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An Analysis of VaR-based Capital Requirements∗ Domenico Cuoco The Wharton School University of Pennsylvania Philadelphia, PA 19104 cuoco@wharton.upenn.edu Hong Liu John M. Olin School of Business Washington University in Saint Louis St. Louis, MO 63130 liuh@olin.wustl.edu This draft: April 2004 Abstract We study the dynamic investment and reporting problem of a financial institution subject to capital requirements based on self-reported VaR estimates, as in the Basel Committee’s Internal Models Approach (IMA), an issue so far unexplored in the banking literature. With constant price coefficients, we show that optimal portfolios display a local three-fund separation property. VaR-based capital requirements induce financial institutions to tilt their portfolios towards assets with high expected return (and high systematic risk), but result nevertheless in a decrease of the overall risk of trading portfolios and of the probability of default. Overall, we find that capital requirements determined on the basis of the IMA can be very effective not only in curbing portfolio risk but also in inducing truthful revelation of this risk. A comparison with capital requirements determined according to the U.S. FED’s Pre-Commitment Approach (PCA) is also provided. Journal of Economic Literature Classification Numbers: D91, D92, G11, C61. Keywords: Capital requirements, Basel Capital Accord, Internal Models Approach, Precommitment Approach, VaR, portfolio constraints. ∗ We thank Viral Acharya, Kerry Back, Phil Dybvig, Bob Goldstein, Karel Janecek, Philippe Jorion and seminar participants at Boston College, University of Brescia, UCLA, Carnegie Mellon, Columbia, Michigan, HEC Montreal, Stockholm School of Economics, USI, Washington University, Wharton, the 2003 AMASES Conference, the 2003 Blaise Pascal Conference on Financial Modeling and the 2004 AFA meeting for helpful suggestions. The usual disclaimer applies. An Analysis of VaR-based Capital Requirements Abstract We study the dynamic investment and reporting problem of a financial institution subject to capital requirements based on self-reported VaR estimates, as in the Basel Committee’s Internal Models Approach (IMA), an issue so far unexplored in the banking literature. With constant price coefficients, we show that optimal portfolios display a local three-fund separation property. VaR-based capital requirements induce financial institutions to tilt their portfolios towards assets with high expected return (and high systematic risk), but result nevertheless in a decrease of the overall risk of trading portfolios and of the probability of default. Overall, we find that capital requirements determined on the basis of the IMA can be very effective not only in curbing portfolio risk but also in inducing truthful revelation of this risk. A comparison with capital requirements determined according to the U.S. FED’s Pre-Commitment Approach (PCA) is also provided. Journal of Economic Literature Classification Numbers: D91, D92, G11, C61. Keywords: Capital requirements, Basel Capital Accord, Internal Models Approach, Precommitment Approach, VaR, portfolio constraints. Financial institutions are required by regulators to maintain minimum levels of capital. This regulation is normally justified as a response to the negative externalities arising from bank failures and to the risk-shifting incentives created by deposit insurance.1 The 1988 Basel Capital Accord imposed uniform capital requirements based on risk-adjusted assets, defined as the sum of asset positions multiplied by asset-specific risk weights. These risk weights were intended to reflect primarily the credit risk associated with a given asset. In 1996 the Accord was amended to include additional minimum capital reserves to cover market risk, defined as the risk arising from movements in the market prices of trading positions (Basel Committee on Banking Supervision, 1996a). The 1996 Amendment’s Internal Models Approach (IMA) determines capital requirements on the basis of the output of the financial institutions’ internal risk measurement systems. Financial institutions are required to report daily their Value-at-Risk (VaR) at the 99% confidence level over a one-day horizon and over a two-week horizon (ten trading days).2 The minimum capital requirement on a given day is then equal to the sum of a charge to cover “general market risk” and a charge to cover “credit risk” (or idiosyncratic risk), where the market-risk charge is equal to a multiple of the average reported two-week VaR’s in the last 60 trading days3 and the credit-risk charge is equal to 8% of risk-adjusted assets. U.S.-regulated banks and OTC derivatives dealers are subject to capital requirements determined on the basis of the IMA. The reliance on the financial institution’s self-reported VaRs to determine capital requirements creates an adverse selection and moral hazard problem, since the institution has an incentive to under-report its true VaR in order to reduce capital requirements. The procedure suggested by the Basel Committee to address this problem relies on “backtesting” (Basel Committee on Banking Supervision, 1996c): regulators should evaluate on a quarterly basis the frequency of “exceptions” (that is, the frequency of daily losses exceeding the reported VaR) for every financial institution in the most recent twelve-month period and the multiplier used to determine the market risk charge should be increased (according to a given scale varying between 3 and 4) if the frequency of exceptions is high.4 Additional 1 See Berger, Herring and Szegö (1995), Freixas and Santomero (2002) or Santos (2002) for a review of the theoretical justifications for bank capital requirements. 2 Simply stated, VaR is the maximum loss of a trading portfolio over a given horizon, at a given confidence level (i.e., a quantile of the projected profit/loss distribution at the given horizon). To avoid a duplication of risk-measurement systems, financial institutions are allowed to derive their two-week VaR measure by scaling up the daily VaR by the square root of ten (see: Basel Committee on Banking Supervision, 1996b, p. 4). 3 More precisely, the market-risk charge is equal to the larger of: (i) the average reported two-week VaR’s in the last 60 trading days times a multiplier and (ii) the last-reported two-week VaR. However, since the multiplier is not less than 3 (see below), the average of the reported VaR’s in the last 60 trading days times a multiplier typically exceeds the last-reported VaR. 4 The reason why backtesting is based on a daily VaR measure in spite of the fact that the market risk charge is based on a two-week VaR measure is that VaR measures are typically computed ignoring portfolio revisions over the VaR horizon. According to the Basel Committee, “it is often argued that value-at-risk measures cannot be compared against actual trading outcomes, since the actual outcomes will inevitably be ‘contaminated’ by changes in portfolio composition during the holding period. [ . . .] This argument is persuasive with regard to the use of value-at-risk measures based on price shocks calibrated to longer holding periods. That is, comparing the ten-day, 99th percentile risk measures from the internal models capital requirement with actual ten-day trading outcomes would probably not be a meaningful exercise. In particular, in any given ten day period, significant changes in portfolio composition relative to the initial 1 corrective actions in response to a high number of exceptions are left to the discretion of regulators. This paper studies the optimal behavior of a financial institution subject to capital requirements determined according to the IMA. We view the system of capital requirements put in place by the 1996 Amendment as a revelation mechanism designed to induce financial institutions to truthfully reveal the risk (VaR) of their trading portfolios and to support this risk with adequate levels of capital, a view consistent with Rochet (1999) and Jorion (2001, p. 65). Accordingly, we consider the simultaneous optimal choice of a reporting and investment strategy. Since the incentives to truthful revelation arise in part from the threat of increased capital requirements in the future (through an increase in the reserve multiplier), we consider a fully dynamic model with discrete reporting and continuous trading. Specifically, we consider a financial institution with preferences represented by a riskaverse utility function defined over the market value of its equity capital at the end of the planning horizon.5 We assume that the institution has fully-insured deposits and limited liability: thus, our model allows for the risk-shifting incentives created by deposit insurance and the option to default. The planning horizon is divided into non-overlapping “backtesting periods”, each of which is in turn divided into non-overlapping “reporting periods”. The institution is required to continuously maintain its capital above a minimum level, which equals the sum of a charge to cover market risk and a charge to cover credit risk. At the beginning of each reporting period, the institution must report to regulators its claimed VaR as well as the actual loss over the previous period. The market-risk charge for the current reporting period is then equal to a multiple k of the reported VaR,6 while the creditrisk charge is equal to the sum of asset positions multiplied by asset-specific credit-risk weights.7 At the end of each backtesting period, the number of exceptions (i.e., the number of reporting periods in which the actual loss exceeded the reported VaR) is computed and this determines the multiple k over the next backtesting period, according to an increasing scale.8 To capture the cost of any additional regulatory action that might be undertaken in positions are common at major trading institutions. For this reason, the backtesting framework described here involves the use of risk measures calibrated to a one-day holding period.” (Basel Committee on Banking Supervision (1996c, p. 3). 5 Risk aversion on the part of financial institutions can be justified by, among other things, the value of the institution’s charter: see Keeley (1990). 6 Consistently with empirical evidence, we find that in our model reported VaR’s display little variation from one reporting period to the next. Thus, making capital requirements proportional to an average of recently-reported VaR’s (rather than proportional to the last-reported VaR) would make little difference in our results. 7 This is consistent with the Basel Capital Accord, which sets the charge to cover credit risk equal to 0.08 times the sum of asset positions multiplied by asset-specific weights ranging from 0 to 1.5 (see: Basel Committee on Banking Supervision (2001)). This corresponds to risk weights between 0 and 0.08×1.5 = 0.12 in our definition. Unrated corporate claims (including equity) are assigned a weight of 100% (0.08 in our definition). 8 Thus, we assume that the VaR measure used for backtesting coincides with that used to determine the capital charge for market risk. This is without loss of generality, as any difference between the two VaR measures can be captured by rescaling the multiplier k. For example, in our numerical calibration we take the reporting period to be one day, as suggested by the Basel Committee, and assume that the multiplier is determined according to the scale suggested by the Basel Committee times the square root of ten: this adjustment captures the fact that the multiplier should be applied to a two-week (rather than one-day) VaR (see footnote 2). 2 response to losses exceeding the reported VaR, we also allow for the possibility of pecuniary sanctions at the end of any reporting period in which an exception is observed. These sanctions are assumed to be proportional to the amount by which the actual loss exceeds the reported VaR. Therefore, the financial institution chooses the level of VaR to report in each period by trading off the cost of higher capital requirements in the current period resulting from a higher reported VaR against the benefit of a lower probability of pecuniary sanctions and a lower probability of higher capital requirements in the future as a result of a loss exceeding the reported VaR. In addition, the institution simultaneously chooses a continuouslyrebalanced trading strategy for its portfolio, subject to the applicable capital requirements. We stress that the problem we consider differs from a standard investment problem with portfolio constraints, since capital requirements are not exogenously fixed, but vary endogenously as a result of the institution’s optimal reporting strategy. We explicitly characterize the solution of the problem described above using martingale duality (as in Cuoco (1997)) and parametric quadratic programming. Even with constant price coefficients, optimal portfolios in the presence of capital requirements do not display two-fund separation: as capital requirements become progressively more binding following losses, financial institutions find it optimal to rebalance their portfolios in favor of assets characterized by high risk-weight-adjusted expected returns (high systematic risks). However, we show that optimal portfolios satisfy a local three-fund separation property, with the three funds being the riskless asset, the mean-variance efficient portfolio of risky assets and a risk-weight-constrained minimum-variance portfolio of risky assets. For no choice of the parameters we find VaR-based capital requirements leading to an increase in overall portfolio risk or to a higher probability of extreme losses and default. In fact, we find that VaR-based capital requirements are effective in completely offsetting the risk-taking incentives generated by deposit insurance and the associated default option, with the risk-taking by a financial institution in the presence of deposit insurance and capital requirements never exceeding that of a similar institution with unlimited liability and no capital requirements. In general, financial institutions may optimally under-report or over-report their true VaR’s, depending on their risk aversion, the current reserve multiplier, the number of exceptions recorded in the current backtesting period, the time remaining to the end of the current backesting period and the level of the pecuniary penalties associated with an exception. Overall, we find that capital requirements determined on the basis of the IMA can be very effective not only in curbing portfolio risks, but also in inducing truthful revelation of these risks. For relative risk aversion coefficients of 0.25 or higher, the threat of a pecuniary sanction equal to 1% of the amount by which the end-of-period loss exceeds the reported VaR is sufficient to ensure that financial institutions never find it optimal to report VaRs that are below true 90% VaRs. The optimal behavior of a financial institution subject to capital requirements determined in accordance to the IMA has been so far unexplored in the banking literature. In a static mean-variance framework, Kahane (1977) and Koehn and Santomero (1980) showed that a more stringent capital requirement (in the form of a lower upper bound on feasible leverage ratios) may induce financial institutions to substitute riskier assets for less risky ones and thus may increase the risk of trading portfolios and the probability of default. Kim and Santomero (1988) established that the same result applies to capital 3 requirements determined on the basis of risk-weighted assets, unless the risk weights happen to be proportional to the assets’ betas. The conclusion that capital requirements could lead to an increase in risk taking and hence in the likelihood of bank failures has been the subject of extensive discussion in the subsequent literature.9 Furlong and Keeley (1989) and Keeley and Furlong (1990) argued that the mean-variance framework is inappropriate to analyze the effect of capital requirements in the presence of deposit insurance and limited liability, because limited liability results in skewed equity return distributions. In particular, Furlong and Keeley (1989) considered a value-maximizing financial institution and showed that stricter leverage limits unambiguously reduce optimal risk-taking. The main reason is that such an institution would always choose the portfolio having the maximum possible risk under the capital requirement (i.e., a corner solution) in order to maximize the value of the deposit insurance (a put option). Gennotte and Pyle (1991) extended the analysis of Furlong and Keeley to allow for investment opportunities having non-zero net present value (NPV) and showed that in this setting tighter capital restrictions can lead financial institutions to increase asset risk. However, this result was obtained under the key assumption that the institution finds it optimal to invest in risky assets having negative NPV: thus, as long as the institution has access to traded securities (zero-NPV investments) with the same or higher risk, the result in Furlong and Keeley would still hold. While all of the above studies considered a static setting, Blum (1999) used a two-period model to show that the incentives to increase the risk of trading portfolios in response to tighter capital requirements are even higher in a dynamic setting. This is because capital requirements increase the marginal utility of a unit of capital tomorrow and thus can lead to an increase in risk in an effort to increase expected return. Differently from these papers, we take into account the impact of capital requirement on the risk-shifting incentives created by deposit insurance and limited liability in a dynamic setting with continuous trading and hence with portfolio return distributions that are not restricted to be normals or truncated normals. In addition, we allow for more general capital requirements that include a charge for market risk in addition to that based on risk-weighted assets. While asset substitution incentives are present in our model, we find that capital requirements never have perverse effects on risk-taking or on the probability of failure. Moreover, differently from Furlong and Keeley (1989), this lack of perverse effects is not due to the fact that the financial institution acts as a risk-lover and is always at a corner solution. Sentana (2001), Emmer, Klüppelberg and Korn (2001), Vorst (2001), Basak and Shapiro (2001) and Cuoco, He and Issaenko (2002) considered the investment problem of a trader subject to an exogenous limit on the VaR of the trading portfolio. None of these papers incorporates limited liability or a realistic model of capital requirements. The first four papers considered the case of a fixed VaR limit, which does not capture the constraint imposed by capital requirements on financial institutions. Cuoco, He and Issaenko considered the case in which the limit varies as a function of the value of the trading portfolio. Their results for the proportional case imply that if the VaR of a financial institution’s trading portfolio were perfectly and continuously observable by regulators and minimum capital requirements at any given point in time were simply equal to a fixed multiple of the contemporaneous VaR (with no penalties for observed exceptions), then, under the assumption of CRRA pref9 See Jackson (1999) for a review of the related empirical evidence. 4 erences and unlimited liability, the optimal portfolio for a financial institution subject to capital requirements would involve a constant proportional allocation to the mean-variance efficient portfolio. Moreover, the capital requirement would either always bind or never bind. Neither of these conclusions holds for the more realistic model of capital regulation considered in this paper.10 Again in a static setting, Chan, Greenbaum and Thakor (1992) and Giammarino, Lewis and Sappington (1993) studied the optimal design of a mechanism that induces truthful risk revelation in a setting in which regulators also provide deposit insurance. By contrast, our focus is not on mechanism design but on the analysis of the specific mechanism implemented by the 1996 Amendment. Ju and Pearson (1999) examined, in a static setting, the bias that arises when the VaR of a portfolio is determined on the basis of the delta-normal method with variances and covariances estimated using past data: in this case, a trader subject to a binding VaR constraint and possessing private information about the relation between current variances and covariances and historical ones, is able to select portfolios whose true VaR exceeds the estimated VaR and hence to assume risks in excess of the stated limit. Ju and Pearson quantified the extent of this bias assuming that the supervisor monitoring the limit provides no incentives for the trader to reveal his information and that the trader has one of three objectives: maximizing the portfolio VaR, maximizing the portfolio expected return, or minimizing the variance of the difference between the return of the chosen portfolio and the return of an exogenously-given reference portfolio. By contrast, because of the penalties associated with exceptions, the 1996 Amendment does provide incentives to financial institutions to reveal private information about risk: these incentives (in addition to a more realistic investment objective) are important features of the model we consider. The rest of the paper is organized as follows. Section I describes our model in detail. Section II characterizes optimal trading strategies in the absence of capital requirements. Section III describes our solution approach to the joint reporting and investment problem in the presence of capital requirements and provides some explicit characterization of optimal trading strategies in this section. Section IV provides a numerical analysis. Section V concludes. Appendix A examines capital requirements determined on the basis of the Pre-Commitment Approach (PCA), an alternative approach proposed by the U.S. Federal Reserve Bank.11 Appendix B contains all the proofs. Appendix C provides some additional auxiliary results. 1 The Model We consider a financial institution with a planning horizon equal to T backtesting periods, where T is a positive integer. Without loss of generality, we normalize the length of a backtesting period to 1. Each backtesting period comprises n non-overlapping reporting 10 Leippold, Trojani and Vanini (2001), using asymptotic approximation techniques, extended the analysis of an exogenous proportional VaR limit in Cuoco, He and Issaenko to incorporate stochastically-varying price coefficients and also examine the equilibrium implications of such a limit. 11 As will be shown in Appendix A, PCA-based capital requirements are a special case of the model of capital requirements studied in Section III. In related work, Kupiec and O’Brien (1997) provide an analysis of PCA-based capital requirements in a static setting. 5 periods of equal length τ = 1/n. At the beginning of each reporting period, the financial institution is required to report to a regulator its current VaR as well as the actual profit/loss over the previous reporting period. As explained later, the reported VaR determines the capital charge to cover market risk for the period. The financial institution has liabilities represented by deposits and (equity) capital. For simplicity, we assume that the face value of deposits D is fixed over the planning horizon and that there are no equity issues or dividend payments over this period. Deposits are fully insured and earn the risk-free interest rate, which is paid out continuously to depositors. The market value of deposits is therefore constant and equal to D. The investment opportunities are represented by m + 1 long-lived assets. The first asset is riskless and earns a constant continuously-compounded interest rate r ≥ 0. The other m assets are risky and their price process S (inclusive of reinvested dividends) follows a geometric Brownian motion with drift vector r1̄ + µ and diffusion matrix σ, i.e., Z S(t) = S(0) + t Z S I (s)(r1̄ + µ) ds + 0 t I S (s)σ dw(s), 0 where I S (t) denotes the m × m diagonal matrix with elements S(t), 1̄ = (1, . . . , 1)> and w is an m-dimensional Brownian motion. We assume without loss of generality that σ has rank m.12 The financial institution can trade continuously and without frictions over [0, T ].13,14 Letting θ be the m-dimensional stochastic process representing the (dollar) investment in the risky assets, the evolution of the value A of the institution’s asset portfolio over any reporting period is then given by dA(t) = (A(t)r + θ(t)> µ) dt + θ(t)> σ dw(t) − rD dt, (1) where the last term reflects interest payments to depositors. We define the institution’s regulatory capital K = A − D as the difference between the value of the institution’s asset portfolio and the value of the institution’s deposits.15 Since 12 If d = rank(σ) < m, some stocks are redundant and can be omitted from the analysis. Moreover, w can be redefined in this case to be a d-dimensional Brownian motion. 13 The assumption of continuous frictionless trading is of course a simplification in the case of a financial institution for which loans constitute a significant portion of investments. However, incorporating illiquidity into the present model would significantly add to its complexity. We view the frictionless case as a reasonable starting point for a first analysis of VaR-based capital requirements, especially in view of the increasing use of loan securitization by financial institutions. 14 While we do not explicitly impose short-sale constraints, our results would be unchanged by these constraints. As it will be shown in Proposition 6, capital requirements never induce financial institutions to short (long) assets that are held long (short) in the unconstrained mean-variance efficient (MVE) portfolio. Thus, assets that are held short in the unconstrained MVE portfolio would never be held in the presence of short-sale constraints (with or without capital requirements) and thus can simply be ignored. On the other hand, assets that are held long in the unconstrained MVE portfolio are also held in non-negative amounts in the presence of capital requirements and thus are unaffected by short-sale constraints. 15 Our definition of regulatory capital is different from the market value of the institution’s equity because, consistently with practice, the market value of deposit insurance is not included in the value of the institution’s assets. Thus, at the terminal date T , the market value of the institution’s equity is equal to K(T )+ = max[0, K(T )]. As it will become clear from equations (2) and (3), we assume that capital requirements are defined in terms of regulatory capital, but that the institution has preferences defined in terms of the market value of capital. 6 the market value of deposits is fixed and there are no new equity issues, it follows from equation (1) that the institution’s regulatory capital satisfies dK(t) = (K(t)r + θ(t)> µ) dt + θ(t)> σ dw(t). The financial institution is required to maintain at all times its regulatory capital above a minimum level equal to the sum of the charge to cover general market risk plus a charge to cover credit (or idiosyncratic) risk. The capital charge to cover market risk equals the VaR reported at the beginning of the current reporting period times a multiplier k. The capital charge to cover credit risk equals the sum of the institution’s trading positions (long and short) multiplied by asset-specific risk weights. Thus, letting β ∈ [0, 1]m denote the vector of asset risk weights,16 the capital charge to cover credit risk at time t equals β > (θ(t)+ +θ(t)− ), where for any x ∈ IR x+ = max[0, x] and x− = max[0, −x].17 Hence, if VaR ≥ 0 denotes the VaR reported to regulators at the beginning of the current reporting period and k is the currently-applicable multiplier, the institution must satisfy the constraint K(t) ≥ kVaR + β > (θ(t)+ + θ(t)− ) (2) at all times during the reporting period.18 The institution is subject to pecuniary sanctions at the end of each reporting period in which the actual loss exceeds the reported VaR.19 We assume that the sanction is proportional to the amount by which the actual loss exceeds the VaR and denote the proportionality coefficient by λ, where λ ≥ 0. At the end of each backtesting period, the number i (i = 0, 1, . . . , n) of reporting periods in which the actual loss exceeded the reported VaR is computed, and the capital reserve multiplier k for the next backtesting period is set equal to k(i), for some given positive numbers k(0) ≤ k(1) ≤ . . . ≤ k(n). We assume that the financial institution’s trading strategy, and hence the financial institution’s true VaR, are unobservable by the regulator. Therefore, the reported VaR can differ from the true VaR. However, the threat of pecuniary sanctions at the end of each reporting period and the revision of the capital reserve multiplier k at the end of each backtesting period represent incentives to not under-report the VaR, while the capital requirement provides an incentive to not over-report. The financial institution has limited liability and preferences represented by a power HARA utility function over the market value of its capital at the end of the planning horizon. Thus, it chooses a reporting and trading strategy over [0, T ] so as to maximize E [u (K(T ))] , 16 It is a straightforward extension to allow β to be different also across long and short positions. Consistently with existing regulation, we are implicitly assuming a zero risk weight for investment in the money market account. 18 The constraint K(t) ≥ β > (θ(t)+ +θ(t)− ) is identical to the one that would arise in the presence of margin requirements if the trader were allowed to earn market interest on the margin: see Cuoco and Liu (2000). Thus, while we assume that trading is frictionless, margin requirements could be easily accommodated and would amount to an increase in the vector of risk weights β by an amount equal to the proportional margin requirement. 19 These sanctions are meant to capture reputation costs or additional disciplinary actions that might be undertaken by regulators. The threat of these sanctions is necessary for institutions to optimally report non-zero VaRs. 17 7 where u(K) = (K + + ε)1−γ 1−γ (3) for some ε ≥ 0 and γ > 0, γ 6= 1.20 2 The Unconstrained Problem We start by deriving an analytical solution for the investment problem of a financial institution not subject to capital requirements: h V U (K, t) = max E u (K(T )) θ K(t) = K i (4) s.t. dK(s) = (K(s)r + θ(s)> µ) ds + θ(s)> σ dw(s) K(s) ≥ −D for all s ∈ [t, T ]. The constraint K ≥ −D reflects the fact that the value of the financial institution’s assets, which equals D + K, cannot be negative. We refer to the problem in (4) as the unconstrained problem. To simplify comparison with the next section,which deals with the constrained problem with capital requirements, we use a martingale duality approach to solve the above problem. Letting ! ! |κ|2 > ξ(t) = exp − r + t − κ w(t) (5) 2 denote the state-price density process, where κ = σ −1 µ, it follows from Cox and Huang (1989), Pliska (1986) or Karatzas, Lehoczky and Shreve (1987) that the problem in (4) can be equivalently written as the static problem V U (K, t) = min max Et [u(x) − ψ (ξ(T )x − ξ(t)K)] , ψ≥0 x≥−D (6) where ψ is a Lagrangian multiplier. Setting Z = ψξ, it then follows from (5) that dZ(t) = −Z(t) (r dt + κ> dw(t)) and (6) implies h i V U (K, t) = min Ṽ U (z, t) + zK , z≥0 where h Ṽ U (z, t) = E ṽ U (Z(T )) | Z(t) = z i (7) and ṽ U (z) = max [u(x) − zx]. x≥−D (8) The next proposition describes the solution of the maximization problem in (8). For convenience, let b = 1 − 1/γ. 20 The case γ = 1 can be treated similarly. 8 Lemma 1 If u(0) > −∞ (that is, if either ε > 0 or γ < 1), there exists a unique zU ∈ (0, ε−γ ] satisfying zb ε1−γ − U + (ε − D)zU − = 0. (9) b 1−γ Lemma 2 Let zU be the constant in Lemma 1 if u(0) > −∞ and let zU = +∞ otherwise. Then the solution of the maximization problem in (8) is given by K = f U (z), where ( U f (z) = z − γ1 −ε if z ≤ zU −D Hence U U (10) otherwise. U ṽ (z) = u(f (z)) − zf (z) =   −  zb b ε1−γ 1−γ + εz if z ≤ zU + Dz otherwise. (11) f U (z) The function in (10) is decreasing for z ∈ (0, +∞) and strictly positive and strictly decreasing for z ∈ (0, zU ), with a single discontinuity at z = zU . The function ṽ U (z) in (11) is continuous and convex. The result in Lemma 2 allows the derivation of an explicit expression for the dual value function in (7) and for the optimal trading strategy. Proposition 1 The dual value function Ṽ U in (7) is given by Ṽ U (z, t) = − z b −b(r+ 12 (1−b)κ2 )(T −t) N b e + εze−r(T −t) N + ε1−γ 1−γ N  −  log( log( zU z  log( zU z )+(r+√ ( 21 −b)|κ|2 )(T −t)  |κ| T −t r− 12 |κ|2 √ )+( )(T −t)  |κ| T −t zU z 1 |κ|2 )(T −t) )+(r+ √ 2 (12)  |κ| T −t  + Dze−r(T −t) N − log( zU z 1 |κ|2 )(T −t) )+(r− √ 2 |κ| T −t  , where N is the standard normal distribution function. In particular, Ṽ U is strictly convex in z for all t ∈ [0, T ). Proposition 2 The optimal trading strategy θU for the unconstrained problem in (4) is given by θ(t) = θU (Z(t), t), where −1 U θU (z, t) = z Ṽzz (z, t) (σσ > ) µ. (13) Moreover, under the optimal investment policy, the financial institution’s capital at time t is given by K(t) = K U (Z(t), t), where K U (z, t) = −ṼzU (Z(t), t). In particular, K U (T ) = f U (Z(T )), where f U is the function in (10). 9 Portfolio weight (θU /(D + K U )) 10 t=1 8 t=0 6 4 2 0 0.01 0.1 1 10 100 U Asset value (D + K ) Figure 1: The graph plots the optimal portfolio allocation to stocks in the unconstrained model at t = 0 and t = 1 as a function of asset value, for the case γ = .25, ε = 0, T = 2, r = .01, κ = .27, σ = .22 and D = 1. Corollary 1 We have lim θU (z, t) = z→0 1 −1 (σσ > ) µK U (z, t), γ lim θU (z, t) = 0, z→+∞ lim K U (z, t) = +∞, z→0 ( lim K U (z, t) = z→+∞ − e−r(T −t) D if u(0) > −∞, 0 otherwise. Proposition 2 implies that, in the absence of capital requirements, the financial institutions has incentives to exploit the option to default by choosing a discontinuous distribution of capital at the terminal date. As long as the state variable Z(T ) is lower than zU , terminal −1 capital equals Z(T ) γ − ε > 0 and default is avoided. However, in states in which Z(T ) is higher than zU , terminal capital equals − D: in other words, the institution completely depletes its assets and defaults for the largest possible amount. This is true even in the case ε = 0 as long as the utility function is finite at zero (that is, as long as γ < 1): infinite marginal utility at zero is not sufficient to prevent default if the institution can default for a non-infinitesimal amount. The incentives to exploit the default option are also evident in Figure 1, which plots the optimal portfolio weight θU /(D + K U ) as a function of asset value at two different points in time (t = 0 and t = 1) for the case γ = .25, ε = 0, T = 2, r = .01, κ = .27, σ = .22 and D = 1.21 As asset value decreases below the default point e−r(T −t) D, the proportional 21 The figure plots asset value on a logarithmic scale, since the portfolio weight decreases steeply to zero as asset value approaches zero. 10 allocation to risky assets in the absence of capital requirements increases significantly and would become unboundedly large as the end of the investment horizon approaches. 3 The Constrained Problem 3.1 Recursion for the Value Function Turning next to the institution’s investment and reporting problem in the presence of capital requirement, let V (K, K − , VaR, i, k, t) denote the value function for the institution’s problem at time t conditional on current capital being K, capital at the beginning of the current reporting period being K − , the VaR reported at the beginning of the current reporting period being VaR, the number of exceptions in the current reporting period being i and the current capital reserve multiplier being k. Without loss of generality, suppose that t is in the h-th reporting period, i.e., that t ∈ [(h − 1)τ, hτ ). Finally, let T = {1, 2, . . . , T } denote the set of backtesting dates. Then it follows from the principle of dynamic programming that h V (K, K − , VaR, i, k, t) = max E v(K(hτ ), K − , VaR, i, k, hτ ) θ K(t) = K i (14) dK(s) = (K(s)r + θ(s)> µ) ds + θ(s)> σ dw(s), s.t. K(s) ≥ kVaR + β > (θ(s)+ + θ(s)− ) for all s ∈ [t, hτ ), for K ≥ kVaR, where v(K, K − , VaR, i, k, hτ ) = (15) max V (K1 , K1 , VaR1 , i1 , k1 , hτ )1{K≥K − −VaR} VaR1 ≥0 + max V (K2 , K2 , VaR2 , i2 , k2 , hτ )1{K 0, then h V (K, K − , VaR, i, k, t) = min Ṽ (ψ, K − , VaR, i, k, t) + ψK ψ>0 i (20) for all K ≥ kVaR and all t ∈ [(h − 1)τ, hτ ), where Ṽ (z, K − , VaR, i, k, t) (21)  = min E ṽ(Zν (τ ), K − , VaR, i, k, hτ ) ν∈N − kVaR Z hτ  Zν (s)ν0 (s) ds Zν (t) = z t   s.t. dZν (t) = −Zν (t) (r + ν0 (t)) dt + κν (t)> dw(t) . While in the unconstrained case it was possible to directly compute the dual value function Ṽ U , an analytic solution is not available in the constrained case. However, the dual value function Ṽ can be computed numerically by solving the associated HamiltonJacobi-Bellman (HJB) equation. Below we denote by ιi the i-th column of the m × m identity matrix. 13 Proposition 4 If β ∈ IRm ++ the dual value function Ṽ in (21) is strictly decreasing and strictly convex in z for all t ∈ [(h − 1)τ, hτ ) and it solves the HJB equation " 1 Ṽz + kVaR 0 = Ṽt − rz Ṽz + z 2 Ṽzz min |σ −1 (µ + ν− − ν0 1̄)|2 − ν0 z Ṽzz ν∈Ã 2 # (22) with terminal condition Ṽ (z, K − , VaR, i, k, hτ ) = ṽ(z, K − , VaR, i, k, hτ ). Moreover, the process ν ∗ attaining the minimum in (22) satisfies 0 ≤ ν0∗ ≤ M and M (1̄ − β) ≤ ν− ≤ M (1̄ + β), > ∗ where M = max{|ι> i µ|/ιi β : i = 1, . . . , m}. Hence, ν ∈ N . Remark 3 The terminal condition for Ṽ in Proposition 4 is given by the function ṽ defined in (19). An explicit expression for ṽ(z, K − , VaR, i, k, hτ ) in terms of the initial value Ṽ (z, K − , VaR, i, k, hτ ) of the dual value function computed over the next reporting period [hτ, (h + 1)τ ) is provided in Appendix C. This expression simplifies the recursive solution of the dual value function. 3.3 Optimal Investment Strategy Once the dual value function Ṽ is known, the optimal trading strategy θ and the process for the institution’s capital K can be recovered as in Proposition 2. To prevent excessively cumbersome notation, we suppress from now on the dependence of the dual value function on the variables (K − , VaR, i, k) which are constant within each reporting period. Proposition 5 If β ∈ IRm ++ , the optimal trading strategy θ for the constrained problem in (14) is given by θ(t) = θ(Zν ∗ (t), t), where ∗ θ(z, t) = z Ṽzz (z, t)(σσ > )−1 (µ + ν− (z, t) − ν0∗ (z, t)1̄). (23) Moreover, under the optimal trading strategy, the institution’s capital at time t is given by K(t) = K(Zν ∗ (t), t), where K(z, t) = −Ṽz (z, t). (24) In particular, K(z, hτ ) = f (z), where f is the function defined in Appendix C. A comparison of equations (13) and (23) shows that the constrained optimal trading strategy coincides with the unconstrained optimal trading strategy in a fictitious economy ∗ − ν ∗ 1̄. The next proposition proin which the vector of stock risk premia equals µ + ν− 0 vides an explicit characterization of the optimal trading strategy in the presence of capital requirements. We denote by Ii the i × m matrix consisting of the first i rows of the m × m identity matrix. 14 Proposition 6 Suppose that β ∈ IRm ++ and that all the components of the unconstrained > −1 mean-variance efficient portfolio (σσ ) µ are different from zero.23 For i, j ∈ {1, 2, . . . , m}, j ≤ i, let > > > −1 ι> j HIi (Ii σσ Ii ) Ii µ ηi,j = > > , ιj HIi (Ii σσ > Ii> )−1 Ii Hβ where    H = diag sign (σσ > )−1 µ and suppose without loss of generality that the assets are sorted so that n o ηi,i = min ηi,j : ηi,j > 0, j = 1, . . . , i . For i = 1, 2, . . . , m, let hi = β > HIi> (Ii σσ > Ii> )−1 Ii (µ − ηi,i Hβ) and let hm+1 = β > H(σσ > )−1 µ. Then 0 = h1 ≤ h2 ≤ . . . ≤ hm+1 . If − Ṽz (z, t) + kVaR ≥ hm+1 , z Ṽzz (z, t) then ν0∗ (z, t) = 0 and θ(z, t) = z Ṽzz (z, t)(σσ > )−1 µ (25) If hi ≤ − Ṽz (z, t) + kVaR < hi+1 , z Ṽzz (z, t) for i = 1, 2, . . . , m then ν0∗ (z, t) = β > HIi> (Ii σσ > Ii> )−1 Ii µ + Ṽz (z,t)+kVaR z Ṽzz (z,t) (26) β > HIi> (Ii σσ > Ii> )−1 Ii Hβ and   θ(z, t) = z Ṽzz (z, t)Ii> (Ii σσ > Ii> )−1 Ii µ − ν0∗ (z, t)Hβ , (27) In particular, Hθ ≥ 0, that is, the components of the constrained optimal portfolio never have the opposite sign of the corresponding components of the mean-variance efficient portfolio. 23 If an asset is not included in the unconstrained mean-variance efficient portfolio, it would not be included in the constrained portfolio and thus can be ignored. 15 The fact that the components of the constrained optimal portfolio never have the opposite sign of the corresponding components of the mean-variance efficient portfolio implies that the nonlinear portfolio constraint in (14) is equivalent to the pair of linear constraints Hθ(s) ≥ 0 > K(s) ≥ kVaR + β Hθ(s). (28) (29) The characterization of the optimal portfolio strategy in the previous Proposition is then quite intuitive: as long as the non-negativity constraint in (28) is not binding, constrained optimal portfolios are combinations of the portfolio that maximizes expected return for a given variance (the mean-variance efficient portfolio) and the portfolio that minimizes the charge for credit risk β > Hθ for a given variance: we refer to the latter portfolio as the constrained minimum-variance portfolio, since it is also the portfolio that minimizes variance subject to a constraint on the charge for credit risk. More generally, for hi < − Ṽz (z,t)+kVaR ≤ hi+1 the non-negativity constraint in (28) binds for the last m − i assets, z Ṽ (z,t) zz so that ι> j θ(z, t) = 0 for j = m − i, . . . , m and (as shown in equation (27)) θ(z, t) = z Ṽzz (z, t)πiMVE − z Ṽzz (z, t)ν0∗ (z, t)πiCMV , where πiMVE = (Ii σσ > Ii> )−1 Ii µ denotes the mean-variance efficient portfolio of the first i risky assets and πiCMV = (Ii σσ > Ii> )−1 Ii Hβ denotes the constrained minimum-variance portfolio of the first i risky assets. Both h(z, t) = − Ṽz (z,t)+kVaR and K(z, t) = −Ṽz (z, t) are monotonically decreasing funcz Ṽ (z,t) zz tions of z.24 Thus, Proposition 6 shows that when capital K(Zν ∗ (t), t) is large (that is, when Zν ∗ (t) is small and h(Zν ∗ (t), t) ≥ hm+1 ), the capital constraint does not bind (ν0∗ (Zν ∗ (t), t) = 0) and the financial institution holds the mean-variance efficient portfolio MVE . For lower levels of capital (that is, when h(Zν ∗ (t), t) < hm+1 ), the of risky assets πm constraint starts to bind (ν0∗ (Zν ∗ (t), t) becomes positive) and the institution is forced to alter its leverage to satisfy the constraint. At the same time, it finds it optimal to rebalance its portfolio of risky assets: this rebalancing is done by shorting the constrained CMV minimum-variance portfolio πm . For lower levels of capital (higher values of Zν ∗ (t)), shorting of the corrective portfolio progressively increases, until the institution reaches a point where its investment in the mth asset is zero (this happens when h(Zν ∗ (t), t) = hm ). Beyond this point, the institution simply drops the m-th asset from its portfolio, since it is never optimal to short (respectively, long) an asset that is held in positive (respectively, negative) amounts in the unconstrained mean-variance efficient portfolio. If the assets are uncorrelated (σ is diagonal) the m-th > asset is the one with the lowest ratio of absolute risk premium |ι> j µ| to risk weight ιj β. The term ν0∗ can be interpreted as the Lagrangian multiplier on the constraint in the primal problem (14): thus ν0∗ is inversely related to capital K. Since h is a decreasing function of ν0∗ (as shown in equation (26)), h must be an increasing function of K (that is, a decreasing function of z). 24 16 Thus, as the institution is forced to reduce its allocation to risky assets to satisfy the capital constraint, it finds it optimal to tilt its portfolio toward assets with high absolute risk premia and low risk weights.25 If the assets are correlated, then correlations are also taken into account in deciding which asset is dropped first from the portfolio, and the m-th asset is the one with the lowest ratio ηm,j . If capital decreases (Zν ∗ (t) increases) even further and the constraint becomes even more severe, the institution sequentially drops other risky assets from its portfolio, concentrating on those characterized by progressively higher absolute risk premia and lower risk weights. This happens each time that h(Zν ∗ (t), t) exceeds a new value hj . Eventually, if h(Zν ∗ (t), t) = h1 = 0 (that is, if K(Zν ∗ (t), t) = kVaR), the institution is forced to invest its entire portfolio in the riskless asset. In general, whenever h(Zν ∗ (t), t) is between hi and hi+1 , the institution only holds the first i risky assets and its portfolio is a combination of the riskless asset and two funds of risky assets: the mean-variance efficient portfolio of the first i assets, πiMVE and the constrained minimum-variance portfolio of the first i assets, πiCMV . Thus, locally (that is, between any pair hi and hi+1 ), optimal portfolios satisfy three-fund separation. 3.4 The One-Dimensional Case Not surprisingly, the results in the previous two subsections take a very simple form in the case of a single risky asset, as shown in the following Corollary. Corollary 2 In the case of a single risky asset (m = 1) and a positive risk premium (µ > 0) the HJB equation (22) reduces to "  2 2 µ σ 0 = Ṽt − rz Ṽz + z Ṽzz min µ ,− β   2  2 1 µ − z 2 Ṽzz min  2 σ , σ β Ṽz + kVaR z Ṽzz (z, t) Ṽz + kVaR z Ṽzz (z, t) !# !2  . Moreover, the optimal investment strategy in (23) reduces to µ 1 , (K(z, t) − kVaR) σ2 β   1 µ 1 = min K(z, t), (K(z, t) − kVaR) , Γ(z, t) σ 2 β   θ(z, t) = min z Ṽzz (z, t) (30) where Γ(z, t) = − Ṽz (z, t) z Ṽzz (z, t) is the primal value function relative risk aversion coefficient.26 25 This substitution effect is similar to the one described by Kohen and Santomero (1980), Kim and Santomero (1988) and Rochet (1992) in a static setting and by Blum (1999) in a two-period setting. 26 By duality, z = VK (K(z, t), t) = VK (−Ṽz (z, t), t) 17 4 Analysis of Optimal Policies For our numerical analysis, we fix throughout the backtesting period to one year, the investment horizon to two years (T = 2) and the reporting period to one day (n = 250, τ = 1/250) and we assume that the reserve multiplier over the second year is determined according to the schedule proposed by the Basel Committee multiplied by the square root of ten,27 that is, √  if i ≤ 4 3.00√10 = 9.49     3.40 10 = 10.75  if i = 5  √    if i = 6  3.50√10 = 11.07 k(i) = 3.65 10 = 11.54 if i = 7 √    3.75√10 = 11.86 if i = 8      3.85 10 = 12.17 if i = 9  √  4.00 10 = 12.65 if i ≥ 10. We consider the cases in which the number m of risky assets equals 1 or 2.28 In either case, we set r = 0 and choose the vector of risk premia µ and the volatility matrix σ so that the risk premium (respectively, the volatility) of the mean-variance efficient portfolio of risky assets equals 0.059 (respectively, 0.22).29 In the case of two risky assets, we assume in addition that the volatility of the first asset (respectively, the second asset) is 25% higher (respectively, 25% lower) than the volatility of the mean-variance efficient portfolio and that the correlation coefficients between the returns on the two assets is 0.50.30 We solve for the dual value function recursively as explained in the previous section by numerically integrating the PDE (22) using the method of lines.31 We also compute the distribution function P of the state variable Zν ∗ at the end of each reporting period by using the method of lines to solve the PDE    ∂ 1 ∂  P (z, t) = |κν ∗ (z, t)z|2 Pz (z, t) + r + ν0∗ (z, t) zPz (z, t), ∂t 2 ∂z P (z, (h − 1)τ ) = 1{z≥ψ∗ } , (31) and hence (differentiating with respect to z and rearranging) − Ṽz (z, t) K(z, t)VKK (K(z, t), t) =− . VK (K(z, t), t) z Ṽzz (z, t) 27 See footnote 8. As noted at the end of the previous section, optimal investment policies satisfy local three-fund separation, so that considering additional risky assets would not affect the analysis. 29 These values correspond to the mean risk premium and the return standard deviation of the market portfolio as estimated by Ibbotson and  Sinquefeld  (1982).   .07225 .27500 .00000 30 These assumptions imply that µ = and σ = in our simulations with .02937 .08250 .14289 m = 2. 31 The method of lines discretizes the spatial variable z and replaces partial derivatives with finite differences to generate a system of first-order ODE’s in the time variable that can be solved backward starting from the terminal condition. 28 18 for t ∈ [(h−1)τ, hτ ), where ψ ∗ is the value of ψ solving (20) with t = (h−1)τ .32 This allows us to compute the distribution of the financial institution’s capital using equation (24), and hence the true VaR of the portfolio, which we compare to the reported VaR. For comparison purposes, we also compute the optimal policies and the true VaR for the unconstrained problem described in Section 2. 4.1 One Risky Asset Table 1 shows the optimal reporting and investment strategy at the beginning of the first and of the last reporting period in the first year (t = 0 and t = 249/250 = .996, respectively) for three different values of the number violations in the current backtesting period (i = 0, i = 5 and i = 9), two different values of the current reserve multiplier (k = 9.49 and k = 12.65) and three different values of the current leverage ratio LR = D/(D + K),33 for the case γ = .25, ε = 0, β = .08,34 λ = .01 and m = 1. For each combination of (t, i, k, LR), the table shows the reported VaR normalized by the total asset value v = VaR/(D + K), the coefficient of relative risk aversion of the primal value function Γ = −KVKK /VK , the maximum possible proportional allocation to risky assets under the capital requirent constraint π̄ = (1 − LR − kv)/β,35 the proportional allocation to risky assets π = θ/(D + K) and the true 1-day 90% and 99% VaRs normalized by the total asset value v.90 and v.99 . For comparison purposes, the table also shows the relative risk aversion coefficient ΓU , the proportional allocation to risky assets π U and the true normalized 1-day 90% and 99% VaRs U and v U in the unconstrained case. Table 2 shows the same information for a higher v.90 .99 value of the institution’s risk aversion coefficient (γ = .50). For the set of parameters considered in Table 1, the proportional allocation to risky assets in the absence of capital requirements varies between 482.7% and 820.2%, increasing when the leverage ratio increases (asset value decreases) and the option to default becomes more valuable (as shown in Figure 1). Capital requirements are effective in curbing the risk of trading portfolios, reducing the range of the proportional allocation to risky assets to between 18.2% and 412.2%. Not surprisingly, this risk reduction is larger when capital requirements are more stringent (that is, when the reserve multiplier k is larger). Moreover, the allocation to risky assets becomes inversely related to leverage ratios, since as capital falls and leverage ratios increase, financial institutions are forced to liquidate their holdings of risky assets to avoid violating capital requirements. As a result of this dynamic behavior, the range of true 99% daily VaRs falls from between 14.60% and 24.15% in the unconstrained 32 The PDE in (31) is obtained by integrating the forward Kolmogorov equation
∂ ∂ 1 ∂2 p(z, t) = |κν ∗ (z, t)z|2 p(z, t) + 2 ∂t 2 ∂z ∂z  (r + ν0∗ (z, t))zp(z, t)  solved by the density function p(z, t) = Pz (z, t). 33 The exact values of the three leverage ratios used for all the tables in the paper are 1/(1 + e3 ) = .0474, 1/(1 + e0 ) = .5000 and 1/(1 + e−3 ) = .9526 before rounding. 34 See footnote 7. 35 The constraint in (14) implies θ K − kVaR 1 − LR − kv ≤ = . D+K β(D + K) β 19 Parameter values: γ = .25, ε = 0, β = .08, λ = .01, τ = .004, m = 1 t i k LR v Γ π̄ π v.90 v.99 ΓU πU U v.90 U v.99 0 0 0 0 0 0 9.49 9.49 9.49 .05 .50 .95 .0663 .0348 .0033 .2862 .2862 .2862 4.243 4.243 4.243 4.041 2.121 0.201 .0663 .0348 .0033 .0663 .0348 .0033 .2357 .0991 .0086 4.927 6.148 6.698 .0853 .1062 .1155 .1488 .1842 .1994 0 0 0 0 0 0 12.65 12.65 12.65 .05 .50 .95 .0515 .0270 .0026 .3133 .3133 .3133 3.952 3.952 3.952 3.707 1.945 0.185 .0515 .0270 .0026 .0515 .0270 .0026 .2357 .0991 .0086 4.927 6.148 6.698 .0853 .1062 .1155 .1488 .1842 .1994 0 0 0 5 5 5 9.49 9.49 9.49 .05 .50 .95 .0667 .0350 .0033 .2920 .2921 .2921 4.197 4.197 4.197 3.976 2.087 0.198 .0667 .0350 .0033 .0667 .0350 .0033 .2357 .0991 .0086 4.927 6.148 6.698 .0853 .1062 .1155 .1488 .1842 .1994 0 0 0 5 5 5 12.65 12.65 12.65 .05 .50 .95 .0505 .0265 .0025 .2914 .2914 .2914 4.126 4.126 4.126 3.930 2.063 0.196 .0505 .0265 .0025 .0977 .0516 .0049 .2357 .0991 .0086 4.927 6.148 6.698 .0853 .1062 .1155 .1488 .1842 .1994 0 0 0 10 10 10 9.49 9.49 9.49 .05 .50 .95 .0656 .0345 .0033 .2781 .2781 .2781 4.327 4.327 4.327 4.122 2.164 0.205 .0656 .0345 .0033 .1012 .0534 .0050 .2357 .0991 .0086 4.927 6.148 6.698 .0853 .1062 .1155 .1488 .1842 .1994 0 0 0 10 10 10 12.65 12.65 12.65 .05 .50 .95 .0504 .0265 .0025 .2913 .2914 .2914 4.127 4.127 4.127 3.931 2.063 0.196 .0504 .0265 .0025 .0976 .0513 .0049 .2357 .0991 .0086 4.927 6.148 6.698 .0853 .1062 .1155 .1488 .1842 .1994 .996 .996 .996 0 0 0 9.49 9.49 9.49 .05 .50 .95 .0656 .0345 .0033 .2781 .2781 .2781 4.327 4.327 4.327 4.122 2.164 0.205 .0656 .0345 .0033 .1016 .0533 .0050 .2405 .0850 .0070 4.827 7.169 8.202 .0836 .1237 .1410 .1460 .2139 .2415 .996 .996 .996 0 0 0 12.65 12.65 12.65 .05 .50 .95 .0504 .0265 .0025 .2913 .2914 .2914 4.127 4.127 4.127 3.931 2.063 0.196 .0504 .0265 .0025 .0860 .0452 .0043 .2405 .0850 .0070 4.827 7.169 8.202 .0836 .1237 .1410 .1460 .2139 .2415 .996 .996 .996 5 5 5 9.49 9.49 9.49 .05 .50 .95 .0667 .0350 .0033 .2915 .2915 .2915 4.202 4.202 4.202 3.984 2.091 0.198 .0667 .0350 .0033 .0667 .0350 .0033 .2405 .0850 .0070 4.827 7.169 8.202 .0836 .1237 .1410 .1460 .2139 .2415 .996 .996 .996 5 5 5 12.65 12.65 12.65 .05 .50 .95 .0517 .0271 .0026 .3171 .3172 .3172 3.923 3.923 3.923 3.662 1.922 0.182 .0517 .0271 .0026 .0517 .0271 .0026 .2405 .0850 .0070 4.827 7.169 8.202 .0836 .1237 .1410 .1460 .2139 .2415 .996 .996 .996 10 10 10 9.49 9.49 9.49 .05 .50 .95 .0656 .0345 .0033 .2781 .2781 .2781 4.327 4.327 4.327 4.122 2.164 0.205 .0656 .0345 .0033 .1129 .0592 .0056 .2405 .0850 .0070 4.827 7.169 8.202 .0836 .1237 .1410 .1460 .2139 .2415 .996 .996 .996 10 10 10 12.65 12.65 12.65 .05 .50 .95 .0504 .0265 .0025 .2913 .2913 .2914 4.127 4.127 4.127 3.931 2.063 0.196 .0504 .0265 .0025 .0976 .0512 .0049 .2405 .0850 .0070 4.827 7.169 8.202 .0836 .1237 .1410 .1460 .2139 .2415 Table 1: The table shows the values of the reported normalized daily VaR (v), the RRA coefficient of the dual value function (Γ), the upper bound on the portfolio allocation to the risky asset (π̄), the portfolio allocation to the risky asset (π), the true normalized 90% VaR (v.90 ) and the true normalized 99% VaR (v.99 ) in the presence of capital requirements, as well as the RRA coefficient (ΓU ), the portfolio allocation U U to the risky asset (π U ) and the true normalized 90% and 99% VaRs (v.90 and v.99 ) in the unconstrained case. 20 Parameter values: γ = .50, ε = 0, β = .08, λ = .01, τ = .004, m = 1 t i k LR v Γ π̄ π v.90 v.99 ΓU πU U v.90 U v.99 0 0 0 0 0 0 9.49 9.49 9.49 .05 .50 .95 .0777 .0408 .0039 .5025 .5024 .5024 2.830 2.830 2.830 2.311 1.213 0.115 .0403 .0213 .0020 .0706 .0370 .0035 .5000 .2078 .0149 2.323 2.933 3.892 .0405 .0513 .0679 .0723 .0923 .1212 0 0 0 0 0 0 12.65 12.65 12.65 .05 .50 .95 .0596 .0313 .0030 .5113 .5113 .5113 2.601 2.601 2.601 2.271 1.192 0.113 .0396 .0209 .0020 .0596 .0313 .0030 .5000 .2078 .0149 2.323 2.933 3.892 .0405 .0513 .0679 .0723 .0923 .1212 0 0 0 5 5 5 9.49 9.49 9.49 .05 .50 .95 .0777 .0408 .0039 .5027 .5026 .5026 2.826 2.826 2.826 2.310 1.213 0.115 .0404 .0213 .0020 .0707 .0370 .0035 .5000 .2078 .0149 2.323 2.933 3.892 .0405 .0513 .0679 .0723 .0923 .1212 0 0 0 5 5 5 12.65 12.65 12.65 .05 .50 .95 .0597 .0313 .0030 .5125 .5124 .5124 2.595 2.595 2.595 2.266 1.189 0.113 .0396 .0209 .0020 .0597 .0313 .0030 .5000 .2078 .0149 2.323 2.933 3.892 .0405 .0513 .0679 .0723 .0923 .1212 0 0 0 10 10 10 9.49 9.49 9.49 .05 .50 .95 .0777 .0408 .0039 .5025 .5024 .5024 2.830 2.830 2.830 2.311 1.213 0.115 .0404 .0213 .0020 .0706 .0370 .0035 .5000 .2078 .0149 2.323 2.933 3.892 .0405 .0513 .0679 .0723 .0923 .1212 0 0 0 10 10 10 12.65 12.65 12.65 .05 .50 .95 .0596 .0313 .0030 .5113 .5112 .5112 2.601 2.601 2.601 2.271 1.192 0.113 .0396 .0209 .0020 .0596 .0313 .0030 .5000 .2078 .0149 2.323 2.933 3.892 .0405 .0513 .0679 .0723 .0923 .1212 .996 .996 .996 0 0 0 9.49 9.49 9.49 .05 .50 .95 .0777 .0408 .0039 .5025 .5024 .5024 2.830 2.830 2.830 2.311 1.213 0.115 .0405 .0213 .0020 .0706 .0372 .0035 .5000 .1922 .0119 2.322 3.172 4.845 .0405 .0557 .0846 .0723 .1010 .1509 .996 .996 .996 0 0 0 12.65 12.65 12.65 .05 .50 .95 .0596 .0313 .0030 .5113 .5113 .5113 2.601 2.601 2.601 2.271 1.192 0.113 .0393 .0207 .0020 .0596 .0313 .0030 .5000 .1922 .0119 2.322 3.172 4.845 .0405 .0557 .0846 .0723 .1010 .1509 .996 .996 .996 5 5 5 9.49 9.49 9.49 .05 .50 .95 .0777 .0408 .0039 .5027 .5026 .5026 2.826 2.826 2.826 2.310 1.213 0.115 .0407 .0213 .0020 .0708 .0372 .0035 .5000 .1922 .0119 2.322 3.172 4.845 .0405 .0557 .0846 .0723 .1010 .1509 .996 .996 .996 5 5 5 12.65 12.65 12.65 .05 .50 .95 .0597 .0313 .0030 .5123 .5123 .5123 2.595 2.595 2.595 2.267 1.190 0.113 .0394 .0208 .0020 .0597 .0313 .0030 .5000 .1922 .0119 2.322 3.172 4.845 .0405 .0557 .0846 .0723 .1010 .1509 .996 .996 .996 10 10 10 9.49 9.49 9.49 .05 .50 .95 .0777 .0408 .0039 .5025 .5024 .5024 2.830 2.830 2.830 2.311 1.213 0.115 .0407 .0215 .0020 .0709 .0374 .0035 .5000 .1922 .0119 2.322 3.172 4.845 .0405 .0557 .0846 .0723 .1010 .1509 .996 .996 .996 10 10 10 12.65 12.65 12.65 .05 .50 .95 .0596 .0313 .0030 .5113 .5113 .5113 2.601 2.601 2.601 2.271 1.192 0.113 .0398 .0209 .0020 .0596 .0313 .0030 .5000 .1922 .0119 2.322 3.172 4.845 .0405 .0557 .0846 .0723 .1010 .1509 Table 2: The