Improving Portfolio Selection Using Option-Implied Moments. This version: October 14, Abstract

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1 Improving Portfolio Selection Using Option-Implied Moments Tzu-Ying Chen *, San-Lin Chung and Yaw-Huei Wang This version: October 14, 2014 Abstract This paper proposes a forward-looking approach to estimate individual stock moments from option prices, including option-implied mean, volatility, beta, and covariance, and uses these inputs to determine optimal portfolios. We find that using forward-looking information to form optimal portfolios leads to a significant improvement in out-of-sample performance. In particular, its superiority is largely due to the use of forward-looking mean. We also show that options can improve asset allocation performance when they serve as a class of investment asset. * Department of Finance, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 106, Taiwan. Tel: d @ntu.edu.tw. Department of Finance, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 106, Taiwan. Tel: chungs@management.ntu.edu.tw. Department of Finance, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 106, Taiwan. Tel: wangyh@ntu.edu.tw. We thank the Ministry of Technology of Taiwan for financial support. 1

2 1. Introduction Based on the mean-variance framework proposed by Markowitz (1952), numerous extensions and applications in finance have become the foundation of modern portfolio theories. 1 When implementing the mean-variance framework, extant studies usually use backward-looking information (i.e., historical data or models) to estimate the expected mean and variance covariance matrix of asset returns. However, empirical evidence indicates that the optimized portfolio does not necessarily perform well. In particular, DeMiguel, Garlappi, and Uppal (2009) question the value of mean-variance optimization by showing that it does not consistently outperform 1/N naive diversification, a portfolio strategy that allocates the fraction 1/N of wealth to each of the N risky assets available for investment at each rebalancing date. 2 In response to DeMiguel et al. s (2009) criticism, one stream of the literature develops more effective strategies that retain the features of the mean-variance optimization (e.g., Tu and Zhou, 2011; Kirby and Ostdiek, 2012). 3 The other approach is to use forward-looking 1 See Grinold and Kahn (1999), Litterman (2003), and Meucci (2005) for practical applications of the mean-variance framework; and see Brandt (2009) for a recent survey of the academic literature. 2 The results of DeMiguel et al. (2009) suggest that the errors in estimating the portfolio means and covariances erode all the benefits gained from the optimization. 3 Tu and Zhou (2011) propose an optimal combination of the 1/N naïve portfolio with the existing theory-based portfolios as a way to improve the performance. Kirby and Ostdiek (2012) find that the poor performance reported by DeMiguel et al. (2009) is largely due to high estimation risk and extreme portfolio turnover and thus propose two alternative mean-variance timing portfolios that are designed to exploit the sample information about the means and variances in a way that mitigates the impact of estimation risk and leads to moderate portfolio turnover. 2

3 (i.e., derivative-implied) rather than historical information to improve the empirical performance of the optimal portfolio. 4 In line with this latter stream, we propose an alternative forward-looking approach in which all moments serving as inputs of optimization, including means and the variance-covariance matrix, are estimated from option prices. In addition to the 1/N naïve portfolio, we provide a comprehensive empirical analysis for 13 different portfolio models across three empirical data sets of monthly returns with two evaluation criteria: Sharpe ratio (SR) and certainty-equivalent return (CER). 5 Our empirical results strongly support the value of the forward-looking approach for optimal asset allocations. Our paper is close to but different from existing studies that support the usefulness of option-implied information for asset allocations. In contrast to our comprehensive data sets and models, Kostakis, Panigirtzoglou, and Skiadopoulos (2011) use only S&P 500 index option prices to extract the stock index implied distributions as inputs to calculate the optimal portfolio consisting of a risky (S&P 500 index) and a riskless asset. Although Kempf, Korn, and Saβning (2014) adopt all equity option prices to extract forward-looking information, they only use them to estimate the variance-covariance matrix and thus focus on the study of global minimum variance portfolio, a special case of our 13 portfolio strategies. To the best of our knowledge, DeMiguel, Plyakha, Uppal, and Vikov (2013) is the only study that uses 4 See, for example, Kostakis et al. (2011), DeMiguel et al. (2013), and Kempf et al. (2014). 5 These empirical data sets and evaluation criteria are similar to those used by DeMiguel et al. (2009). 3

4 option-implied information to estimate the expected returns of individual stocks for asset allocation purposes. However, they simply fine-tune the mean returns based on the grand mean return across all stocks and only for the stocks in the top and bottom deciles (i.e., with those of the remaining 80% of stocks having the same grand mean return). Different from DeMiguel et al. (2013), Kempf et al. (2014), and other previous studies on optimal portfolios with forward-looking approaches, we estimate all the required moments from option prices and comprehensively test a wide range of optimal portfolio strategies under the mean-variance framework of Markowitz (1952). In addition to confirming that the mean-variance optimization portfolio is not necessary to outperform the 1/N naive portfolio based on the moments estimated from historical data, we further explore why our forward-looking approach in general outperforms naive diversification. Specifically, we identify which one of the forward-looking moments is the most useful to improve the out-of-sample performance. Finally, we investigate to what extent options further enhance portfolio performance when they serve as a class of investment asset in addition to providing valuable information. This study makes three main contributions to the literature. First, we reaffirm the usefulness of the mean-variance portfolio theory by providing a useful approach to eliminate doubt on the estimation of portfolio moments reported by DeMiguel et al. (2009). Based on the analyses of U.S. stocks and options for the sample period from January 1996 to December 4

5 2012, we find that using forward-looking information from option prices leads to a significant improvement of out-of-sample performance and that mean-variance optimization portfolios outperform the 1/N naïve portfolio. Specifically, 80.77% of optimization portfolios constructed with forward-looking information outperform their corresponding portfolios constructed with historical data, and 58.97% of the differences are statistically significant at the 5% level. In addition, 75.64% of optimization portfolios constructed with forward-looking information outperform naïve diversification, and 52.56% of the differences are statistically significant at the 5% level. Second, we identify the source of improvements and find that the superiority of our forward-looking approach is mainly due to the use of the option-implied means. Our empirical results suggest that the estimation of the expected returns is the most critical task of asset allocation. This finding is consistent with Merton (1980), Chopra and Ziemba (1993), and Kostakis et al. (2011) who also report that expected returns estimation is the most crucial input for mean-variance optimization portfolios. By improving the mean returns estimation, the performance of our forward-looking approach is statistically superior to that suggested by DeMiguel et al. (2013). Third, we show that, beyond providing valuable information for portfolio optimization, options serve as an important class of assets that further enhance asset allocation performance. For instance, we find that all forward-looking stock-option portfolios 5

6 outperform the naïve stock-option portfolio, and % of the differences are statistically significant at the 5% level. In addition, all forward-looking stock-option portfolios significantly outperform the forward-looking stock only portfolio at the 5% level. The remainder of the paper is organized as follows. Section 2 outlines the methodology of moment estimates using the forward-looking information from option prices. Section 3 introduces various asset-allocation models from the portfolio choice literature that we consider. Section 4 describes the data on the stocks and options. Section 5 presents the main empirical results of the optimal portfolio allocation for each portfolio strategy and discusses the relative performance between the use of forward-looking and backward-looking moments. Section 6 extends to evaluate the portfolio performance when options are included as a class of investment asset. Finally, Section 7 provides our conclusions. 2. Method Following the nonparametric methods developed by Bliss and Panigirtzoglou (2002, 2004), we first generate the risk-neutral and real-world densities of the stock market index from the prices of index options to estimate the first two moments of the index return. Given the estimated two moments of the market index return, we then adopt the approach proposed by Buss and Vilkov (2012) with the prices of options on individual stocks to obtain the option-implied moments of individual stock returns. Finally, we construct the mean-variance 6

7 optimization portfolio using the forward-looking moments of individual stocks. 2.1 Option-Implied Distributions and the Moments of the Market Index We first estimate the option-implied risk-neutral density (RND) of the market index using the nonparametric method developed by Bliss and Panigirtzoglou (2002) and then follow the approach proposed by Bliss and Panigirtzoglou (2004) to determine the corresponding real-world density (RWD). Breeden and Litzenberger (1978) show that the RND of the underlying asset price at time T, g(s T ), is related to its European option prices, C(S t, K, T, t), as g(s T ) = e r(t t) 2 C(S t,k,t,t) K 2 K=ST, (1) where S t is the current price of the underlying asset, K is the strike price, and T t is the time to maturity. In practice, options are traded only for a discrete set of strike prices, and we must apply thus a smoothing continuous price function from the available option contract prices to compute the twice-differentiable function in Equation (1). We adopt the implied volatility smoothing method suggested by Bliss and Panigirtzoglou (2002) to estimate the RND for the market index. 6 Specifically, we first collect the prices of at-the-money and out-of-the-money index options, compute their Black Scholes implied volatilities, and use the natural cubic spline interpolation for the curve fitting of the implied volatility function in 6 Bliss and Panigirtzoglou (2002) provide strong evidence of the superior stability of the smoothed implied volatility estimation method. This method is currently used by Bank of England. 7

8 the delta-space. 7 Finally, we estimate the RND numerically with the option prices converted from the fitted implied volatility function with the Black Scholes formula. RND is generated under the risk-neutral framework. However, we must consider investors risk preference and risk premium when constructing portfolios. Therefore, we follow the procedure suggested by Bliss and Panigirtzoglou (2004) to transform RND to its corresponding RWD with a single-parameter utility as f(s T ) = g(s T ) U (S T ) g(x) U (x) dx, (2) where g(s t ) is the RND, f(s t ) is the corresponding RWD, and U( ) is a utility function. We consider two most commonly used utility functions in the finance literature: (i) the exponential utility function and (ii) the power utility function. The exponential utility function is defined as U(S T ) = exp( ηs T ), η 0, (3) η where η is the coefficient of absolute risk aversion (ARA). The power utility function is defined as U(S T ) = S T 1 γ 1, γ 1, (4) 1 γ where γ is the coefficient of constant relative risk aversion (RRA). We use the parameter values estimated in Bliss and Panigirtzoglou (2004) to implement the transformation 7 Malz (1997) shows that smoothing the implied volatility function in the delta-space performs better than in the strike-space. 8

9 numerically, that is, RRA = 3.96 for exponential utility function and RRA = 4.08 for power utility function. 8 Given the RWD of the market index, we numerically calculate the implied moments of market returns at time t, including mean E(R mt ) and variance Var(R mt ). In the following subsection, we detail how to compute the moments of individual stocks using both the corresponding stock option prices and the implied moments of market returns. 2.2 Option-Implied Moments of Individual Stocks According to the capital asset pricing model of Sharpe (1964) and Lintner (1965) and the market model of Sharpe (1963), the expected return and the covariance between two stock returns are given, respectively, as E(R it ) = R ft + β it [E(R mt ) R ft ], (5) Cov(R it, R jt ) = β it β jt Var(R mt ), (6) where E(R it ) is the expected return on the ith stock, R ft is the risk-free rate, E(R mt ) is the expected return on the market index, and β it and β jt are the beta coefficients for stocks i and j, respectively. To compute E(R it ) and Cov(R it, R jt ) from Equations (5) and (6), we 8 For the power utility the RRA is simply γ, while for the exponential utility the RRA is dependent on both η and the level of the underlying asset. At each time t, we employ the estimate of RRA and the index realizations to determine the corresponding values for η when the exponential utility is used. In the later section, the empirical results reported are for the case of exponential utility function only. The main insights of our analyses do not change when using a power utility function to describe the preferences of the individual investors. 9

10 need to identify the beta coefficients from the prices of options written on individual stocks, because the mean and variance of market return can be estimated from market index option prices as described in the previous subsection. We adopt the seminal method of Buss and Vilkov (2012) to obtain option-implied betas. By definition, the beta estimate can be written as β it = Cov(R it,r mt ) Var(R mt ) = N j=1 w jt Var(R it )Var(R jt )Corr(R it,r jt ), (7) Var(R mt ) where w jt is the proportion of the market-mimicking portfolio invested on the jth stock and Corr(R it, R jt ) is the pairwise stock correlations. To identify the option-implied correlation, Buss and Vilkov (2012) propose the following parametric form: Corr Q (R it, R jt ) = Corr P (R it, R jt ) α t [1 Corr P (R it, R jt )], (8) where Corr Q (R it, R jt ) is the expected correlation under the risk-neutral probability measure Q, Corr P (R it, R jt ) is the expected correlation under the real-world (physical) probability measure P, and α t denotes the parameter that is derived as N N Var Q (R mt ) i=1 j=1 w it w jt Var Q (R it )Var Q (R jt )Corr P (R it, R jt ) α t =. N N w it w jt Var Q (R it )Var Q (R jt ) (1 Corr P (R it, R jt )) i=1 j=1 The risk-neutral correlation is assumed to differ from the real-word correlation by a negative proportion, α t, of the distance between the maximum correlation and the real-world correlation. 9 Thus, substituting the option-implied correlation in Equation (7) with that in 9 The real-world correlation is estimated from historical data in this study. 10

11 Equation (8), we obtain the option-implied beta and plug it into Equations (5) and (6) to yield the mean and covariance estimates of the individual stock returns Portfolio Construction and Performance Evaluation We consider several asset-allocation models frequently adopted in the literature of portfolio construction. The performance of these models is evaluated with two criteria. 3.1 Asset-Allocation Models Let r t = R t 1 N R ft denotes the N-dimensional vector of excess returns (over the risk-free asset) on the N risky assets, where R t is an N 1 vector of returns on the N risky assets, R ft is the return on the risk-free asset at time t, and 1 N is an N 1 vector of ones. In addition, the N 1 vector of μ t denotes the expected excess returns on the N risky assets, and Σ t denotes the corresponding N N variance-covariance matrix of excess returns at time t. We consider the portfolio including risky assets only with w t denoting the vector of portfolio weights invested in the N risky assets. Table 1 provides the details for the models that we consider. [Insert Table 1 about here] 10 As suggested by Taylor, Yadav, and Zhang (2010), we use the implied volatility of the at-the-money option to estimate the variance of the individual stock returns because it outperforms the model-free implied volatility of Jiang and Tian (2005) for volatility forecasting. 11

12 The first approach that we consider is the naïve portfolio that allocates an equal weight across assets available for investment. This simple model does not involve any optimization or estimation and completely ignores all the data-implied information. The vector of portfolio weights for the N risky assets at time t is w t 1N = 1 N N. (9) This naïve diversification across the N risky assets is also referred as the 1/N naïve portfolio. The second type of approach attempts to reach the optimal trade-off between return and risk or the minimization on risk in a single objective function, which is referred as the mean-variance portfolio and the minimum-variance portfolio, respectively. Under the standard mean-variance framework, the objective function is written as the following expected utility maximization: max x t x t T μ t γ 2 x t T Σ t x t, (10) where γ is the representative investor's coefficient of RRA, x t is the vector of portfolio weights for risky assets, and x t T is the transpose of x t. The solution of Equation (10) provides a vector of portfolio weights x t MV for risky assets, and the remainder (1 1 N T x t MV ) is invested in the risk-free asset. Because this portfolio considers risky assets only, we normalize the weights of the optimal risky portfolio as w t MV = = x MV t 1 T N xt MV Σ 1 t μt 1 T N Σt 1 μt (11) so that the sum of normalized weights equal to 1. 12

13 Because estimating the expected means of asset returns precisely is very difficult, the minimum-variance model avoids the estimation risk by focusing on the minimization of risk only. To implement the minimum-variance portfolio, we choose the vector of portfolio weights that minimize the variance of portfolio returns as min w t w T t Σ t w t s. t. 1 T N w t = 1. (12) Solving this minimization problem, the optimal risky portfolio weights are obtained as w t Min = Σ t 1 1 N 1 T N Σt 1 1N. (13) This model completely ignores the estimation of the expected returns, and we thus need only to estimate the variance-covariance matrix of asset returns. Kan and Zhou (2007) introduce a three-fund portfolio that optimally combines the risk-free asset, the mean-variance portfolio, and the minimum-variance portfolio to maximize out-of-sample performance. A three-fund portfolio performs better than the optimal two-fund portfolio because adding another risky portfolio can help to diversify away the estimation risk of the two-fund portfolio. The mixture portfolio can be expressed as a combination of the sample-based mean-variance portfolio and the minimum-variance portfolio: x tmv Min = 1 γ (cσ t 1 μ t + dσ t 1 1 N ), (14) where c and d are chosen optimally to maximize the expected utility of a mean-variance investor. Because we consider the portfolio with risky assets only, the portfolio weights are normalized as w tmv Min = x tmv Min 1 T N x tmv Min. 13

14 One of the advantages of the naïve portfolio is that it does not involve any parameter estimation. As reported by Tu and Zhou (2011), the naïve portfolio can significantly improve the performance of any one of the optimization-based portfolios. Therefore, we also use the naïve portfolio as a reference portfolio in combination with the mean-variance portfolio, the minimum variance portfolio, or both to form an optimally combined portfolio. First, to combine the naïve portfolio and the mean-variance portfolio, we follow Kan and Zhou (2007) to construct the portfolio weights as x t1n MV = 1 γ (c 1 N 1 N + dσ t 1 μ t), (15) where c and d are chosen optimally to maximize the expected utility of a mean-variance investor. The concepts of the parameters c and d are similar to those of the optimally chosen parameters described in three-fund portfolio of Kan and Zhou. The normalized weights of the portfolio with risky assets only are expressed as w t1n MV = x t1n MV 1 T N x t1n MV. Second, to create the naive portfolio and the minimum-variance portfolio, we follow the same procedure as before and optimally combine portfolio weights as w t1n Min = c 1 N 1 N + dσ t 1 1 N, s. t. 1 N T w t1n Min = 1, (16) where c and d are chosen optimally to maximize the expected utility of a mean-variance investor. Finally, to combine the naïve portfolio, the mean-variance portfolio, and the minimum-variance portfolio, we again follow the same procedure as previously detailed and 14

15 give the portfolio weights as x t1n MV Min = 1 γ (c 1 N 1 N + dσ t 1 μ t + eσ t 1 1 N ), (17) where c, d, and e are chosen optimally to maximize the expected utility of a mean-variance investor. The normalized weights of the portfolio with risky assets only are given as w t1n MV Min = x t1n MV Min 1 T N x t1n MV Min. Because in practice individual stocks are often restricted from short selling, we also consider a number of portfolios that restrict short selling. We impose an additional nonnegative constraint on the portfolio weights, w t 0, to each of the aforementioned models in Equations (10), (12), (14), (15), (16), and (17); the solutions of the corresponding optimization problems are denoted by w t MV C, w t Min C, w tmv Min C, w t1n MV C, w t1n Min C, and w t1n MV Min C, respectively. Finally, we consider the minimum variance with generalized constraints portfolio proposed by DeMiguel et al. (2009). This portfolio can be interpreted as a simple generalization of the short sale-constrained minimum-variance portfolio. By imposing an additional constraint on the portfolio weights, w t a1 N with a [0, 1 N], to the model in Equation (12), the solution of the corresponding optimization problems is denoted as w t G Min C. 11 Therefore, in addition to the 1/N naïve portfolio, we consider 13 different 11 Following the same procedure as DeMiguel et al. (2009), we consider the case with a = 1 analyses. 2N in our empirical 15

16 portfolio models in total for our empirical analysis Portfolio Performance Evaluation To evaluate the performance of all portfolios, we consider the following two performance measurements: Sharpe ratio and certainty-equivalent return. The Sharpe ratio is designed to measure the reward obtained for each unit of risk taken. For an investment, reward is measured by the sample mean of excess returns, and risk is measured by the sample standard deviation of excess returns. Thus, the Sharpe ratio is defined as SR i = μ i, (18) σ i where SRi is the Sharpe ratio of portfolio i, μ i is mean excess returns of portfolio i, and σ i is the standard deviation of excess returns of portfolio i. This approach evaluates portfolio performance in a risk-adjusted framework. To test whether the Sharpe ratio of a particular portfolio are statistically distinguishable from that of another portfolio that serves as a benchmark, we compute the p-values of the differences, using the approach suggested by Jobson and Korkie (1981) after making the correction pointed out in Memmel (2003) Best and Grauer (1991) find that the mean-variance portfolio weights are extremely sensitive to small changes in the estimation of parameters. Thus we follow the same procedure as suggested by Kostakis et al. (2011) and constrain the portfolio weights to the interval [ 1, 2]. 13 Specifically, consider two portfolios i and bench, with μ i, μ b, σ i, σ b, σ i,b, as their estimated means, variances, and covariances over a sample size T. We want to test the hypothesis that the Sharpe ratio of portfolio i is worse (smaller) than that of the benchmark portfolio bench; that is, H 0 : μ i σ i μ b σ b 0. To do this, the test statistic is obtained by z JK, which is asymptotically distributed as a standard normal: 16

17 The certainty-equivalent return is measured as the risk-free rate of return that an investor is willing to accept for forgoing the investment in a particular risky portfolio. Specifically, certainty-equivalent return is computed as CER i = μ i γ 2 σ i 2, (19) where CER is the certainty-equivalent return of portfolio i, μ i and σ i2 are the sample mean and sample variance, respectively, of the excess returns for portfolio i, and γ is the coefficient of constant RRA. 14 The higher the certainty-equivalent return is, the better the portfolio performance is. To test whether the certainty-equivalent returns from two portfolios are statistically different, we also compute the p-value of the difference, using the approach suggested by Greene (2002) Data Description z JK = σ bμ i σ iμ b, with θ = 1 θ T (2σ i 2 σ b2 2σ iσ bσ i,b μ i 2 1 σ b2 + 2 μ b 2 σ i2 μ iμ b 2 σ i,b ). σ iσ b 14 As before, we adopt the RRA parameter estimated in Bliss and Panigirtzoglou (2004), that is, RRA = 3.96 for the exponential utility function and RRA = 4.08 for the power utility function. 15 Specifically, consider two portfolios i and bench, with the vector of moments ν = (μ i, μ b, σ i 2, σ b 2 ) and ν is the empirical counterpart obtained from a sample of size T. We want to test the hypothesis that the certainty-equivalent return of portfolio i is worse (smaller) than that of the benchmark portfolio bench; that is, H 0 : (μ i γ 2 σ i 2 ) (μ b γ 2 σ b 2 ) 0. To do this, we denote that f(ν) = (μ i γ 2 σ i 2 ) (μ b γ 2 σ b 2 ) and the asymptotical distribution of f(ν)is T(f(ν ) f(ν)) N (0, f ν T Θ f ν 2 σ i σ i,b σ i,b σ b 0 0 Θ = σ i 2σ i,b 2 4 ( 0 0 2σ i,b 2σ b ) ), in which 17

18 We include all optioned stocks traded in the United States from January 1996 to December The data on stock returns are obtained from the Center for Research in Securities Prices database. Table 2 describes the three empirical data sets considered in our study, which are similar to those used by DeMiguel et al. (2009). These data sets consist of monthly excess returns on (i) 10 S&P sector portfolios and the market portfolio, 16 (ii) 10 industry portfolios and the market portfolio, 17 and (iii) 25 size- and book-to-market portfolios and the market portfolio. [Insert Table 2 about here] The data set of the S&P 500 index and equity options is obtained from the OptionMetrics Ivy DB database. In this empirical study, we focus on the options with a constant (one-month) maturity. Only at-the-money and out-of-the-money options are retained in the data set due to the liquidity concern. We collect daily closing bid and ask quotes over our sample period and take the averages of bid and ask quotes to represent the market prices of options. Following the standard data-filtering procedure adopted in the literature, we filter 16 The S&P Sector data set consists of monthly excess returns on 10 value-weighted industry portfolios formed by using the Global Industry Classification Standard developed by Standard & Poor s and Morgan Stanley Capital International. The 10 industries considered are energy, material, industrials, consumer-discretionary, consumer-staples, healthcare, financials, information-technology, telecommunications, and utilities. 17 The Industry data set consists of monthly excess returns on 10 industry portfolios formed using SIC codes. The 10 industries considered are consumer-discretionary, consumer-staples, manufacturing, energy, high-tech, telecommunication, wholesale and retail, health, utilities, and others. 18

19 option contracts with the following three criteria to ensure that the selected options contain the most reliable information. First, we discard options with an ask quote less than or equal to the bid quote or with a bid quote less than or equal to zero. Options with a missing bid or ask quotes are also omitted from the sample. Second, we eliminate options with prices violating Merton s (1973) arbitrage restrictions. For example, the call option prices must be lower than the underlying stock price but higher than the underlying stock price minus the present values of the strike prices and the dividends. Third, we discard deep-moneyness contracts with a delta greater than 0.99 or less than 0.01 due to the liquidity concern. 18 A smoothed implied volatility curve is therefore constructed if at least three usable implied volatilities exist, with the lowest delta less than or equal to 0.25 and the highest delta greater than or equal to We construct and evaluate portfolios with the following procedure. On the first trading day right after the most nearby contracts expire, we use the prices of the new most nearby contracts to compute the forward-looking moments as our inputs to construct the optimal portfolios on the following trading day. We hold the portfolios until the expiration of the new most nearby contracts and then evaluate the performance of the portfolios. For example, given that the January and February 1996 contracts expire on January 20 and February 17, respectively, we construct portfolios on January 23 with the forward-looking moments computed on January 22 from the February contracts and then evaluate the 18 We transform the at-the-money and out-of-the-money put delta values (negative) into the corresponding call delta values (positive) using the put call parity. Thus all the delta values are positive and in the interval (0,1). 19

20 portfolios on February 17. We repeat this procedure every month until the end of December In total, we construct and evaluate portfolios 204 times in our analyses. 5. Empirical Results Based on Stock Portfolios In this section we present our main results with 13 different portfolio models across three empirical data sets of monthly returns in the case where an exponential utility function describes the preferences of individual investors. We first revisit the framework adopted by DeMiguel et al. (2009) to examine how the mean-variance optimization portfolios perform against the 1/N naive portfolio based on the moments estimated from historical data. Then, we investigate whether the use of forward-looking moments obtained from option prices are more informative than those provided by historical time-series (backward-looking) data for the asset allocation purpose. Furthermore, we identify which forward-looking moments are the most useful for improving out-of-sample performance. Finally, we provide a comprehensive comparison across our empirical results and those reported in the literature Portfolios with Historical Moments The first part of our empirical analyses investigates the performance of the mean-variance optimization portfolios against the 1/N portfolio. We use the rolling-window approach, with a 19 The main insights of our analyses do not change when using a power utility function to describe the preferences of the individual investors. 20

21 60-month estimation period. Table 3 reports the performance of all portfolio strategies (in terms of the Sharpe ratio and certainty-equivalent return); p-values measure the statistical significance of the differences from the 1/N naïve portfolio. [Insert Table 3 about here] Table 3 shows that the Sharpe ratios (certainty-equivalent returns) of the 1/N naïve portfolio are (0.025), (0.026), and ( 0.013) for the S&P Sectors, Industry, and 25 Size/BTM data sets, respectively. Most of the mean-variance optimization portfolios with inputs estimated from historical data do not perform well. Specifically, 38.46% of the optimization portfolios outperform the naïve diversification, and only 15.38% of the differences are statistically significant at the 5% level. Hence, when using the estimates of the historical moments, the out-of-sample performance of the mean-variance optimization portfolios is worse than the naïve diversification, which, by and large, confirms the findings of DeMiguel et al. (2009). 5.2 Portfolios with Option-Implied Moments As reported in the literature, the unsatisfactory performance of the mean-variance optimization portfolios is likely to be caused by estimation errors resulting from the use of historical data to estimate the expected moments. The second part of our empirical analysis investigates whether the performance of mean-variance portfolios can be improved by using 21

22 forward-looking information implied in option prices to estimate the expected moments. We consider the use of forward-looking moments obtained from option prices with risk-adjusted mean, option-implied covariance, and option-implied beta detailed in Equations (5), (6), and (7), respectively, for each individual stock. Table 4 presents the empirical results with the option-implied information for the optimal portfolio allocation. We report two p-values: compared to the naïve benchmark portfolio and compared to the corresponding optimization portfolios produced by the historical moments. The null hypothesis is that the evaluated portfolio based on forward-looking estimates performs no better than the compared portfolio. If the p-value is smaller than or equal to 5%, we reject the null hypothesis. [Insert Table 4 about here] Comparing the performance of the mean-variance portfolios constructed with the forward-looking information with that of the 1/N naïve portfolio, Table 4 shows that 75.64% of the optimization portfolios outperform the naïve diversification and that 52.56% of the differences are statistically significant at the 5% level. In addition, in the comparison between the performance of the mean-variance portfolios constructed with the forward-looking information and the corresponding portfolios with historical data, 80.77% of the forward-looking portfolios outperform their corresponding historical portfolios and 58.97% of the differences are statistically significant at the 5% level. These findings indicate that the 22

23 mean-variance optimization portfolios constructed with the forward-looking information from option prices not only lead to a significant improvement in the out-of-sample performance but also significantly outperform the 1/N naïve portfolio. As reported by DeMiguel et al. (2009), the errors in estimating the portfolio moment may erode all the benefits gained from optimization. Therefore, we suggest that the performance of the mean-variance portfolios can be effectively raised by improving the estimation quality of the expected moments of the portfolios and that using the forward-looking information implied in option prices has the potential to improve estimation quality and therefore out-of-sample performance. 5.3 Sources of Improvement Because our previous empirical findings support the value of option-implied moments for the construction of a mean-variance optimization portfolio, we next identify which forward-looking moment plays the most important role in improving out-of-sample performance. We reimplement portfolio construction with the following three information combinations: (i) historical mean and the historical variance-covariance matrix, (ii) historical mean and the forward-looking variance-covariance matrix, and (iii) forward-looking mean and the forward-looking variance-covariance matrix. Existing studies largely focus on improving the estimation of the variance-covariance 23

24 matrix that uses forward-looking information from a cross-section of option prices. 20 Thus we first investigate the improvement of the forward-looking variance-covariance matrix for the construction of mean-variance optimization portfolios. We then investigate whether the forward-looking mean can improve out-of-sample performance even further. Table 5 reports the source of the improvement of the forward-looking approach. 21 Panels A to C present the out-of-sample performance with three different types of information combinations for the optimal portfolio allocation. We report two p-values in parenthesis. Panel B shows p-values evaluated against historical mean and the variance-covariance matrix. Panel C provides p-values evaluated against historical mean and the forward-looking variance-covariance matrix. The null hypothesis is that the evaluated portfolio performs no better than the compared portfolio. If the p-value is smaller than or equal to 5%, we reject the null hypothesis. [Insert Table 5 about here] The results in Table 5, Panel B, shows that when improving the estimation of variance-covariance matrix individually, most of the mean-variance optimization portfolios do not perform well. Specifically, 41.67% of the optimization portfolios outperform the corresponding portfolios with historical information, and only 18.75% of the differences are 20 For instance, Kempf et al. (2014) focus on global minimum variance portfolio with a family of the estimators of the covariance matrix that relies on forward-looking information. 21 We consider 8 out of 13 portfolio models that depend jointly on mean and variance listed in Table 1. 24

25 statistically significant at the 5% level. 22 However, Panel C shows that when considering both forward-looking mean and the variance-covariance matrix, 81.25% of the optimization portfolios further improve the performance from that of benchmark portfolios using only the forward-looking variance-covariance matrix and 70.83% of the differences are statistically significant at the 5% level. 23 Because the historical data tend to be limited and noisy, these findings indicate that the outperformance is particularly strong when considering forward-looking mean and variance-covariance matrix simultaneously. 24 Overall, we find that the superiority of the forward-looking approach largely comes from the use of the forward-looking mean. Our finding confirms that the estimation of expected returns is a crucial task for the implementation of mean-variance optimization, which has been neglected in almost all previous studies utilizing the option-implied variance-covariance matrix only. 22 DeMiguel et al. (2013) examine the ability of forward-looking information implied in option prices to improve the out-of-sample performance of various minimum-variance portfolios in terms of portfolio volatility, Sharpe ratios, and turnover. They conclude that using option-implied volatility only helps to reduce the portfolio volatility while using option-implied correlation does not improve any of the metrics. 23 When comparing the results shown in Panels A and C of Table 5, we find that 89.58% of the forward-looking portfolios outperform their corresponding historical portfolios and 60.42% of the differences are statistically significant at the 5% level. 24 DeMiguel et al. (2009) provide the simulated results and find that the estimation window required for the sample-based mean-variance portfolios to outperform the naïve benchmark portfolio is around 3,000 months for a portfolio with 25 assets and about 6,000 months for a portfolio with 50 assets. They conclude: There are still many miles to go before the gains promised by optimal portfolio selection can actually be realized out of sample. 25

26 5.4 DeMiguel et al. (2013) Revisited DeMiguel et al. (2013) propose an alternative forward-looking approach that uses option-characteristic-adjusted means in the construction of optimal portfolios. They use option-based characteristics to rank stocks and then adjust the mean of benchmark returns with those characteristics as a scaling factor. More precisely, they specify the linear function E(R i,t+1 ) = μ i,t = μ bench,t (1 + K k=1 δ k,t x ik,t ), where μ bench,t is the expected benchmark return, x ik,t is the sorting index of stock i with respect to characteristic k, and δ k,t is the effect of characteristic k on the conditional mean at time t. Note that δ k,t is not asset-specific but constant across assets. Given the positive relation between characteristic k and the stock returns, 25 they define x ik,t equal to 1 if stock i is located in the top decile (decile 10) in the cross-section of all stocks, equal to 1 if stock i is located in the bottom decile (decile 1) in period t, and otherwise zero. In their empirical analysis, they consider the grand mean return across all stocks as the benchmark return and choose the parameter δ k,t = δ k = Both the variance risk premium and implied skewness have been shown in the literature with a significant predictive power to explain the cross-section of stock returns. Bali and Hovakimian (2009) show that the implied-realized volatility spread is significant positively linked to the expected stock returns. Bali and Hovakimian, Cremers and Weinbaum (2010), and Xing, Zhang, and Zhao (2010) find a positive relation between various measures of implied skewness and future stock returns. 26 Instead of inferring the mean estimates from the risk-adjusted probability density function, DeMiguel et al. (2013) use the option-based characteristics to adjust mean returns based on the grand mean return across all stocks directly. They adjust the mean returns for the stocks in the top and bottom deciles only and regard the mean returns of the remaining 80% of stocks as the same as the benchmark. As they let the intensity of the scaling factor δ k,t equal to 0.10, they inevitably restrict the returns of the input components with the range 26

27 Following DeMiguel et al. (2013), we use three option-implied characteristics to adjust mean returns individually to isolate the effect of each option-based characteristic: model-free implied volatility (MFIV), model-free implied skewness (MFIS), and call-put volatility spread (CPVS). We obtain both MFIV and MFIS from the model-free approach developed by Bakshi, Kapadia, and Madan (2003). Using the approach of Bali and Hovakimian (2009), we compute CPVS as the difference between the current Black-Scholes implied volatilities of the one-month at-the-money call and put options. In addition to evaluating the benefits of using option-based characteristics to form mean-variance portfolios, we also examine whether the improvements in the out-of-sample performance are largely determined by the information content from option prices. Thus we consider the other case (BENCH) in which the mean returns make no adjustment to that of benchmark. That is, for each stock i, E(R i,t+1 ) = μ i,t = μ bench,t. This simple case does not rely on option-implied information and assumes that all assets have the same expected return. We first revisit the framework adopted by DeMiguel et al. (2013) to examine how the mean-variance optimization portfolios perform when using option characteristic-adjusted means. 27 Table 6 reports the out-of-sample performance for various short-sale constrained from 90% to 110% of the benchmark and thus the adjusted mean is still very close to that of the grand mean across all stocks. 27 DeMiguel et al. (2013) analyze the portfolio selection problem among a large set of stocks and provide evidence on the mean-variance portfolios that restrict short selling. In other words, they only examine portfolio 7 listed in Table 1. 27

28 mean-variance portfolios. Panel A provides the results based on the moments estimated from historical data. Panel B provides the results using the option-implied characteristics to adjust the mean returns of benchmark. Panel C makes no adjustment to that of benchmark. We report two p-values: compared to the naïve 1/N benchmark portfolio and compared to the corresponding optimization portfolios produced by the historical moments. The null hypothesis is that the evaluated portfolio performs no better than the compared portfolio. If the p-value is smaller than or equal to 5%, then we reject the null hypothesis. [Insert Table 6 about here] Panel B of Table 6 shows that the examined mean-variance portfolios with option-characteristic-adjusted means not only lead to a significant improvement in the out-of-sample performance but also significantly outperform the 1/N naïve portfolio. For example, for the S&P Sectors data set, the Sharpe ratio for the naïve portfolio is and that of the mean-variance portfolio with historical moments is For the portfolios using MFIV, MFIS, and CPVS, the Sharpe ratio is 0.525, 0.534, and 0.530, respectively. This result is consistent with the findings of DeMiguel et al. (2013). However, Panel C shows that the mean-variance optimization portfolio performs well, even when we do not consider those option-based characteristics. For example, for the S&P Sectors data set, the Sharpe ratio for the portfolio using BENCH is 0.525, which is very close to the Sharpe ratios of the portfolios with option-characteristic-adjusted means. Therefore, this finding indicates that using 28

29 option-based characteristics to adjust mean returns is not an effective way to improve the estimation errors of expected moments for portfolio construction. Next, we compare the relative performance of our forward-looking approach and the alternative proposed by DeMiguel et al. (2013) for the optimal portfolio allocation. Tables 7 and 8 report the performance of the mean-variance portfolios generated in terms of the Sharpe ratio and the certainty-equivalent return, respectively, following our approach and the approach proposed by DeMiguel et al. 28 In each table, we report the results for the S&P Sectors, Industry, and 25 Size/BTM data sets in Panels A, B, and C, respectively. P-values are evaluated for our forward-looking approach against the option-characteristic-adjusted approach. The null hypothesis is that the evaluated portfolio constructed from our approach is no better than that of the alternative approach. If the p-value is smaller than or equal to 5%, we reject the null hypothesis. [Insert Table 7 about here] [Insert Table 8 about here] The results of Tables 7 and 8 compared with the results of Table 3 shows that the mean-variance portfolios based on option-characteristic-adjusted means significantly outperform the corresponding portfolios with historical moments at the 5% level. When using MFIV, MFIS, and CPVS (BENCH) as the adjusting (non-adjusting) characteristic, 64.58%, 28 To investigate the benefits of using option-characteristic-adjusted means for forming portfolios, we consider 8 out of 13 portfolio models that take into account the mean of returns listed in Table 1. 29

30 58.33%, and 75.00% (70.83%) of the mean-variance portfolios outperform the traditional portfolios with historical moment estimates, respectively, and 41.67%, 35.42%, and 41.67% (39.58%) of the performance differences are statistically significant at the 5% level, respectively. In addition, comparing these findings to those in Table 4 for our forward-looking approach, we find that the gains from incorporating the option-characteristic-adjusted approach are much smaller than those provided by our approach for the asset allocation purpose. As indicated by the p-values of the differences between these two approaches, 77.08%, 79.17%, 77.08%, and 72.92% of the optimization portfolios constructed from our approach perform better than those formed based on the MFIV, MFIS, CPVS, and BENCH characteristic, respectively, and 45.83%, 56.25%, 52.08%, and 47.92% of the differences are statistically significant at the 5% level, respectively. In sum, although the option-characteristic-adjusted means approach of DeMiguel et al. (2013) improves the performance of the mean-variance portfolios based on historical information, the improvements in the out-of-sample performance are not fully attributed to the use of forward-looking information from option prices. In addition, the empirical results suggest that our forward-looking approach further significantly improves the alternative approach proposed by DeMiguel et al. for the asset allocation purpose. 6. Empirical Results Based on Stock-Option Portfolios 30

31 Goyal and Saretto (2009) find that a trading strategy based on the deviation between historical realized volatility and option-implied volatility generates an economically and statistically significant option portfolio return. This evidence suggests that the information contained in these two volatility measures allows the construction of a profitable option portfolio. However, the existing literature pays little attention to the inclusion of stock options in optimal portfolio allocation. In the previous section, we show that the option prices serve as a useful information source to improve portfolio construction. In this section, we extend our approach to examine whether adding options into the stock portfolio can further improve out-of-sample performance. We consider the zero-cost option portfolio proposed by Goyal and Saretto (2009) and demonstrate how to construct the long-short delta-hedged portfolio as follows. 29 First, we sort individual stock options into deciles based on the difference between historical volatility and option-implied volatility. Decile 1 (decile 10) consists of stock options with the lowest (highest) difference between these two volatility measures. Second, for stock options in each decile, we form delta-hedged option portfolios by buying one call contract and short-selling the delta shares of the underlying stock. For each stock option and each month in the sample, we select the call contract that is closest to at-the-money with one month to maturity. Finally, we construct a zero-cost option portfolio by taking a long position on the options in the top 29 As a robustness check, we repeat the analysis using the long-short straddle portfolio, and the results remain the same. 31