Optimal portfolios with minimum capital requirements

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Abstract

We propose a novel approach to active risk management based on the recent Basel II regulations to obtain optimal portfolios with minimum capital requirements. In order to avoid regulatory penalties due to an excessive number of Value-at-Risk (VaR) violations, capital requirements are minimized subject to a given number of violations over the previous trading year. Capital requirements are based on the recent Basel II amendments to account for the ‘stressed’ VaR, that is, the downside risk of the portfolio under extreme adverse market conditions. An empirical application for two portfolios involving different types of assets and alternative stress scenarios demonstrates that the proposed approach delivers an improved balance between capital requirement levels and the number of VaR exceedances. Furthermore, the risk-adjusted performance of the proposed approach is superior to that of minimum-VaR and minimum-stressed VaR portfolios.

Highlights

► We propose a portfolio selection approach to optimize capital requirements. ► The policy finds the optimal balance between VaR measures and VaR violations. ► We confirm empirically that the proposed approach outperform competing ones.

Introduction

The Basel II framework (Bank for International Settlements, 2006) requires banks to set aside a minimum amount of regulatory capital to cover potential losses arising from their exposure to market risk, credit risk, and operational risk. Market risk is the risk of losses on positions in equities, interest rate related instruments, currencies and commodities due to adverse movements in market prices. The capital requirement (CR) for market risk is based upon estimates of the Value-at-Risk (VaR), defined as the maximum loss on the bank’s positions in these assets that could occur over a given holding period with a specified confidence level. Recent changes in the Basel II regulations establish an additional CR based upon a stressed VaR (sVaR), which reflects the risk on the bank’s current portfolio if the relevant market factors were experiencing a period of stress; see Bank for International Settlements (2009).

Basel II allows banks to use ‘internal’ models to measure their VaR and sVaR, as an alternative to the standardized approach described in the accord (Hendricks and Hirtle, 1997). This standardized approach is known to render conservative VaR estimates, leading to excessively high CR. From the banks’ perspective this is undesirable given that regulatory capital involves an opportunity cost as it cannot be used for other, profitable purposes. Hence, it is attractive for banks to attempt to lower their capital charges using their own risk management system. The empirical evidence presented by Pérignon et al. (2008) suggests that the use of internal models indeed is widespread among large financial institutions.

Although internal risk measurement systems are subject to supervisory approval based on qualitative and quantitative standards, banks enjoy a large degree of freedom in devising the precise nature of their models. This flexibility does not, however, imply that banks are tempted to pursue the lowest possible VaR estimates. This is due to the fact that the relation between the VaR estimates and capital requirements is non-monotonic, as it takes into account not only the magnitude of the VaR but also the number of VaR violations (i.e. actual losses exceeding the VaR) in the recent past. Specifically, the regulatory capital required to be held on day t + 1 is determined as the maximum of the current VaR estimate and the average VaR over the preceding 60 business days multiplied by a scaling factor, that is,CRt+1=maxVaR$t(h,α),(3+k)×VaR$t,60(h,α)¯,where VaR$t(h, α) is the estimate at day t of the VaR for a holding period of h days at confidence level α  (0, 1) and VaR$t,60(h,α)¯=160j=059VaR$t-j(h,α). Note that these VaR estimates are expressed in dollar terms, representing the loss that might be incurred on the current portfolio; that is, VaR$t(h,α)=Vt(1-eVaRt(h,α)) with Vt being the current portfolio value and VaRt(h, α) the VaR in terms of returns. Usually it is the latter VaR that is first obtained from a model for the portfolio return distribution, and we follow this practice here. The Basel II accord requires the use of VaR estimates for a holding period h of 10 days at confidence level α of 1%. Moreover, the accord allows the 10-day VaR estimates to be computed from VaR estimates for shorter periods by using the square-root-of-time-rule, that is VaRt(10,α)=10/hVaRt(h,α) for some h < 10, see Bank for International Settlements (2006, paragraph 718(Lxxvi)).1 The penalty or “plus” k in the multiplication factor in (1) ranges between 0 and 1. Its exact value is determined by the number of VaR exceedances during the last 250 business days, as shown in Table 1.

During the financial crisis of 2007/2008, losses in most banks’ trading books have been substantially larger than the VaR-based minimum CR determined according to (1). In response, the Bank of International Settlements (BIS) released a set of modifications to the existing regulatory framework regarding market risk; see Bank for International Settlements (2009). Among the main adjustments are the introduction of the sVaR and a corresponding new CR formula that leads to higher CR levels. The new CR formula isCRt+1=maxVaR$t(h,α),(3+k)×VaR$t,60(h,α)¯+maxsVaR$t(h,α),(3+k)×sVaR$t,60(h,α)¯,where sVaR$t(h, α) is the estimate at day t of the sVaR for a holding period of h days at confidence level α  (0, 1) and sVaR$t,60(h,α)¯=160j=059sVaR$t-j(h,α). The new regulations state that the backtesting results applicable for calculating the penalty parameter k are based upon estimates of the VaR only and not on the sVaR. Finally, no particular methodology is prescribed for computing the sVaR, except that it should reflect the VaR of the bank’s current portfolio under extreme adverse market conditions. We discuss in Sections 2.1 VaR and sVaR estimation, 3.4 Stress scenarios alternative approaches to obtain the sVaR.

The expressions for the CR in (1), (2) seemingly suggest that lower capital charges could be achieved by lower VaR (and sVaR) estimates. This, however, need not be the case as lower VaR estimates are possibly violated more often, thus increasing the regulatory capital through the effects of the penalty factor k. Apart from direct costs due to the larger amount of capital that needs to be put aside, this may also bring indirect costs by damaging the bank’s reputation. Both types of costs become particularly severe when the red zone is entered, that is, when ten or more VaR violations occur during a period of 250 business days. In that case, the bank may be forced to adopt the Basel accord’s standardized approach for VaR estimation. As noted before, this approach is known to render conservative VaR estimates, leading to excessively high CR. In addition, the ban of the bank’s internal models obviously has detrimental effects on its reputation.

In practice, banks appear to be wary of being overly optimistic about their level of market risk during tranquil periods. In fact, empirical evidence presented by Berkowitz and O’Brien, 2002, Pérignon et al., 2008, PTrignon and Smith, 2010 suggests that they systematically overestimate their VaR. For instance, Berkowitz and O’Brien (2002) document that the number of violations of VaR estimates of six large US banks is usually lower than expected. Similarly, Pérignon et al. (2008) report that for VaR estimates at the 1% level of the six largest Canadian banks there are only two violations during the 7354 trading days analyzed, whereas the expected number is 74. The opposite situation seems to occur in times of stressed market conditions. During the 2007/2008 financial crisis, banks systematically underestimated their VaR and their level of market risk; see Bank for International Settlements (2009). This alternation of over- and underestimation of market risk levels may, at least to some extent, be due to the fact that VaR measures typically are calibrated using historical data. Following a period of calm in financial markets, the VaR estimates and the accompanying CR can decline to low levels, but then they might underestimate risk during a period of stress that lies ahead. In fact, one of the main motivations for the introduction of the sVaR in the amendments to the Basel II accord is to reduce the procyclicality of the minimum CR. In addition, European regulatory institutions performed a number of stress tests to assess the resilience of the banking system to absorb shocks on credit, market and sovereign risks and introduced further modifications in the regulatory framework; see Committee of European Banking Supervisors (2010).

The exaggeration of banks’ own level of risk during normal times implies an excessive amount of regulatory capital, directly affecting the profitability of the bank. Another, at least as undesirable consequence is that such banks appear more risky than they actually are, thus generating reputational concerns about their risk management systems. Similarly, the underestimation of banks’ own level of risk during times of stressed market conditions may lead to insufficient amount of regulatory capital to cover potential losses in the trading book, thus increasing the risk of bankruptcy. This affects investors’ perception and can induce underinvestment in VaR-overstating and VaR-understating banks. Indeed, Jorion (2002) shows that VaR disclosures are informative about the future variability in trading revenues, thus corroborating the idea that analysts/investors may be using the VaR measures to support investment decisions.

In this paper we put forward a novel portfolio construction methodology to overcome the drawbacks of both over- and understatement of a bank’s VaR. Specifically, we propose to determine optimal portfolio weights by directly minimizing the CR subject to a restriction on the number of VaR violations during the preceding year (250 business days). Implicitly, our approach aims to find the optimal balance between the level of VaR measures and the number of VaR violations, thus leading to the lowest possible level of CR.

Although minimizing CR is an important criterion to take into account, in real world situations portfolio managers and investors traditionally decide upon their asset allocations by considering standard performance measures, such as expected returns or Sharpe ratios. In addition, portfolio weights often are restricted in order to avoid shortselling or to limit the exposure to individual assets. For this reason, we consider a general formulation of the portfolio construction problem in which the optimal portfolio composition is found by minimizing the level of CR subject also to a given (i.e. user specified) target performance and to direct constraints on the portfolio weights.

We apply the proposed methodology to two different asset portfolios: (i) a mixed portfolio composed of 30 futures on a variety of assets including equities, bonds, commodities and currencies, and (ii) an equity portfolio comprising 48 US industry indices. The minimum capital requirement (MCR) portfolio is compared to various benchmark portfolios, including the minimum-VaR portfolio (Alexander and Baptista, 2002), the minimum-sVaR portfolio, and the equally weighted (1/N) portfolio. In our empirical analysis we pay particular attention to the consequences of the introduction of the sVaR-based CR. For this purpose, in addition to ‘normal’ market conditions we consider several alternative, realistic scenarios in which expected returns, volatilities and cross-correlations are modified to reflect a stressed environment. Furthermore, we consider different models for obtaining forecasts of expected returns, volatilities and correlations, which are crucial inputs for the asset allocation decisions. We also examine the robustness of our results to the specific restrictions imposed on the portfolio’s target rate of return and the re-balancing frequency.

The results for the futures portfolio indicate that our approach delivers lower CR levels in comparison to the benchmark portfolios. For the portfolios of sector indices, the novel portfolio construction approach delivers a better balance between CR levels and the number of VaR violations, as it yields a lower average number of VaR exceedances. For both data sets, we find that the number of VaR violations under the MCR portfolio policy does not enter the red zone in any of the (normal and) stress scenarios regardless of the specifications used for forecasting volatilities and correlations. This is in sharp contrast to the benchmark portfolios, for which we frequently find more than ten VaR violations. Finally, the performance of the MCR portfolios in terms of risk-adjusted returns and portfolio turnover is generally superior to the minimum-VaR and minimum-sVaR portfolios.

Our proposed methodology differs in important ways from previous, related research. First, in order to achieve the goal of lower CR, one possibility is to develop a VaR model that delivers lower levels of capital charges, as proposed recently by McAleer et al. (2010), for instance. Using the terminology of Christoffersen (2009), this approach can be considered a risk measurement or passive risk management approach, since it is applied to a given (i.e. predetermined) portfolio composition. Alternatively, in this paper, we propose to perform active risk management by deciding on the portfolio allocations themselves to attain lower levels of CR.

Second, portfolios with low levels of CR may be obtained by imposing constraints on the amount of CR or on the portfolio VaR, as in Sentana, 2003, Cuoco and Liu, 2006, Alexander et al., 2007. In our approach, the level of CR plays a much more central role as it is taken to be the objective function that should be minimized.

The remainder of the paper is organized as follows. In Section 2 we describe the procedure to obtain optimal portfolios with minimum CR subject to restrictions on the number of VaR violations. In Section 3 we present the empirical applications. We conclude in Section 4.

Section snippets

Portfolios with minimum capital requirements

The main ingredient required to obtain optimal portfolios with minimum capital requirements is a measure of the VaR and of the sVaR. Therefore, in Section 2.1, we first describe the procedure for obtaining VaR and sVaR estimates considered in this paper. In Section 2.2, we then develop the optimization problem that leads to the construction of MCR portfolios.

Empirical application

We evaluate the performance of the MCR strategy for two portfolios with different types of assets. In the optimization problem in (14), we adopt a daily target portfolio return of Ξ = 4 bp, corresponding to an annual target return of 10%. Furthermore, we focus on the case in which only long positions are allowed by imposing a no-shortselling restriction, i.e. w  0.

Concluding remarks

Banks and other large financial institutions tend to overestimate the VaR of their asset portfolios during tranquil times and to underestimate their risk in times of stressed market conditions, as documented in previous empirical studies. The VaR overestimation during quiet periods results in prohibitive amounts of regulatory capital requirements, thus generating opportunity costs and giving rise to reputational concerns. On the other hand, the underestimation of risk levels is unattractive as

Acknowledgements

We thank an anonymous referee for detailed comments and suggestions, which have led to substantial improvements in the paper. We are also grateful to Kees Bouwman, Newton da Costa Jr., Victor DeMiguel, Andreas Heinen, Erik Kole, Marcio Laurini, Michael McAleer, Roberto Meurer, Alfonso Novales, Pilar Poncela, Antonio Sanvicente, Timo Teräsvirta, Helena Veiga, Pedro Valls, Bernardo da Veiga, and seminar participants at Universidad Carlos III de Madrid, Econometric Institute – Erasmus University

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