Executive Summary: A CVaR Scenario-based Framework For Minimizing Downside Risk In Multi-Asset Class Portfolios Axioma, Inc. by Kartik Sivaramakrishnan, PhD, and Robert Stamicar, PhD August 2016 In this paper, which is an executive summary of an Axioma technical report [1], we show how to minimize downside risk in multi-asset class (MAC) portfolios. By comparing the scenario-based Conditional Value at Risk (CVaR) approach with parametric Mean-Variance Optimization (MVO) approaches that linearize all the instruments in the MAC portfolio, we show that (a) the CVaR approach generates MAC portfolios with better downside risk statistics, and that (b) the CVaR hedges return more attractive risk decompositions and stress-test numbers tools commonly used by risk managers to evaluate the quality of hedges. MAC portfolios comprise investments in equities, fixed-income, commodities, foreign-exchange, credit, derivatives, and alternatives, such as real-estate and private equity. Such portfolios often have substantial nonlinear exposure to risk factors, particularly in the presence of derivatives or over longer time horizons. The return for such nonlinear portfolios is asymmetric with significant tail risk. The traditional Markowitz MVO framework, which linearizes all the assets in the portfolio and uses the standard deviation of return as a measure of risk, does not accurately measure risk for such portfolios. To mitigate this problem, we use a CVaR approach. We seek to minimize downside risk in a MAC portfolio by adding appropriate overlays. The CVaR approach uses: (a) (b) Monte Carlo simulations to generate asset return scenarios, and, Scenario-based convex optimization to generate overlay holdings. The calculation returns an optimal portfolio of holdings in the overlay portfolio that minimizes the expected losses beyond a specified probability threshold. The authors would like to thank Diana Rudean, PhD, for her contribution to this paper.
1. Introduction Multi-asset class (MAC) portfolios are an integral component of asset-manager, asset-owner and hedge-fund investment, and comprise a broad set of assets that include equities, fixed income, commodities, foreign exchange, credit, and alternatives, such as real estate and private equity (Figure 1). MAC instruments provide a more diversified set of asset allocation opportunities that are based on a wide spectrum of risk and return profiles. For example, adding investment grade bonds to an equity portfolio can improve its risk profile. Figure 1: Multi-asset class (MAC) portfolio composition With a broad set of assets, portfolio construction is challenging in a MAC setting. In Figure 2 below, on the left side, we schematically represent the investment process. This figure is by no means exhaustive but serves to illustrate the complexity of portfolio construction for MAC portfolios. We do not consider portfolio construction in the traditional sense where we adjust all portfolio holdings so as to optimize an objective function in this paper. Instead, we consider the risk management problem of mitigating tail risk from a hedging perspective. Tail risk is a rare event or outcome with a small probability of occurring. The tails are the end portions of a distribution. In this study we concentrate on the left tail of the portfolio return distribution, which reflects the probabilities of worst-case scenarios with severe losses. More specifically, we present a methodology for mitigating downside (left tail) risk of MAC portfolios via multi-asset class overlays. There has been considerable interest in downside risk protection, especially in the wake of the 2008 financial crisis. Our starting point is a MAC portfolio with nonlinear assets that has been constructed, say, from an investment process in a traditional manner or using a sophisticated optimization algorithm. The downside risk of the existing portfolio is then minimized with the addition of MAC hedging overlays over a specified time horizon. A CVaR Scenario-based Framework page 2
Figure 2: Moving parts in MAC portfolio construction The Markowitz parametric mean-variance optimization (MVO) framework (see Markowitz [5]), which uses the standard deviation of returns as a risk measure and is widely applied in equity portfolio management, is not suitable for MAC portfolios. In particular, positions with nonlinear payoffs introduce asymmetry between positive and negative returns. MVO is not suitable because linear models do not capture asymmetric returns. Moreover, parametric higher moment optimization, which also incorporates the skew and the kurtosis of the returns, is fairly limited in the size of problems that it can handle. To illustrate how standard deviation understates the tail risk in a MAC portfolio, consider Figure 3 that shows the cumulative return distributions for the S&P 500 and a covered call on the S&P 500 (long 100 shares in index and short a call contract on index). Note that the cumulative distribution for the S&P 500 is normal and symmetric about the origin. This implies that a positive return for the index of between 6% and 10% is equally likely as a negative return between -10% and -6%. The upside of the covered call portfolio, however, is capped at 5% by the strike of the call. This is because the counterparty will exercise the call option once the index price exceeds the strike price. As we see in Figure 3, the covered call has a zero probability of a positive return between 6% and 10%, while it has a nonzero probability of a negative return between -10% and -6%. In particular, the covered call has a nonlinear asymmetric return distribution with a long left tail. In this study we present a scenario-based hedging framework to mitigate the tail risk for MAC portfolios that contain nonlinear instruments with asymmetric payoffs. The framework consists of two phases. In the first phase, we generate Monte Carlo simulations for asset and hedging position returns. These scenarios are generated via appropriate pricing engines for the different instruments. The pricing engine captures the nonlinearity of the instrument returns. For example, we use the Black-Scholes pricing engine to price an equity index option that reflects the nonlinear relationship between the index option returns and risk factors, such as the underlying and the implied volatility returns that drive the option price. In the second phase, the simulations from the first phase are fed into a scenario-based convex optimization problem where tail risk is minimized by selecting overlay A CVaR Scenario-based Framework page 3
hedges, which are constrained by a budget. The convex optimization can conceivably be extended to include other objectives and constraints that model manager preferences and institutional mandates. Figure 3: Cumulative return distribution for a covered call portfolio and underlying stock index It is important to highlight two key points. First, the simulation-based approach is only as good as the pricing engines and the risk factors that are used to generate instrument return scenarios. Second, we acknowledge that the scenario-based approach, however useful, should not be a standalone analysis. It is a complement to existing tools in the arsenal of the risk manager, such as stress testing, risk decomposition, delta-hedging, what-if analysis, etc. 2. The CVaR Downside Risk Measure Downside risk measures are one-sided risk measures meant to identify potential negative outcomes. They better reflect the left-sided tail risk in MAC portfolios. The most popular downside risk measures include Value at Risk (VaR) and Conditional Value at Risk (CVaR), also known as Expected Shortfall. VaR at confidence level ε is the (1 ε) percentile of the portfolio return distribution. VaR has a simple interpretation: A portfolio VaR at the 95% confidence level over a 10 day period of $10 million implies that we are 95% confident that the portfolio will not suffer losses greater than $10 million over a 10 day period. CVaR at confidence level ε is the expected value of the loss exceeding VaR. These concepts are graphically illustrated in Figure 4, where (a) VaR at the 95% confidence level is the 5% percentile of the portfolio return distribution, and (b) CVaR at the 95% confidence level is the VaR at the 95% confidence level plus the area of the shaded region divided by the length of the axis under the shaded region (excess loss exceeding VaR). A CVaR Scenario-based Framework page 4
CVaR was introduced to overcome some of the shortcomings of VaR. First, CVaR is a coherent risk measure encouraging diversification (see Artzner et al. [2]). Second, it is a tail statistic that measures the length of the left tail of the portfolio return distribution. Third, it is easy to optimize. Optimizing CVaR can be done by solving a scenario-based linear optimization problem (see Rockafellar and Uryasev [6]). CVaR will replace VaR as the risk standard in early 2018, and it will be used for all risk and capital calculations under the Basel Committee s FRTB (fundamental review of the trading book; see FTRB link [7]). Figure 4: VaR and CVaR Graphical Representation 3. CVaR Scenario-Based Framework and Results Our approach involves a scenario-based CVaR framework for minimizing the downside risk of an existing MAC portfolio. In the first step, we generate return scenarios for the different MAC instruments in the portfolio using Monte Carlo simulations (see Glasserman [3]). In the second step, we employ a scenario-based convex optimization model that takes the Monte Carlo scenarios as inputs and then generates the overlay hedges. We refer the interested reader to the extended version of this paper for more details on the two phases of the scenario-based framework. We must emphasize that there is a good deal of flexibility in the choice of risk factors and the pricing engines in Phase 1 (Monte Carlo scenario generation) of the algorithm. We analyze several MAC case studies in the extended version of the paper (see Sivaramakrishnan and Stamicar [1]) where we keep the holdings of the original or base portfolio fixed and adjust the weights of hedging/overlay positions to minimize the CVaR at the 95% confidence level. We discuss the second example from the paper below. In this test we want to hedge a callable bond portfolio by purchasing interest rate caps with different strikes and times to expiration, with 1% to 5% budgets (current market value of the base portfolio) in increments of 1%. Most of the risk in the callable bond portfolio is interest rate (IR) risk. We want to hedge this portfolio against an increase in interest rates with over-the-counter (OTC) interest rate caps. Note that both the base portfolio and the hedging overlays are nonlinear instruments. Callable bonds have fat left tails since their upside is capped when interest rates decrease. This is because the issuer of the callable bond may call, i.e., redeem the bond A CVaR Scenario-based Framework page 5
when its value increase beyond the call price. We consider a fixed-income portfolio with 48 US callable bonds on May 2, 2006. We chose this analysis date since the Fed had increased the key rate 15 straight times until then, and there was a high probability that the Fed would raise rates in the near future. Our strategy is as follows: 1. Hedge callable bond portfolio with OTC interest rate caps over a six-month horizon. There are 10 OTC interest rate caps with varying strikes and times to expiration: (a) Floating rate is based on the six-month LIBOR rate. The prevailing six-month LIBOR rate on May 2, 2006 is 5.34%. (b) Strikes are chosen to the prevailing six-month LIBOR rate. (c) Short-term caps expire on June 30, 2008 and long-term caps expire on June 30, 2010. (d) Each caplet tenor is six months. 2. Maintain the callable bond holdings in the portfolio. 3. Experiment with budgets of 1% and 5% of the total market value of the callable bond portfolio. 4. Compare the CVaR and MVO hedges over a six-month hedging horizon. Here is a brief description of the Monte Carlo pricing engine used to generate the asset scenarios: 1. The single factor Hull-White engine (see Hull [4]) is used to price the callable bonds, where the pricing factors include the US sovereign, swap, issuer credit, and swaption volatility factors. 2. Black s formula (see Hull [4]) is used to price the caps as a sequence of caplets, where the pricing factors include the US sovereign, swap, issuer credit, and the cap volatility surface factors. The MVO approach linearizes both the callable bonds (base portfolio) and the IR caps (hedging overlays) to arrive at a covariance matrix for the portfolio that is minimized in a parametric risk term. Figure 5 shows the smoothed kernel density plots of 20,000 realizations of the CVaR and the MVO hedged portfolio returns at the end of the six-month hedging horizon, when the option budget is 1% and 5% of the reference size of the callable bond portfolio, respectively. Notice that the unhedged callable bond portfolio has a fat left tail since the upside of this portfolio is capped when the bonds are redeemed by the issuer. Importantly, both the CVaR and the MVO approaches reduce the tail risk but the CVaR approach offers better downside protection. A CVaR Scenario-based Framework page 6
Figure 5: Density plots of portfolio returns at end of rebalancing period Table 1 presents the downside risk statistics for the different portfolios. Notice that the CVaR portfolio has superior downside risk statistics, even for the small 1% budget. In particular, the worst-case return, CVaR and VaR of the left tail of the return distribution, and standard deviation are much smaller than that of the MVO portfolio. The MVO portfolio does not use its entire budget of 1% and so the results are unchanged when one increases the budget to 5%. On the other hand, the downside risk statistics for the CVaR portfolio get better when the budget is 5%. Statistic CVaR 1% Scenario-based CVaR 5% Scenario-based MVO Unhedged CVaR 3.58% 2.45% 4.41% 5.08% VaR 2.86% 1.93% 3.49% 3.85% Worst Loss -7.30% -4.92% -8.10% -9.13% StdDev 1.71% 1.18% 1.99% 2.25% Right CVaR 3.48% 2.40% 3.85% 4.27% Table 1: Downside risk statistics Figure 6 plots the profit and losses (P/Ls) for the CVaR, MVO, and unhedged portfolios, when the portfolios are subject to instantaneous interest rate shocks ranging from -1% to 1%. An interest rate shock of 1% implies that we move the entire yield curve up by 1%. When a shock of 1% is applied, the base portfolio loses 6%, the MVO portfolio loses 4%, the CVaR portfolio with 1% budget loses 2%, and the CVaR portfolio with 5% budget loses nothing. On the other hand, when the entire yield curve moves down by 1%, the base portfolio gains 5%, the MVO and the CVaR portfolios with 1% budget gain 4%, and the CVaR portfolio with 5% budget gains 2%. Notice that the CVaR portfolio with 1% budget (shown by the dashed blue line in Figure 6) has a smaller downside than the MVO portfolio (shown by the solid red line in Figure 6) while both portfolios have the same upside. A CVaR Scenario-based Framework page 7
Figure 6: Interest Rate Stress Tests In conclusion, when comparing the CVaR hedging approach with the MVO approach, which linearizes both the callable bonds (the underlying instruments) and the IR caps (hedging overlays), we find that CVaR hedge returns have the best downside risk statistics: smallest CVaR, Worst-Case Loss, VaR, and Standard Deviation. The CVaR hedge also returns the best statistics when all the portfolios are subjected to instantaneous interest rate stress tests. We refer the reader to the extended version of this paper for two other MAC hedging use-cases. In general, we expect these results to extend to other MAC portfolio hedges as well. That is, the CVaR approach generates portfolios with better downside risk statistics than the traditional parametric MVO approaches, without taking much away from the upside. The CVaR hedges also return more attractive statistics on stress tests and on other tools commonly used by risk managers to evaluate the quality of hedges. 4. Conclusions We present a two-phase scenario-based CVaR hedging approach for minimizing the downside risk in MAC portfolios. The base portfolio is fixed and the hedging approach determines the overlay holdings by minimizing the CVaR of the overall portfolio. The first phase of the hedging approach uses a Monte Carlo framework for generating the scenarios for the different instruments in the portfolio. The second phase incorporates these scenarios in a scenario-based convex optimization problem to generate the overlay holdings. We compare the CVaR approach with MVO-based approaches that linearize all the instruments in the portfolio on three examples and show that (a) the CVaR portfolio has better downside risk statistics, and (b) the CVaR portfolio returns more attractive stress-test and risk-decomposition statistics tools commonly used by risk managers to evaluate the quality of hedges. A CVaR Scenario-based Framework page 8
We must emphasize that the CVaR hedging approach is flexible. This includes (a) the choice of the risk factors and the pricing models in the Monte Carlo framework, and (b) the setup of the scenario-based convex optimization, including how it is regularized to return stable optimal solutions. We refer the interested reader to the extended version of the paper for more details on the two-phase CVaR approach and other use-cases. Ultimately, the CVaR hedging approach is not a standalone analysis and the flexibility should be used wisely and in conjunction with other tools in the arsenal of the risk manager, such as risk decomposition, deltahedging and stress testing. References [1] K. Sivaramakrishnan and R. Stamicar (2016). A CVaR Scenario-based Framework: Minimizing Downside Risk of Multi-asset Class Portfolios, Technical Report 66, Axioma, May 2016. [2] P. Artzner, F. Delbaen, J.M. Eber, and D. Heath (1999). Coherent Measures of Risk, Mathematical Finance, 9(3), 203 228. [3] P. Glasserman (2003). Monte Carlo Methods in Financial Engineering, Springer. [4] J.C. Hull (2008). Options, Futures, and Other Derivatives, 7th edition, Prentice Hall. [5] H. Markowitz (1959). Portfolio Selection: Efficient Diversification of Instruments, Wiley. [6] R.T. Rockafellar and S. Uryasev (2000). Optimization of Conditional Value-at- Risk, Journal of Risk, 2, 493 517. [7] FTRB Regulations: http://frtb.info/summary-of-the-frtb-regulations/ A CVaR Scenario-based Framework page 9
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