Covered Calls and Their Unintended Reversal Bet

Covered Calls and Their Unintended Reversal Bet Roni Israelov, Ph.D. May 2014 Vice President Equity index covered calls have historically Lars N. Nielsen Principal provided attractive risk-adjusted returns largely because of their joint exposures to the equity and volatility risk premia. However, they also embed exposure to an uncompensated risk, a naïve equity market reversal strategy. This paper presents a novel performance attribution methodology, which deconstructs the strategy into its three identified exposures, in order to measure each s contribution to the covered call s return. We provide evidence that the reversal exposure is responsible for about one quarter of the covered call s risk, but provides very little reward. Finally, we show that hedging the strategy in order to eliminate the uncompensated risk could improve the covered call s Sharpe ratio and reduces its volatility, maximum drawdown, equity beta, and downside equity beta. We thank Jacob Boudoukh, Brian Hurst, Ronen Israel, Bradley Jones, Michael Katz, Ari Levine, John Liew, Andrew Sterge, and Daniel Villalon for helpful comments and suggestions; and Matthew Klein and Harsha Tummala for data and analysis. We also thank Jennifer Buck for design and layout. AQR Capital Management, LLC Two Greenwich Plaza Greenwich, CT 06830 p: +1.203.742.3600 f: +1.203.742.3100 w: aqr.com

Payoff Payoff Payoff Covered Calls and Their Unintended Reversal Bet 1 Introduction Equity index covered calls have historically realized returns not much less than their underlying index with less volatility. Although the covered call s higher risk-adjusted returns have been widely established, the sources of these returns are less well understood. In this paper, we shed light on the exposures that contribute to the strategy s risk and expected return. In order to do so, we first present a novel performance attribution methodology for portfolios holding options, such as the covered call strategy. We then employ our performance attribution methodology to deconstruct the portfolio s return into its independent components, allowing us to analyze the properties of each piece in isolation. We show that covered calls generate their returns by collecting equity and volatility risk premia. 1 Covered calls are also exposed to equity timing, a third source of uncompensated risk that has not been previously discussed in the academic literature. This important exposure to equity market timing is a consequence of selling options. While portfolio managers typically focus on a covered call s exposure to volatility, equity timing may contribute more than three times the risk of the volatility risk premium exposure and nearly half the risk of its passive equity risk premium exposure. In fact, over a quarter of a covered call s risk may be attributed to equity timing. This paper seeks to further the understanding of covered call strategies by Exhibit 1 Covered Call Payoff Diagram Long Half Equity + Long Half Cash $200 $175 $150 $125 $100 $75 $50 $25 $0 $0 $25 $50 $75 $100 $125 $150 $175 $200 Asset Price Short Half Straddle $100 $75 $50 $25 $0 -$25 -$50 -$75 -$100 $0 $25 $50 $75 $100 $125 $150 $175 $200 Asset Price Source: AQR. For illustrative purposes only. Covered Call $150 $125 $100 $75 $50 $25 $0 -$25 -$50 $0 $25 $50 $75 $100 $125 $150 $175 $200 Asset Price 1 Bakshi and Kapadia (2003) report evidence of the volatility risk premium by analyzing delta-hedged index option returns. Hill, Balasubramanian, Gregory, and Tierens (2006) show the impact on covered calls of selling options at implied volatility instead of at the volatility realized over the life of the option.

2 Covered Calls and Their Unintended Reversal Bet shining a light on their embedded timing of the equity market. Our paper s final contribution demonstrates that a risk-managed covered call strategy, which hedges to remove dynamic equity exposure, has improved risk-adjusted returns versus the conventional covered call approach. Index covered calls may be attractive because they have similar total returns as the equity index, but with lower volatility. Taking it one step further, hedged covered calls may be attractive because they have similar total returns as conventional covered calls, but with lower volatility. Hedged covered calls have also had lower maximum drawdown, equity beta, and downside equity beta. constructs the payoff diagram for the covered call strategy using this approach. The long equity position provides the equity exposure and the short straddle position provides the volatility exposure with no equity exposure on average. In this example, selling the half straddle generates $25 in option premium. Covered Calls Bet on Equity Reversals When an ATM call option is written, it has approximately 0.5 delta to its underlying security. At the same time, the covered call strategy, which is long the equity and short the straddle, also has a 0.5 delta to its underlying security. However, this equity exposure changes as soon as the equity s price moves. Once hedged, we believe it is less helpful to describe the strategy as long equity and short a call option. The description is too opaque and can lead to many misunderstandings and myths. See Israelov and Nielsen (2014) for a debunking of eight of these myths. A preferable and more transparent description of the hedged strategy is that it provides pure exposure to two compensated and desirable risk premia: equity and volatility risk premia. Covered Call Definition A covered call is a combined long position in a security and a short position in a call option on that security. The combined position caps the investor s upside on the underlying security at the option s strike price in exchange for the option premium. It is possible to construct an identical exposure to At-The-Money (ATM) covered calls by investing half of the NAV in the underlying equity and selling short half of a straddle (a call option and put option at the same strike and maturity). This representation of the covered call is convenient because, unlike a short call option, the short straddle position is delta-neutral. Exhibit 1

Delta Covered Calls and Their Unintended Reversal Bet 3 Exhibit 2 plots this relationship for hypothetical long call and covered call strategies. 2 As the index price increases, the call option s delta increases to reflect the higher probability that it will expire in the money. When the equity price falls, the call option s delta declines to reflect the higher probability that it will expire out of the money. As a result, the covered call s equity exposure is negatively related to the index price. A falling market leads to larger equity exposure and a rising market leads to smaller, but still positive equity exposure. On average, the ATM covered call s delta is approximately 0.5, matching its value at the time the option is written. Deviations from this average exposure may be considered an active exposure to the equity index. The ATM covered call s active exposure ranges from 0.5 to +0.5, is zero on average, and is close to its extremes near option expiration. The covered call s active equity exposure mimics a reversal strategy because it is negatively related to the equity s historical return. changing delta. Six days later, the covered call strategy bets on one-week reversal and the day prior to option expiration it bets on one-month reversal. When the option expires, the covered call starts afresh with no reversal exposure. Exhibit 2 also shows that the relationship between moneyness and equity exposure is often stronger for nearer-dated options. For example, if one month until option expiration the index appreciates from 100 to 103, the covered call s exposure will decline from 0.50 to 0.25. If the same index move happened with one week until expiration, the exposure would have declined to 0.08 instead. As expiration nears, so long as the index value remains above 100, the covered call s equity exposure will converge to zero. A covered call s equity exposure must converge to either zero or one on its expiration date. However, this reversal strategy is path dependent because it is tied to the return since the date the option was sold. The day after the call option is written, the covered call effectively bets on oneday reversal because of the short call option s Exhibit 2 Strategy Index Exposure vs. Index Value for ATM Call Strategies 1.0 0.8 0.6 0.4 0.2 0.0 90 92 94 96 98 100 102 104 106 108 110 Index Value Covered Call (Month) Long Call (Month) Covered Call (Week) Long Call (Week) Source: AQR. For illustrative purposes only. 2 This example assumes a 100 strike price, 50bps financing rate, 2% annual dividend yield, and 15% implied volatility.

4 Covered Calls and Their Unintended Reversal Bet Exhibit 3 CBOE S&P 500 BuyWrite Index s Delta 1.00 0.75 0.50 0.25 0.00 Source: AQR. Delta computed according to Black-Scholes model on the hypothetical CBOE S&P 500 BuyWrite Index. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. This convergence is observable in Exhibit 3, which shows the evolution of an ATM covered call s delta across four recent expiration cycles. The CBOE S&P 500 BuyWrite Index s delta is slightly above 0.5 on option initiation dates, which is when the options have been sold. The covered call s delta begins to fluctuate and by the time the call option expires, the strategy s delta has converged to either zero or one, depending on whether the index has appreciated or depreciated, respectively. These characteristics are shown in Exhibit 4, which plots the distribution of the covered call s delta against the number of days since the call option was sold. Immediately after the call option is sold, the strategy s delta is tightly distributed around 0.5. As time passes, the covered call s delta disperses and by the time the option expires, the delta has settled on either zero or one. The active exposure is smallest immediately after the call option is sold, largest immediately prior to the option s expiration, and the average absolute active exposure is approximately 0.25. Exhibit 5 plots the distribution over all days. An Unintended Bet For those who explicitly want to express a reversal view in their portfolio, covered calls may be ineffective due to the path dependence described in the previous section. Selling a call option is unnecessary for betting on equity reversals since the desired dynamic equity exposure may be obtained directly by trading the underlying security. However, selling a call option provides short volatility exposure, something that cannot be attained by trading the underlying security. For this reason, we view the equity reversal exposure as unintended. A potential counterargument is that the reversal exposure is the wrong lens in which to view the strategy. Investors don t necessarily want the reversal exposure per se; they want the covered call payoff profile. If the reversal exposure is how the covered call payoff profile is obtained, then so be it. We do not view this argument as credible. Who are the investors that specifically desire the covered call payoff profile (as depicted in Exhibit 1)? A covered call has full downside participation and limited upside participation. If any nonlinear profile is desirable, it is the opposite of the covered call. A long call has limited downside participation and full upside participation. This is why there is a volatility risk premium. Those who want the long call profile are willing to pay an insurance premium to those who are willing to accept the covered call profile in return for the

Covered Call Delta Covered Calls and Their Unintended Reversal Bet 5 Exhibit 4 Range of CBOE S&P 500 BuyWrite Index s Deltas 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Business Days Since Last Call Option Sale 10th & 90th Percentiles 25th & 75th Percentiles Median Source: AQR. Delta computed according to Black-Scholes model on the hypothetical CBOE S&P 500 BuyWrite Index. Chart and statistics are based on the period from March 25, 1996 to December 31, 2013. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. premium. Investors in covered calls are not seeking strategies with full downside and limited upside; they are attracted to the higher riskadjusted returns afforded by the volatility risk premium. Covered Calls Equity Timing is a Significant Risk If the static equity exposure is 0.5 and the average absolute active exposure is 0.25, then a back-ofthe-envelope calculation suggests that equity timing has approximately half the risk allocation of the equity risk premium. This is a significant allocation to an unintended risk. To formalize this analysis, we re-arrange the covered call equation to isolate the three underlying exposures: 3 Covered Call = Equity Risk Premium Exposure + Volatility Risk Premium Exposure + Equity Timing Exposure Exhibit 5 CBOE S&P 500 BuyWrite Index s Delta Distribution 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 0.0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1.0 Covered Call Delta Source: AQR. Delta computed according to Black-Scholes model on the hypothetical CBOE S&P 500 BuyWrite Index. Chart and statistics are based on the period from March 25, 1996 to December 31, 2013. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. 3 Covered Call = Equity Call = Equity (Call CallDelta*Equity) CallDelta*Equity = (1 InitCallDelta)*Equity DeltaNeutralCall + (InitCallDelta CallDelta)*Equity = EquityRiskPremium + VolatilityRiskPremium + EquityTiming

6 Covered Calls and Their Unintended Reversal Bet The equity risk premium comes from the long equity position. The volatility risk premium and equity timing exposures are supplied by the short call option position. The short call option also reduces the exposure to the equity risk premium provided by the long equity position. We decompose a simple overwriting strategy, which mimics the industry standard covered call benchmark the CBOE BuyWrite Index into these three components. results. Table 1 reports the The equity risk premium realized approximately 8% annualized volatility, while the volatility risk premium realized a modest 2.0% volatility. At 4.9% volatility, equity timing does indeed realize about half the risk of the equity risk premium and is responsible for four times the risk contribution of the volatility risk premium. 4 Table 1: Simple Overwriting Return Decomposition (Annualized) Equity Risk Premium Volatility Risk Premium Equity Timing Excess Return 3.2% 1.9% 0.6% Volatility 8.3% 2.0% 4.9% Sharpe Ratio 0.39 0.95 0.12 Risk Contribution 64% 7% 28% Alpha to S&P 500 -- 1.7% 0.1% Source: AQR. Statistics are calculated over the period March 25, 1996 to December 31, 2013. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. Based on our analysis, we believe adding volatility risk premium exposure to an equity portfolio is desirable. It has 1.7% alpha to the S&P 500 and more than double its Sharpe ratio. In our analysis, shorting volatility provides onethird of the covered call s average return even though it is only responsible for less than 10% of its risk. Although equity timing has also realized moderately positive returns over our sample, the 0.6% annualized return is not statistically significant given its 4.9% annualized volatility. More importantly, it is unclear why this method of equity timing would be a compensated risk premium. Hedged Overwriting It is possible to invest in an S&P 500 index covered call strategy without equity timing. The call option s delta is known in advance and hedging away its active exposure by trading S&P 500 index futures or an index ETF is not particularly expensive. The resulting strategy, which we denote Hedged Overwriting, provides purer exposure to the equity and volatility risk premia. Table 2 reports performance statistics for the S&P 500 Index (SPX), the CBOE BuyWrite Index (BXM), and a hedged overwriting strategy. By hedging the equity timing risk, which has nearly 5.0% annualized volatility, the hedged overwriting strategy has 2.6% lower annualized volatility than BXM. Further, the hedged overwriting strategy has smaller drawdowns than BXM because BXM s equity timing component provides increasing equity exposure during S&P 500 drawdowns. The difference in upside and downside betas for BXM provides additional evidence in favor of its non-linear S&P 500 index exposure; it has nearly twice the exposure to negative S&P 500 returns. While the hedged overwriting strategy also has higher exposure to negative S&P 500 returns, hedging away the active equity timing component significantly reduces its downside exposure to S&P 500 Index. 4 Risk contribution is defined as the covariance of the component with the BuyWrite Index divided by the variance of the BuyWrite Index.

Covered Calls and Their Unintended Reversal Bet 7 Table 2: Summary Statistics SPX BXM Hedged Overwriting Excess Return 4.6% 4.9% 4.7% (Geometric) Volatility 16.8% 11.6% 9.0% Sharpe Ratio 0.28 0.42 0.51 Worst Drawdown -62% -43% -34% Beta to S&P 500 Index 1.00 0.62 0.53 - Upside Beta 1.00 0.46 0.46 - Downside Beta 1.00 0.85 0.59 Source: AQR, S&P 500 Total Return Index and CBOE S&P 500 BuyWrite Index reported over the period April 1, 1996 to December 31, 2013. Returns are excess of cash (US 3-Month LIBOR). Annualized volatility and Betas are computed using 21-day overlapping returns. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. Historical Evidence Exhibit 1 showed that a covered call embeds a reversal exposure in a hypothetical example. Stepping away from the hypothetical, Exhibit 6 plots simple overwriting s active exposure as it relates to the S&P 500 s return since the date the call option was written. When the index return has been positive, the active equity exposure tends to be negative and vice versa. The relationship is non-linear, in part due to the fact that the active exposure is bounded by ±½. Visual inspection of Exhibit 6 shows a significant negative relationship between overwriting s equity exposure and the S&P 500 return since the prior option sale. Table 3 confirms the relationship, as estimated via regression, between simple overwriting s equity exposure and the S&P 500 Index return since the last option trade. The regression confirms the negative relationship, which is statistically significant with a 30.6 t- statistic. 5 The 0.5 intercept represents the simple overwriting s static exposure to the S&P 500 Index. We report the results of this regression in order to test the relationship. We do not 5 Because of the unusual overlapping nature of the data, we use the bootstrap resampling techniques to estimate standard errors for the three regressions reported in this paper. Reported coefficients are the average regression coefficients from the bootstrapped samples. recommend using this model to determine hedging quantities. Those are better ascertained with an option pricing model, such as Black- Scholes. Table 3: Regression of Simple Overwriting Delta on S&P 500 Return Since Last Option Expiration Intercept Index Return R 2 Coefficient 0.50-6.68 74.2% (t-statistic) (241.0) (-30.6) Source: AQR. Regression is estimated over the period March 25, 1996 to December 31, 2013. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. Testing this from another angle, we form a simple reversal strategy in which the S&P 500 position is equal to the negative of the index s return since the prior option s expiration. 6 Table 4 reports the regression of simple overwriting s equity timing component on our reversal strategy s return. Table 4: Regression of Equity Timing Return on Reversal Strategy Return Intercept Reversal Strategy Return R 2 Coefficient 0.0 +10.0 42.0% (t-statistic) (0.9) (16.0) Source: AQR. Regression is estimated over the period March 25, 1996 to December 31, 2013. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. In our analysis, the simple reversal strategy explains 42.0 percent of equity timing s variance. The coefficient is both positive and statistically significant. 7 The strength of the reversal bet is a function of the equity index return (Table 3), but it also depends on how much time is left until the option expires (Exhibit 2). The interaction between the index return and time until option expiration is why Table 3 shows that the index return explains 74.2% of simple overwriting s active equity 6 We winsorize returns at ±7.5% in order to cap and floor the strategy s position in a manner similar to the covered call which has absolute active positions no larger than ½. 7 The magnitude of the coefficient is not important because choice of leverage is arbitrary.

Equity Exposure 8 Covered Calls and Their Unintended Reversal Bet Exhibit 6 Simple Overwriting Equity Index Exposure vs. S&P 500 Return 1.0 0.8 0.6 0.4 0.2 0.0-40% -30% -20% -10% 0% 10% 20% 30% 40% S&P 500 Return Since Last Call Option Sale Source: AQR. Returns are for the period from March 25, 1996 to December 31, 2013. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. exposure and yet Table 4 shows that using the same index return to linearly construct a reversal strategy explains 42.0% of the simple overwriting s equity timing return. Shorting a call option does not provide clean exposure to equity index reversals. It s difficult to make a compelling case that the reversal strategy should be a compensated risk premium, but it is not implausible that it could be a source of alpha. To test whether simple overwriting s active exposure predicts equity returns, we regress the daily S&P 500 index returns on simple overwriting s active exposure. Our finding, reported in Table 5, confirms that although there is a positive relationship, the coefficient is not statistically significant and the R 2 requires a second decimal place to be measured as non-zero. Hence, we conclude that the active equity exposure does not increase expected returns because it does not forecast equity returns. Table 5: Regression of S&P 500 Return on Simple Overwriting s Active Equity Exposure Intercept Active Equity Exposure R 2 Coefficient 2.6 bps 5.60 0.01% (t-statistic) (1.4) (0.6) Source: AQR. Regression is estimated over the period March 25, 1996 to December 31, 2013. Broad-based securities indices are unmanaged and are not subject to fees and expenses typically associated with managed accounts or investment funds. Investments cannot be made directly in an index. Past performance is not a guarantee of future performance. Conclusion While many reasons are provided in support of equity index covered calls, Israelov and Nielsen (2014) show that ultimately the strategy is attractive because of its exposure to the equity and volatility risk premia. However, covered calls also provide exposure to equity timing via an embedded reversal bet. Due to the complexity and opacity of the covered call strategy, this equity timing exposure, which is typically responsible for over 25 percent of the covered call s risk, is hidden in plain sight. It is important that investors and portfolio managers understand the risks they are taking when writing a covered call. We believe the portfolio can and should be hedged against exposures to market timing so that the limited risk budget may be allocated to compensated risk premia.

Covered Calls and Their Unintended Reversal Bet 9 Related Studies Bakshi, G. and N. Kapadia (2003), Delta-Hedged Gains and the Negative Market Volatility Risk Premium Hill, J.M., Gregory, K.B. and I. Tierens (2006), Finding Alpha via Covered Index Writing, Financial Analysts Journal 62(5), 29-46. Israelov, R. and L.N. Nielsen (2014), Eight Myths and One Fact about Covered Calls AQR Whitepaper.

10 Covered Calls and Their Unintended Reversal Bet Biographies Roni Israelov, Ph.D., Vice President Roni oversees AQR s short-term systematic futures trading strategy and the management of related portfolios. Separately, he also manages AQR s volatility trading strategies. Prior to AQR, he was a research analyst in the quantitative equities strategies group at Lehman Brothers. He shared the Graham & Dodd Award for the paper International Diversification Works (Eventually) published in Financial Analysts Journal. Roni earned a B.S. in mechanical engineering from Georgia Institute of Technology, an M.S. in mathematical risk management from Georgia State University, and an M.S. in finance and a Ph.D. in financial economics from Carnegie Mellon University. Lars N. Nielsen, Principal Lars oversees research in AQR s Global Stock Selection and Global Asset Allocation teams, and is a part of the portfolio management teams for a number of AQR s multistrategy hedge funds as well as longonly equity portfolios. Prior to AQR, Lars was a visiting graduate student at Cornell University, where his research interests were finance and econometrics. Before that, he was a quantitative equity analyst at Danske Invest, the largest asset-management firm in Denmark. Lars earned a B.Sc. and an M.Sc. in economics from the University of Copenhagen.

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AQR Capital Management, LLC Two Greenwich Plaza, Greenwich, CT 06830 p: +1.203.742.3600 I f: +1.203.742.3100 I w: aqr.com