
3. THE AAA “COST OF CAPITAL METHOD” VERSUS ACTUARIAL “RISK LOADS”
The cost of capital method, as presented in the AAA monograph, provides a risk-adjusted discount rate for an
insurance liability. Most practicing property/liability actuaries tend to think more in terms of “risk loads” than in
terms of risk-adjusted discount rates. Thus, in order to further clarify the cost of capital method, and the issues
involved in credit risk, let’s first translate the formulas on page 8 of the AAA monograph into the more-traditional
P/L risk load formulas.
In a 1990 article in the Proceedings of the Casualty Actuarial Society, Rodney Kreps [4] described the reinsurer’s
risk load problem as follows:
The underlying economic point of view taken is that of a reinsurer considering a new contract.
The reinsurer has committed surplus to support the variability of his existing book; the new
contract will require additional surplus to support its variability. The return on this marginal
surplus required must be at least as much as is available in the capital markets; otherwise the
reinsurer might just as well invest directly.
In effect, Kreps is dealing with the same problem that we are confronting when we try to determine the fair value of
an existing insurance liability: namely, we need to estimate how much a reinsurance company would charge to
assume the liability – and, importantly, this premium must provide the reinsurer with an adequate return on the
marginal surplus required to support the reinsurance contract. In his paper, Kreps went on to derive the following
formula for the premium on a reinsurance contract (see formulas 1.5, 1.9, and 2.1 in his paper):
Premium = Present Value of Expected Loss Amount + Present Value of Reinsurer’s Underwriting Expenses
+ Risk Load,
where “Risk Load” was defined as the product of marginal surplus and required return on marginal surplus.
The major insight of Kreps’ paper was to provide a separate adjustment for time and risk. That is, the adjustment for
the time value of money was accomplished by discounting the expected loss amount at the risk-free rate; the
adjustment for risk was accomplished by adding an additional term for the “risk load”.
Unfortunately, Kreps was a little loose in his terminology and his formulas. Glenn Meyers has subsequently cleaned
up some of Kreps’ basic formulas; importantly, Meyers pointed out that the risk load is actually determined as the
product of the following two items: (1) the marginal surplus, and (2) the discounted value of the difference between
the required return on marginal surplus and the investment return that the insurer earns on its asset portfolio.
Specifically, if we ignore underwriting expenses and assume that the reinsurer invests both the premium and the
contributed surplus in risk-free assets, then Meyers would provide the following formula for the premium on a
reinsurance contract – assuming that the expected loss amount on the contract is payable at the end of one year:
Premium = [L / (1 + Rf)] + E * [(Re – Rf) / (1 + Rf)],
Where L is the expected loss amount, Rf is the risk-free interest rate, E is the marginal surplus required to support
the policy, and Re is the required return on marginal surplus. For more information on this formula, see Meyers’
1999 paper “Underwriting Risk.” [5]
But how does this formula compare to the AAA’s “cost of capital method” formulas shown on page 8 of the
monograph? In contrast to the actuarial “risk load” approach, the AAA monograph is making a combined
adjustment for both time and risk via a “risk-adjusted discount rate” for expected losses. The AAA monograph
provides the following formula for the risk-adjusted discount rate (see the bottom of page 8):
Risk-Adjusted Discount Rate for Expected Loss Amount = Ra – e * (Re – Ra),
Where Ra is the return on the insurer’s asset portfolio, and Re is the required return on marginal surplus. In
addition, the AAA monograph defines “e” as the ratio of marginal surplus to the expected loss amount, or E / L.
Unfortunately, they are speaking a little loosely here. In fact, “e” should actually be defined as the ratio of marginal
surplus to the present value of the expected loss amount, or E / PV(L), as demonstrated in the finance literature by
Miller & Modigliani’s famous Proposition II [6].
In order to compare the AAA formula to the version of Meyer’s formula provided above, let’s again ignore
underwriting expenses, assume that the insurer invests solely in risk-free assets, and assume that the expected loss
amount is payable at the end of one year. Under these assumptions, the AAA monograph provides the following
formula for the risk-adjusted discount rate: Rf – (E / PV(L)) * (Re – Rf).
The present value of the expected loss is then determined by discounting the expected loss at the risk-adjusted
discount rate:
PV(L) = L / [1 + Rf – (E / PV(L)) * (Re – Rf)].
Solving this formula for PV(L) gives us the following:
PV(L) = [L / (1 + Rf)] + E * [(Re – Rf) / (1 + Rf)].
Also, under the AAA method there is no need for a separate “risk load” term in the premium formula, since we are
incorporating risk via the risk-adjusted discount rate. Therefore, the PV(L) is also equal to the premium on the
policy (keeping in mind that we are ignoring underwriting expenses in this example). Hence, we can see that the
AAA formula is actually equivalent to the actuarial risk load formula, assuming that we are using the same
assumptions for the variables.
Once we see things in this light, it clarifies a couple of issues. First, when we are estimating the fair value of a
liability, we need to determine how much a reinsurance company would require to assume a given expected loss
amount. In order to do this, we can use the standard actuarial risk load formula for the premium on the reinsurance
contract (which, as we have shown, is consistent with the “cost of capital method” in the AAA monograph!):
Premium = [L / (1 + Rf)] + E * [(Re – Rf) / (1 + Rf)]
In framing the problem in this manner, it is entirely clear that the marginal surplus, or E, and the required return on
marginal surplus, or Re, should pertain to the reinsurance company – not the ceding insurance company. As such, it
is unlikely that we will make the same mistakes that the AAA monograph falls into on pages 21-22 – namely,
confusing the enterprise and the third party and incorrectly concluding that the credit risk of the enterprise is
relevant. Second, this formulation points out that the ISC definition of fair value does not produce a closed
axiomatic system. That is, the fair value of the liability depends on the relevant assumptions regarding the marginal
surplus and required return on marginal surplus of the reinsurance company. Consequently, in determining fair
value, one will still need to utilize actuarial judgment regarding these parameters, in consideration of the types of
reinsurance companies who may be willing to complete a contract with the enterprise. This issue is discussed in
further detail in the final section of this paper.
4. FAIR VALUE OF INSURANCE LIABILITIES AND ACTUARIAL JUDGMENT
As discussed earlier, the current ISC definition of fair value is incomplete in the sense that it does not specify the
credit standing of the third party guarantor. In addition, there is not much that economic/financial theory can
contribute to the specification of this credit standing. In general, one would suppose that it would be desirable to
specify that the third party guarantor possess a negligible probability of insolvency. This would have the added
benefit of simplifying the fair value calculations. For instance, the actuary utilizing the financial method would no
longer need to estimate the “present value of the default option”.
As alluded to at the close of the prior section, however, there are still practical hurdles associated with the ISC
definition -- even if we can amend it to clearly specify the credit standing of the third party. For instance, let's say
that the ISC definition is amended to specify that the third party guarantor has a negligible probability of ruin. To
make things specific, let's say that Insurance Company A has a liability with an expected value of $100 that will be
paid at the end of one year. According to the ISC definition, the fair value of this liability is the amount that a third
party would charge to take over the liability. If we assume that the third party has no chance of default, the financial
formula for the corresponding reinsurance premium (and, hence, the "fair value" of the liability) would be as
follows:
Reinsurance Premium = PV of Expected Loss (Assuming no risk of default) + PV of Surplus Costs
Now, the PV of Expected Loss (assuming no risk of default) depends on the characteristics of the underlying loss
distribution. For instance, let's assume that the random variable representing the loss amount is uncorrelated with
the random variable representing the return on the "market portfolio". In this case, the Sharpe/Lintner CAPM would
indicate that the risk-free rate is the correct discount rate. Let's say that the risk-free rate is 5%. Thus, PV of
Expected Loss (assuming no risk of default) is $100 / 1.05 = $95.24.
The second term, however, depends on the particular reinsurance company (even if we only consider reinsurance
companies with a negligible probability of ruin). In other words, let's consider two reinsurance companies:
Reinsurance Company B and Reinsurance Company C, both of which have a negligible probability of ruin. How do
we know for a fact that Company B and Company C will each require the same marginal surplus to write the policy?
For instance, it may that Reinsurance Company B already has a book of existing liabilities that are highly correlated
with this one, whereas Reinsurance Company C has a book of existing liabilities that are highly uncorrelated with
this one. In this case, you would expect that Reinsurance Company B would require a higher marginal surplus to
assume this liability than Reinsurance Company C would. Moreover, the PV of Surplus Costs are proportional to
the marginal surplus amount. For instance, let's assume that Reinsurance Company B requires a marginal
surplus of $100 to assume the liability, and that Reinsurance Company C requires a marginal surplus of $10. Also,
assume that PV of Surplus Costs are equal to 10% of marginal surplus. In this case, Reinsurance Company B
would require $95.24 + $10 = $105.24 to assume the liability, whereas Reinsurance Company C would require
$95.24 + $1 = $96.24. So is the "fair value" of the liability under the ISC definition $105.24 or $96.24?
We would encounter similar problems using the actuarial risk load methods discussed in the previous section:
namely, one would still need to specify the marginal surplus and cost of capital of the relevant third party. These
considerations emphasize an important point regarding the practical determination of fair value: that is, determining
the fair value of a liability cannot be reduced to a “cookbook” approach. In order to determine the fair value of an
insurance liability, in accordance with the ISC definition, we will need to utilize actuarial judgment regarding the
types of reinsurance companies that may be willing to complete a transaction with the enterprise, as well as the
actual market prices that have been observed in a relevant “arms length” transaction.
5. CONCLUSION
As the accounting community continues its inexorable drive toward the requirement for fair valuation of liabilities, it
is critical that the actuarial voice be heard. We need to ensure that any such requirements are sound and in
accordance with generally accepted actuarial principles. In particular, the issue of credit risk needs to be clearly
understood in the fair valuation context. As this paper demonstrates, both from first principles and by using the
formulas in the AAA monograph, the fair value of a liability does not depend on the credit risk of the enterprise.
This fair value does depend on the credit risk of the third-party guarantor, however. Finally, the determination of
fair value will always necessitate actuarial judgment regarding the relevant methodology and parameters, with
consideration of any relevant “arms-length” reinsurance market prices that may be available.
REFERENCES
[1] American Academy of Actuaries, Fair Valuation of Insurance Liabilities: Principles and Methods, September
2002.
[2] Brealey, Richard A., and Stewart C. Myers, Principles of Corporate Finance, Sixth Edition, McGraw-Hill, 2000.
[3] Butsic, Robert P., “Determining the Proper Interest Rate for Loss Reserve Discounting: An Economic
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[4] Kreps, Rodney E., “Reinsurer Risk Loads from Marginal Surplus Requirements,” PCAS LXXX, 1990, pp. 196203.
[5] Meyers, Glenn G., “Underwriting Risk,” CAS Forum, Spring 1999, pp. 185-220.
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