Stress tests and risk capital

Transcription

1 Stress tests and risk capital Paul H. Kupiec Freddie Mac, 8200 Jones Branch Drive, McLean, Virginia , USA For many nancial institutions, ``stress tests'' are an important input into processes that set risk capital allocations. In the current regulatory environment, two distinct modelbased approaches for setting regulatory capital requirements include stress test components. The Basle internal models approach used by US banking regulators requires stress tests to augment VaR-based capital requirements. The US Congress has mandated that the regulatory risk-based capital requirements of the governmentsponsored housing enterprises be based upon a stress test alone. This paper investigates the formal link between stress tests and risk capital in the context of an equilibrium model of a rm's capital structure. Two common approaches used to allocate capital under a stress test are examined. The results demonstrate that the stress testing approaches do not solve for an equity capital allocation that protects against default in stress scenarios. One approach produces biased estimates of the optimal debt±equity funding mix. The alternative approach estimates the minimum value of a portfolio of risk-free bonds that will ``hedge'' or fully insure stress scenario exposures. Neither approach estimates the equity component of an optimal capital structure. Because stress test procedures do not estimate the optimal debt±equity funding mix, stress test and VaR-based estimates of risk capital are not comparable. A modi ed stress test procedure can be used to set optimal equity capital allocations using stress test techniques that are comparable with VaR-based capital measures. 1. INTRODUCTION Modern nancial theory suggests that capital structure should be selected to balance the bene ts of debt nancing in the form of tax advantages and the attenuation of principal agent problems against the costs of leverage that include moral hazard and nancial distress costs. In practice, nancial institutions often set capital allocations using value-at-risk VaR) techniques when their objective is to limit the probability of default below a target threshold. Alternatively, institutions may use stress testing methods that require that the rm remain solvent in speci c stress test scenarios. The use of VaR capital allocation techniques is widespread in the banking industry. Stress testing approaches for determining capital adequacy are common in the insurance industry 1 and are the basis for the regulatory riskbased capital requirements of the government-sponsored housing enterprises. 2 1 See, e.g., Moody's Investors Service Insurance Guide 2000), accessible through Moody's Investors Service internet home page, 2 Federal Housing Enterprises Regulatory Reform Act 1992), 12 USC, Section

2 28 P. H. Kupiec While these alternative approaches historically have had distinct clienteles within the nancial services industry, the 1996 Market Risk Amendment to the Capital Accord mandates that internationally active banks allocate market risk capital using VaR techniques that are to be supplemented by regular stress testing exercises that ``evaluate the capacity of the bank's capital to absorb potential large losses''. 3 Given their importance to management and regulators, it is of interest to formally analyze stress testing approaches for setting capital and to investigate the comparability of these alternative capital allocation techniques. Kupiec 1999) investigates the ability of VaR capital allocation models to identify a capital structure consistent with the objective of limiting the probability of default to a target threshold. His results show that VaR-based capital allocation techniques for market risk have signi cant biases, but these biases are easily corrected, and VaR techniques can be used to estimate the capital structure debt±equity funding mix) that is consistent with a target default rate. This paper complements the results in Kupiec 1999) and investigates the relationship between stress test measures of risk exposure and the capital allocations that are necessary to ensure that management stress test risk limits are satis ed. The results of this study demonstrate that common stress test approaches used to assess capital adequacy produce biased estimates of the equity capital that is needed to ensure solvency in stress scenarios. The bias arises because the common approaches for assessing equity capital do not solve a traditional capital allocation problem. Instead of solving for an optimal debt±equity funding mix, one approach solves for the minimum amount of risk-free collateral that is necessary to ensure performance in stress test scenarios. The alternative approach fails to properly account for debt interest payments and thereby produces inaccurate equity capital estimates. While typical stress tests cannot be used to estimate equity capital, it is shown that one of the approaches can be modi ed to produce accurate capital measures that are comparable with VaR capital measures. The modi ed stress test must, however, recognize debt liabilities that are excluded in market risk VaR calculations. The implication is that one cannot stress test a market risk VaR model and estimate equity capital requirements. If capital estimates are to be comparable, the stress test and VaR estimates require separate models. These results highlight a potential inconsistency in the Basle regulatory framework. While the Basle Supervisors have never speci ed how banks are to conduct stress tests, the regulations convey the impression that the mandatory stress tests can be performed using the bank's market risk VaR models, and indeed this is how stress testing is done in many banking institutions see Kupiec 1998 for further discussion). The results of this study show that VaR-based stress test are not complementary tools for VaR-based capital allocation models. These results echo the Berkowitz 2000) discussion of the conceptual challenge of interpreting stress testing results in the context of a probabilistic VaR framework. An outline of the paper follows. Section 2 de nes the alternative approaches 3 Amendment to the Capital Accord to Incorporate Market Risks Basle Committee on Banking Supervision 1996, p. 46). Journal of Risk

3 Stress tests and risk capital 29 for assessing capital adequacy in a stress testing framework. Section 3 develops an equilibrium model of a rm's capital structure. The model is used to characterize an optimal stress test constrained debt±equity funding mix. Section 4 compares the capital allocations recommended by the alternative stress testing procedures with the true optimal stress test capital allocation. Section 5 shows that one common stress test measure of capital estimates the market value of risk-free collateral that could be used to ensure solvency in a stress scenario; alternative solutions that minimize the equity needed for such insurance are discussed. Section 6 presents an algorithm for estimating an optimal stress test constrained equity capital allocation that is comparable with a VaR-based capital estimate and discusses the compatibility of this approach with the market risk VaR models of banks. Section 7 concludes the paper. 2. COMMON STRESS TEST BASED APPROACHES FOR ASSESSING CAPITAL ADEQUACY There are at least two approaches that are commonly used to assess the adequacy of equity capital using a stress test analysis. One of these approaches is more commonly used when a rm attempts to set an initial capital structure for an investment project. The objective of this procedure is to produce an estimate of the amount of risk capital that is required by a rm or an investment project in order to satisfy a set of stress test constraints. An alternative approach is often used to assess the capital adequacy of an existing capital structure. This approach attempts to estimate the capital surplus or de cit if any) that would exist in stress scenarios under an existing nancing structure. The conceptual issues addressed in this study can be analyzed in a singleperiod model. To illustrate the alternative stress testing approaches, consider a single-period setting in which the end-of-period value of the rm can be decomposed into cash revenues and both debt-related and non-debt-related cash expenses. If the cash ows of the rm are multiperiod, as they are in most practical applications, the end-of-period market value of the rm's assets, nondebt liabilities, and debt liabilities should be substituted for their singleperiod cash ow counterparts. In the multiperiod setting, there are signi cant issues surrounding the propriety of using market or book accounting) values. While economic arguments generally favor the use of market values, there are issues associated with valuing non-market-traded assets e.g., individual bank loans), and many rm capital allocation models are based upon book accounting) values. 4 While the choice of an accounting-based or market-based valuation model has de nite implications for the character of risk capital estimates, such issues are not addressed in this study. The single-period cash ow analysis that follows extends to the multiperiod cash ow setting if the end-of-period market 4 For example, the stress testing models of the government-sponsored enterprises and those used by some ratings agencies project accounting balance sheet and income statement values. See, for example, the Notice of the O ce of Housing Enterprise Oversight OFHEO 1999). Volume 2/Number 4, Summer 2000

4 30 P. H. Kupiec values of the future cash ows are substituted for the single-period cash ows of the model. 5 Assume that the end-of-period cash revenues and expenses are functions of the realized values of x, y, andz, and let r x; y; z represent the cash in ows revenues) and c x; y; z the contractual expenses costs) exclusive of any costs related to debt nancing. Let b x; y; z represent the contractual end-of-period cash payments associated with debt nance. This decomposition of cash ows is not restrictive as to the types of investment that it allows. For example, under some random variable realizations, some types of nancial contracts could contribute to either revenues or expenses. A swap contract would generate revenues in some states of nature and costs in other states. Consistent with their de nitions as revenues and expenses, assume that the respective functions' values are nonnegative on their support, i.e., r x; y; z > 0, c x; y; z > 0, b x; y; z > 0, for all x, y, z. For simplicity, assume that the term structure is at and the continuously compounded riskfree rate of interest over the single period is r f. Assume either that a regulatory capital adequacy directive or that the rm's management wants to ensure that the rm is nanced in such a way that it will survive a selected set of stress scenarios. Let fx s i; y s i; z s ig represent the ith stress scenario that management or a regulator has selected as a survivorship constraint and let S represent the set of all stress test constraints that must be satis ed. One approach to assessing capital adequacy examines the investment's revenues and non-debt-related expenses for the stress test scenarios and arrives at an estimate of the required level of risk capital. For ease of reference, de ne r x; y; z c x; y; z as ``net operating income''. De ne the maximum operating loss in the set of stress scenarios by K ˆ max c x s i ; y s i; z s i r x s i; y s i; z s i ; 0 : i2s We de ne e r f K to be the operating income stress test OIST) equity capital requirement for the project. If the OIST procedure is used to assess the adequacy of an existing capital position, then e r f K is compared with the value of the rm's existing equity. 6 If the value of the rm's initial equity exceeds e r f K, the rm is said to be ``adequately'' capitalized; 7 and if e r f K exceeds the rm's equity value, the rm is undercapitalized. The di erence between the rm's current equity value and e r f K is taken as a measure of risk capital surplus or shortfall. Under an alternative approach for assessing capital adequacy, the cash ow 5 In the multiperiod cash ow setting, one would still be assessing capital adequacy on a market-tomarket basis for a one-period horizon. 6 Alternatively, e rf K can be compared with the di erence between the market value of the rm's balance sheet assets and the market value of all long positions in derivative contracts and its balance sheet liabilities and the market value of all short positions in derivative contracts. 7 Some rms may gross up e rf K by a xed percentage to account for so-called ``operational'' and ``legal'' risks. Journal of Risk

5 Stress tests and risk capital 31 shortfall test CFST), the rm ``tests'' to see whether or not the existing level of debt service payments can be met in the stress scenarios selected by management or a regulator. 8 De ne the maximum cash ow shortfall in the set of stress scenarios by F ˆ max i2s c x s i ; y s i; z s i b x s i; y s i; z s i r x s i; y s i; z s i : If F < 0, then the largest cash ow shortfall is a pro t and the rm is overcapitalized. In this case, e r f F is interpreted as an estimate of the overcapitalized amount. If F > 0, then the rm is not adequately capitalized and e r f F is interpreted as the risk capital shortfall. The apparent intuition behind these approaches to capital adequacy assessment is as follows. If the rm were to use the OIST approach and choose an amount of risk capital equal to e r f K, then there would be no chance that in any of the stress scenarios of interest, end-of-period operating losses would ever exceed the value of the rm's equity. The OIST approach mirrors a market risk VaR calculation in that its risk exposure measure excludes from consideration the cash ows value) of the rm's funding debt. Alternatively, if the rm were to use the CFST approach, it would consider debt-service payments in the exposure measure. Under the CFST approach, the intuition is that, should the rm augment its existing equity capital by e r f F, no stress event cash out ow, including out ows due to debt service payments, would ever exceed the value of the rm's augmented equity. The next section investigates the issue of setting optimal risk capital under stress test solvency constraints in the context of an equilibrium model of a rm's capital structure. After formally addressing the capital allocation problem, these simple capital adequacy tests are revisited, and their prescriptions for capital allocations are analyzed. The analysis will demonstrate that neither of the capital adequacy assessment tests will produce a completely accurate prescription for the optimal amount of risk capital that is necessary to satisfy the stress test based leverage constraint. 3. ESTIMATING RISK CAPITAL REQUIREMENTS UNDER STRESS TEST LIMITS An important assumption that underlies a stress test approach to limiting leverage is that, within limits, leverage is bene cial, but excessive leverage generates costs that should be avoided. Further, it must be assumed that the costs associated with excessive leverage can be attenuated by controlling the probability of default in the speci c stress scenarios articulated by management 8 The stress testing procedures speci ed by OFHEO for the housing GSE's regulatory capital requirement are roughly similar to this approach, with a key di erence that the OFHEO stress test uses simulated accounting balance sheet and income statement values in a stress scenario to determine solvency and not economic market values. The Basle internal models approach requires stress testing to supplement VaR calculations, but does not specify how to stress test or how stress test results should alter VaR-based regulatory capital requirements. See Berkowitz 2000) for further discussion. Volume 2/Number 4, Summer 2000

6 32 P. H. Kupiec or a regulator). A stress test implicitly assumes that the optimal capital structure is one with maximum leverage subject to the constraint of avoiding default in any of the speci c stress scenarios of interest. The details of a stress test capital calculation will be illustrated in the context of a simple one-period model. If there are no taxes, transactions are costless, short sales are possible, trading takes place continuously, borrowers and savers have access to the debt market on identical terms, and investors in asset markets act as perfect competitors, then Merton 1974) established that the Modigliani± Miller capital structure irrelevance theorem holds when rms may issue risky debt. That is, the market value of the rm is equal to the market value of its underlying assets and is completely independent of the capital structure the rm should choose to nance those assets. The environment that is analyzed in this study is consistent with the assumptions of Merton 1974) and therefore does not admit an optimal capital structure. For purposes of the analysis, however, it is assumed that the rm's management wants to maximize leverage subject to the constraint that the rm be solvent in the designated stress test scenarios. That is, it is assumed management has a speci c target capital structure objective for reasons outside the scope of the model, and we investigate the ability of common stress test related capital allocation schemes to ``deliver'' the target capital structure. Let ~x, ~y, and ~z represent random variables with a joint physical probability distribution represented by f x; y; z and an equivalent martingale probability density given by f x; y; z. Assume that the end-of-period cash in ows r x; y; z and nondebt expenses c x; y; z are nonnegative functions of the realized values of x, y, and z. Assume that the term structure is at with a continuously compounded risk-free rate of interest is r f. Assume that the rm's owners bene t from the limited liability o ered by the corporate form of organization. In a competitive market, the ``fair market value'' for the rm project), i.e., its equilibrium present value V 0, is the expected value of the project's net cash ows under the equivalent martingale measure, discounted at the risk-free rate, V 0 ˆ e r f E max r x; y; z c x; y; z ; 0 ˆ e r f zˆ 1 yˆ 1 xˆ 1 max r x; y; z c x; y; z ; 0 f x; y; z dx dy dz: If the rm is totally nanced with equity, the fair market value of the equity is given by V 0. The limited liability of the corporate structure makes the equity payo on a debtless rm equivalent to that of a call option with a strike price of 0 written on the rm's net cash ows. 9 9 The rm's labor, raw materials, and other service providers bear the residual default risk. As long as the factors include a competitive credit risk premium in their respective compensation terms, no moral hazard problems are created by the limited liability. The Merton and Black±Scholes models avoid this complication as the geometric Brownian motion that governs the dynamics of the rm's assets' value prohibits negative realizations. Journal of Risk

7 Stress tests and risk capital 33 While more complicated debt instruments could be analyzed, for simplicity assume that the rm wants to partially fund itself with a zero-coupon debt issue. Moreover, assume that the rm's management wants to maximize the rm's leverage subject to the condition that it remain solvent in all managementspeci ed stress scenarios. This optimal capital structure objective is achieved by maximizing the par value of the discount debt issue subject to the constraint that the rm must be able to satisfy the payment due to debt holders in all speci ed stress scenarios. 10 Let fx s i; y s i; z s ig be the ith stress scenario that management has selected to be a constraint on leverage, and let S represent the set of all stress test constraints that must be satis ed. Let the par value of the rm's zero-coupon debt issue be represented by B. Under a stress test limited leverage policy, the par value of the optimal debt issue B satis es the following condition: n B o ˆ max min r x s i ; y s i; z s i c x s i; y s i; z s i ; 0 : fx s i ;ys i ;zs i g2s Notice that if r x s i; y s i; z s i c x s i; y s i; z s i 6 0 for any fx s i; y s i; z s ig2s, then B ˆ 0; or, alternatively, net operating revenues must be positive in all stress test scenarios or no leverage will be optimal under the stress test limits. Let V D be the initial market value of the rm's zero-coupon debt issue and V E be the initial equilibrium market value of the rm's equity. If B > 0, then the optimal initial equilibrium values of the rm's debt and equity will be V D ˆ e r f E max minfb ; r ~x; ~y; ~z c ~x; ~y; ~z g; 0 ; V E ˆ e r f E max r ~x; ~y; ~z c ~x; ~y; ~z B ; 0 : Here V E is the optimal value for the rm's equity and its optimal equity risk capital allocation under the stress test limited leverage policy. 11 As in Black and Scholes 1973) and Merton 1974), the value of the rm's equity is equivalent to the value of a European call option with a strike price of B written on the rm's end-of-period net revenues. Owing to limited liability, the equilibrium market value of the rm's debt is equivalent to the market value of a portfolio with three assets: a long position in a risk-free zero-coupon bond with a par value of B and two put options written on the value of the rm's net operating revenues a short put with a strike price of B and a long put with a strike price of 0). The second long position) put option arises because the rm's debt holders are protected by limited liability. 10 Because risk-neutral probabilities are nonnegative, maximizing the par value of the zero-coupon debt issue subject to the stress test constraint will also maximize the constrained market value of the zero-coupon debt issue. 11 The capital structure is optimal because it meets the rm's assumed objectives. For any choice of leverage, however, the Modigliani±Miller theorem holds: V 0 ˆ V E V D. Volume 2/Number 4, Summer 2000

8 34 P. H. Kupiec 4. ASSESSING STRESS TEST CAPITAL ALLOCATION PROCEDURES Under the OIST approach, the recommended level of equity capital is e r f K, where K is the worst operating loss in the stress scenarios examined: K ˆ max c x s i ; y s i; z s i r x s i; y s i; z s i ; 0 : i2s Notice that if there are operating losses in any stress scenario, then e r f K > 0 and the rm will not be able to issue any zero-coupon debt and still satisfy the stress test leverage constraint. Thus, if e r f K > 0 and the rm's optimal debt issuance policy is one that requires solvency in stress scenarios, then the rm must be totally equity nanced. The results in Section 3 establish that, for an allequity nanced rm, the market value of the rm's equity is V 0 6ˆ e r f K. Alternatively, if K ˆ 0, the maximum stress scenario operating loss is a pro t and debt issuance is optimal. The OIST approach, however, suggests an optimal capital structure of 100% debt nancing. Clearly, there is no guarantee that such an issuance policy would preclude default in stress scenarios, as the OIST approach completely ignores the magnitude of debt payments. Thus the OIST approach for setting capital does not produce an accurate estimate of the optimal equity capital allocation needed to maintain solvency in stress scenarios. As the OIST method does not produce an accurate benchmark against which an existing capital structure can be compared, it is not a suitable approach for assessing an existing capital structure. The CFST approach is perhaps more promising as it tests the rm's ability to meet both operating costs and required debt payments out of the revenues received in the selected stress environments. If the largest cash ow shortfall is 0, then F ˆ 0 and it is straightforward to prove that the CFST procedure accurately indicates that the rm's current nancial structure is optimal. While the CFST approach can identify an optimally capitalized rm, its estimates of the magnitude of a capital surplus or shortfall are, however, not accurate. To identify the bias, assume that the rm issues zero-coupon debt in an amount ^B 6ˆ B, i.e. ^B ˆ b x; y; z for all x, y, andz. Then the true amount of under- or overcapitalization is given by VE ˆ e r f E max r ~x; ~y; ~z c ~x; ~y; ~z B ; 0 E max r ~x; ~y; ~z c ~x; ~y; ~z ^B; 0 : In this setting, V E is equal to the di erence in the value of two European call options written on the rm's net operating income: a long call option with a strike price of B and a short position in a call option with a strike price of ^B. A positive value of V E is the true amount of additional equity capital that is required to satisfy the stress test leverage constraint under the condition that the rm issues only zero-coupon debt; a negative value of V E is the rm's true Journal of Risk

9 Stress tests and risk capital 35 excess equity capital. Notice that VE 6ˆ e r f F, and thus the CFST does not produce an accurate estimate of an equity capital surplus or shortfall. That VE 6ˆ e r f F formally demonstrates the inaccuracy of the CFST recapitalization signals; intuition is enhanced, however, by considering an alternative proof. When the CFST procedure detects a capital surplus, i.e., e r f F < 0, the procedure indicates that the rm could optimize its capital structure by executing a debt-for-equity swap issue debt, retire equity) with a market value of je r f Fj. Unless the recapitalized rm's debt is risk free, however, the par value of the zero-coupon debt issued in the swap would have to exceed F and, post recapitalization, the rm would no longer satisfy the stress test solvency constraint. 12 Thus the CFST procedure overstates the risk capital surplus. When the CFST procedure suggests a capital shortfall, i.e., e r f F > 0, the shortfall is overstated. Again, unless the rm's debt is risk free, the par value of the debt that must be retired to execute an equity-for-debt swap with a market value je r f Fj will exceed F, and the rm will be underleveraged after the recapitalization. Thus the CFST procedure overstates the amount of additional equity that is required for an optimal capital allocation. 5. ``OPERATING INCOME'' STRESS TESTS AND THE DEMAND FOR INSURANCE CAPITAL If the OIST method of assessing capital adequacy does not accurately estimate risk capital requirements, what exactly does this approach estimate? The OIST procedure requires that ``operating income'' should never be negative in a stress scenario. As operating income excludes debt payments, it should be apparent that this approach to testing for capital adequacy is at best only tangentially related to the problem of ensuring debt performance or estimating ``debt capacity''. The OIST method does not estimate an equity capital requirement; rather it estimates the present value of a risk-free bond portfolio that, if held in conjunction with the initial investment project or rm) being evaluated, will ensure that all operating costs could be paid in the stress scenario that generates the worst operating loss. If the worst operating loss is 0, then K ˆ 0 and some use of debt nance is possible under a stress test leverage constraint. If, however, K > 0, the rm should not issue any debt if its objective is to satisfy the stress test requirement. Notice also that the value of the risk-free bond portfolio estimated by the OIST approach, e r f K, is not the value of the equity capital needed both to fund the project and to satisfy the OIST constraints. If risk-free bonds are purchased to ``insure'' stress scenario net income shortfalls, the amount of risk capital required is e r f K V 0, an amount which exceeds the all-equity nancing requirement by e r f K. While the OIST method can be used to estimate jointly 12 It is easiest to see this under the physical measure. If the rm's debt is not risk free, it will be discounted at a premium over the risk-free rate. If the total debt issue is to have a market value of e rf F, then the par value of the issue must exceed F if the discount rate exceeds r f. Volume 2/Number 4, Summer 2000

10 36 P. H. Kupiec a risk capital allocation and a risk-free bond portfolio allocation that will ``hedge'' out the negative operating income cash ow realizations that violate the stress test constraint, the bond portfolio hedge will not minimize the use of equity capital. 5.1 Minimizing Risk Capital and Insuring a Stress Test Constraint The OIST constraints can be satis ed by augmenting the initial investment with an appropriate investment in a portfolio composed of any set of assets that have positive payo s in the stress scenarios that produce operating income de cits or indeed by short-selling assets that have negative payo s in the critical stress scenarios of interest). As a consequence, the demand for operating income insurance does not uniquely de ne an equity funding requirement. While it is certainly not uncommon to nd practitioners adopting a Treasury bond hedging/insurance convention, there is no theoretical preference, and indeed little practical reason, for adopting this method of ``hedging''. 13 In fact, if the rm's underlying objective is to minimize the use of equity capital, a risk-free bond portfolio is necessarily a suboptimal hedging vehicle as, even in the absence of taxation issues, this portfolio requires the shareholders to purchase additional cash ows in states when the stress test constraint is satis ed. If the objective of the rm is to augment the original investment project with assets that would ensure nonnegative operating income in the designated stress scenarios, but to do so with a minimum investment of equity capital, then it will never be optimal to ``hedge'' the investment project with a portfolio of risk-free bonds. 14 Rather, a specialized portfolio of options provides the desired protection with the minimum required equity capital allocation. To de ne the minimum equity capital hedge portfolio that can be used to satisfy the OIST constraints, let there be N stress test scenarios of interest. De ne the individual scenario operating income shortfalls K i to be K i ˆ c x s i; y s i; z s i r x s i; y s i; z s i : Let f x s i; y s i; z s i represent the equivalent martingale probability associated with stress scenario i. The minimum cost collateral portfolio includes N digital options, each of which has a unique payo of K i when ~x; ~y; ~z ˆ x s i; y s i; z s i 13 In a more complete model with corporate and personal taxes, the double taxation of interest income would make Treasury bonds a relatively expensive hedge vehicle. 14 The desire to minimize the equity share in nancing is consistent with commonly encountered practitioner claims that equity capital is a relatively ``costly'' source of nancing. The higher relative costs associated with equity nance may be due to tax advantages a orded by debt nance, implicit or explicit government guarantees, principal agent problems, or other market or information imperfections. We do not explore these issues, but instead assume that rms wish to maximize the use of leverage. Journal of Risk

11 Stress tests and risk capital 37 and a payo of 0 for all other ~x; ~y; ~z realizations. 15 The initial cost O of this option portfolio in a competitive market is given by O ˆ e r f X N iˆ1 f x s i; y s i; z s i K i ; and the minimum risk capital that is necessary to satisfy the operating income stress test is O V 0, an amount that is always less than e r f K V 0, the amount of risk capital that is required using the Treasury bond hedge portfolio. The e ciency of the option-based hedge portfolio is, at a theoretical level, beyond dispute. One may argue that the result requires the existence of a complete set of nancial markets in which credit risks can be traded e ciently at competitive market prices. While one may also object that the use of digital options ``games'' the stress test and allows for more leverage than intended by management or a regulator, it should be noted that if the set of stress scenarios is comprehensive with respect to the conditions that cause unacceptable distress costs, it is impossible to ``game'' the test with digital options. Gaming issues may only arise when the stress test speci cation is incomplete, and it is unclear how to allocate capital optimally when the constraint set is misspeci ed. When markets are not complete or competitive, the theoretical minimum cost hedge portfolio will generally not exist, and the rm will be required to use hedges that are less than perfectly aligned with the operating income shortfalls. Lack of competition may make the hedging vehicles more expensive than the theoretical minimum cost benchmark. Moreover, it could be argued that customized derivative hedges may entail their own credit risk exposures that, as a practical matter, signi cantly complicate estimation of the hedge portfolio's payo s. Regardless of the aforementioned ``practical'' objections to existence of the minimum risk capital hedge portfolio, it is possible to approximate the ideal hedge portfolio as long as the Treasury market and its associated derivatives markets are competitive. Under these conditions, a close substitute to the hedge portfolio can be constructed by purchasing the stress test designated Treasury hedge portfolio, and selling derivative contracts which pay out the cash ows that are not needed to satisfy the stress test constraint. In practice, the contracts could, for example, take the form of bonds with promised payments that mirror the underlying Treasury portfolio's, but with a knockout feature that coupon and principal payments are suspended under speci ed stress test conditions. If these derivative contracts are collateralized using the Treasury bill portfolio itself, credit risk is eliminated. The net cash ows from this portfolio will satisfy the stress test constraint, and, provided that the derivatives are close to competitive and designed appropriately, the net equity needed to fund the hedge portfolio will approximate O. 15 The K i could be negative, in which case the rm could sell the rights to the excess cash ow and still satisfy the stress test. Volume 2/Number 4, Summer 2000

12 38 P. H. Kupiec 6. AN ALGORITHM FOR SETTING RISK CAPITAL UNDER A STRESS TEST CONSTRAINT An optimal equity capital allocation can be estimated using an algorithm that modi es the CFST approach for setting capital. If the objective is to maximize leverage subject to the constraint that debt payments cannot exceed ``operating income'' in any relevant stress scenario, then the CFST can be used to determine the optimal amount of leverage. Recall that, given a capital structure composed of zero-coupon debt and equity, F ˆ 0 is an indication that the given capital structure is optimal, F < 0 indicates that the capital structure is not su ciently leveraged, and F > 0 indicates that the capital structure includes excessive debt. While it has been demonstrated that the magnitude of F is not an accurate guide to the magnitude of the optimal recapitalization, the sign of F is an accurate guide as to whether leverage should be increased or decreased to meet management goals. These characteristics make F useful as an indicator variable and suggest the following iterative algorithm for setting risk capital. Choose an initial capital structure and calculate the value of F. If F ˆ 0, the initial capital structure is optimal; if F < 0, increase the par value of the zero-coupon debt issue and recalculate F; and if F > 0, reduce the par value of the debt issue and recalculate F. Iterate until F ˆ 0. Note that a possible result is F > 0, even when no debt is used to nance the project. Such an outcome is an indication that the choice of a capital structure alone cannot ensure that the rm's operating earnings remain positive in the stress scenarios examined. After arriving at an estimate of the optimal par value of debt, the initial market value of this debt contract must be determined. The optimal amount of equity capital is the total funding requirement or market value of the rm's assets) less the initial equilibrium market value of the optimal debt contract, or V D. It is interesting to note that debt service payments must be recognized in the stress test exposure measure if the test is to be a useful vehicle for assessing equity capital needs. While Kupiec 1998) discusses how to incorporate stress testing in a VaR framework, market risk VaR models typically focus on exposure measures for the assets of a bank. Indeed, the Basle market risk regulatory capital requirement speci cally excludes any consideration of a bank's own debt liabilities in the VaR calculation. Given the results of this paper, it appears that the Basle Supervisors' requirement for supplementing VaR-based market risk capital estimates with the results of stress test exercises is mandating the use and comparison of incompatible measures. The results of this paper suggest that, at a minimum, bank capital stress tests have to be formulated to recognize bank debt liabilities before any meaningful comparison can be made between stress test and VaR measures of a bank's market risk capital. 7. CONCLUSIONS While appropriately speci ed stress tests can uncover exposures that are relevant for the capital allocation process, the analysis in this paper demonstrates that Journal of Risk

13 Stress tests and risk capital 39 some of the common approaches that use stress tests to set risk capital allocations produce severely biased estimates of the optimal amount of equity capital. Common stress testing capital allocation procedures produce either 1) estimates of the value of risk-free collateral that must be held in addition to the assets and liabilities examined in the stress test) to ensure performance in all relevant stress scenarios, or 2) biased measures of the equity capital surplus or shortfall necessary to achieve an optimal leverage target. Thus, while an appropriately constructed stress test can illuminate whether or not a given capital structure ``passes'' a stress test scenario, the stress testing procedures do not themselves suggest an appropriate equity capital allocation. As a consequence, the equity capital allocation suggested by common stress testing procedures is not comparable with those recommended by VaR techniques. This noncompatibility highlights an important inconsistency in the Basle Supervisors' market risk capital regulations. If capital is to be allocated optimally using stress test procedures, the rm's debt liabilities must be recognized in the stress test exposure measure and the optimal use of leverage must be determined with an iterative algorithm. REFERENCES Basle Committee on Banking Supervision 1996). Amendment to the Capital Accord to Incorporate Market Risks. Bank for International Settlements, Basle, Switzerland. Berkowitz, J. 2000). A coherent framework for stress testing. Journal of Risk, 2 2), 5±15. Black, F., and Scholes, M. 1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637±654. Federal Housing Enterprise Regulatory Reform Act 1992), 12 USC, Section Kupiec, P. 1998). Stress testing in a value-at-risk framework. Journal of Derivatives, 6 1), Fall, 7±24. Kupiec, P. 1999). Risk capital and VaR. Journal of Derivatives, 7 2), Winter, 41±52. Merton, R. 1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 29, 449±470. Merton, R., and Perold, A. 1993). Theory of risk capital in nancial rms. Journal of Applied Corporate Finance, 6 3), Fall, 16±32. OFHEO, Department of Housing and Urban Development 1999). Notice of Proposed Rulemaking, Risk Based Capital, 12 CFR Part Federal Register, 64 70) 13 April). Volume 2/Number 4, Summer 2000