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M A T R N Y FO Implied Expected Returns and the Choice of a Mean–Variance Efficient Portfolio Proxy A TH K RIS BOUDT IT IS IL LE G A L TO R EP R O D U C E is associate professor of finance and econometrics at Vrije Universiteit Brussel and VU University Amsterdam and research partner at Finvex Group in Brussels. [email protected] 68 can be more accurately predicted than mean returns.2 Second, if w is a mean–variance efficient portfolio with respect to a universe of assets with known expected returns vector μ and covariance matrix Σ, then there exists a linear relation between the assets’ expected return μ and their covariance with that efficient portfolio, i.e., Σw . Hence, if a mean– variance efficient weight vector is known and a covariance matrix has been accurately estimated, the above-mentioned linear relation can be exploited to create implied expected returns which may have a higher accuracy than the time series model based return forecasts. This idea of reverse engineering for expected returns goes back to Sharpe [1974], Fisher [1975] and Best and Grauer [1985]. Its application in practice was boosted by the research of Black and Litterman [1991, 1992]. An elementary feature in the implementation of reverse engineering of expected returns is the choice of proxy for the mean– variance efficient portfolio. Traditionally the standard choice has been to use the market capitalization weighted portfolio. The proponents of this approach tend to emphasize the result that under the capital asset pricing model (CAPM) the market capitalization portfolio is mean–variance efficient. However, for almost all portfolio managers, the R TI growing amount of capital is attracted by smart beta equity solutions, such as the equalweighted, equal-risk-contribution, fundamental indexation, and maximum diversification portfolio.1 The determination of portfolio weight does not require any return forecast, but from the portfolio weight and the covariance matrix, an implied expected return can be computed. We investigate the usefulness of this alternative implied expected return, in comparison with the standard approach of deriving implied expected returns from the market capitalization weighted portfolio. The implied expected return methodology is founded on two fundamental insights. First, the availability of higher frequency data allows estimates of covariances at a higher accuracy than expected returns. In fact, as shown by Merton [1980], using higher sampling frequencies improves significantly the accuracy of the covariance estimator, but not the estimation of the mean parameter. As it is commonly done in practice, we use weekly observations to estimate the covariance matrix over a monthly horizon. Another reason is that the mean and covariance matrix are time-varying parameters. Since the timevariation in the covariance matrix is much more persistent than the time-variation in the expected return, covariance matrices IS is assistant professor of finance at Laval University in Québéc City, QC, Canada. [email protected] C A DAVID A RDIA LE IN A DAVID ARDIA AND KRIS BOUDT I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY SUMMER 2015 Copyright © 2015 JPM-ARDIA.indd 68 7/21/15 2:13:08 PM investable universe does not correspond to all tradable assets and hence, the CAPM conditions are by construction not verified. We therefore advocate that the market capitalization weighted portfolio is only one possible proxy for a mean–variance efficient portfolio and that other proxies may lead to more accurate expected returns. We test this hypothesis empirically for monthly returns of the S&P 100 stocks over the 1987–2012 period. Importantly, the economic benefits of the choice of proxy for computing implied expected returns are examined not only in terms of forecast precision (as measured by the hit ratio and root-mean-squared forecast error), but also in terms of stability, both crosssectionally (in order to avoid extreme investment bets, as described for instance in Michaud [1989]) and from a time series perspective (to reduce portfolio turnover). In addition, we compare those implied expected returns with the return forecasts obtained from various time series models. The analysis is done for a broad set of alternative mean–variance portfolio proxies, namely the smoothed market capitalization portfolio (Chen et al. [2007]), the fundamentally weighted portfolio (Arnott et al. [2005]), the equally weighted portfolio (DeMiguel et al. [2009]), the inverse volatility weighted portfolio (Leote De Carvalho et al. [2012]), the equalrisk-contribution portfolio (Maillard et al. [2010]), the maximum diversification portfolio (Choueifaty and Coignard [2008]) and the risk-eff icient portfolio (Amenc et al. [2011]). We refer the reader to Lee [2011] for a recent overview on smart beta portfolios and Leote De Carvalho et al. [2012] and Clarke et al. [2013] for empirical evidence on the risk factor exposures of those strategies. We conclude that, for all proxies considered, the implied expected returns outperform the time series model based forecasts in terms of forecast precision and stability. We find that the choice of a mean–variance portfolio proxy matters and that the largest gains are in terms of stability of the return forecasts. The use of a maximum diversification or equal-risk-contribution portfolio as proxy reduces significantly the cross-section and time series dispersion in the implied expected returns and leads to a small improvement in forecast precision, compared with using a market capitalization, fundamental value or equal weighting scheme. This result is shown to be robust with respect to the choice of the SUMMER 2015 JPM-ARDIA.indd 69 level of risk aversion as well as the length of the estimation window. REVERSE ENGINEERING EXPECTED RETURNS FROM MEAN–VARIANCE EFFICIENT PORTFOLIOS A general description of the implied expected return methodology starts by specifying the investment universe, the first and second moments of the asset returns, and the portfolio weights that will be assumed to be mean–variance efficient. We consider a market with N risky securities and denote a generic portfolio in this market by the (N × 1) vector w. The expected arithmetic returns at the desired holding horizon are denoted by the ( 1) vector μ, and the corresponding ( ) covariance matrix of arithmetic returns is denoted by Σ. We drop the time dimension for simplicity but consider the conditional moments as time-dependent. Let ι be the ( 1) vector 1) vector of ones and 0 the ( of zeros. A mean–variance efficient portfolio is defined as a portfolio that for some value of the risk-aversion parameter γ , 0 < γ < ∞ , maximizes the mean–variance utility function under a full investment constraint: w ∗ ≡ argmax w ∈CFFI { } (1) where C FI ≡ {w {w R N | w ′ 1} is the set of portfolio weights satisfying the full-investment constraint. While this problem is easy to solve explicitly, it still remains that accurate estimates of the expected return vector μ are difficult to obtain. At the same time, investors often have a good proxy available for a mean–variance efficient portfolio. A common approach in practice is therefore to turn the problem around and instead of optimizing the portfolio weights with a set of expected returns and covariance matrix, extract from a given portfolio structure the set of implied expected returns for which the portfolio is mean–variance efficient. To derive these implied expected returns formally, we need the first order conditions of mean–variance efficiency. The Lagrangian corresponding to (1) is L( w l ) w ′μ γ w ′ w l( w ′ 2 1) THE JOURNAL OF PORTFOLIO M ANAGEMENT 69 7/21/15 2:13:09 PM with l ∈R . The corresponding first order conditions are μ − γΣw lι 0 w ′ι 1 (2) (3) From (2)–(3), it follows that the generic solution to the mean–variance optimization problem in (1) is w= 1 −1 Σ (μ l ) γ (4) It also follows from the first order condition in (2) that there exists a linear relationship between μ and Σw: μ = lι + γ Σw (5) Note that as γ is finite, (5) excludes the minimum variance portfolio. The linear relationship in (5) is well known, and for given values of l, γ, Σ and w, it can be used to compute the implied expected returns. The intercept parameter l is a consequence of imposing (3), i.e., the budget constraint, in the calculation of implied expected returns, as recommended by Herold [2005]. He shows that this is necessary to obtain meaningful implied expected returns. As in the original publication of Black and Litterman [1991], we set l to the risk-free rate and γ = 2.4. In the sensitivity study, we also present results for γ = 1 and γ = 5. OVERVIEW OF PROXIES FOR MEAN– VARIANCE EFFICIENT PORTFOLIOS Our main analysis pertains to the choice of proxy for the mean–variance efficient portfolio weights w. The most popular one is the market capitalization weighted portfolio, which we denote by w mkt . It has the mean– variance efficiency property under the CAPM assumptions, but in most practical cases, these assumptions are unrealistic (Goltz and Le Sourd [2011]). This claim is supported by an increasing body of literature that criticizes the mean–variance efficiency of the market capitalization weighted portfolio and proposes alternatives that (under different assumptions) are mean–variance efficient. The first strand of research that we consider is the fundamental indexing literature (Arnott et al. [2005], 70 JPM-ARDIA.indd 70 Treynor [2005]). The main motivation for this approach is that in noisy markets, overpriced stocks receive a higher weight in the market capitalization weighted portfolio than in the mean–variance efficient one, and vice versa for underpriced stocks. If pricing errors are corrected, this leads to a drag on returns for market-capitalization-based portfolios. Perold [2007] and Graham [2012] emphasize, however, that this result is sensitive to the assumption that stock prices will systematically revert to fair value, which is not directly testable since the fair value is not observable. Chen et al. [2007] argue that by smoothing the market capitalization weights, pricing errors are reduced and hence a more efficient estimate of the fundamental market capitalization weights are obtained. We take the same setup as in Chen et al. [2007] and consider the median capitalization weights over a five-week period. We denote this portfolio by w smkt . An alternative is to proxy the firm’s fair value using fundamental metrics of company size such as book value of common equity, dividends, net operating cash f low and sales. This leads to the fundamentally weighted portfolios popularized by Arnott et al. [2005]. We follow their framework and compute the fundamental portfolio weights as the average value of the normalized version of the fundamental size proxies, that is, w fund ≡ 4 1 4 ⎛ ∑ ⎜⎝ ∑ k =1 max{xk ,1, 0} N j =1 max{{xk j , 0}} , …, ⎞′ ∑ j 1 max{xk j, 0} ⎟⎠ max{xk ,N , 0} N with x1,i the most recently observed end of fiscal year value of the firm i ’s book value of common equity and x 2,i , x 2,i , x 4,i five-year trailing averages of the yearly value of dividends, net operating cash f low, and sales, respectively. In our application, the yearly fundamental data are retrieved from the Compustat database and are lagged by one quarter to avoid any possible look-ahead bias. Net operating cash f low is taken as the difference between the operating income before depreciation and total accruals (Kothari et al. [2005]). A second strand of research relates to the general objective of achieving diversified portfolios, whereby the focus of diversification implicitly leads to a portfolio that is mean–variance efficient under some assumptions. The most popular mean–variance efficient portfolio proxies that achieve efficiency by focusing on diversification are the following. I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY SUMMER 2015 7/21/15 2:13:10 PM Equally weighted portfolio. The equally weighted portfolio, which we denote by w ew , spreads its capital investment equally over all N assets, i.e., w ew (1 (1 )ι. DeMiguel et al. [2009] show that, for various popular datasets, the 1/N allocation rule outperforms several optimized portfolios. Plyakha et al. [2014] benchmark the equal weighting approach against value and price weighted strategies. They find an outperformance in approximately 65% of the investigated periods. Part of the outperformance is explained by higher exposure to the market, size, value and momentum factors. From (4) it follows that this portfolio is mean– variance efficient when the expected excess returns of the stocks, i.e., (μ − lι), are proportional to the sum total of their covariances, i.e., Σι. Inverse volatility weighted portfolio. The inverse volatility weighted portfolio, called the equalrisk-budget portfolio by Leote De Carvalho et al. [2012], allocates to the N stocks with volatility σ 1 ,…,σ N the weight ⎛ ⎞′ 1/σ w iv ≡ ⎜ 1N/σ1 ,…, N N ⎟ ∑ j 11/σ j ⎠ ⎝ ∑ j =111/σ j As noted by Leoto De Carvalho et al., the inverse volatility weighted portfolio is mean–variance efficient with the highest Sharpe ratio if the Sharpe ratio for each stock is the same and all pairwise correlations are equal. Equal-risk-contribution portfolio. The equalrisk-contribution portfolio is the portfolio for which all assets contribute equally to the overall portfolio volatility. Or equivalently, it is the portfolio for which the percentage volatility risk contribution of all N assets equals 1/N, where percentage volatility risk contribution of the ith asset is given by %RC i ≡ w i [ Σw ]i w ′Σ ′ w We denote this portfolio by w erc. Numerically, it is computed by solving the following optimization problem 3 : ⎧N ⎫ w erc ≡ argmin ⎨∑ (%RC i − N1 )2 ⎬ w ∈CFFI ⎩ i =1 ⎭ SUMMER 2015 JPM-ARDIA.indd 71 with CFI≡{ w N w 1}, the set of weights guaranteeing that the portfolio is fully invested. Maillard et al. [2010] and Roncalli [2013] derive the theoretical properties of the equal-risk-contribution portfolio and show that its volatility is located between those of the minimum variance and the equally weighted portfolio. Bruder and Roncalli [2012] show that the equal-risk-contribution portfolio is mean– variance efficient when all assets contribute equally to the portfolio excess return w ′μ − l . Maximum diversif ication portfolio. Let σ ≡ diag( Σ ) be the ( 1) vector of standard deviations of arithmetic returns of the N assets in the universe. An important property of the portfolio standard deviation is its sub-additivity, meaning that, because of diversification effects, the portfolio’s standard deviation is always less or equal to the weighted average volatility: w ′′Σw ≤ w ′σ or equivalently that the ratio between the weighted average volatility and the portfolio volatility exceeds one: DR( w ) ≡ w ′σ ≥1 w ′Σ ′ w (6) Choueifaty and Coignard [2008] call (6) the portfolio’s diversification ratio and define the maximum diversification portfolio as the portfolio that has the highest diversification ratio. From (6) it is clear that the maximum diversification portfolio coincides with the maximum Sharpe ratio portfolio in the special case where the stocks’ expected returns (in excess of the riskfree rate) are proportional to their volatility. This special case is also discussed by Martellini [2008]. We denote this portfolio by w md . Risk-eff icient portfolio. Amenc et al. [2011] recommend to construct a maximum Sharpe portfolio under the assumption that the stock’s expected return is a deterministic function of its semi-deviation and the cross-sectional distribution of semi-deviations, and where portfolio weights are constrained to be between 1/(2N ) and 1/N. More precisely, they sort stocks by their semi-deviation, form decile portfolios, and then compute the median semi-deviation of stocks in each decile portfolio, which we denote by ξ j ( j = 1 …,10). The risk-efficient portfolio is then given by THE JOURNAL OF PORTFOLIO M ANAGEMENT 71 7/21/15 2:13:11 PM w ref ≡ argmax w ∈C FI∗ { } where J is a ( 10) matrix of zeros whose (i,j )th element is 1 if the semi-deviation of stock i belongs to decile j, ξ ≡ ( ξ1 , ξ10 )′ and CFFI∗ ≡ {w {w R N | w ′ 1,(1 / (2N ))ι ≤ w ≤ (2 ( / N ) } is the full-investment constraint with bound restrictions on the portfolio weights. Results The differences in allocation between the alternative mean–variance efficient portfolio proxies and the market capitalization weighted portfolio can be interpreted as active allocation bets, which lead to bets in the implied returns. The translation of these allocation bets into implied returns bets interacts with the correlation and variance properties of the stocks in the mean– variance efficient portfolio. This effect can be seen in more detail from the definition of the implied return for stock i that follows from the general expression in (5): ⎛ ⎜ N ⎞ ⎟ ⎝ j =1 ⎠ μ i = λ + γ ( Σw )i = λ + γ cov ⎜⎜ ri , ∑ w j r j ⎟⎟ =λ+γΣ ⎛ ⎜ 1/2 ⎜ ⎜ ii ⎜ ⎜ ⎜⎝ N Σ w i + ∑ ρij Σ w 1/2 ii j=1 j =i / 1/2 jj ⎞ ⎟ ⎟ ⎟ j⎟ ⎟ ⎟⎠ (7) (8) with ρij the correlation between the returns of stock i and j. We thus see that the implied expected return for stock i is proportional to the covariance between the stock’s return and the return on the mean–variance efficient portfolio. Ceteris paribus, this covariance will tend to be higher for stocks that have: (i) a relatively larger weight in the efficient portfolio proxy, (ii) a high volatility, and (iii) a high correlation with the stocks in the efficient portfolio. The propagation of the weights bets to implied returns bets thus interacts with the variance and correlation properties of the individual stock returns. The aim of this section is to provide a better understanding of the sensitivity of the stability and forecast precision of the monthly implied expected returns to the choice of a mean–variance efficient portfolio proxy, and compare those with common time series model based 72 JPM-ARDIA.indd 72 forecasts of expected returns. The empirical analysis provides a better insight in the conclusion of Lee [2011] that, if risk-based allocation approaches outperform the market portfolio, it must be that the implied expected returns are a better set of return forecasts than the market’s implied expected returns. The analysis is done for a sample of large U.S. stocks over the period January 1984 to December 2012. The expected return forecast horizon that we consider is one month and the prediction is made at the end of the month. The baseline results assume a rolling estimation window of three years, such that the out-of-sample evaluation period ranges from January 1987 to December 2012 (312 monthly observations). Data and Estimation Methods For each month between January 1984 and December 2012, the universe is restricted to the 100 end-of-month largest S&P 500 equities for which there are no missing observations in the estimation sample used to forecast the covariance matrix and expected returns. For convenience, we call this henceforth the S&P 100 universe. Results are presented for the overall universe and for each of the ten GICS sectors separately. The adjusted price data, market capitalization, and firm fundamentals have been retrieved from COMPUSTAT. The accounting data used to compute the fundamental weights have a yearly frequency. We follow Arnott et al. [2005] by using five-year trailing averages of the yearly value of dividends, net operating cash f low, and sales to avoid excessive volatility in the index weights due to year-to-year variability in the accounting data. We obtained the Fama and French [1993] or Carhart [1997] factors computed for the U.S. market from Kenneth French’s website.4 The risk-free rate is the three-month U.S. Treasury bill. All figures are in U.S. dollars. In total, we are comparing twelve return-forecast methods, eight of which are obtained using the implied expected return methodology. The others are classified as time series model based forecasts because they explicitly model the dynamics in the stock returns. We consider the constant mean model, the autoregressive model of order one (AR(1)), as well as the Fama and French [1993] and Carhart [1997] three- and four-factor models, whereby the factors also follow an AR(1) process. All models are estimated on a rolling estimation sample of three years (156 weeks), which is common practice in the financial I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY SUMMER 2015 7/21/15 6:43:07 PM industry. As shown by Merton [1980], using higher sampling frequencies improves significantly the accuracy of the covariance estimator, but not the estimation of the mean parameter. As it is commonly done in practice, we use monthly returns to estimate the constant mean, AR(1), and factors models by ordinary least squares. For the covariance matrix, we use weekly log-returns and the exponentially weighted moving average (EWMA) estimator with a decay parameter of 0.94 as advocated by RiskMetrics Group [1996]. The covariance matrix is initialized at the Ledoit and Wolf [2003] shrinkage estimate of the unconditional covariance matrix over the estimation window. The covariance matrix is then projected into a monthly arithmetic returns covariance matrix estimator as in Meucci [2004]. In the sensitivity analysis, we also consider rolling samples of two and four years. Performance and Sector Allocation of Mean–Variance Efficient Portfolio Proxies tribution portfolio and the lowest for the fundamental value portfolio (0.35). The performance of the market capitalization weighted portfolio is in between those two extremes, with a volatility of 15.6%, a VaR95 of −7%, and a Sharpe ratio of 0.41. We thus find that the allocation style has only a minor effect on the out-of-sample average return but matters for the portfolio risk: risk-optimized portfolios are characterized by a significantly lower risk and a similar return as the market capitalization weighted portfolio. The observed difference in risk-adjusted performance of the various mean–variance efficient portfolio proxies can be partly attributed to underlying sector bets, which become visible when aggregating the portfolio weights at the sector level. Panel B of Exhibit 1 reports the average end-of-month sector allocation for each of the mean–variance efficient portfolio proxies. Note that the sector bets are intuitive and can be explained by the cardinality per sector and the average risk (correlation and volatility) of the stocks per sector, as reported in Columns 2–4 of Exhibit 2. We see for instance in Exhibit 2 that there are proportionally fewer stocks in the IT sector, explaining why the equally weighted portfolio has a relatively lower weight for that sector compared with the The intrinsic motivation to compute implied expected returns from alternative proxies for a mean– variance efficient portfolio than the market capitalization weighted portfolio is that other portfolios show a better out-of-sample performance in term of risk-adjusted returns. Exhibit 1 verifies that this prediction, observed by E X H I B I T 1 various authors on different datasets, also Out-of-Sample Performance of the Various Mean–Variance Efficient holds for the sample under investigation. Portfolio Proxies and Their Average End-of-Month Sector Allocation The mean–variance efficient portfolio proxies that we consider are rebalanced at the end of the month, when the return forecast is made.5 In Panel A of Exhibit 1, we see that the annualized average return of all proxies is around 10% for all portfolios. The portfolio risk is thus the distinctive factor. The out-of-sample annualized portfolio volatility and monthly value-at-risk (at the 95% risk level, denoted by VaR95) are the lowest for the risk-based portfolios wiv, werc, wmd, and wref (around 14% and −6%, respectively) and the highest for the fundamental value portfolio (around 16% and −8%, respectively). These differences in volatility are statistically signifi- Notes: Mean: annualized mean (in percent). Vol: annualized volatility (in percent). Sharpe: annualized Sharpe ratio. VaR95: empirical monthly value-at-risk at the 95% risk level (in cant at the 1% level.6 The Sharpe ratio is percent). The evaluation period ranges from January 1987 to December 2012 for a total of 312 the highest (0.50) for the equal-risk-con- monthly observations. SUMMER 2015 JPM-ARDIA.indd 73 THE JOURNAL OF PORTFOLIO M ANAGEMENT 73 7/21/15 2:13:14 PM EXHIBIT 2 Average Value of Stock Characteristics per Sector Notes: N: (rounded) average number of stocks per sector. Vol: average annualized volatility (in percent). ρ: average correlation (in percent). The last eight columns report the average value per sector of the stock returns’ covariance with the various mean–variance efficient portfolio proxies (in permil). Row All refers to the overall S&P 100 universe. All numbers reported are averages of stock characteristics per sector and then averaged over the 312 months in the out-of-sample evaluation period. market capitalization weighted portfolio. The largest sector bets compared with the market capitalization weighted portfolio are made by the most diversified portw md ), which is under-invested especially in Finanfolio (w cial and Industrial stocks (5.7% and 6.6% compared with 13.4% and 9.8%) and over-invested in Consumer Goods, Energy, and Healthcare stocks. The under-investment in Financial and Industrial stocks is for our sample due to the relatively lower diversification potential of Financial and Industrial stocks (their correlation with other stocks is on average 35% compared with 29% for the Healthcare sector for instance). The other risk-based portfolios (wiv and wref ) make similar but less extreme sector bets. They under-invest in the relatively higher volatility sectors (such as Communications, Financials, and IT) and over-invest in the low-risk sectors, such as Consumer Staples, Healthcare, and Utilities. Descriptive Analysis of Expected Return Forecasts Let us now study how the differences in the mean– variance efficient portfolio allocations translate into differences in implied expected returns. Exhibit 3 reports the average annualized implied expected return (in percent) per sector for each benchmark portfolio (Panel A), and compares them with the average return prediction under the constant mean, AR(1), Fama–French threefactor, and Carhart four-factor models (Panel B). We see 74 JPM-ARDIA.indd 74 that the implied expected return predictions are, overall, more pessimistic than the time series model predictions in the sense of predicting a lower return. The implied expected return and time series model forecasts also disagree in terms of ranking sectors by expected returns. The time series model based return forecasts predict the highest return performance for the IT sector, while the implied expected return based forecasts predict the highest return for the Financial sector. A further interesting result in Exhibit 3 is that the most diversified portfolio (w w md ) implies the least pronounced bets of all mean–variance efficiency portfolio proxies. This can be attributed to the fact that the implied expected return for stock i is proportional to the covariance between the returns of that stock with the returns of the mean–variance efficient portfolio proxy; see (7). Since the maximum diversif ication portfolio explicitly seeks portfolio diversification, it comes as no surprise that the stocks have a relatively lower covariance with this portfolio than with the other mean–variance efficient portfolios. This is further confirmed by Columns 5–12 of Exhibit 2, which report the average covariance of the stock returns with the various mean–variance efficient portfolio proxies; we see that the lowest covariance is systematically observed for the maximum diversification portfolio. The most pronounced bets in terms of implied expected returns are observed for the market capitalization weighted I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY SUMMER 2015 7/21/15 2:13:15 PM EXHIBIT 3 Averages of Annualized Monthly Expected Return Predictions (in percent) per Sector for Implied Expected Returns and Time Series Model Based Approaches Notes: ier-: implied expected returns forecasts. Constant: forecasts under constant mean model. AR(1): autoregressive model forecasts. Fama–French and Carhart: three- and four-factor model forecasts. Each entry is the (equally weighted) mean end-of-month stock return forecast per sector and then averaged over all months in the sample. For convenience, results are reported in percentages and annualized. Column All refers to the overall S&P 100 universe. portfolio, the fundamental weighting portfolio, and the equally–weighted portfolio. Since implied expected returns are proportional to the stock’s covariance with the efficient portfolio proxy, we also see that implied expected returns tend to be higher for sectors with stocks that have: (i) a large weight in the eff icient portfolio, (ii) a high volatility and (iii) a high correlation with other stocks. This is for instance the case for the Financial sector, as can be seen from Exhibits 1 and 2. Overall, we find substantial (but not extreme) differences in implied expected return predictions depending on the mean–variance portfolio proxy used. In general, it seems that using the (smoothed) market capitalization weighted or equally weighted portfolio, more optimistic return predictions are obtained than using the risk-based portfolios. The fact that the differences in implied expected returns at the sector level are not extreme suggests that, while Lee [2011] is right to point out that smart beta strategies can be seen as views on expected returns, these views can be seen as being reasonable from a practical perspective. The analysis in Exhibit 3 is static and hides the strong time variation in the predicted return. This is shown in Exhibit 4, which displays the monthly time SUMMER 2015 JPM-ARDIA.indd 75 series of the average annualized predicted return (across all stocks). The solid lines show the minimum, median, and maximum value across all implied expected return methods, while the dashed lines correspond to the time series of the minimum, median, and maximum forecasts obtained by the various time series model forecast approaches. It is interesting to see that the two families disagree especially in periods of high volatility. The implied expected return methods impose the risk–return tradeoff in the prediction such that higher volatility is associated with a higher expected return. The time series model forecasts extrapolate past returns into the future, which, in bear markets, can lead to the (unrealistic) prediction of negative returns. Forecast Precision Comparison of Expected Return Forecasts The ultimate question is which expected return prediction is most useful? We have essentially portfolio applications in mind, where expected returns need to be not only precise but also stable and not excessively dispersed. The results of this analysis are reported in Exhibit 5 where we zoom in on the statistical properties of the implied expected returns (panel A) compared with the time series forecasts (panel B). THE JOURNAL OF PORTFOLIO M ANAGEMENT 75 7/21/15 2:13:15 PM E X HIBI T 4 Evolution of the Annualized Monthly Average Expected Return Forecasts across All S&P 100 Constituents Solid lines are for implied expected returns approaches and dashed lines for time series model based return forecasts. The three lines per family of estimation methods correspond to, for each time point, the minimum, median, and maximum value (across methods) of the average forecasts (across stocks). The precision of the forecasts is analyzed in Columns 2–3 of Exhibit 5. We consider two precision measures, namely the hit ratio (HR), which reports the percentage of times the sign of a stock return is correctly predicted by the corresponding forecast method, and the root-mean-squared forecast error (RMSFE) of the onemonth-ahead predicted return compared with the actual return. Both measures are computed for each month across the 100 predicted returns for the S&P 100 constituents and then averaged over time. We see that the hit ratio is 55.5% for all implied expected return methods and slightly higher than the hit ratios obtained using the time series models (between 53.3% and 53.8%). Results for RMSFE indicate a better performance for implied expected returns methods compared with the time series models. For instance, the forecasts implied by the market capitalization weighted portfolios lead to a significantly lower RMSFE than each of the time 76 JPM-ARDIA.indd 76 series model based forecasts (at the 1% level). Also, we find that within the family of implied expected returns approaches, the most precise predictions are obtained using the risk-based portfolios (wiv, werc, wmd, and wref ). The RMSFE accuracy gains of switching from the use of the market capitalization weighted portfolios to the inverse volatility weighted portfolios are statistically significant at the 5% level. They are significant at the 10% level for the risk-efficient portfolio. For other risk-based portfolios, the difference is not statistically significantly different. Next, we study in Column 4 of Exhibit 5 the magnitude of the cross-sectional bets as measured by the time series average of the monthly standard deviation of return forecasts across the S&P 100 constituents (σ i ). The higher this dispersion is the more extreme views are in terms of differences in future performance of stocks. We see that the standard deviation of the time series I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY SUMMER 2015 7/21/15 2:13:16 PM EXHIBIT 5 Precision, Dispersion, and Instability of Monthly Return Forecasts for Implied Expected Returns and Time Series Approaches Notes: Precision measures: HR (hit ratio, in percent) and RMSFE (root–mean–squared forecast error, in permil). Dispersion measure: σi (cross-sectional dispersion, in percent). Instability measures: σi (monthly time series volatility, in percent) and ρi ( first-order autocorrelation, in percent). Precision and dispersion measures are computed at each time point for the one hundred stocks in the portfolio, and then averaged over time. The instability measures are computed first for the individual predicted return series, for which at least twelve consecutive observations are available, and then averaged over the stocks in the sample. ier-: implied expected returns forecasts. Constant: forecasts under constant mean model. AR(1): autoregressive model forecasts. Fama-French and Carhart: three- and four-factor model forecasts. model based return predictions is more than four times the one of the implied expected return prediction. These differences are economically and statistically significant (at the 1% level). When used in portfolios, this implies that the time series model based forecasts of return have a higher risk to act like an error maximizer, as described by Michaud [1989], and to exhibit a much larger portfolio turnover. The lowest dispersion is observed for the maximum diversification portfolio. Finally, we analyze in Columns 5–6 of Exhibit 5 the time series instability of the return forecasts, by considering the volatility (σ t ) and the first-order autocorrelation (ρ1 ) of the forecasts, computed for the union of all stocks included in the universe over the backtest period (and provided at least twelve consecutive observations are available). These quantities are then averaged over stocks. We see that the standard deviation of the expected return predictions is more than two times the one of the implied expected return prediction. These differences are economically and statistically significant (at the 1% level). The first-order autocorrelation shows that the implied expected return based forecast are much more persistent. This is expected, since the time series based return forecasts are a linear combination of the SUMMER 2015 JPM-ARDIA.indd 77 past stock and factor returns, while the implied expected returns are a linear combination of the risk-free rate and the assets’ covariances with the portfolio proxy ( Σw ). Because stock returns tend to be more volatile and more dispersed than the assets’ covariances with the portfolio proxy, we expect to find a higher stability in the implied expected return forecasts compared with the time series model based forecasts. This is confirmed by our results. Note also that the implied expected returns based on pure risk-optimized portfolios are more stable than those using a market capitalization, fundamental value, or equal weighting scheme. Robustness of Results The results hitherto obtained are for a particular choice of estimation window (three years) and a specific value of the the risk-aversion parameter γ (set to 2.4, as in the original publication of Black and Litterman [1991]). In Exhibit 6, we present the results of our robustness analysis with respect to those two choices. Our analysis confirms the robustness of the higher forecast accuracy of implied expected returns compared with the time series based model forecasts considered and the gains of THE JOURNAL OF PORTFOLIO M ANAGEMENT 77 7/21/15 2:13:16 PM E X HIBI T 6 Sensitivity of Forecast Precision (RMSFE, in permil) of Implied Expected Returns and Time Series Model Based Return Forecasts to the Estimation Window and the Risk-Aversion Parameter γ Notes: RMSFE (in permil) for various value of risk-aversion parameter γ and estimation window (two, three, and four years). ier-: implied expected returns forecasts. Constant: forecasts under constant mean model. AR(1): autoregressive model forecasts; Fama–French and Carhart: three- and four-factor model forecasts. The time series model based approaches are insensitive to the choice of the risk-aversion parameter. using a risk-based portfolio proxy compared with the capitalization weighted portfolio. We find that for the time series model based forecasts, the use of a longer time span (four instead of three years) improves the RMSFE, but it remains substantially higher than the one obtained using the implied expected return estimates. The length of the estimation sample has almost no impact on the precision of the implied expected return estimates and the observed differences in precision are statistically insignificant. Moreover, increasing or decreasing the value of γ to 1 or 5 does not lead to a statistically significant difference in RMSFE (at the 1% level). As such, our results are robust to the choice of γ. It is however, interesting, to note that for the risk-based approaches, the accuracy is the highest for γ = 2.4, while for the (smoothed) market capitalization weighted portfolio, the fundamental value portfolio, and the equally weighted portfolio, the accuracy is improved by reducing the value of γ, and thus giving less weight to the portfolio weights in the implied return calculation. Forecast Combination We conclude our empirical analysis by studying the gains of forecast combination for predicting expected returns. From the literature on forecast combination (see, 78 JPM-ARDIA.indd 78 e.g., Tu and Zhou [2011]), it can be expected that combinations of the different forecasts improve further the stability of the forecasts. We study the impact of four types of forecast combinations in Exhibit 7 and consider combinations over: (i) all time series model based forecasts, (ii) all implied expected return based forecasts, and (iii) all pure risk-based forecasts. Panel A corresponds to the results obtained by simple averaging of the return forecasts. Panel B takes a more outlier-robust approach by setting the combination forecast equal to the median value of the individual forecasts. Following Stock and Watson [2004] and Genre et al. [2013], we also report performance-based combination forecasts in which the predicted return is a weighted average of the individual forecasts, with weights proportional to the one-month lagged hit ratio (Panel C) or inversely proportional to a 12-months average of the mean squared forecast error (Panel D).7 Compared with the individual component forecasts, we find that the combination-based forecast tends to be the second best and has the advantage of not requiring the hindsight information of the ex post best forecast method. The forecast combination method (simple averaging, median aggregation, or performancebased aggregation) has only a minor impact on the performance. I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY SUMMER 2015 7/21/15 2:13:17 PM EXHIBIT 7 Precision, Dispersion, and Instability of Monthly Return Forecasts for Combination-Based Approaches Notes: ier-all: equally–weighted implied expected return based forecasts; ier-risk-based: equally– weighted risk-based implied expected return based forecasts. See Exhibit 5 for more details. CONCLUSION We study the performance of implied expected return predictions obtained using various mean–variance efficient portfolio proxies and compare them with time series model based forecasts. Our motivation is that, even though traditionally the market capitalization weighted portfolio dominates in terms of reference portfolio from which the implied expected returns are derived, there is growing empirical evidence that risk-based portfolios may be a better proxy for a mean–variance efficient portfolio. In fact, in his important review on risk-based asset allocation, Lee [2011] concludes that if risk-based allocation approaches outperform the market portfolio, it must be that the implied expected returns are a better set of return forecasts than the market’s implied expected returns. We investigate empirically this prediction for the monthly returns of the S&P 100 stocks over a period ranging from January 1987 to December 2012. While Lee [2011] is right to point out that smart beta strategies can be seen as views on expected returns, we additionally show that these implied returns are rea- SUMMER 2015 JPM-ARDIA.indd 79 sonable and potentially useful from a practical perspective. We find that there is only a small forecast accuracy gain in using implied expected returns from risk-based portfolios compared with the market capitalization weighted portfolio but conclude that those forecasts are indeed better in the sense of being substantially more stable and less dispersed. When used in dynamic portfolio optimization, we expect them to yield optimized portfolio weights that are more stable and less extreme, implying lower turnover costs and a reduced sensitivity to the sampling variability in the return data. Our current work on the use of implied expected returns from riskbased portfolios in a mean–variance portfolio optimization framework seems to confirm this intuition. ENDNOTES The authors thank the Editor, an anonymous referee, Marie-Hélène Gagnon, Georges Hübner, Thierry Roncalli, Enrico Schumann, Piet Sercu, Marjan Wauters and participants at the IAF seminar (University of Neuchâtel 2013), the R/Finance 2013, the SSES 2013 and the AFFI 2014 conference for useful suggestions. Financial support from the Dutch science foundation, the IFM2 Montréal and the Hercules Foundation (Project No. AKUL/11/02) is gratefully acknowledged. 1 According to Flood [2013], global assets held in advanced beta strategies grew from USD 58 billion at the end of 2010 to USD 142 billion at the end of March 2013. 2 We restrict our attention to time series methods that extrapolate observed return dynamics into the future. The predictability of asset returns using state variables such as dividend-price or dividend-earnings ratios is a matter of controversy in the literature and beyond the scope of this article. See, e.g., Campbell and Thompson [2007]. 3 If a numerical solution is not found, we follow the recommendation in Maillard et al. [2010] and slightly modify the problem and optimize over the N-dimensional vector u such that w ≡ u / ( u ′ ) , under the constraint that u ≥ 0 and u ′ι 0. This new optimization problem is easier to solve numerically as an inequality constraint is less restrictive than the full investment equality constraint. THE JOURNAL OF PORTFOLIO M ANAGEMENT 79 7/21/15 2:13:17 PM 4 http://mba.tuck.dartmouth.edu/pages/faculty/ken. french/. 5 As noted by a referee, the actual implementation of smart beta portfolios typically uses a quarterly or annual rebalancing to reduce the high costs associated with portfolio turnover. Our focus is on the implied expected return prediction, which requires the use of portfolios for which the rebalancing is aligned with the return prediction horizon. 6 Throughout the article, statistical significance of differences in performance are computed using a t-test with heteroskedasticity and autocorrelation consistent (HAC) standard errors. 7 For more details, see Equation (1) in Genre et al. 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