the Equity Insurance

Transcription

1 Harvesting the Risk Premium Equity Insurance Hewitt EnnisKnupp, An Aon Company 1

2 Introduction Attractive risk-adjustedd returns can be achieved whenever a behavioral or regulatory factor enters into one investor s decision making process and not another s: these opportunities can be considered a socially driven risk-premium arbitrage. Some socially drivenn opportunities are long-lasting, while others eventually disappear. Convertible bond arbitrage was an opportunity that existed for almost a decade until the analytics used to identify the opportunities became more widespread, and the knowledge advantage was arbitraged away. High yield bonds rated BB have shown attractive risk-adjusteon investments rated below investment grade. Stock price behavior following a stock splitt or inclusion into the S&P 500 Index is returns versus BB BB and B rated bonds due predominately to guidelinee restrictions another example of another opportunity that has been quite persistent. In this white paper, we will discuss another investment opportunity that might be reasonably expected to persist, through harvesting what we will describee as the Equity Insurance Risk Premium. Discussion Equity options 1 can be a useful tool to create asymmetrical return profiles on the underlying equity. The owner of a European call option benefits from price appreciation above the strike price, but incurs no loss other than the premiumm paid if the price of the underlying equity is below the strike price at expiration. The owner of a European put option has the right to sell the underlying equity at the strike price of the option. If the price declines below the strike price at expiration, a gain will result (less the premium paid to acquire the option) though if the price remains above the strike price, the option would not be exercised and the net loss would be fixed at the premium paid. Unlike owning the equity directly, and being exposed to symmetrical upside and downside returns, the owner of an option is exposed to return in only one direction. There is value in obtaining this asymmetry andd it is reflected in the price of the option. The price of any equity option is a function of the current price of the underlying equity, the strike price of the option, interest rates, dividends, time to expiration, and the volatility of the underlying equity. As volatility increases, the price of the option will increase. When pricing an option, the price of the underlying instrumentt can be observed, the strike price is known, as is the interest rate and expected dividends corresponding to the maturityty of the option. Future volatility, on the other hand, is the big unknown and must be estimated. The equity index options markets are deep, liquid and efficient markets, therefore finance theoryy would suggest the implied volatility in option prices would be the best estimate of the actual subsequent realized volatility. The likelihood that realized volatility will be above the estimate is expected to be the same as the likelihood it will be below the estimate. However, studies have shown implied volatility is an upwardly biased estimator of realized volatility ( Neely, 2002; Christensen and Prabhala, 1998, Hill, et al, 2006 ). That is, implied volatility tends to overestimate subsequent realized volatility leading to apparent mispricing of the options. Given the efficiency of the equity options market, one would not expect implied volatility in options pricing to exhibit this upwards bias. 1 In this paper, we use the term Equity Options generically in reference to the use of Equity Index Options as opposed to options on individual equity securities. 1

3 [Chart 1] % 80.00% 60.00% 40.00% 20.00% 0.00% Implied vs. Realized Volatility VIX Realized [Chart 2] 40.00% 30.00% 20.00% 10.00% 0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 01/02/90 01/02/91 01/02/92 01/02/93 01/02/94 01/02/95 01/02/96 01/02/97 01/02/98 01/02/99 01/02/00 01/02/01 01/02/02 01/02/03 01/02/04 01/02/05 01/02/06 01/02/07 01/02/08 01/02/09 01/02/10 01/02/11 01/02/12 VIX less Realized Average Finance scholars have attempted to explain these anomalies in option pricing by modifying the distributional properties of volatility within the Black and Scholes Model with limited success (Cox, Ross, 1976; Heston, 1993). An implicit assumption underlying the Black and Scholes Model is the ability to create a risk-free replicating portfolioo for the option. Bergmann (1981) and Omberg (1991) have shown 2

4 this not to be the case: the Black-Scholes replicating portfolioo is not riskless. With the additional consideration that investors are risk-averse, behavioral aspects of investors will impact option pricing. The Prospect Theory was developed by two psychologists (Kahneman and Tversky, 1979) incorporating general behavioral biases in investor s decision making process. Three key aspects of the prospect theory are: Individuals place different values on losses and gains relative to a reference point, rather than on final wealth Loss aversion: individuals value losses greater than an equivalent gain (relative to the reference point). They note individuals will be more willing to accept a gamble with uncertainty (and an expected loss) than a guaranteed loss of the same amount. Individual s subjective perception of probability, called decision weights, differs from objective probabilities. Decision weights are generally lower than the corresponding objective probabilities, except in the range of low probabilities. The value function can be graphically described as follows: Value Losses Gains To see the implications of the Prospect Theory on option pricing, consider the following diagram: Profit Stock Option Stock Price Loss 3

5 The green shaded areaa is the amount of loss avoidance achieved by the owner of a call option versus the owner of the stock. The red shadedd area is the amount of gain sacrificed by owning an option. Under Prospect Theory, the value ascribed to loss avoidance is greater than the value to the gain sacrificed, and the subjective probabilities of large price moves are overestimated resulting in a value ascribed to the option which is greaterr than that under the risk-free arbitrage approach of the Black-Scholes Model. This description of investor behavior is consistent with the observation thatt implied volatility in option pricing is a biased estimator of realized volatility. Long term capital providers such as pension funds, endowments, foundations are in a unique position to capitalize on the behavioral impacts on short-dated option pricing. Options can be considered a form of insurance on a financial asset. We will call the prospect theory induced pricing premiumm the Equity Insurance Risk Premium. This paper will demonstrate how investors can capture the Equity Insurance Risk Premium. Investment Strategy If there is indeed an equity insurance risk premium, a strategyy designed to capture this risk premium should be superior to a strategy that is not exposed to this risk. To fully capture the insurance risk premium in a symmetrical fashion we will utilize both puts andd calls. In order to avoid leverage in the portfolio a strategy can be developed as follows 2 : Invest 50% in the S&P 500 and sell out of the money calls with a fixed, pre-determined delta (the equity sensitivity of the option is constant at the time of purchase, resulting in strike prices, degree of out of the moneyness, that change due to changes in implied volatility). Invest 50% in cash and sell out of the money puts with the same fixed delta as in #1 above. We test the existence and magnitude of the equity insurance risk premium by comparing the returns of this strategy to a fully invested equity portfolio, as well as a portfolio invested 50% in equity and 50% in cash, which approximately equates the market sensitivity (or beta) of the strategy. 2 See Hill, et al, Finding Alpha via Covered Index Writing, Financiall Analysts Journal, 2006 for a description of dynamic strike strategies 4

6 Figure 1 Portfolio Return Profile The strategy outlined above reduces the exposure to the equity risk premium by 50% relative to a fully invested equity position because equity exposuree has been cut in half, yet exposes the portfolio to 100% of the equity insurance risk premiumm because overpriced options have been sold on the entire portfolio. If there is a premium to be captured from insurance risk, one would expectt a return from this strategy to be in excess of the 50% equity/50% cash portfolio and to havee less volatility than the fully invested portfolio since risk premiums have been diversified. Figure 2 Diversification of Risk Premiumm Hypothesis Covered Option Writing 100% Equity Add Insurance Risk Reducee Equity Risk 50% Equity/50% Bills 100% Bills 5

7 A qualitative benefit of this approach occurs as a result of the quasi-rebalancing nature of the strategy. After a significant rally during the month, the exposure to the market is reduced by the short calls. Conversely, after theree is a significant decline in the market, exposure is increased by the short puts. In other words, buy low and sell high. If one were to buy insurance on the market as opposed to selling insurance, one would be doing the opposite: selling low and buying high. Historical Returns The option strategy was simulated by allocating the portfolio too two equally weighted components: 1) writing one month, CBOE listed covered calls on the S&P 5000 Index with a fixed delta strike price, and 2) writing one month, CBOE listed covered puts on a T-Bill position with a fixed delta strike price. The portfolio was rebalanced monthly and estimated transactions costs were deducted from the portfolio. The simulated returns, standard deviations and Sharpe ratios are displayedd in the following table: (1/1/90-6/30/12) Annualized Return Option Strategy S& &P % S&P 500/ 50% Tbills 1yr 3yr 5yr 10yr 20yr Inception 10.28% 14.00% 6.65% 9.14% 10.40% 10.35% 5.45% 16.40% 0.22% 5.33% 8.34% 8.71% 3.10% 8.30% 0.99% 3.88% 6.04% 6.28% Annualized Standard Deviation Option Strategy S& &P % S&P 500/ 50% Tbills 1yr 3yr 5yr 10yr 20yr Inception 9.06% 7.95% 10.74% 8.71% 8.34% 8.26% 17.40% 15.88% 19.05% 15.78% 15.11% 15.16% 8.70% 7.94% 9.50% 7.89% 7.58% 7.60% 6

8 Sharpe Ratio Option Strategy S&P % S&P 500/ 50% Tbills 1yr 3yr 5yr 10yr 20yr Inception (0.03) As one would expect, the volatility of the option strategy is less than the S& &P 500 due to the reduction in equity market beta, and for the same reason, the strategy underperforms in large market rallies. Though over longer time periods, the presence of the insurance risk premium more than compensates for the lower exposure to the equity risk premium. When one adjustss for the reduced beta of the strategy, and compares performancee to the 50% S&P/50% T-bill strategy, one finds consistent outperformance due to the exposure and diversification benefits of the insurance risk premium. The return enhancement and risk reduction is also evident when the Sharpe Ratio is compared to either of the market based strategies. [Chart 3] 40.00% Rolling 3 Year Returns 30.00% Return 20.00% 10.00% 0.00% 10.00% 12/01/92 11/01/93 10/01/94 09/01/95 08/01/96 07/01/97 06/01/98 05/01/99 04/01/00 03/01/01 02/01/02 01/01/03 12/01/03 11/01/04 10/01/05 09/01/06 08/01/07 07/01/08 06/01/09 05/01/10 04/01/11 Option Strategy S&P 500 Equity/Bill Blend 20.00% 7

9 We also reviewed average annual returns to measure the consistency of the insurance risk premiumm and to ensure the compounded returns shown above are not influenced by any one large occurrence which is then compounded forward over the testing horizon. 8

10 Excess Return vs. 50% S&P 500/ 50% T-bill (1/1990-3/2012) Average Standardd Deviation Information Ratio Annual 3.82% 2.81% 1.36 One can infer the insurance risk premium is consistent over this period since the average annual excess return is similar to the average compounded excess return of 4.05%. In addition, the positive average annual excess return is statistically significant, and the correlation of excess return, driven by the insurance risk premium, is 0.15 to the beta adjusted market return. The insurance risk premium, driven by investor preferences, can be thought of as an alpha or quasi-active strategy exhibiting much more consistency than most traditional active strategies. Moreover, long-term returns can be achieved that are consistent with equity market returns with substantially less volatility through the inclusion of the insurance risk premium. We reviewed option pricing on other equity index options (for example, the R2000 and FTSE 100 Indices) and found similar persistence between implied volatility and realized volatility. The equity insurance risk premium is a global phenomenon across capitalization ranges. Riskss The absolute risks to this strategy are straightforward: the portfolio remains exposed to the equity market, albeit at a lower level, and will be vulnerable to lossess in the event t of significant equity market declines. Secondly, it is uncertain whether the equity insurance risk premium will continue to manifest itself as we have seen historically. The strategy of selling out of the money covered calls and covered puts will underperform the blended equity/bill benchmark under two predominant market environments. First, large short-term equity moves upwards or downwards may result in short-term underperform mance versus the blended benchmark. This is due to the potential exercise or cash settlement of an in-the-money option at exercise. Secondly, a trending market with low volatility may also result in relative underperformance. These two risks are mitigated to some extent by utilizing the dynamic strike price strategy. Conclusion Utilizing a relatively passive strategy, we have shown the existence of a consistent equity insurance risk premium which can be harvested by selling short-dated covered puts and calls. The insurance risk premium is uncorrelated to the risk most long-term portfolios are exposed to: equity risk. The addition of this strategy to the portfolio will therefore provide more diversification and improve the risk/return characteristics of the total portfolio. Unlike some other exotic risk factors investors have added to the portfolios, exposure to the equity insurance risk premium can be created utilizing exchange traded options. That is, there is little limitation to the amount of capital that can be exposed to the equity insurance risk premium, the strategy is liquid, and there is a lower cost of implementation depending upon the amount of active management the investor chooses to employ. 9

11 Will the magnitude of the equity insurance risk premium decline in the future? Certainly, as more investors become aware of the attractive attributes of this strategy, one would expect the risk premium to decline. Yet, academics have documented and studied this anomaly since the mid-80 s, and the premium still persists. The developments in behavioral finance suggest this risk premium will continue into the future. In either case, theree is certainly a first mover advantage to this strategy. For investors searching to diversify the equity sensitivity of their portfolio, and enhance the overall risk/return characteristics of the portfolio, we recommend pursuing a strategy to capture the equity insurance risk premium. 10

12 References: Arnold, T.., Crack, T., Schwartz, A. ( 2007), Inferring Risk-Averse Probability Distributionss From Option Prices using Implied Binomial Trees, Unpublished manuscript. Benth, F., Groth, M., Lindberg, C., 2010, The implied risk aversion from utility indifference option pricing in a stochastic volatility model, International Journal of Applied Mathematics & Statistics, 16(10), Christensen, B.J., Prabhala, N.R., 1998, The relation betweenn implied and realized volatility, Journal of Financial Economics, 50(1998), Cox, J., Ross, S., 1976, The Valuation of Options for Alternative Stochasticc Processes, Journal of Financial Economics, 3(1), Heston, S., 1993, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6, Hill, J., Balasubramanian, V., Gregory, K., Tierens, I., 2006, Finding Alpha via Covered Index Writing, Financial Analysts Journal, 62(5), Kahneman, D., Tversky, A., 1979, Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47(2), Neely, C., 2002, Forecasting Foreign Exchange Volatility: Why is Implied Volatility Biased and Inefficient? And Does It Matter, Federal Reservee Bank of St. Louis, Working Paper Series. Pena, A., Alemanni, B.., Zanotti, G., 2010, On the role of behavioral finance in the pricing of financial derivatives: the case of the S&P 500 options, Centre for Applied Research in Finance, Working Paper Series. Ruas, J., Curto, J., Nunes, J., 2012, The Implied Volatility Bias: A No-Arbitrage Approach for Short-Dated Options, Manuscript submitted for publication. Subrahmanyam, M., Franke, G., Stapleton, R., 1998, Why aree Options Expensive, New York University Working Paper. Versluis, C., Lehnert, T., Wolff, C., 2010, A Cumulative Prospect Theory Approach to Option Pricing, Working Papers of Centre of Research in Finance-Luxembourg School of Finance, University of Luxembourg. 11