Optimal Portfolio Allocation with Option-Implied Moments: A Forward-Looking Approach

Transcription

1 Optimal Portfolio Allocation with Option-Implied Moments: A Forward-Looking Approach Tzu-Ying Chen National Taiwan University, Taipei, Taiwan Tel: (+886) d @ntu.edu.tw San-Lin Chung National Taiwan University, Taipei, Taiwan Tel: (+886) chungs@management.ntu.edu.tw Yaw-Huei Wang National Taiwan University, Taipei, Taiwan Tel: (+886) wangyh@ntu.edu.tw This version: September 10, 2014 Abstract This paper proposes a forward-looking approach to estimate the individual stock moments from option prices, including option-implied mean, volatility, beta, and covariance, and use them as the inputs to determine the optimal portfolios. We find that using the forward-looking information to form the optimal portfolios leads to a significant improvement in the out-of-sample performance. In particular, its superiority is largely due to the use of forward-looking mean. Moreover, we show that options can further improve the asset allocation performance when they serve as a class of investment asset. JEL classification: C13, G11, G13, G17 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, one usually uses the backward-looking information (i.e. historical data and/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 the mean-variance optimization by showing that it does not consistently outperform the 1/N naive diversification, i.e. a portfolio strategy that allocate a fraction 1/N of wealth to each of the N risky assets available for investment at each rebalancing date. 2 To respond to the criticism of DeMiguel, Garlappi, and Uppal (2009), one stream of the literature develops more effective strategies that retain the features of the mean-variance optimization, e.g. see Tu and Zhou (2011) and Kirby and Ostdiek (2012). 3 The other approach for solving their criticism is to use forward-looking (i.e. derivative-implied) instead of historical information to improve the empirical 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, Garlappi, and Uppal (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 documented by DeMiguel, Garlappi, and Uppal (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 performance of the optimal portfolio. 4 In line with the later stream, we propose an alternative forward-looking approach in which all moments serving as inputs of the optimization, including means and variance-covariance matrix, are estimated from option prices. We provide a comprehensive empirical analysis for thirteen extensions of the mean-variance model across three empirical datasets of monthly returns with two evaluation criteria: (1) Sharpe ratio (SR) and (2) 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 the existing papers that support the usefulness of option-implied information for asset allocations. In contrast to our comprehensive datasets and models, Kostakis, Panigirtzoglou, and Skiadopoulos (2011) only use 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 Sassning (2014) also adopt all the equity option prices to extract the 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 (portfolio 3 in our Table 1) of 4 See, for example, Kostakis, Panigirtzoglou, and Skiadopoulos (2011) and DeMiguel, Plyakha, Uppal, and Vikov (2013), and Kempf, Korn, and Sassning (2014). 5 These empirical datasets and evaluation criteria are similar to those used by DeMiguel, Garlappi, and Uppal (2009). 3

4 our thirteen portfolio strategies. To the best of our knowledge, DeMiguel, Plyakha, Uppal, and Vikov (2013) is the only paper that uses option-implied information to estimate the expected returns of individual stocks for asset allocation purpose. 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, Plyakha, Uppal, and Vikov (2013), Kempf, Korn, and Sassning (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 portfolios 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 generally outperforms the 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 the options further enhance the portfolio performance when they serve as a class of investment asset, in addition to providing valuable information. 4

5 The three main contributions made by this our study are as follows. First, we reaffirm the usefulness of the mean-variance portfolio theory by providing a useful approach to eliminate the doubt on the estimation of portfolio moments documented by the DeMiguel, Garlappi, and Uppal (2009). Based on the analyses with the U.S. stocks and options for the sample period from January 1996 to December 2012, we find that using forward-looking information from option prices not only leads to a significant improvement on the out-of-sample performance, but also makes the mean-variance optimization portfolios outperform the 1/N naïve portfolio. Specifically, 80.77% of the optimization portfolios constructed with the forward-looking information outperform their corresponding portfolios constructed with the historical data and 58.97% of the differences are statistically significant at the 5% level. Moreover, 75.64% of the optimization portfolios constructed with the forward-looking information outperform the 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 for the asset allocation purpose. This finding is consistent with Merton (1980), Chopra and Ziemba (1993), and Kostakis, Panigirtzoglou, and Skiadopoulos 5

6 (2011) who also document 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, Plyakha, Uppal, and Vikov (2013). Third, we show that beyond providing valuable information for portfolio optimization, options also serve as an important class of assets to further enhance the asset allocation performance. For instance, we find that all the forward-looking stock-option portfolios outperform the naïve stock-option portfolio and % of the differences are statistically significant at the 5% level. Moreover, all the 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. 6

7 Finally, we draw our conclusions in Section Methodology Following the non-parametric 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 in order 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 (2013) with the prices of options on individual stocks to obtain the option-implied moments of individual stock returns. Finally, we construct the mean-variance 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 non-parametric 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, ( ), is related to its European option prices, ( ), as follows: 7

8 ( ) ( ) ( ), (1) where is the current price of the underlying asset, is the strike price, and is the time to maturity. In practice, options are traded only for a discrete set of strike prices and thus one has to apply 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 In specific, 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 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. The RND is generated under the risk-neutral framework. However, we have to consider investors risk preference and risk premium when constructing portfolios. Therefore, we follow the procedure suggested by Bliss and Panigirtzoglou (2004) to transform the RND to its corresponding RWD with a single-parameter utility as: ( ) ( ) ( ) ( ) ( ), (2) 6 Bliss and Panigirtzoglou (2002) provide strong evidence of the superior stability of the smoothed implied volatility estimation method. Moreover, this method is currently used by Bank of England. 7 Malz (1997) shows that smoothing the implied volatility function in the delta-space performs better than in the strike-space. 8

9 where ( ) is the RND, ( ) is the corresponding RWD, and ( ) is a utility function. We consider two most commonly used utility functions in the finance literature: (1) exponential utility function and (2) power utility function. The exponential utility function is defined as ( ) ( ) (3) where is the coefficient of absolute risk aversion (ARA). The power utility function is defined as ( ) (4) where is the coefficient of constant relative risk aversion (RRA). We take the parameter values estimated in Bliss and Panigirtzoglou (2004) to implement the transformation numerically, i.e. for exponential utility function and for power utility function. 8 Given the RWD of the market index, we numerically calculate the implied moments of market returns at time, including mean ( ) and variance ( ). In the following subsection, we detail how to compute the moments of individual stocks using both the corresponding stock option prices and the implied 8 For the power utility the RRA is simply, while for the exponential utility the RRA is dependent on both 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 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 moments of market returns. 2.2 Option-Implied Moments of Individual Stocks According to the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) and the Market Model (MM) of Sharpe (1963), the expected return and the covariance between two stock returns are given by: ( ) [ ( ) ], (5) ( ) ( ), (6) where ( ) is the expected return on the i th stock, is the risk-free rate, ( ) is the expected return on the market index, and and are the beta coefficients for stocks i and j, respectively. In order to compute ( ) and ( ) from Equations (5) and (6), what we need to deal with is 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. In this paper, we adopt the seminal method of Buss and Vilkov (2012) to obtain option-implied betas. By definition, the beta estimate can be written as: ( ) ( ) ( ) ( ) ( ) ( ) (7) where is the proportion of the market-mimicking portfolio invested on the j th 10

11 stock and ( ) is the pairwise stock correlations. To identify the option-implied correlation, Buss and Vilkov (2012) propose the following parametric form: ( ) ( ) [ ( )], (8) where ( ) is the expected correlation under the risk-neutral probability measure, ( ) is the expected correlation under the real-world (physical) probability measure, and denotes the parameter that is derived as: ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) The risk-neutral correlation is assumed to differ from the real-word correlation by a negative proportion,, 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 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 9 The real-world correlation is estimated from historical data in this study. 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 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 denotes the -dimensional vector of excess returns (over the risk-free asset) on the N risky assets, where is an vector of returns on the N risky assets, is the return on the risk-free asset at time, and is an vector of ones. Moreover, the vector of denotes the expected excess returns on the risky assets and denotes the corresponding variance-covariance matrix of excess returns at time. We consider the portfolio including risky assets only with denoting the vector of portfolio weights invested in the risky assets. The list of the models we consider is summarized in Table 1 and detailed as follow. [Insert Table 1 about here] The first approach 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 risky assets at time is: 12

13 (9) This naïve diversification across the N risky assets is also referred as the 1/N naïve portfolio. The second type of approach aims 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 (MV) portfolio or the minimum-variance (Min) portfolio, respectively. Under the standard mean-variance framework, the objective function is written as the following expected utility maximization: (10) where is the representative investor's coefficient of relative risk aversion, is the vector of portfolio weights for risky assets, and is the transpose of. The solution of Equation (10) provides a vector of portfolio weights for risky assets and the remainder ( ) is invested in the risk-free asset. Because of considering the portfolio with risky assets only, we normalize the weights of the optimal risky portfolio as: (11) so that the sum of normalized weights equal to one. Because it is very difficult to precisely estimate the expected means of asset returns, the minimum-variance model avoids the estimation risk by focusing on the 13

14 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: (12) Solving this minimization problem, the optimal risky portfolio weights are obtained as: (13) This model completely ignores the estimation of the expected returns and we thus needs to estimate the variance-covariance matrix of asset returns only. 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 the out-of-sample performance. The reason why a three-fund portfolio performs better than the optimal two-fund portfolio is motivated by the fact that 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 ( ), (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. 14

15 One of the advantages of the naïve portfolio is that it does not involve any parameter estimation. As documented 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 the reference portfolio to combine with the mean-variance portfolio, the minimum variance portfolio, or both to form an optimally combined portfolio. The first one is the combination of the naïve portfolio and the mean-variance portfolio. In specific, we follow the thoughts mentioned in Kan and Zhou (2007) to construct the portfolio weights of the combination under the structure ( ), (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 (2007). The normalized weights of the portfolio with risky assets only are expressed as. The second one is the combination of the naive portfolio and the minimum-variance portfolio. Following the same procedure as before, we then optimally combine portfolio weights as:, (16) 15

16 where c and d are chosen optimally to maximize the expected utility of a mean-variance investor. The third is the combination of the naïve portfolio, the mean-variance portfolio, and the minimum-variance portfolio. Based on the same procedure detailed above, the portfolio weights are given as: ( ), (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. Because it is often seen in practice that individual stocks are restricted from short selling, we also consider a number of portfolios that restrict short selling. With imposing an additional non-negativity constraint on the portfolio weights,, 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,,, and, respectively. Finally, we consider the minimum variance with generalized constraints (G-Min-C) portfolio proposed by DeMiguel, Garlappi, and Uppal (2009). This portfolio can be interpreted as a simple generalization of the shortsale-constrained minimum-variance portfolio. With imposing an additional constraint on the portfolio 16

17 weights, with [ ], to the aforementioned model in Equations (12), the solution of the corresponding optimization problems are denoted by. 11 Therefore, we consider fourteen portfolios in total for our empirical analysis Portfolio Performance Evaluation To evaluate the performance of all portfolios, we consider the following two performance measurements: (1) Sharpe ratio (SR) and (2) Certainty-equivalent return (CER). 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 SR of portfolio is defined as:. (18) This is an approach to evaluate portfolio performance in a risk-adjusted framework. To test whether the SRs 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 11 Following the same procedure as DeMiguel, Garlappi, and Uppal (2009), we consider the case with in our empirical analyses. 12 Best and Grauer (1991) document that the mean-variance portfolio weights are extremely sensitive to small changes in the estimation of parameters. Thus we follow the the same procedure as suggested by Kostakis, Panigirtzoglou, and Skiadopoulos (2011) that the portfolio weights are constrained in the interval [ 2]. 17

18 the correction pointed out in Memmel (2003). 13 The CER 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. In specific, the CER of portfolio can be computed as: (19) where and are the sample mean and sample variance of the excess returns for portfolio, and is the coefficient of constant relative risk aversion (RRA). 14 The higher the CER is, the better the portfolio performance is. To test whether the CERs from two portfolios are statistically different, we also compute the p-value of the difference, using the approach suggested by Green (2002) Data Description 13 Specifically, consider two portfolios i and bench, with,,,,, 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,. To do this, the test statistic is obtained by, which is asymptotically distributed as a standard normal:, with ( ). 14 As before, we adopt the RRA parameter estimated in Bliss and Panigirtzoglou (2004), i.e., for exponential utility function and for power utility function. 15 Specifically, consider two portfolios i and bench, with the vector of moments ν ( ) 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, ( 2 ) ( 2 ). To do this, we denote that f(ν) ( ) ( ) and the asymptotical distribution of f(ν)is T(f(ν ) f(ν)) N ( ν T Θ ν ), in which Θ ( ) 18

19 We include all optioned stocks traded in the U.S during the period from January 1996 to December The data on stock returns are obtained from the Center for Research in Securities Prices (CRSP) database. Table 2 describes the three empirical datasets considered in our study, which are similar to those used by DeMiguel, Garlappi, and Uppal (2009). These datasets consist of monthly excess returns on (1) 10 S&P sector portfolios and the market portfolio, 16 (2) 10 industry portfolios and the market portfolio, 17 and (3) the 25 size- and book-to-market portfolios and the market portfolio. [Insert Table 2 about here] The dataset 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 (ATM) and out-of-the-money (OTM) options are retained in the dataset 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 dataset consists of monthly excess returns on 10 value-weighted industry portfolios formed by using the Global Industry Classification Standard (GICS) developed by Standard & Poor s (S&P) and Morgan Stanley Capital International (MSCI). The 10 industries considered are Energy, Material, Industrials, Consumer-Discretionary, Consumer-Staples, Healthcare, Financials, Information-Technology, Telecommunications, and Utilities. 17 The Industry dataset consists of monthly excess returns on 10 industry portfolios formed by using the Standard Industrial Classification (SIC). The 10 industries considered are Consumer-Discretionary, Consumer-Staples, Manufacturing, Energy, High-Tech, Telecommunication, Wholesale and Retail, Health, Utilities, and Others. 19

20 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 the Merton (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 the 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 there are at least three usable implied volatilities, 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 We transform the ATM and OTM 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). 20

21 contracts expire on 20 January and 17 February, respectively, we construct portfolios on 23 January with the forward-looking moments computed on 22 January from the February contracts and then evaluate the portfolios on 17 February. This procedure is repeated every month until the end of December In total, portfolios are constructed and evaluated 204 times in our analyses. 5. Empirical Results based on Stock Portfolios In this section, we present our main results with 13 extensions of the standard mean-variance model across three empirical datasets of monthly returns in the case where an exponential utility function describes the preferences of the individual investors. We first revisit the framework adopted by DeMiguel, Garlappi, and Uppal (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 devote to demonstrate whether the use of forward-looking moments obtained from option prices are more informative than those provided by historical time series (backward-looking) for the asset allocation purpose. We proceed to identify which one of the forward-looking moments is the most useful for improving the out-of-sample performance. Finally, we provide a comprehensive comparison across 21

22 our empirical results and those documented 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 the length of the estimation period being 60 months. Table 3 reports the performance of all portfolio strategies (in terms of the SR and CER) with the p-values measuring the statistical significance of the differences from the 1/N naïve portfolio. [Insert Table 3 about here] As shown in Table 3, the SRs (CERs) of the 1/N naïve portfolio are (0.025), (0.026), and (-0.013) for the S&P Sectors, Industry, and 25 Size/BTM datasets, respectively. We find that most of the mean-variance optimization portfolios with inputs estimated from historical data do not perform well. In specific, totally 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, we conclude that the 19 The main insights of our analyses do not change when using a power utility function to describe the preferences of the individual investors. 22

23 out-of-sample performance of the mean-variance optimization portfolios is worse than the naïve diversification, which by and large confirms the findings documented by DeMiguel, Garlappi, and Uppal (2009). 5.2 Portfolios with Option-Implied Moments As documented in the literature, the unsatisfactory performance of the mean-variance optimization portfolios is likely to be caused by the estimation errors resulted from using historical data to estimate the expected moments. The second part of our empirical analyses investigates whether the performance of mean-variance portfolios can be improved by using the 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) 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 with the first ( vs. 1/N p-val ) against the naïve benchmark portfolio and the second ( vs. HIS p-val ) against 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 23

24 to 0.05, then 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 totally 75.64% of the optimization portfolios outperform the naïve diversification and 52.56% of the differences are statistically significant at the 5% level. Moreover, when comparing the performance of the mean-variance portfolios constructed with the forward-looking information to that of the corresponding portfolios with historical data, we find that 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 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 documented in DeMiguel, Garlappi, and Uppal (2009), the errors in estimating the portfolio moment may erode all the benefits gaining 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 using the forward-looking information implied in option prices has a superior potential for the improvement. 24

25 5.3 Sources of Improvement Since our previous empirical findings support the value of option-implied moments for the construction of a mean-variance optimization portfolio, we proceed to identify which forward-looking moment plays the most important role in improving the out-of-sample performance. We re-implement the portfolio construction with the following three information combinations: (1) historical mean & historical variance-covariance matrix, (2) historical mean & forward-looking variance-covariance matrix, and (3) forward-looking mean & forward-looking variance-covariance matrix. The existing studies are largely focused on improving the estimation of the variance-covariance matrix that uses forward-looking information from a cross-section of option prices. 20 Thus we first investigate the improvement of using forward-looking variance-covariance matrix for the construction of mean-variance optimization portfolios. Then we show that whether the forward-looking mean can benefit for the out-of-sample performance even further. Table 5 reports the source of improvement of the forward-looking approach For instance, Kempf, Korn, and Sassning (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 the eight of fourteen portfolio models which depend jointly on mean and variance listed in Table 1. 25

26 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. One of the p-values ( vs. PA p-val ) is evaluated against the backward-looking information with historical mean and variance-covariance matrix (i.e. the results shown in Panel A). The other one ( vs. PB p-val ) is evaluated against the second type of information with historical mean and forward-looking variance-covariance matrix (i.e. the results shown in Panel B). 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 0.05, then we reject the null hypothesis. [Insert Table 5 about here] When improving the estimation of variance-covariance matrix individually (Panel B, Table 5), we find that most of the mean-variance optimization portfolios do not perform well. In specific, totally 41.67% of the optimization portfolios outperform the corresponding portfolios with historical information, and only 18.75% of the differences are statistically significant at the 5% level. 22 However, when considering both forward-looking mean and variance-covariance matrix (Panel C, Table 5), we find that 81.25% of the optimization portfolios further improve the performance from 22 DeMiguel, Plyakha, Uppal, and Vikov (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. 26

27 that of benchmark portfolios using only forward-looking variance-covariance matrix and 70.83% of the differences are statistically significant at the 5% level. 23 Because the historical data tends 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 forward-looking mean. Our finding confirms that the estimation of the expected returns is a crucial task for the implementation of the mean-variance optimization, which has been neglected in almost all of previous studies utilizing option-implied variance-covariance matrix only. 5.4 A Revisit to DeMiguel et al. (2013) Results DeMiguel, Plyakha, Uppal, and Vikov (2013) propose an alternative forward-looking approach by using option-characteristic-adjusted means in the construction of optimal portfolios. They use the option-based characteristics to rank stocks and then adjust the mean of benchmark returns with those characteristics as a scaling factor. More 23 When comparing the results shown in Panels A and C, 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, Garlappi, and Uppal (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 3000 months for a portfolio with 25 assets and about 6000 months for a portfolio with 50 assets. They conclude that there are still many miles to go before the gains promised by optimal portfolio selection can actually be realized out of sample. 27

28 precisely, they specify the linear function ( ) ch ( k δ k k ), where ch is the expected benchmark return, k is the sorting index of stock with respect to characteristic, and δ k is the effect of characteristic on the conditional mean at time. Note that δ k is not asset-specific, but constant across assets. Given the positive relation between characteristic and the stock returns 25, they define k equal to 1 if the stock is located in the top decile (decile 10) in the cross-section of all stocks, equal to -1 if the stock is located in the bottom decile (decile 1) in period, and otherwise equal to 0. In their empirical analysis, they consider the grand mean return across all stocks as the benchmark return and choose the parameter δ k δ k. 26 Following DeMiguel, Plyakha, Uppal, and Vikov (2013), we use three option-implied characteristics to adjust mean returns individually to isolate the effect of each option-based characteristic; the first one is model-free implied volatility (MFIV), the second one is model-free implied skewness (MFIS), and the last one is call-put volatility spread (CPVS). We obtain both measures, MFIV and MFIS, from 25 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 expected stock returns. Bali and Hovakimian (2009), 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, Plyakha, Uppal, and Vikov (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 equal to 0.10, it is inevitable to restrict the returns of input components with the range 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. 28

29 the model-free approach developed by Bakshi, Kapadia, and Madan (2003). Using the approach in Bali and Hovakimian (2009), we compute CPVS measure 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 further examine that 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, ( ) ch. 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, Plyakha, Uppal, and Vikov (2013) to examine how the mean-variance optimization portfolios perform when using the option-characteristic-adjusted means. 27 Table 6 reports the out-of-sample performance for various short-sale-constrained mean-variance portfolios. Panel A provides the results based on the moments estimated from historical data. Panel B provides the results by using the option-implied characteristics to adjust the mean 27 DeMiguel, Plyakha, Uppal, and Vikov (2013) analyze the portfolio selection problem among a large set of stocks and provide evidence on the mean-variance portfolios that restrict on short selling. Thus DeMiguel, Plyakha, Uppal, and Vikov (2013) only consider one of the thirteen extensions of the standard mean-variance model that we analyze in our study. 29

30 returns of benchmark while Panel C makes no adjustment to that of benchmark. We report two p-values with the first ( vs. 1/N p-val ) against the naïve 1/N benchmark portfolio and the second ( vs. HIS p-val ) against the corresponding optimization portfolios produced by the historical moments (i.e. the results shown in Panel A). 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 0.05, then we reject the null hypothesis. [Insert Table 6 about here] As shown in Panel B, we can see that the examined mean-variance portfolios with option-characteristic-adjusted means can 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 dataset, 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, it is 0.525, 0.534, and 0.530, respectively. This result is consistent with the findings of DeMiguel, Plyakha, Uppal, and Vikov (2013). However, as shown in Panel C, we find 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 dataset, the Sharpe ratio for the portfolio using BENCH is 0.525, which is very close to the Sharpe ratios 30

31 of portfolios with option-characteristic-adjusted means. Therefore, this finding indicates that using 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 examine the relative performance between our forward-looking approach and the alternative proposed by DeMiguel, Plyakha, Uppal, and Vikov (2013) for the optimal portfolio allocation. Tables 7 and 8 reports the performance of the mean-variance portfolios generated with the approach proposed by DeMiguel, Plyakha, Uppal, and Vikov (2013) and ours in terms of SR and CER, respectively. 28 In each table, we report the results for the S&P Sectors, Industry, and 25 Size/BTM datasets in Panels A, B, and C, respectively. The p-values are evaluated for our forward-looking approach against the option-characteristic-adjusted ones. The null hypothesis is that the evaluated portfolio constructed from our approach is no better than that from the alternative approach. If the p-value is smaller than or equal to 0.05, then we reject the null hypothesis. [Insert Table 7 about here] [Insert Table 8 about here] Comparing the results of Tables 7 and 8 with those of Table 3, the mean-variance 28 In order to investigate the benefits of using option-characteristic-adjusted means for forming portfolios, we consider the eight of thirteen portfolio models which care about the mean of returns listed in Table 1. 31

32 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, we find that 64.58%, 58.33%, and 75.00% (70.83%) of the mean-variance portfolios outperform the traditional portfolios with historical moment estimates and 41.67%, 35.42%, and 41.67% (39.58%) of the performance differences are statistically significant at the 5% level, respectively. Moreover, comparing these findings to those shown 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 ours 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 and 45.83%, 56.25%, 52.08%, and 47.92% of the differences are statistically significant at the 5% level, respectively. In summary, although the option-characteristic-adjusted means approach of DeMiguel, Plyakha, Uppal, and Vikov (2013) has improved 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 32

33 information from option prices. Moreover, the empirical results suggest that our forward-looking approach further significantly improve the alternative approach proposed by DeMiguel, Plyakha, Uppal, and Vikov (2013) for the asset allocation purpose. 6. Empirical Results based on Stock-Option Portfolios 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 one to construct a profitable option portfolio. However, the existing literature has paid little attention to the inclusion of stock options in the optimal portfolio allocation. In the previous section, we have shown that the option prices serves as a useful information source to improve the portfolio construction. In this section, we extend our approach to examine whether adding options into the stock portfolio can further improve the out-of-sample performance. We consider a zero-cost option portfolio proposed by Goyal and Saretto (2009) and demonstrate how to construct the long-short delta-hedged portfolio as follows As a robustness check, we repeat the analysis using the long-short straddle portfolio and the results remain the same. 33

34 First, we sort individual stock options into deciles based on the difference between historical volatility (HV) and option-implied volatility (IV). Decile 1 consists of stock options with the lowest difference while decile 10 consists of stock options with the highest difference between these two volatility measures. Second, for stock options in each decile, we form the delta-hedged option portfolios by buying one call contract and short-selling 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 decile (decile 10) and a short position on the options in the bottom decile (decile 1). Table 9 presents the out-of-sample performance of the optimal stock-option portfolios with the use of option-implied moments. We report two sets of p-values. The first ( vs. 1/N p-val ) set of p-values shown in the first row of Table 9 is for the case where the benchmark portfolio is the 1/N naïve portfolio. The second ( vs. SP p-val ) set of p-values is evaluated against the optimal stock portfolios with forward-looking moments obtained from option prices (i.e. the results shown in Table 4). 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 0.05, then we reject the null hypothesis. 34