A. Market Risk VaR

The “textbook” formulation for market risk VaR assumes that assets’ returns are normally distributed over the interval of interest. In this setting, if the asset of interest has a present value of V₀, and a single-period normally distributed return, $\tilde{r}$, with a mean of μ and a variance of σ², then the 1 percent VaR measured relative to the asset’s initial market value, V₀, over the single period horizon, VaR (0.01), is given by,

This definition of VaR is consistent with the common description of VaR as a measure of potential loss exposure relative to the portfolio’s initial value.

Under a common alternative definition, VaR is a measure of the distance between a selected left-hand critical value and the mean of the end-of-period value (or return) distribution. When losses are measured relative to the asset’s expected value, the 1 percent VaR for an asset with normally distributed returns with a mean of μ and a variance of σ² is given by,

where the notation VaR^μ is used to indicate that potential losses are being measured relative to the expected end-of-period asset value, i.e., relative to V₀(l +μ). VaR^μ is, for example, the measurement basis for the Basel Internal Models Approach for setting market risk capital requirements.

In typical short-horizon applications that assume normally distributed returns, VaR is intended to measure one-day exposures and μ is either approximately 0 or intentionally set to 0 to minimize the effects of errors associated with the estimation of short-horizon expected returns and there is no difference in the alternative measures. Capital allocation decisions, however, require VaR calculations for holding periods substantially longer than a day, and the differences in these alternative VaR measures can become substantial.⁵

B. Credit Risk VaR

In contrast to the market risk setting in which VaR methods were initially developed for monitoring trading book exposures over short horizons, credit VaR techniques were developed to measure risks over relatively long horizons primarily for use in capital allocation and Risk-Adjusted Return On Capital (RAROC) decisions. It has been widely presumed that an appropriate approach for setting the equity share of funding for a credit portfolio is to set equity capital equal to an estimate of a portfolio’s so-called unexpected credit loss. The credit risk modeling techniques used to estimate unexpected credit losses are generically called credit VaR models.⁶

In anticipation of the discussion that follows, it is worth reviewing the “current best practices” recommendations for credit risk capital allocation. The CreditMetrics Technical Document recommends using unexpected credit losses (the CreditMetrics so- called “percentile level” measure) to measure credit risk and set credit risk capital.⁷Wilson (1997b) and Saunders (1999) argue that unexpected credit losses are a measure of credit risk and an appropriate capital benchmark. While the Basel Bank Supervisors voice concerns regarding the empirical implementation of credit VaR models, they too subscribe to the view that the unexpected credit loss metric is, in principle at least, a sound measure of credit risk and an appropriate gauge of credit risk capital.⁸ Beyond the credit risk measurement and capital literature, the view that credit VaR measures of credit risk are an appropriate benchmark for capital is also common in the literature on risk- adjusted performance or so-called RAROC analysis.⁹

A stylized credit VaR unexpected credit loss measure is illustrated in Figure 1. The probability distribution pictured in Figure 1 represents the true probabilities associated with all potential end-of-period values that may be realized on an asset (portfolio) with credit risk. The potential profit and loss distribution of interest is generated by potential changes in the value of credit risk sensitive exposures to individual counterparties over the horizon that has been selected to measure credit risk and set capital.¹⁰ Credit VaR models attempt to estimate unexpected credit losses in either a mark-to-market (MTM) or a held to maturity (HTM) setting. If the asset has yet to mature in the horizon of interest, the end-of-period value distribution represents the asset’s potential MTM values or its range of potential values in an early default. If the end of the period in question corresponds to the maturity of the asset, the variation in end-of-period values owes entirely to variation in default severity.

Stylized Credit VaR

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Stylized 1 percent unexpected credit loss calculation over a one-year horizon for a discount bond with a maturity of one-year and a par value of 105.

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Stylized Credit VaR

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Stylized 1 percent unexpected credit loss calculation over a one-year horizon for a discount bond with a maturity of one-year and a par value of 105.

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Stylized Credit VaR

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Stylized 1 percent unexpected credit loss calculation over a one-year horizon for a discount bond with a maturity of one-year and a par value of 105.

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The unexpected credit loss measure is defined as the difference between the mean of the end-of-period value distribution and the loss associated with a user-selected critical value in the loss tail of the distribution.¹¹ In Figure 1, for example, the unexpected credit loss measure is 31.43 when measured using the distribution’s 1 percent critical value.

A. Market Risk Capital

In the BSM model, the firm’s underlying assets evolve in value according to geometric Brownian motion and have future values that exhibit “market risk” in the vernacular of risk managers. In this setting, the selection of the firm’s debt-equity funding mix under an objective of achieving a target default rate on its funding debt is a market risk capital allocation problem. In the market risk setting, the VaR calculation is applied to the physical probability distribution for the firm’s asset value at a horizon equal to the desired maturity of the firm’s funding debt.

Under the assumptions of the BSM model, the value of the firm’s assets evolve following,

where dz is a standard Weiner process. If A₀ represents the initial value of the firm’s assets, and A_T the value of the firm’s assets at time T, Ito’s lemma implies,

where ϕ[a, b] represents the normal density function with a mean of “a” and a standard deviation of “b”. Equation (5) defines the physical probability distribution for the end-of- period value of the firm’s assets,

where $\overline{\epsilon }~{\varphi }[0,1].$

Let Φ(x) represent the cumulative density function for a standard normal random variable evaluated at x, and Φ⁻¹ (α), the inverse of this function evaluated at 0 ≤ α ≤ 1. The market risk VaR measure that is appropriate for calculating an equity capital allocation consistent with a target default rate of α for a funding debt maturity of T is given by,

$A_0-\mathit{VaR}({\alpha })=A_0{e}^{({\mu }-\frac{{{\sigma }}^2}{2})T+{\sigma }\sqrt{T}{{\Phi }}^{-1({\alpha })}}$is the maximum par value of discount debt that can be issued without violating the firm’s target default rate. The BSM debt pricing condition (expression (3)) can then be used to determine the initial market value of this debt issue. The difference between the initial market value of the debt and its par value is the equilibrium interest compensation that must be offered to the firm’s debt holders. In the BSM model setting, the interest payments are,

This interest amount must be added to VaR(α)to calculate the true equity capital needed to achieve the target default rate on funding debt. The true amount of equity required to achieve a target default rate of α on funding debt of maturity T is given by,

The components of the equity capital allocation are instructive. The first component of expression (9) increases equity over VaR(α) to allow funding debt holders to receive a risk free return on their investment. The second term in expression (9) further increases equity capital above VaR(α) to ensure that the funding debt holders receive the proper credit risk interest spread on their investment.

B. Credit Risk Capital

In order to illustrate the buffer stock capital allocation technique that is appropriate for assets with credit risk, it necessary to introduce a modified version of the BSM model in order to value the funding debt of a firm that purchases credit risky assets. Consider the case in which a firm’s only asset is a risky BSM discount debt issue. Assume that the firm will fund this bond with its own discount debt and equity issues. In this setting, the firm’s funding debt issue is a compound option.

Let ${\tilde{A}}_T$ represent the time T value of the assets that support the purchased discount debt. Let Par_p represent the par value of the purchased discount bond and Par_F represent the par value of the discount bond that is used to fund the asset purchase. If the maturity of the firm’s funding debt matches the maturity of the firm’s asset (both equal to M), then the end-of-period cash flows that accrue to the firm’s debt holders are given by,

If the funding debt is of a shorter maturity (T) than the purchased discount bond (M), then the end-of-period cash flows that accrue to the firm’s funding debt holders are given by,

Equilibrium absence of arbitrage conditions impose restrictions on the underlying asset’s Brownian motion’s drift term, μ, = r_f + λσ, where λ is the market price of risk associated with the firm’s assets. Define dA^η =(μ–λσ)A^ηdt + A^ησ dz to be the “risk neutralized” process that is used to value derivative claims after an equivalent martingale change of measure. The probability distribution of the underlying end-of-period asset values after the equivalent martingale change of measure, ${\tilde{A}}_M^{{\eta }}$ is,

When the maturity of the firm’s funding debt matches the maturity of the firm’s asset (both equal to M), the equivalent martingale probability distribution of the end-of-period asset’s value, ${\tilde{A}}_M^{{\eta }}$, is used to calculate the initial market value of the funding discount bond by discounting (at the risk free rate) the expected value of (10) taken with respect to probability density of ${\tilde{A}}_M^{{\eta }}$.¹⁴

In the alternative case in which the funding debt is of a shorter maturity (T) than the purchased discount bond (M), because the value of the put option in the purchased discount bond depends only on the risk free rate, the time to maturity on this bond (both deterministic), and the underlying value of the supporting assets, is straight forward to calculate the initial equilibrium value of the funding debt as the discounted value (at the risk free rate) of the expected value of expression (11), where the expectation is taken with respect to the equivalent martingale probability density ${\tilde{A}}_T^{{\eta }}$.

Given the equilibrium valuation relationships that must be satisfied by the firm’s funding debt, we now consider the buffer stock capital allocation process for assets with credit risk. Assume that the firm’s objective is to maximize the use of debt funding subject to limiting the default rate on its funding debt to a maximum acceptable rate. Recall that the firm’s asset is a BSM risky discount bond of maturity M. We consider initially capital allocation when the maturity of the funding debt is equal to the maturity of the purchased bond.

C. Held-to-Maturity Credit VaR

At maturity, the payoff on the firm’s purchased bond is given by,

Min[Par_p, Ã_M]. The credit risk VaR measure appropriate for credit risk capital allocation is given by,

where B₀ is the initial market value of the purchased discount debt given by expression (3), and α is the target default rate on the funding debt. If α is sufficiently small (which will be assumed), the expression $\mathit{Min}[{\mathit{\text{Par}}}_P,A_0{e}^{[{\mu }-\frac{{{\sigma }}^2}{2}]M+{\sigma }\sqrt{M}{{\Phi }}^{-1}({\alpha })}]$ simplifies to $A_0{e}^{[{\mu }-\frac{{{\sigma }}^2}{2}]M+{\sigma }\sqrt{M}{{\Phi }}^{-1}({\alpha })}$, and consequently, the expression for credit VaR is,

The notation for credit VaR includes three arguments, the target default rate α, the maturity of the funding debt issue, M (the second argument), and the maturity of the credit risky asset, M (the third argument). The utility of this unusual notation will become clear in the subsequent section.

$B_0-{\mathit{VaR}}_{\mathit{\text{Credit}}}({\alpha },M,M)=A_0{e}^{[{\mu }-\frac{{{\sigma }}^2}{2}]M+{\sigma }\sqrt{M}{{\Phi }}^{-1}(\alpha )}$, is the maximum par value of the funding debt that is consistent with the target default rate. The initial market value of this funding debt issue is given by,

where the notation E^η [ ] denotes the expected value operator with respect to the probability density for ${\tilde{A}}_M^{{\eta }}$. Using these relationships, it should be clear that the equilibrium required interest payment on the funding debt is given by,

Expressions (14) and (16) imply that the initial equity allocation consistent with the target default rate α is given by,

D. Mark-to-Market Credit VaR

The credit VaR profit-and-loss distribution differs according to whether the VaR horizon corresponds to the maturity of the credit risky asset or a shorter period of time. When calculating credit VaR for buffer stock capital purposes, the credit VaR horizon must be equal to the maturity of the funding debt issue. Any other credit VaR horizon will produce capital allocations with default rates that differ from the intended target.¹⁵

In some instances in which the end-of-period value of the purchased bond is less than the par value of its funding debt, it is possible for the firm to refinance its debt and avoid default without an equity injection. In such instances, however, the implied default rate on the new debt issue is necessarily much higher than the firm’s original target default rate because, in order to refinance, the firm must dilute its equity value by promising a greater share of the end-of-period cash flows to the new bond investors.¹⁶ Regardless of whether the firm is actually forced to default when the value of the purchased debt falls below the funding debt’s par value at maturity, the firm’s capital allocation objective has been violated and the firm cannot avoid default and continue in business while funding at its optimal target default rate unless the shareholders inject new equity capital.

When the firm’s funding debt matures at date T before the purchased risky discount bond’s maturity, M, T < M, the α level credit VaR is given by,

Again, B₀ - VaR_Credit(α, T,M) detennines the maximum par value of the funding debt that satisfies the target default rate constraint; this value is given by,

where the arguments in the notation for Par_F (α, T, M) conform with those in VaR_Credit (α, T, M).

Using the expression for Par_F(α,T, M), the initial market value of the funding debt issue is,

and the equilibrium required interest payment on the funding debt is,

Expressions (18), (19), and (21) imply that the equity allocation consistent with a target default rate of a is given by,

E. Remarks

Equations (17) and (22) are respectively the equity capital allocations necessary to achieve the target default rates of the firm’s funding debt in the HTM and the MTM credit risk cases. In both of these expressions, the equity capital requirement is determined by the α critical value of the risky discount debt’s supporting asset distribution. Notice that the underlying capital allocation credit VaR measures (expressions (14) and (18)) are not measures of the credit risk of the risky asset. Credit risk measures are defined relative to the promised maturity payment on a fixed income asset, not its initial value. In contrast to credit VaR measures of unexpected loss, accurate equity capital allocation requires that VaR be measured relative to the fixed income security’s (or portfolio’s) initial market value, not its promised, expected maturity, or expected future MTM value. These results challenge the long-standing tradition of linking the processes of credit risk measurement and capital allocation. These results also highlight the importance of establishing an accurate estimate of the initial MTM value for a credit portfolio. Accurate buffer stock capital estimates require that initial market values can be accurately measured. Such a task is often thought to be particularly complicated in the case of bank loans.

Another implication of the analysis is that capital allocation credit VaR measures (expressions (14) and (18)) can be, and indeed frequently are negative. The negative credit VaR measures that are appropriate for credit risk capital allocation are unlikely to conform with the intuition of many risk mangers who typically expect to find a positive relationship between VaR measures, risk, and capital allocations.

A. Market Risk Capital Allocation

Assume that a firm wishes to fund a risky asset with characteristics given by expression (23) using the maximum amount of one-year discount debt possible subject to the constraint that the probability of default on the funding debt cannot exceed 1 percent. The probability distribution for the asset’s value after one-year is pictured in Figure 2. The asset’s future value is distributed log-normally, with a left-hand 1 percent critical value of 66.63. The 1 percent, one-year market risk VaR measure appropriate for capital allocation is 33.37, or the difference between the asset’s initial market value (100) and the 1 percent critical value of its future value distribution. 66.63 is the maximum par value of the funding debt that can be issued while constraining the funding debt’s default rate to 1 percent. Using the BSM valuation model, this discount debt issue will sell for 63.32. The difference between 66.63 and 63.32 is the equilibrium interest compensation required by the funding debt holders. The 1 percent equity capital requirement is equal to 36.68, the sum of VaR(0.0l) = 33.37, and the equilibrium interest payments to funding debt holders, 3.31. In contrast to these estimates, the “traditional” VaR capital allocation recommendation—setting equity equal to the difference between the mean of the future value distribution (108.33) and its 1 percent critical value (66.63)—would imply equity capital of 41.70.

Market Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Market risk capital allocation example for an asset with an initial value of 100 and a future value that evolves according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous return standard deviation of 20 percent. The risk free rate is assumed to be 5 percent.

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Market Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Market risk capital allocation example for an asset with an initial value of 100 and a future value that evolves according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous return standard deviation of 20 percent. The risk free rate is assumed to be 5 percent.

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Market Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Market risk capital allocation example for an asset with an initial value of 100 and a future value that evolves according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous return standard deviation of 20 percent. The risk free rate is assumed to be 5 percent.

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B. Held-to-Maturity Credit Risk Capital Allocation

Figure 3 illustrates the HTM credit risk capital allocation for a risky BSM discount bond that has a par value of 66.63, an initial market value of 63.32, and is supported by assets with an initial market value of 100 and future values that evolve according to expression (23). In this example, a firm purchases the risky discount bond that is sold by the firm that funded the asset acquisition illustrated in the prior MTM example. Since the purchased discount bond is the firm’s only asset, if this bond is completely funded with debt, the funding debt will have a probability of default identical to that of the purchased bond (1 percent). If this bond is in part funded with equity, the funding debt’s probability of default must be less than 1 percent.

Held-to-Maturity Credit Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Held-to-maturity credit risk capital allocation for a one-year BSM risky discount bond with a par value of 66.63 that is supported by assets that have an initiation market value of 100 and future values that evolve according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous standard deviation of 20 percent. The initial market value of the bond is 63.32 and the risk free rate is assumed to be 5 percent.

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Held-to-Maturity Credit Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Held-to-maturity credit risk capital allocation for a one-year BSM risky discount bond with a par value of 66.63 that is supported by assets that have an initiation market value of 100 and future values that evolve according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous standard deviation of 20 percent. The initial market value of the bond is 63.32 and the risk free rate is assumed to be 5 percent.

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Held-to-Maturity Credit Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Held-to-maturity credit risk capital allocation for a one-year BSM risky discount bond with a par value of 66.63 that is supported by assets that have an initiation market value of 100 and future values that evolve according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous standard deviation of 20 percent. The initial market value of the bond is 63.32 and the risk free rate is assumed to be 5 percent.

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Assume that the objective of the firm that purchases the risky BSM discount bond is to maximize the use of debt finance, subject to limiting the probability of default on its funding debt to ½ of 1 percent. The ½ of 1 percent critical value of the underlying asset’s future value distribution is 63.38. Under this target default rate objective, the capital allocation credit VaR measure is given by 63.32–63.38 = –0.06, or negative 6 cents. The par value of the funding debt consistent with the ½ percent target default rate is 63.38, and using expressions (15) and (24), the initial value of the funding debt is calculated to be 60.26. The required interest payment on the funding debt is 3.12, and so the required equity funding—the credit VaR estimate plus funding debt interest—is 3.06. Notice that in this example the capital allocation credit VaR measure is negative. For comparison purposes, we note that if credit risk capital were set equal to unexpected credit losses—the difference between the mean of the end-of-period value distribution (66.59) and its ½ i percent critical value (63.38)—equity capital would be set equal to 3.21.

C. Mark-to-Market Credit Risk Capital Allocation

Suppose that the bond issued in the prior market risk example was purchased by another firm and funded for only six-months. Assume that the purchasing firm wanted to fund the issue with as much six-month debt as possible while limiting the default rate on its debt to ½ of 1 percent. Figure 4 illustrates the capital allocation calculations in this example.

Mark-to-Market Credit Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Six-month mark-to-market credit risk capital allocation for a one-year BSM risky discount bond with a par value of 66.63 that is supported by assets that have an initial market value of 100 and future values that evolve according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous standard deviation of 20 percent. The initial market value of the bond is 63.32 and the risk free rate is assumed to be 5 percent.

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Mark-to-Market Credit Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Six-month mark-to-market credit risk capital allocation for a one-year BSM risky discount bond with a par value of 66.63 that is supported by assets that have an initial market value of 100 and future values that evolve according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous standard deviation of 20 percent. The initial market value of the bond is 63.32 and the risk free rate is assumed to be 5 percent.

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Mark-to-Market Credit Risk Capital Allocation

Citation: IMF Working Papers 2002, 099; 10.5089/9781451852288.001.A001

Source: Six-month mark-to-market credit risk capital allocation for a one-year BSM risky discount bond with a par value of 66.63 that is supported by assets that have an initial market value of 100 and future values that evolve according to geometric Brownian motion with an instantaneous drift rate of 8 percent, and an instantaneous standard deviation of 20 percent. The initial market value of the bond is 63.32 and the risk free rate is assumed to be 5 percent.

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The end-of-period bond valuation distribution is generated using the BSM discount bond valuation condition (expression(3)), and the distribution for the bond’s supporting assets (expression (23)) setting $T=\frac{1}{2}$ Using this future asset value distribution, the ½ of 1 percent critical value of the BSM bond’s end of period value distribution is 63.56, and its corresponding VaR measure, is ${VaR}\text{measure},\mathit{VaR}(0.005,\frac{1}{2},1)\text{is}-0.24$ or negative 24 cents.

The maximum par value of the funding debt that can be issued without violating the target default rate constraint is 63.56, and expressions (20) and (24) are used to calculate the initial equilibrium value of this debt which is 61.99. The equilibrium interest payment required by funding debt investors is 1.57, and so the required amount of equity funding needed to achieve the firm’s funding objective is 1.33, the VaR amount, -0.24, plus the required interest on funding debt, 1.57. If alternatively, capital were set equal to unexpected losses has been traditionally recommended, equity capital would be set equal to 3.06.