Can Tail Risk Be Hedged?

Can Tail Risk Be Hedged? Summary of Empirical Results Lisa R. Goldberg Ola Mahmoud (Marie Curie Fellow) 1 Peter Shepard Kurt Winkelmann August 2011 1 Ola Mahmoud was a Marie Curie Fellows at MSCI. The research leading to these results has received funding from the European Community s Seventh Framework Programme FP7-PEOPLE-ITN-2008 under grant agreement number PITN-GA-2009-237984 (project name: RISK). The funding is gratefully acknowledged.

Contractual Tail Risk Hedging Strategies The cost of insurance We examine the cost of passive tail risk hedging via zero-cost collars. These are contractual option strategies limiting both the potential downside and upside to a specific range. Collars are created by purchasing an out-of-the-money put, aimed at capping losses, together with selling a call to finance the purchase of the put. We backtest the zero-cost collar strategy using the S&P 500 as the underlying. Market prices of 3- month-to-maturity call and put options on the S&P 500 going back to 2003 are matched up into five zero-cost put/call pairs, ranging from deep out-of-the-money ( widest collar) to very near the money ( narrowest collar). Figure 1 shows the performance of each of these collar strategies compared to the underlying. Transaction costs are taken into account by buying at the bid and selling at the ask price. Collars as downside protection strategies come at the cost of potential upside, with their success on limiting extreme losses coming into effect only during times of extreme market losses. As is well appreciated, when limiting both losses and gains to values that are close to the center of the returns distribution, the distribution tails at either ends narrow down considerably, resulting in stable but minimal cumulative performance ( narrowest collar of Figure 1). On the contrary, cutting out only the most extreme losses and gains via a very wide collar yields a typical high risk/high return performance comparable to that of the market itself ( widest collar of Figure 1). To obtain a clearer picture of how much of the potential upside is sacrificed for the sake of insurance, we look at the ratio of downside to upside return thresholds (Figure 2). Clearly, all of the zero-cost collars are very expensive downside protection instruments, with their downside generally exceeding the upside. We also see from Figure 2 that the wider the collar, the larger the ratio of downside to upside. Figure 1: Performance of zero-cost collar hedging strategies using 3-months-to-maturity put and call options on the S&P 500. Figure 2: Ratio of downside to upside returns of 4 of the zero-cost collar strategies on the S&P 500 of Figure 1. 2 of 12

Tail risk hedging and asset allocation We investigate the impact of hedging tail risk on long-term asset allocation. Our benchmark portfolio consists of 65% allocated in global equity (represented by the MSCI All Country World Index) and 35% in fixed income (represented by a CRSP Fama portfolio of bonds for maturities between 60 and 120 months). In the absence of option price data, we synthetically construct collars as fixed-return cutoffs. Our hypothetical symmetric collar caps total portfolio returns at 5%, and the asymmetric collar caps returns at 10% and 5%. To incorporate the effect of transaction costs, we subtract a constant from insured portfolios. Table 1 summarizes the main risk and return characteristics of some of the portfolio/insurance combinations we consider. Evidently, the market exhibits higher return and volatility and more negative skewness compared to its fixed income counterpart. In our benchmark 65/35 portfolio, some of these risks are offset. The effect of adding protection to the overall returns of this benchmark via synthetic collars is presented as portfolios 2a and 2b. In the table, 1-month (resp. 1-year) skewness and kurtosis refer to the skewness and kurtosis at the specified return horizon (i.e. monthly and yearly returns). Generally, the symmetric collar displays better return distribution qualities: the Sharpe ratio is slightly higher, and so is the overall 1-month and 1-year skew. On the other hand, the kurtosis is reduced, indicating that both insurance strategies indeed succeeded at slimming down the tails. The overall returns of both collar strategies are slightly lower than those of the benchmark. As it is no surprise that insurance strategies come at a cost, one may ask what asset allocation an investor could have chosen to achieve the same performance of an insured portfolio, and what the corresponding impact on its risk would have been. We see that an equal-weighted allocation (portfolio 3 in Table 1) yields a return and volatility comparable to those of the symmetric collar strategy. Note, however, that both the Sharpe ratio and the overall skewness have enhanced, indicating that changes in asset allocations may effectively lead to an implicit, non-contractual type of insurance. Portfolio Return Std Sharpe 1M Skew 1Y Skew 1M Kurtosis 1Y Kurtosis MSCI All World 0.10 0.15 0.69-0.56-0.32 4.64 3.64 CRSP 60-120m Bonds 0.08 0.07 1.22 0.43 0.85 6.45 4.07 1: 65/35 0.10 0.10 0.93-0.37-0.28 4.25 3.92 2a: 65/35 w/(-10%,+5%) collar 0.08 0.10 0.87-0.72-0.56 3.77 3.24 2b: 65/35 w/(-5%,+5%) collar 0.09 0.09 1.03-0.33-0.37 2.57 2.65 3: 50/50 0.09 0.09 1.07-0.20-0.10 4.18 3.88 4a: Short (-10%,+5%) insurance 0.01 0.02 0.70 4.84 2.17 35.45 8.91 4b: 65/35, short (-10%,+5%) insurance 0.11 0.11 0.95 0.11 0.20 5.59 5.33 Table 1: Summary statistics With the view of an economic equilibrium in mind, we investigate the effect of providing insurance. To this end, we study the return and risk attributes of portfolios which sell the collar hedging strategies. Purely selling the asymmetric collar (portfolio 4a) results in minimal returns and volatility which can be seen as a steady but limited source of income, similar to what a traditional long-term investor might seek. The safety of such a strategy is further reflected in its skew, which is significantly positive. It may 2 of 12

therefore be intriguing to consider our initial 65/35 asset allocation and sell rather than buy protection on it. Curiously, when comparing portfolios 4b and 2a of Table 1, we observe a better Sharpe ratio and a positive skewness when adding a short collar-hedge position. This outperformance naturally comes at the cost of larger excess kurtosis, as the extreme gains and losses taken over from the insurance buying counterpart amplify the size of the tails of the insurance-selling investment portfolio. Volatility-based insurance Volatility possesses certain characteristics which make it an attractive tradable asset. Just like interest rates, it appears to be mean reverting, hence tends to drop after periods of high volatility and rise after calm periods. It furthermore tends to increase during times of uncertainty. This seemingly regime dependent behavior allows investors to speculate on future levels of volatility. On the other hand, investors with portfolios that have natural exposure to volatility may want to hedge it away. A more intriguing use of volatility viewed as an asset class is portfolio diversification. The underlying assumption is the existence of significant negative correlation between the market returns and their volatility, especially during turbulent times; an empirical observation first pointed out by Black [] and since referred to as the leverage effect (see Figure 3). From this point of view, being long volatility can be a beneficial hedge against the tail risk of a long-only equity portfolio. Until recently, options were the only products giving exposure to volatility. The common strategy is to hedge the exposure of the option to the underlying in order to isolate the exposure to volatility. One of the main disadvantages to this approach is the constant need to rehedge large market moves will cause the option to be a directional bet on the underlying. An option s sensitivity to volatility will diminish when the underlying moves away from the strike price. Over the past few years, variance and volatility based investment products have emerged as alternative hedging vehicles, examples of which are variance swaps and futures on volatility indices. These are meant to provide pure exposure to volatility or variance, independent of the other risks that would accompany an option-based strategy. According to the Chicago Board Options Exchange (CBOE) [], trading of volatility has roughly doubled over the past few years. The volume of trading one such instrument, namely futures on the VIX, has almost grown to its ten-fold between 2007 and 2011, as illustrated in Figure 4. Figure 3: The Leverage Effect Figure 4: Total volume of VIX futures traded (2007 2011) 3 of 12

In the context of Black s leverage effect, we next investigate the premia of trading variance for the purpose of insurance. Passive option-based insurance strategies such as zero-cost collars are meant to provide explicit protection against extreme events that affect the tail. Volatility-based strategies can implicitly provide the same kind of insurance against extreme down movements in the market, by explicitly hedging volatility. The variance risk premium Implied volatilities are a forward looking measure of the market s perception of its own variability. These are normally more reactive to anticipated changes and tend to differ from what has occurred recently. Indeed, traders use implied volatilities (of the most actively traded options) to monitor the market s opinion about the expected volatility level. Position-taking in pure volatility is therefore largely based on implied volatility forecasts. Indices of implied volatilities, such as the VIX, are meant to provide timely reflections of future market variability. They are calculated as a weighted average of option implied volatilities for various strikes and, in the context of the Black-Scholes framework, can be interpreted as the risk neutral estimate of future volatility [Derman]. These are generally higher than realized volatility levels. One aspect of this is the embedded market premium on out-of-the-money options used for insurance. Implied volatilities therefore contain important information regarding investors demand for portfolio insurance. The volatility (variance) risk premium refers precisely to this difference between realized and implied volatility (variance). It reflects the payoff an investor would get by betting on future volatility levels, for instance by purchasing variance swaps. Figure 5a illustrates the variance risk premium over time for the S&P 500. The estimates are obtained by subtracting, at each point in time, the observed VIX level from the 30-day forward looking realized one-day volatility. The distribution of the payoffs from variance swaps is displayed in Figure 5b. We see that the payoff is generally negative, and exhibits large positive jumps during periods of turbulence. The negative variance risk premium reflects intrinsic investor behaviours and expectations when it comes to buying and selling insurance. The highly negative premium combined with its strong positive skewness suggests that investors who buy insurance are willing to endure losses on average, in the hope of being rewarded when realized market volatility increases. The other transaction party, the insurance seller, is thereby able to generate a regular and stable stream of positive income, but has the occasional exposure to extreme losses. (relate to long and short horizon investment strategies) Figure 5a: The variance risk premium Figure 5b: Variance swap return distribution 2 of 12

The cost of hedging volatility Volatility and variance based strategies aim at exploiting information about the expected payoff embedded in variance risk premia. By entering a variance swap for instance, an investor bets on the future variance risk premium from the perspective of buying or selling downside protection. A VIX futures contract offers an alternative yet equivalent opportunity to gain the same exposure to variance as variance swaps. We test an insurance strategy on the S&P 500 which is long a futures contract on the VIX. We use a history of daily market prices for VIX futures for various expiries, with our focus on two maturities (1 month and 1 year). A contract is assumed to be settled daily and held until its expiry, at which point a new contract of the given maturity is purchased. Figure 6 shows how this strategy performs relative to the S&P 500. It displays the 1-month- and 1-yearto-expiration VIX futures strategies with and without incorporating transaction costs. Costs are accounted for by buying at the bid price. When incorporating costs, we see that none of the insurance strategies outperforms the market, except for the few months during the peak of the financial crisis. Comparing the short and long horizon strategies, the 1-year horizon clearly outperforms its shorter horizon counterpart (except for a steep dip in early 2009, which is quickly recovered from). However, the summary statistics of Table 2 show that the 1-month strategy possesses superior risk characteristics in terms of all its other moments (smaller volatility and kurtosis, and larger positive skewness). In fact, the market itself beats the 1-year insured portfolio not only in its performance, but also in its other moments. The dotted lines in Figure 6 show how both strategies would perform if we were to neglect transaction costs. The effect of the costs is most visible in the cumulative performances, which are enhanced significantly. On the other hand, the risk attributes are comparable to those of the corresponding strategies that account for the cost. Figure 6: Hedging the S&P 500 with VIX futures. Portfolio Volatility 1D Skewness 1M Skewness 1D Kurtosis 1M Kurtosis S&P 500 0.28-0.35-1.12 7.45 6.22 S&P + 1M VIX Futures (with cost) 0.19 0.05-0.68 11.43 3.38 S&P + 1M VIX Futures (no cost) 0.21 0.07-0.93 16.88 5.46 S&P + 1Y VIX Futures (with cost) 0.31-5.81-1.29 122.81 9.87 S&P + 1Y VIX Futures (no cost) 0.27-5.44-0.77 105.11 8.07 Table 2: Summary statistics for the portfolios of Figure 6.

On the Importance of Higher Moments Insights into the skew It is generally acknowledged that investors dislike highly negative skewness and large excess kurtosis in their portfolios. At a short-term return horizon, the market usually exhibits such higher-moment characteristics, indicating that short-term investors may either expect higher compensation for these higher-moment risks, or that they may seek to hedge against the skew and kurtosis. From the Central Limit Theorem, we know that aggregating independent and identically distributed (i.i.d.) daily returns to obtain longer horizon returns would approach normality the longer the horizon. Both skewness and excess kurtosis would therefore be expected to converge to their Gaussian values of zero when the return horizon increases. This may point towards a less pressing need for long-horizon investors to account for higher moments in their investment and risk management. To understand these effects, we study high moments of the market at increasing horizons. Figure 1 displays skewness and kurtosis of logarithmic returns to the S&P 500 as a function of return horizon, ranging from daily to yearly returns. Our empirical setting uses a sufficiently large data set (1950 2011) for the values of the skew and particularly the kurtosis to be meaningful. For comparison, we plot the same for simulated time series of returns that are assumed to be log-normal and i.i.d. These display the expected decay behavior: for 1 and 1 the 1-day skewness and kurtosis, respectively, daily i.i.d. returns at an -day horizon would have 1 / and 3 1 3 / skewness and kurtosis, respectively. Observe that the kurtosis of the market does indeed fall off at longer horizons, though with some noise and at a decay rate that is naturally slower than its i.i.d. counterpart. Market skewness, however, does not converge to normality with longer horizons. In fact, it persists and roughly fluctuates around a value of 0.9 no matter what return horizon is taken. Figure 7: Skewness (left) and kurtosis (right) of logarithmic returns to the S&P 500 (January 1950 April 2011) as a function of return horizon ranging from 1-day to 1-year returns. How can an investor use these insights to decide on tail risk hedging strategies? On the one hand, the decaying of the kurtosis confirms that the long horizon investor is less affected by extreme events and should therefore be selling rather than buying insurance. On the other hand, the persistence of the skew implies that a long-horizon investor needs to worry about the skew as much as the short-horizon investor. Therefore, one conclusion to be drawn for the long-horizon investor is to only sell very deep insurance to short-term investors the higher moment risk which does not affect the long-term investor is kurtosis; i.e. only events that occur far in the tail do not matter in the long horizon. However, skewness matters. Selling any other level of insurance (protection against not so severe events) can lead

to significant losses, since one is just as much affected by the overall negative asymmetry as the shorthorizon investor. The persistence of the skew is a phenomenon mainly observable in markets or equity indices; the picture is quite different when looking at single name stocks, and therefore care needs to be taken as to the underlying one seeks insurance for. Appendix A replicates the graphs of Figure 1 for two singlenames, namely Coca Cola and BP. Higher moments & asset allocation strategies The passive/contractual insurance strategies we considered so far did not account for higher moments, but the resulting skew and kurtosis statistics of Table (?) suggest that an investor may need to explore other dimensions that are complementary to volatility. Consider again the simple setting of a two-asset portfolio fixed income and equity. We know that increasing the fixed income component decreases both volatility and expected return, and a straightforward mean-variance optimization will yield the optimal such combination according to a given risk appetite. A standard such combination is a 65/35 portfolio. If, as investors holding such an equity-dominant portfolio, we are now only interested in its skewness and kurtosis, we may ask ourselves: how much fixed income should we add to reduce higher moment risk? With the US market as the allocation universe (represented by MSCI USA and the Merrill Treasury Index), Figure 2 shows the volatility, skewness and kurtosis of portfolios representing a range of allocations to each asset, ranging from 100% in equity (far left) to 100% in fixed income (far right) at three return horizons. Note that the risks associated with skewness and kurtosis decrease significantly only when one goes beyond a 50% allocation to fixed income, and therefore, in this very simplified setting, one may conclude that a 35/65 allocation is preferred over a 65/35 allocation from the point of view of higher moment risks.

Figure 8: Volatility, skewness and kurtosis of an equity/fi portfolio as a function of allocation to fixed income Active higher-moment strategies A general comment on active hedging. First, active compared to passive makes more sense, as allocations should be dynamic and based on current market regimes. Some prerequisites: (i) good judgment of short-term future market conditions, either through subjective opinion or through statistical methods; (ii) feasibility of underlying methodology; (iii) limited costs. Traditional portfolio theory does not account for the impact of higher moments on decision making. Under the Capital Asset Pricing Model (CAPM) particularly, investors are mean-variance optimizers and do not have any preference regarding the skewness and kurtosis of their portfolios. Different asset-pricing models that account for higher moments have been developed as an attempt to extend the traditional CAPM. The intertemporal capital asset pricing model (ICAPM) is one such alternative proposed by Merton [cite], under which the state variables reflect changes in the distribution of future returns. In this factor model, one may take market volatility, skewness and kurtosis as state variables reflecting changes in investment opportunity set [Christofferson]. The equilibrium prices of risky assets are then determined by the conditional covariances between asset returns and changes in state variables. Note that investors remain mean-variance optimizers under the ICAPM, and higher moments affect their decision only indirectly at the level of their relationship with asset returns. A conceptually simpler approach is given by the four-moment CAPM [cite]. This model is based on the hypothesis that investors care about the skewness and kurtosis of their portfolios as much as they care about their returns and variance. Under this setting, they are mean-variance-skewness-kurtosis optimizers, and their utility function is derived by expanding it to the fourth instead of the second order. The main difficulty in higher moment optimization lies in the third- and fourth-order terms appearing in the objective function, making the optimization no longer convex [cite]. Moreover, the dimensionality of

the problem increases, making estimation of parameters such as co-skewness and co-kurtosis more challenging (possible third challenge: lack of good enough risk model?). Many approximations to optimization routines that incorporate higher moments have been investigated, most recently by Deutsche Bank [cite], who developed a so-called polynomial goal programming algorithm which aims at balancing the conflicting goals of minimizing variance and kurtosis and maximizing return and skewness. We study a simple, yet mathematically stable, alternative to accounting for higher moments in portfolio optimization, namely by replacing skewness and kurtosis with the convex measure of shortfall [Minimizing Shortfall]. Depending on the confidence level under consideration, expected shortfall can be regarded as a descriptive statistic of a portfolio s distribution. For example, by considering a 95% shortfall confidence level, we are measuring the fatness of the loss tail, whereas a 60% shortfall describes overall asymmetry [to be elaborated on]. Minimizing expected shortfall would thereby result in minimizing the risk associated with kurtosis and skewness, respectively. This framework would also enable a generalization of the mean-variance objective function to include all four moments by simply adding additional shortfall terms at the desired confidence levels. Portfolio Allocation Volatility Sharpe Ratio Skewness Excess Kurtosis 95% Shortfall MSCI USA 100-0 20.71% 0.64-1.432 7.57 3.08% Merrill US Treasury 0-100 4.78% 1.12-0.161 2.37 0.68% Minimum Variance 11.25-88.75 4.20% 0.978-0.166 2.46 0.63% Minimum Skewness 11.6-88.4 6.33% 0.960-0.147 2.1 0.61% Minimum Kurtosis 7.8-92.2 7.40% 0.966-0.154 1.73 0.45% 65/35 65-35 13.12% 0.89-0.91 7.51 1.90% 50/50 50-50 10% 0.91-0.954 7.25 1.42% Risk Parity 20-80 4.84% 0.94-0.163 4.01 0.67% Table 3: Summary statistics for portfolios of various asset allocation strategies. Table 3 summarizes the statistics of portfolios containing equity (MSCI USA) and fixed income (Merrill US Treasury), with the allocation to each decided by an optimization routine. For comparison, we also show what a typical 65/35 portfolio looks like, together with an equal-weighted allocation, and a portfolio under a risk parity strategy. The allocations to the minimum risk portfolios resemble that of a risk parity strategy, with the majority assigned to the safe fixed income asset. Their return and volatility attributes are all very similar. Note, however, that each minimum risk strategy did indeed minimize the risk under consideration. Most notable is the kurtosis optimization, which did indeed minimize tail risk in terms of kurtosis and 95% shortfall. [remark on: one risk minimized, at the expense of others increasing?] The allocations we obtain from the various optimizations lead us to consider that, just as is common with a risk parity strategy, a minimum-variance-kurtosis/maximum-return-skewness portfolio may be combined with leverage to yield maximum returns but minimum high-moment risks. Of course, the main concern with such a strategy is the extent to which taking on leverage is allowed.

Appendix A Single-stock Skewness and Kurtosis

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