Out-of-Sample Testing of Risk-Minimizing
Portfolios Using a Variety of Shrinkage Methods.
Out-of-Sample Testing of Risk-Minimizing
Portfolios Using a Variety of Shrinkage Methods.
Cody Wegen Abstract
This paper examines the out-of-sample performance of risk-minimizing portfolios using a variety of shrinkage methods. It finds that risk-minimizing portfolios earned higher returns at lower risk relative to the capitalization-weighted FTSE developed index in the period lasting from April 1999 to July 2017. The minimum variance approach is found to be better at reducing volatility than the simple approach based on historical volatilities, whilst maintaining a comparable level of return. The low volatility of the risk-minimizing portfolios is attributed to their relatively low exposures to the market factor. The excess return of the simple low volatility portfolio is explained by the profitability, investment and betting against beta factors, while the excess return of all but the linear shrunk minimum variance portfolios is attributable to the size, investment and betting against beta factors. Several flaws in the design of risk-minimizing portfolios are also identified, namely their highly concentrated positions and high turnover rates.
1
Introduction
Pension funds are major players in the global asset management industry. In 2018 they man-aged over $41,355 billion in 22 pension markets worldwide, up from $36,571 billion the previous year according to a study by Willis Towers Watson (2018). In a tightening regulatory environ-ment and with the economy heavily dependent on investenviron-ment returns, efficient asset portfolio allocation is of great importance to pension funds. Wholesale active fund managers are man-dated to outperform passive investment portfolios whilst maintaining an acceptable level of risk (Chaudhry and Johnson, 2008). Historically, risk-minimizing portfolios have satisfied both conditions simultaneously. This study examines the performance of risk-minimizing portfolios in a portfolio selection application suitable for real world active pension fund managers. In particular, the effectiveness of risk-minimizing portfolios using a variety of shrinkage methods is evaluated.
shrinkage by Ledoit and Wolf (2004b) and nonlinear shrinkage by Ledoit and Wolf (2012).
To understand potential outperformance of risk-minimizing portfolios it is fruitful to decompose their returns. It turns out that the capital asset pricing model and the three factor model of Fama and French (1993) only seem to offer a partial explanation, while Blitz and Vliet (2007) find statistically significant positive alphas for simple risk-minimizing portfolios even after cor-recting for the value and size factor. Extended models may provide more insights into returns. For example, Chow, Hsu, Kuo, and Li (2014) observe insignificant alphas for risk-minimizing portfolios in the four factor model by Carhart (1997) with the betting against beta factor. The latter factor is introduced by Frazzini and Pedersen (2014) and captures the anomaly that investors bid up high-beta assets when faced by leverage and margin constraints. The factor explains the excess returns of risk-minimizing portfolios, which typically contain low-beta stocks. This paper further considers the capital asset pricing model and five factor model by Fama and French (2006). In both cases the betting against beta factor is added to the model.
The risk-minimizing portfolios are tested in an out-of-sample application similar to that used by Hommes (2013). The estimation period consists of the years preceding the rebalancing moment, and the estimates are used to initialize or update the weights. The respective strategies’ perfor-mances are compared in the next rebalancing period. This procedure continues throughout the investment period. To reflect the reality faced by pension fund managers, all available capital is invested and short selling is prohibited. The risk-minimizing strategies are compared with the FTSE developed index. After a general analysis, the Sharpe ratio, Treynor ratio, Information ratio, Sortino ratio, Value at Risk and Expected Shortfall are computed for the risk-minimizing portfolios. This enables any possible drawbacks in the various portfolio construction methods to be identified. Lastly, the sources of the return variation of the risk-minimizing portfolios are identified by fitting factor models. The capital asset pricing model, capital asset pricing model including the betting against beta factor, the three factor model by Fama and French (1993), the five factor model by Fama and French (2006), and five factor model including the betting against beta factor are estimated using time series regressions.
The data consists of historical weekly firm-level returns of stocks in the FTSE developed index between February 1994 and July 2017. The constituents of the FTSE developed index and their respective market values are collected for each month between December 1995 and December 2017. Returns of the global factors by Fama and French (2006) and the betting against beta factor between April 1999 and June 2017 are obtained from the Fama and French and AQR databases, respectively. For all risk-minimizing portfolios the estimation period is set to five years; the sample covariance matrix estimated at each monthly rebalancing moment is therefore based on 260 weekly observations. The portfolios’ performances are based on the 952 weekly returns realized in the investment period lasting from April 1999 to July 2017.
The analysis shows that the risk-minimizing portfolios provided higher returns at lower risk relative to the FTSE developed index in the period under observation. The difference in re-lated Sharpe ratios is statistically significant at the 5% level. Moreover, the corrected minimum variance portfolio and the portfolios based on the five factor model and constant correlation model shrinkage achieve a higher Sharpe ratio than the simple low volatility portfolio based on historical stock return volatilities. The differences are also significant at the 5% level. It ap-pears from these results that taking correlations into account and selecting an efficient shrinkage method results in statistically significant higher Sharpe ratios.
be explained by their factor loadings. The relatively low historical volatility of risk-minimizing portfolios is justified by reduced market beta. The profitability, investment and betting against beta factor capture the excess return of the simple portfolio based on historical stock volatil-ity. The excess return of all but the linear shrunk minimum variance portfolio is explained by significantly and economically large loadings on the size, investment and betting against beta factor. The betting against beta factor significantly improves the adjusted R-squared of the models under consideration. The results suggest that the returns of risk-minimizing portfolios can be best explained by the betting against beta factor.
This paper makes the following key contributions to the literature. First, the results indi-cate that historically risk-minimizing portfolios earn higher returns at lower risk relative to the FTSE developed index. Second, the results suggest that taking the whole covariance matrix into account, rather than focusing on the diagonals alone, leads to the construction of better performing portfolios which successfully capture the benefits of diversification. The associated large dimensional sample variance matrix problems are solved by means of various shrinkage methods. Finally, the sources for outperformance of the risk-minimizing portfolios are identi-fied. The significance of the value factor observed by Chow et al. (2014) is captured by the profitability and investment factor. The betting against beta combined with the market factor are enough to explain the excess return of risk-minimizing portfolios, while Blitz and Vliet (2007) find significant positive alphas even after correcting for the size and value factors.
The paper is structured as follows. Section 2 reviews the literature most relevant to the portfolio selection problem under consideration. Section 3 elaborates on how the portfolios are constructed, reviews the shrinkage methods, and concludes by outlining the performance measurements. Section 4 elaborates on the study’s research data and Section 5 presents the analytical results. Section 6 sets out concluding observations and suggestions for further re-search.
2
Related Literature
Risk and return are the two key factors which influence efficient portfolio design. The con-ventional asset pricing literature predicts a positive relationship between risk and return, and holding that the only possible way to obtain higher returns is to invest in riskier stocks. How-ever, a number of empirical studies have found that risk-minimizing portfolios can outperform capitalization-weighted benchmarks in terms of risk and return. This low volatility anomaly has been documented by Clarke et al. (2006), Blitz and Vliet (2007), Blitz, Falkenstein, and Vliet (2014), Baker, Bradley, and Wurgler (2011), and Dutt and Humphery-Jenner (2013), among others.
Two main approaches to creating risk-minimizing portfolios appear in the literature. The first involves simply constructing portfolios based on stocks with the lowest historical volatility. A leading example of this is Blitz and Vliet (2007) who create decile portfolios based on the diagonals of the historical covariance matrix. They find that low risk stocks exhibit significantly higher risk-adjusted returns than the market portfolio. In particular, the top decile portfolio shows statistically significant higher Sharpe ratios relative to the market portfolio. The alpha of this top decile portfolio remains significant after correcting for the value and size factors. The authors call this phenomenon the low volatility effect, and attribute it to leverage restrictions.
mean-variance optimization. This is an improvement on the simple approach as it successfully cap-tures diversification benefits. Neither approach requires estimates for expected future returns, but the key advantage of the latter is that estimates of the covariance matrix converge faster to their underlying true parameter values. Clarke et al. (2006) show empirically that minimum variance portfolios can achieve volatility reduction whilst delivering comparable, or even higher, average returns than the market portfolio.
The minimum variance optimizer involves estimating the population covariance matrix. The traditional method is to use the sample covariance matrix, but this has two serious drawbacks. The first is that the matrix contains estimation error. Minimum variance optimizers latch onto the extreme individual (co)variances estimates, a phenomenon described by Michaud (1989) as error maximization. Indeed, Kan and Zhou (2007) find that this can lead to poor out-of-sample performance. The second drawback is that when the sample covariance matrix is based on fewer time observations than stocks, the resulting estimator is non-positive definite. Minimum vari-ance optimizers can only find weights for invertible matrices.
The estimation error can be reduced and positive definiteness achieved using a variety of shrink-age methods. One form of shrinkshrink-age adjusts the sample covariance matrix. With this method, new estimates are constructed as a convex combination of sample estimates and shrinkage targets. The most extreme coefficients are pulled toward more central values and this sys-tematically reduces estimation error where it matters most. The efficiency of shrinkage to a statistical target in portfolio selection problems is confirmed by Jobson and Korkie (1981); Jorion (1986); Frost and Savarino (1988); and DeMiguel, Garlappi, and Uppal (2009), among others. Theoretically motivated shrinkage targets are justified by Black and Litterman (1992); Kandel and Stambaugh (1996); Pastor (2000); and Pastor and Stambaugh (2000).
Ledoit and Wolf (2004a) propose the linear shrinkage estimator of the population covariance matrix. The same shrinkage intensity is applied to all sample eigenvalues. It follows that the estimator can be derived by finding the optimal convex combination of the sample covariance matrix and the identity matrix. When the dimension to observation ratio is large, and/or the population eigenvalues matrix are close to one another, linear shrinkage captures most of the potential improvement. By thorough Monte Carlo simulations the authors confirm that the asymptotic results of the linear shrinkage estimator hold well in finite sample.
In the paper by Ledoit and Wolf (2004b) the shrinkage target is constructed using the constant correlation model. As with the linear shrinkage, the estimator is a convex combination of the sample covariance matrix and the shrinkage target. The crux of the method is that by properly combining the sample covariance matrix and structured matrix, one is able to obtain an ad-justed estimator which performs better than either extreme. Using real stock market data the authors show that the shrinkage method reduces tracking error relative to a benchmark index, and substantially increases the Information ratio.
Risk-minimizing portfolios based on shrinkage methods and simple heuristics have been further tested by Chow et al. (2014). The authors find empirically that risk-minimizing portfolios can provide reduced volatility and higher returns than traditional capitalization-weighted index-ing. The relatively low volatility comes from reduced exposure to the market factor, while the excess return follows from accessing high exposures to the value, betting against beta and du-ration factors. Chow et al. (2014) additionally show that minimum variance portfolios perform better in terms of risk against the investigated heuristics approaches. However, the heuris-tic based-portfolios are compensated with higher returns. The resulting Sharpe ratios of the risk-minimizing portfolios are statistically similar.
3
Methodology
This section sets out the various portfolio construction methods that are relevant to the study at hand. It also describes the shrinkage methods used to find positive definite covariance matrix estimators in the minimum variance portfolios. The section concludes with a discussion of the performance measurements that extend the general analysis.
3.1
Portfolio constructions
The out-of-sample test is explained first, as a clear understanding of this is a necessary precursor to the ensuing analysis. The benchmark portfolio and related investment universe are discussed, followed by a review of the low volatility and minimum variance portfolios.
Out-of-sample test
The out-of-sample test consists of three periods: estimation, rebalancing and investment. The estimation period precedes the moment of rebalancing and is used to initialize or update the weights of the portfolios. The benchmark portfolio is based on capitalization-weighted index-ation and hence does not require an estimindex-ation period. For the risk-minimizing portfolios the period is initially set to five years, in keeping with the estimation period used by Clarke et al. (2006).
After the weights have been set based on the estimation period, the rebalancing period is initiated. Merton (1969) states that portfolios should be continuously rebalanced. As the mar-ket changes over time, so too should the weights in order to maintain the portfolio’s target weight and desired level of risk aversion. In reality however, it is impractical to continuously update the portfolio. Frequent rebalancing is accompanied by large transaction and managing costs. To reflect the real-world portfolio selection challenge faced by pension funds, the rebal-ancing period is therefore set to one month. During this period the portfolios’ performance is measured based on returns. In the rebalancing period the weights are updated by
wi,t =
wi,t−1(1 + ri,t)
PN
i=1wi,t−1(1 + ri,t)
, (1)
where ri,t is the return realized over week t − 1 to t, and t is in the rebalancing period. Keeping
the weights constant during the rebalancing period is an active strategy. The money invested in every stock should be altered according to the firm’s changing market value. The rebalancing period ends at the next rebalance moment, at which point the weights are updated using the new estimation period. Thereafter, the next rebalancing period starts.
Figure 1: Out-of-sample testing procedure for the risk-minimizing portfolios
This figure displays the out-of-sample test for the risk-minimizing portfolios. The estimation period is used to initialize or update the portfolio weights. Note that for the initial weights the estimation period precedes the investment period. In the rebalancing period the portfolios’ returns are used to measure their performance. The process is repeated throughout the investment period.
performance is determined by the return realized in this period. For an illustration of the out-of-sample test for the risk-minimizing portfolios the reader is referred to Figure 1.
Benchmark portfolio
The benchmark portfolio is based on the FTSE developed index. The market capitalization-weighted index represents the performance of large- and mid-capitalization firms in developed markets. At each rebalance moment the investment universe of the benchmark portfolio con-tains the constituents of the FTSE developed index. The weights are based on the market values of these constituents. The weights follow from
wi,t =
Mi,t
PNt
i=1Mi,t
, (2)
where Mi,t is the market value for firm i = 1, .., Ntat rebalance moment t. The number of stocks
in the index changes over time and is given by Nt. The resulting portfolio closely approximates
the FTSE developed index. Hence, it is referred to as the FTSE developed throughout this paper.
Investment universe
The investment universe of the benchmark portfolio consists of the constituents of the FTSE developed index, numbering approximately 2000 stocks. This paper seeks to compare the risk-minimizing portfolios to the benchmark portfolio, and hence the investment universe of the risk-minimizing portfolios also contains the constituents of the FTSE developed index. Stocks with a return history shorter than five years are excluded from the investment universe to avoid data gaps. The resulting investment universe of the risk-minimizing portfolios consists of approximately 1800 stocks of the FTSE developed universe.
Low volatility portfolio
as shown in equation 2. The top decile portfolio is called the low volatility portfolio and has the highest risk-adjusted return relative to all other decile portfolios; see Appendix A.
Minimum variance portfolios
The minimum variance portfolios improve the low volatility portfolio by considering the whole covariance matrix rather than only its diagonals. The objective is as follows:
z = min w0Σwˆ (3)
subject to
w0ι = 1, (4)
wi ≥ 0 ∀i = 1, ..., Nt. (5)
Where ι is the column vector of appropriate dimension whose entries are one. w =w1, ..., wNt’
is the column vector of portfolio weights. Moreover, ˆΣ is the covariance matrix estimator. The number of stocks in the investment universe of the risk minimizing portfolios is given by Nt at
moment t. Equation 4 is the capital constraint, while constraints 5 are the long-only constraints.
At each rebalance moment the weights follow from a quadratic solver with the covariance matrix estimator as input. The estimator depends on the minimum variance portfolio variant and the solver is subject to the capital and non-negative weight constraints. The pseudo-code for the minimum variance portfolio constructions is given in Appendix B
3.2
Shrinkage methods
To find positive definite covariance matrix estimators it is necessary to apply shrinkage methods. First, Highman’s algorithm and the five factor structure method are explained. The linear, constant correlation model, and nonlinear shrinkage methods are then discussed.
Higman’s algorithm
The corrected minimum variance portfolio ensures positive definiteness in the covariance matrix estimator using the algorithm by Higham (2002). The algorithm finds the nearest positive definite covariance matrix in the squared Frobenius norm to the sample covariance matrix, using the correction by Dykstra (1983). Negative sample eigenvalues are replaced by positive eigenvalues. In contrast to other shrinkage methods, the algorithm does not focus on reducing estimation error. Let S0 be a random symmetric matrix. The algorithm applies numerical
optimization to find
ˆ
Σ = min
Sk
||Sk− S||2F, (6)
where Sk is the adjusted matrix at iteration k, and k is sufficiently large. Moreover, S is the
sample covariance matrix. The squared Frobenius norm is defined as ||A||2F = ΣNt i=1Σ Nt j=1a 2 i,j. (7)
The squared Frobenius norm is the easiest norm to work with for this type of problem and the natural choice from a statistical point of view. Sk follows by omitting negative eigenvalues in
the previous found matrix Sk−1. The optimization continues until the convergence criteria is
Five factor model structure
Positive definiteness in the covariance matrix estimator can be established by decomposing stock returns using principal component or factor analysis. In this paper, factor analysis is conducted using the five factor model by Fama and French (2006). The model consists of the market (MKT), size (SMB), value (HML), profitability (RMW) and investment (CMA) factors. The model is estimated using time series regression
ri,t− rf,t = αi+ βiMKTt+ γiSMBt+ δiHMLt+ κiRMWt+ θiCMAt+ i,t, (8)
for firm i = 1, ..., Nt over the estimation period. The conventional asset pricing literature
holds that no abnormal returns can be earned. In theory, ˆαi should therefore be zero for all
i = 1, ..., Nt. Let the ordinary least squares coefficients for all firms be given by the 5 x Nt
matrix ˆξ = ( ˆβ, ˆγ, ˆδ, ˆκ, ˆθ)0. For estimation period length T , the T x 5 matrix of historical factor returns are given by F = (MKT, SMB, HML, RMW, CMA). It is assumed that the estimated error terms are uncorrelated. Now, the covariance matrix estimator is given by
ˆ
Σ = V( ˆα + ˆξ0F0 + ˆ) (9)
= ˆξ0V(F)ˆξ + V(ˆ), (10)
where V stands for the covariance matrix. Moreover, V(ˆ) is the diagonal matrix of firm-specific variance. The non-diagonal elements are set to zero because of the uncorrelated error terms assumption. The relatively small dimensional factor covariance matrix is positive definite since T > K. Hence, the covariance matrix estimator is structured and positive definite.
Linear shrinkage
The linear shrinkage estimator of the population covariance matrix is proposed by Ledoit and Wolf (2004a). The general idea is to pull down the largest eigenvalues and push up the smallest ones toward the grand mean of all sample eigenvalues. The same shrinkage intensity is applied uniformly to all eigenvalues. It follows that the covariance matrix estimator can be derived by finding the optimal convex combination of the sample covariance matrix and identity matrix. The non-diagonal elements of the sample covariance are pulled downwards. Thus, the individual stock variances play a larger role in determining the optimal portfolio’s weights. The estimator is defined as
ˆ
Σ = ˆρI + (1 − ˆρ)S, (11)
where I and S are the identity and sample covariance matrix, respectively. Moreover, ˆρ is the estimated shrinkage intensity. The asymptotic quantity corresponds to the optimal intensity. The optimal intensity follows by minimizing the squared Frobenius norm of the difference between the estimator and the population covariance matrix. The optimal shrinkage intensity is given by
ρ∗ = min
ρ ||ρI + (1 − ρ)S − Σ|| 2
F, (12)
where Σ is the population covariance matrix. For the formal definition of the squared Frobenius norm the reader is referred to equation 7.
Shrinkage using the constant correlation model
matrix used for linear shrinkage. This method properly combines the sample covariance matrix and structured shrinkage target to obtain an adjusted estimator that performs better than either extreme. The covariance matrix estimator is given by
ˆ
Σ = ˆδF + (1 − ˆδ)S, (13)
where F and S are the shrinkage target and sample covariance matrix. Moreover, ˆδ is the estimated shrinkage intensity. The shrinkage target results from the vector of sample variances and the common constant correlation. The latter is estimated by taking the average of all sample correlations. The diagonal elements of shrinkage target F are given by
fi,i = si,i ∀i = 1, ..., Nt, (14)
and the non-diagonal elements by fi,j = ¯r
√
si,isj,j ∀i, j = 1, ..., Nt, i 6= j, (15)
where si,iis the sample variance for i = 1, ..., Nt. Moreover, ¯r is the common constant correlation
defined as ¯ r = 2 (Nt− 1)Nt ΣNt−1 i=1 Σ Nt j=i+1ri,j, (16)
where ri,j is the sample correlation for i, j = 1, ..., Nt. As with the linear shrinkage method,
the estimated shrinkage intensity is asymptotic to the optimal shrinkage intensity. The latter results from minimizing the squared Frobenius norm of the difference between the estimator and population covariance matrix. Formally, the optimal shrinkage intensity is given by
δ∗ = min
δ ||δF + (1 − δ)S − Σ|| 2
F, (17)
where Σ is the population covariance matrix. The formal definition of the squared Frobenius norm is given by equation 7.
Nonlinear shrinkage
Mar˘cenko and Pastur (1967) investigated the relationship between sample and population eigen-values under large dimensional asymptotics and found that linear shrinkage is the first order approximation to an essentially nonlinear problem. More recently, Ledoit and Wolf (2012) studied the nonlinear shrinkage estimation of large-dimensional covariance matrices. Nonlinear shrinkage is a marked improvement on the linear shrinkage method, for instead of a single shrinkage intensity the nonlinear method computes an individualized shrinkage intensity for each sample eigenvalue. The method uses the consistent estimates of the nonlinear shrinkage intensities of Ledoit and P´ech´e (2011), in a uniform sense. Let the rotation-invariant estimator be given by
ˆ
Σ = U ˆDU0, (18)
where U is the matrix whose i-th column is the sample eigenvector ui and ˆD = diag( ˆd1, ..., ˆdN)
is the estimated diagonal matrix. The sample eigenvectors follow by solving
(S − λiI)ui = 0, ∀i = 1, ..., Nt, (19)
where λiis the i-th eigenvalue of sample covariance matrix S, for i = 1, ..., Nt. Moreover, I is the
Figure 2: Effect of the shrinkage methods on the sample eigenvalues
the optimal diagonal matrix. The optimal diagonal matrix results from minimizing the squared Frobenius norm of the distance between the covariance matrix estimator and the population covariance matrix. It is given by
D∗ = min
D ||UDU
0− Σ||2
F, (20)
where Σ is the population covariance matrix. The interpretation of d∗i is that it captures how the i-th sample eigenvector ui relates to the population covariance matrix as a whole. To see
how the shrinkage methods affect the eigenvalues the reader is referred to Figure 2.
3.3
Performance measurements
The general results alone provide only a partial insight into portfolio construction; for the analysis to be comprehensive, additional measurements must also be considered. The various performance measurements available to a researcher have different rationales behind their com-putation. In this study, investors’ market belief and preferences determine the relevance of the ratios. The performance measurements utilised in this research are the Sharpe ratio, Treynor ratio, Information ratio, Sortino ratio, Value at Risk and Expected Shortfall.
Sharpe ratio
The conventional asset pricing literature holds that higher returns can only be achieved by investing in riskier stocks, and the Sharpe ratio encapsulates this trade-off between risk and return. Previously known as the reward-to-variability ratio, the Sharpe ratio was introduced by Sharpe (1966) and is derived from the capital market line. It measures the realized return per unit of risk for a zero-investment strategy (Sharpe, 1994). The formula for the ratio is as follows: Sp = ¯ rp− ¯rf ˆ σp , (21)
where ¯rp and ˆσp are respectively the average realized return and estimated standard deviation
of portfolio p. Moreover, ¯rf is the average realized return of the risk-free rate.
Although widely used, the Sharpe ratio has its limitations. It measures only one dimension of risk, namely variance, so the ratio is designed to be applied to investment strategies that have normal expected return distributions. However, for return distributions this assumption is often violated. To compare portfolios, the ratio is used on a relative basis. In particular, portfolios with higher Sharpe ratios are preferred to portfolios with lower ratios. This perfor-mance measurement assesses absolute risk, which is convenient for risk-minimizing portfolios as they are generally measured based on absolute risk.
The Sharpe ratio test by Wright, Yam, and Yung (2014) is used to test for equality of mul-tiple Sharpe ratios. The authors extend the multivariate Sharpe ratio statistic of Leung and Wong (2006) by making the test hold under the more general assumption that the returns are stationary and ergodic.
Treynor ratio
the risk of an asset in the portfolio relative to the movements of the market portfolio (Fama and MacBeth, 1973). The Treynor ratio is given by
Trp = ¯ rp− ¯rf ˆ βp , (22)
where ¯rp and ˆβp are the average realized return and estimated market beta of portfolio p,
re-spectively. The estimated market beta follows from fitting the capital asset pricing model on the return. Moreover, ¯rf is the average return of the risk-free rate.
The Treynor ratio is sensitive to the denominator. It provides unstable and imprecise per-formance measures for market-neutral funds due to measurement error risk (H¨ubner, 2005). Risk-minimizing portfolios commonly have low market beta by design, so these portfolios have particularly high Treynor ratios. The ratio incorporates the possible investors’ belief that low beta portfolios are preferable to high beta portfolios given the former’s historic outperformance.
Sortino ratio
A further measure to consider when evaluating portfolio performance is the Sortino ratio. This determines the relationship between the realized average excess return and the downside deviation. The excess return and downside deviation are determined according to a given threshold. In this study, the risk-free rate is treated as the threshold since risk-minimizing portfolios are typically measured on an absolute basis. The Sortino ratio is calculated as
Sorp = ¯ rp − ¯rf ˆ σdd , (23)
where ¯rp and ¯rf are the average realized returns of portfolio p and the risk-free rate, respectively.
Moreover, ˆσdd is the estimated downside deviation given by
ˆ σdd = v u u t 1 T T X t=1 (rp,t− rf,t)2Irp,t<rf,t. (24)
where the indicator is defined as
Irp,t<rf,t =
n 1 if r
p,t < rf,t
0 otherwise. (25)
rp,t and rf,t are the realized returns of portfolio p and the risk-free rate at moment t = 1, ..., T ,
respectively.
The normality assumption of return distributions essential for the Sharpe and Information ratios is not required for the Sortino ratio. This is convenient as the normality assumption is often violated for stock returns. Portfolios with positively skewed distribution of alpha are rewarded with a higher Sortino ratio, implying less vulnerability against a downside adjustment of the portfolio.
Information ratio
2005). The ratio is calculated by dividing the portfolio’s average excess return relative to its benchmark by the variability of that excess return. The ratio is defined by
IRp = ¯ rp − ¯rb ˆ σp−b , (26)
where ¯rp and ¯rb are the average realized return of portfolio p and benchmark b, respectively.
Furthermore, ˆσp−b is the estimated tracking error calculated as
ˆ σp−b= r 1 TΣ T t=1(rp,t− rb,t)2, (27)
where rp,tand rb,tare the realized return of respectively portfolio p and benchmark b at moment
t = 1, ..., T .
The ratio reveals the outperformance of the portfolio relative to the benchmark on a risk-adjusted basis, but does not tell us how this is established. As with any statistic based on historical data, a high Information ratio in the past is no guarantee of a high one in the future (Goodwin, 1998). In keeping with the Sharpe ratio, the Information ratio assumes normal return distributions. However, the Information ratio is based on relative risk; it follows that investors who want to maximize returns subject to total risk should use the Sharpe ratio over the Information ratio (Baker et al., 2011). This study therefore places greater emphasis on the Sharpe ratio since risk-minimizing portfolios are ordinarily measured according to absolute risk.
Value at Risk
According to McNeil, Frey, and Embrechts (2015), the Value at Risk (VaR) is probably the single most widely used risk measure in financial institutions. The notion of the VaR is to locate the minimum return that is not exceeded with a given high probability. This measurement is computed by the Gaussian and historical methods. The Gaussian method assumes normal portfolio return distributions, that is rp,t ∼ N (µp, σp2) for t = 1, .., T . Under this method the
VaR is given by
VaRp,α = µp+ σpΦ−1(α), (28)
for portfolio p with confidence level α. Second, the VaR is based on historical data. This method assumes that history repeats itself from a risk perspective. If, for example, one has a sample of 100 return observations and the historical VaR with a 5% confidence level is re-quested, the 5th smallest return is the answer.
The VaR has its own set of drawbacks. First, the Gaussian method produces unreliable out-comes when return is not normally distributed. Moreover, the efficacy of the historical method may be called into question since it is debatable whether historical performance is a reliable indicator of future risk. Perhaps the most serious drawback is that the VaR includes no infor-mation about the severity of losses if the threshold is exceeded.
Expected Shortfall
Figure 3: Relationship between the Value at Risk and Expected Shortfall
This figure illustrates the relationship between the Value at Risk and Expected Shortfall. The notion of the Value at Risk is to locate the minimum return not exceeded with a given high probability. The Expected Shortfall is the average loss given this Value at Risk is exceeded.
the return distribution. Figure 3 illustrates the relationship between the Value at Risk and Expected Shortfall. The Expected Shortfall can be formally defined by
ESp,α = E(rp|rp < VaRp,α), (29)
for portfolio p with confidence level α. An Expected Shortfall of -3.0 with a confidence level of 5% implies that over a 20-year period the lowest observed return is approximately -3.0.
As with to the Value at Risk, the Gaussian and historical method are applied to find the Expected Shortfall. The same drawbacks of the Value at Risk therefore also apply to the Expected Shortfall, namely that the normal distribution assumption may be violated in the Gaussian method, and that history does not necessarily reflect the future as assumed by the historical method.
4
Data
The investment universe for the benchmark and risk-minimizing portfolios is based on the FTSE developed index, which consists of approximately 2000 stocks. The total market value is the sum of the market values of all the companies in the FTSE developed index. For this study, the constituents of the FTSE developed index are identified and their respective market values collected at the end of each month between December 1995 and December 2017.
Table 1: Descriptive statistics of the factor returns
This table presents the descriptive statistics of the 952 weekly factor returns in the investment period lasting from April 1999 to July 2017. The market factor returns follow from a long-only portfolio, while all other factor returns are based on self-financing long-short portfolios. The betting against beta is the only not dollar-neutral factor of the self-financing ones since it is constructed to produce beta of zero’s. The average return and standard deviation are annualized after computation. The former represents the compounded annual growth rate. Only the upper triangle of the correlation model is presented since all pairwise correlations are identical. The market (MKT), size (SMB), value (HML), profitability (RMW) and investment (CMA) factor returns are collected from the Fama and French database, while the betting against beta (BAB) factor returns follow from the AQR database.
Factors MKT SMB HML RMW CMA BAB
Average Return (%) 5.5 2.9 4.9 4.1 4.1 11.7 Standard Deviation (%) 16.9 6.7 7.7 4.9 6.2 9.6 Correlations MKT SMB HML RMW CMA BAB MKT 1 -.33 -.06 -.39 -.45 -.48 SMB 1 .02 -.05 .09 .27 HML 1 .07 .61 .20 RMW 1 .26 .53 CMA 1 .42 BAB 1
For this research, the risk-free rate by Fama and French is used. The rate is based on the returns of a four week Treasury bill. The daily returns of the market, size, value, profitability and investment factor of the developed market between July 1990 and March 2018 are also ob-tained from the Fama and French database. The daily betting against beta factor returns follow from an updated and extended version of the paper data by Frazzini and Pedersen (2014), with the dataset collected from the AQR database. The daily factor returns are used to determine the accumulative weekly returns by
rweek+ 1 = Πf ridayday=monday(rday+ 1). (30)
The weekly factor return dates are set to coincide with the weekly stock return dates for analytic purposes.
The descriptive statistics of the factor returns are presented in Table 1. The market factor re-turns follow from a long-only portfolio, while all other factor rere-turns are based on self-financing long-short portfolios. The betting against beta is the only not dollar-neutral factor of the self-financing ones since it is constructed to produce beta of zero’s. As expected, the standard deviation of the market portfolio is the highest at 16.9%, while all other standard deviations of the long-short portfolios range between 4.9% and 9.6%. The betting against beta average factor return is the highest at 11.7%, while all other returns range between 2.9% and 5.5%. This can be attributed to the betting against beta not being dollar-neutral. From the result it seems that the betting against beta efficiently takes long positions in undervalued stocks and short positions in overvalued stocks.
correlations are sufficiently lower than one.
5
Results
To assess the efficiency of the risk-minimizing portfolios, empirical research is performed based on the constituents of the FTSE developed index. The results are drawn from an out-of-sample application. In this scenario, the investor invests all available capital in stocks on a long-only basis. The general analysis and additional performance measurements are presented first. Then a factor attribution analysis provides insights in the return profiles of the risk-minimizing portfolios. Last, the various shrinkage methods are evaluated using a shorter estimation period and with a relaxation of the long-only constraint.
5.1
General analysis
Table 2 presents the general analysis of the FTSE developed and risk-minimizing portfolios. The annualized average return and standard deviation of the portfolios are illustrated in a risk-return graph, see Figure 4. The risk-minimizing portfolios earned a higher risk-return at lower risk relative to the FTSE developed index during the time period under observation. This result is at odds with the conventional asset pricing literature which holds that a higher return can only be achieved by taking greater risks. The outperformance of the risk-minimizing portfolios is in line with the higher risk-adjusted returns observed by Blitz and Vliet (2007) and Chow et al. (2014).
All but the linear shrunk minimum variance portfolios show a significant reduction in volatility while earning at least as much return as the simple low volatility portfolio. Moreover, the linear shrunk minimum variance portfolio has a standard deviation of 13.1% which is similar to the standard deviation of the low volatility portfolio of 12.8%. In comparison to other minimum variance portfolios, the linear shrunk portfolio focuses less on correlations and more on variance in individual stocks. The results suggest that taking the whole covariance matrix into account, rather than focusing on the diagonals alone, leads to the construction of better performing portfolios which can further reduce the volatility without sacrificing return.
As expected, the risk-minimizing portfolios have a lower drawdown than the FTSE developed. The portfolios are constructed to follow the market less tightly in a downswing. The relatively high tracking error of all risk-minimizing portfolios indicates that the portfolios deviate strongly from the FTSE developed. For some investors, a high tracking error is undesirable since the portfolio earns relatively lower returns in bull markets. In this context however, high tracking errors may be less relevant because risk-minimizing portfolios are typically measured on an absolute rather than a relative basis.
One problem associated with risk-minimizing portfolios is that they are highly concentrated, as too much emphasis can be put on stocks with relatively low volatility and/or good diver-sification properties in the recent past. The effective N known as the inverse Herfindahl score assesses the concentration over the measured dimension. Notably, the maximum effective N of the low volatility portfolio is significantly lower since it only assigns weight to the 10% best performing stocks with respect to their historical volatility. In line with the existing literature, the effective N of the risk-minimizing portfolios is relatively low.
Table 2: General analysis of the FTSE developed and risk-minimizing portfolios
This table presents the general analysis of the FTSE developed and risk-minimizing portfolios. The analysis is based on the 952 realized weekly returns over the investment period from April 1999 to July 2017. The average return, standard deviation, maximum drawdown and tracking error are annualized after calculation. The average return represents the compounded annual growth rate. The effective N and one-way turnover rate depend on the estimation and rebalancing period, which are respectively set to five years and one month. The FTSE developed (FTSE), low volatility portfolio (Low Vol), corrected minimum variance portfolio (MVPc),
and minimum variance portfolio based on the five factor model by Fama and French (2006) (MVP5f), linear
shrinkage (SMVPl), constant correlation model shrinkage (SMVPc) and nonlinear shrinkage (SMPVnl) are
denoted in order of sophistication.
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Average Return (%) 4.6 8.3 10.2 10.5 9.1 9.1 9.1
Standard Deviation (%) 16.8 12.8 9.5 10.1 13.1 9.0 9.6
Maximum Drawdown (%) 56 45 33 23 55 32 36
Tracking Error (%) - 9.7 12.0 14.3 10.2 12.7 12.2
Effective N 349.2 35.7 19.2 39.8 23.7 18.1 26.2
One-Way Turnover Rate (%) 3.4 11.0 29.4 14.7 26.4 37.9 27.9 Correlations
FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
FTSE 1 .82 .71 .53 .79 .67 .70 Low Vol 1 .74 .61 .77 .75 .76 MVPc 1 .87 .85 .96 .95 MVP5f 1 .75 .88 .90 SMVPl 1 .83 .85 SMVPc 1 .94 SMVPnl 1
Figure 4: Risk-return plot of the FTSE developed and risk-minimizing portfolios
rebalancing period which is set to one month. The minimum variance portfolio based on the five factor model has a significantly lower turnover rate of 14.7% relative to the other mini-mum variance portfolios. This turnover rate is only roughly 3% higher than that of the low volatility portfolio. Using the five factor model provides more weight stability, which may be explained by the relatively small dimensional factor covariance matrix that has to be estimated.
As expected, the correlations between all minimum variance portfolios are high given their related constructions. The correlations of the risk-minimizing portfolios with the FTSE de-veloped relate to the tracking error. The highest correlations with the FTSE dede-veloped are achieved by the low volatility portfolio and linear shrunk minimum variance portfolio, with correlations of 0.82 and 0.79 respectively. These portfolios exhibit characteristics more sim-ilar to the FTSE developed than the other risk-minimizing portfolios. Each risk-minimizing portfolio can be considered unique as their correlations are sufficiently lower than one.
5.2
Performance measurements analysis
The performance ratios are denoted in Table 3. The Sharpe ratio can be considered the risk-adjusted return. The respective Sharpe ratios of the risk-minimizing portfolios are higher than the Sharpe ratio of the FTSE developed. The differences are significant at the 5% level. Moreover, the corrected minimum variance portfolio and the portfolios based on the five factor model and constant correlation model shrinkage achieve a higher Sharpe ratio than the low volatility portfolio. The differences are also significant at the 5% level. It appears from these results that taking correlations into account and selecting an efficient shrinkage method results in statistically significant higher Sharpe ratios.
As with the Sharpe ratio, the risk-minimizing portfolios achieve higher Treynor and Sortino ratios relative to the FTSE developed. The relative difference between the Treynor ratios in-creases for all but the linear shrunk minimum variance portfolios and the low volatility portfolio. This can be attributed to minimum variance portfolios typically having low market beta. The Sortino ratio is measured using downside risk, and does not generate any unexpected findings. The Information ratio compares the risk-minimizing portfolios to the FTSE developed, and for this the portfolios show similar results. In practice however, risk-minimizing portfolios are measured on an absolute basis. Efficient risk-minimizing portfolios reduce absolute volatility while keeping at least the same return as the benchmark. The results support the view that if properly constructed, minimum variance approaches can offer an improvement on the simple historical volatility approach as they capture diversification benefits.
Table 3: Performance ratios of the FTSE developed and risk-minimizing portfolios
This table presents the performance ratios of the FTSE developed and risk-minimizing portfolios. The Sharpe, Treynor, Information and Sortino ratio are denoted. The Information ratio uses the FTSE developed as bench-mark, and hence is only denoted for the risk-minimizing portfolios. The ratios are computed using annualized numbers. The FTSE developed (FTSE), low volatility portfolio (Low Vol), corrected minimum variance portfolio (MVPc), and minimum variance portfolio based on the five factor model of Fama and French (2006) (M V P5f),
linear shrinkage (SM V Pl), constant correlation model shrinkage (SM V Pc) and nonlinear shrinkage (SM P Vnl)
are denoted in order of sophistication.
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Sharpe ratio (%) 17 52 89 87 57 82 77
Treynor Ratio (%) 3 11 21 28 12 21 19
Sortino Ratio (%) 23 70 122 124 77 111 104
Table 4: Value at Risk and Expected Shortfall of the FTSE developed and risk-minimizing portfolios
This table presents the Value at Risk and Expected Shortfall of the FTSE developed and risk-minimizing port-folios. The performance measurements are based on weekly realized returns, but annualized after computation. The Gaussian method assumes normal portfolio return distributions, while the historical method assumes that history repeats itself from a risk perspective. The FTSE developed (FTSE), low volatility portfolio (Low Vol), corrected minimum variance portfolio (MVPc), and minimum variance portfolio based on the five factor model of
Fama and French (2006) (M V P5f), linear shrinkage (SM V Pl), constant correlation model shrinkage (SM V Pc)
and nonlinear shrinkage (SM P Vnl) are denoted in order of sophistication.
Gaussian method
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Value at Risk -1.9 -1.4 -1.0 -1.1 -1.5 -1.0 -1.0
Expected Shortfall -2.4 -1.8 -1.3 -1.4 -1.8 -1.2 -1.3
Historical method
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Value at Risk -1.9 -1.3 -0.9 -1.0 -1.3 -0.9 -1.0
Expected Shortfall -2.9 -2.2 -1.6 -1.6 -2.3 -1.5 -1.7
The Value at Risk and Expected Shortfall of the FTSE developed and risk-minimizing port-folios are given in Table 4. The Expected Shortfall estimates using the historical method are lower than those produced by the Gaussian method. In keeping with the literature, the return distributions do not seem to satisfy the normality distribution. In particular, The Shapiro-Wilk normality test by Shapiro and Wilk (1965) is rejected at the 1% level. The left tail of the return distribution is longer than the fitted normal distribution; see Appendix C. However, under both methods the findings are in line with the performance ratios. All return distributions suffered from their negative skewness. From the results it appears that the performance measurements provide reliable outcomes, despite non-normal return distributions.
5.3
Factor attribution analysis
The returns are decomposed using factor models to understand the return profiles of the risk-minimizing portfolios. The capital asset pricing model, three factor model by Fama and French (1993), five factor model by Fama and French (2006), augmented capital asset pricing model and augmented five factor model, with the latter two models both including betting against beta factor. The estimated models are presented in Table 5. Since the FTSE developed index represents the performance of large and mid-capitalization firms in developed markets, a substantial part of the FTSE developed may naturally be explained by the market factor. As expected, the market factor loadings are significantly lower for the risk-minimizing portfolios. The low volatility and linear shrunk minimum variance portfolio have a beta coefficient of 0.72 or less, while all other risk-minimizing portfolios have a beta coefficient of 0.54 or less in every model. The relatively low beta explains the reduced volatility, since it is computed as the correlation between the portfolio and market portfolio times the volatility of the portfolio divided by the volatility of the market portfolio ( ˆβp = cor(rp, rm)σσmp).
Table 5: Return decomposition of the FTSE developed and risk-minimizing portfolios using a variety of factor models
This table presents the estimated capital asset pricing model, three factor model by Fama and French (1993), five factor model by Fama and French (2006), augmented capital asset pricing model and augmented five factor model are considered, with the latter two models both including betting against beta factor. The market (MKT), size (SMB), value (HML), profitability (RMW) and investment (CMA) factor returns follow from the Fama and French database, while the betting against beta (BAB) factor returns are collected from the AQR database. The excess portfolio and market return relative to the risk-free rate are denoted as re
p,t= rp,t− rf,t and rem,t=
rm,t− rf,t, respectively. The FTSE developed (FTSE), low volatility portfolio (Low Vol), corrected minimum
variance portfolio (MVPc), and minimum variance portfolio based on the five factor model of Fama and French
(2006) (M V P5f), linear shrinkage (SM V Pl), constant correlation model shrinkage (SM V Pc) and nonlinear
shrinkage (SM P Vnl) are denoted in order of sophistication.
Capital asset pricing model
Regression re
p,t= αp+ βprem,t+ p,t
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
ˆ
α −0.01 0.04∗ 0.06∗∗∗ 0.07∗∗∗ 0.05∗ 0.05∗∗∗ 0.05∗∗
ˆ
β (MKT) 0.97∗∗∗ 0.60∗∗∗ 0.40∗∗∗ 0.32∗∗∗ 0.61∗∗∗ 0.36∗∗∗ 0.40∗∗∗
Adj. R2 0.94 0.61 0.50 0.28 0.61 0.44 0.48
Three factor model
Regression re
p,t= αp+ βprem,t+ γpSMBt + δpHMLt+ p,t
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
ˆ α −0.01 0.04∗ 0.05∗∗ 0.06∗∗ 0.03 0.04∗∗ 0.04∗∗ ˆ β (MKT) 0.95∗∗∗ 0.56∗∗∗ 0.42∗∗∗ 0.35∗∗∗ 0.64∗∗∗ 0.37∗∗∗ 0.42∗∗∗ ˆ γ (SMB) −0.15∗∗∗ −0.34∗∗∗ 0.16∗∗∗ 0.23∗∗∗ 0.20∗∗∗ 0.10∗∗ 0.17∗∗∗ ˆ δ (HML) 0.08∗∗∗ 0.31∗∗∗ 0.12∗∗∗ 0.13∗∗∗ 0.21∗∗∗ 0.15∗∗∗ 0.14∗∗∗ Adj. R2 0.95 0.67 0.52 0.31 0.64 0.46 0.50
Five factor model
Regression re
p,t= αp+ βprem,t+ γpSMBt+ δpHMLt+ κpRMWt+ θpCMAt+ p,t
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
ˆ α −0.01 −0.01 0.03∗ 0.03 0.01 0.02 0.02 ˆ β (MKT) 0.96∗∗∗ 0.72∗∗∗ 0.50∗∗∗ 0.44∗∗∗ 0.69∗∗∗ 0.46∗∗∗ 0.51∗∗∗ ˆ γ (SMB) −0.14∗∗∗ −0.22∗∗∗ 0.21∗∗∗ 0.29∗∗∗ 0.26∗∗∗ 0.16∗∗∗ 0.22∗∗∗ ˆ δ (HML) 0.08∗∗∗ 0.06 −0.03 −0.03 0.24∗∗∗ 0.01 -0.04 ˆ κ (RMW) 0.07∗ 0.63∗∗∗ 0.23∗∗∗ 0.30∗∗∗ 0.43∗∗∗ 0.30∗∗∗ 0.24∗∗∗ ˆ θ (CMA) 0.01 0.50∗∗∗ 0.29∗∗∗ 0.32∗∗∗ −0.08 0.29∗∗∗ 0.36∗∗∗ Adj. R2 0.95 0.75 0.54 0.34 0.66 0.50 0.54
Capital asset pricing model + BAB
Regression re
p,t= αp+ βprem,t+ φpBABt+ p,t
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
ˆ α -0.01 -0.01 0.02 0.01 0.00 0.01 0.01 ˆ β (MKT) 0.97∗∗∗ 0.70∗∗∗ 0.49∗∗∗ 0.43∗∗∗ 0.70∗∗∗ 0.46∗∗∗ 0.49∗∗∗ ˆ φ (BAB) 0.01 0.37∗∗∗ 0.34∗∗∗ 0.42∗∗∗ 0.35∗∗∗ 0.37∗∗∗ 0.36∗∗∗ Adj. R2 0.94 0.67 0.59 0.41 0.66 0.56 0.58
Five factor model + BAB Regression re
p,t= αp+ βprem,t+ γpSMBt+ δpHMLt+ κpRMWt+ θpCMAt+ φpBABt+ p,t
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
ˆ α −0.01 −0.03 0.01 0.01 -0.01 0.00 0.00 ˆ β (MKT) 0.96∗∗∗ 0.74∗∗∗ 0.53∗∗∗ 0.47∗∗∗ 0.71∗∗∗ 0.49∗∗∗ 0.54∗∗∗ ˆ γ (SMB) −0.14∗∗∗ −0.29∗∗∗ 0.11∗∗∗ 0.17∗∗∗ 0.18∗∗∗ 0.05 0.12∗∗∗ ˆ δ (HML) 0.08∗∗∗ 0.04 −0.04 −0.05 0.22∗∗∗ −0.01 −0.06 ˆ κ (RMW) 0.08∗ 0.45∗∗∗ −0.04 −0.03 0.21∗∗∗ 0.02 −0.02 ˆ θ (CMA) 0.02 0.43∗∗∗ 0.20∗∗∗ 0.20∗∗∗ −0.16∗∗ 0.19∗∗∗ 0.27∗∗∗ ˆ φ (BAB) -0.01 0.22∗∗∗ 0.31∗∗∗ 0.39∗∗∗ 0.27∗∗∗ 0.33∗∗∗ 0.32∗∗∗ Adj. R2 0.95 0.77 0.60 0.42 0.68 0.57 0.60
indexation. The results suggest that large companies provide more stable returns and thus ap-pear in the low volatility portfolio. Moreover, all minimum variance portfolios have statistically significant positive coefficients for the size factor. This result indicates that the inclusion of smaller companies may yield significant diversification benefits.
Historically, high beta stocks are generally found in growth industries. Portfolios with lower market beta may therefore contain more value stocks. This is line with the significant positive coefficients for the value factor for the risk-minimizing portfolios in the three factor model. Chow et al. (2014) even find significant positive coefficients for the value factor in the four factor model by Carhart (1997). However, the factor becomes insignificant in the five factor models. The results suggest that the profitability and investment factors better explain the return and capture the part first explained by the size factor for risk-minimizing portfolios.
The profitability factor has statistically significant and economically large loadings for all port-folios in the five factor model. However, in the augmented five factor model the profitability factor becomes insignificant for all but the linear shrunk minimum variance portfolios. The betting against beta factor appears to encapsulate the part first explained by the profitability factor for these portfolios. Moreover, the profitability factor is significant in both models for the low volatility portfolio. From the results it seems that firms with stable cash flows and a strong track record are generally less volatile.
The coefficient for the investment factor is statistically significant and economically large for all but the linear shrunk risk-minimizing portfolios. In contrast to the profitability factor, the significance remains after including the betting against beta factor. From the results it appears that conservative firms are less volatile and may provide more diversification benefits relative to growth firms.
The augmented models, incorporating the betting against beta factor, have significantly im-proved adjusted R-squared and statistically and economically large loadings on the betting against beta factor for the risk-minimizing portfolios. The risk-minimizing portfolios assign relatively large weights to low beta companies. The significance of the betting against beta is also observed by Chow et al. (2014). The alphas become insignificant when the betting against beta is included. In particular, the betting against beta factor in combination with the market beta are enough to explain the relatively high realized returns of the risk-minimizing portfolios. These results support the reasoning by Clarke et al. (2011) that minimum variance portfolio performance is attributable largely to low beta stocks. It also seems to satisfy the hypothesis by Blitz and Vliet (2007) that the low volatility effect observed in their research may be explained by outperformance of low beta stocks due to leverage restrictions. The results suggest that the returns of risk-minimizing portfolios can be best explained by the betting against beta factor.
Table 6: Return decomposition of the excess return of the risk-minimizing portfolios against the FTSE developed and low volatility portfolio using the augmented five factor model including the betting against beta factor
This table presents the estimated augmented five factor model including the betting against beta factor. The market (MKT), size (SMB), value (HML), profitability (RMW) and investment (CMA) factor returns follow from the Fama and French database, while the betting against beta (BAB) factor returns are collected from the AQR database. The excess market return relative to the risk-free rate is denoted as rem,t= rm,t− rf,t. Moreover,
the excess return of the risk-minimizing portfolios against the FTSE developed and low volatility portfolio are denoted as re∗
p,t = rp,t− rF T SE,t and re∗∗p,t = rp,t− rLowV ol,t, respectively. The FTSE developed (FTSE), low
volatility portfolio (Low Vol), corrected minimum variance portfolio (M V Pc), and minimum variance portfolio
based on the five factor model of Fama and French (2006) (M V P5f), linear shrinkage (SM V Pl), constant
correlation model shrinkage (SM V Pc) and nonlinear shrinkage (SM P Vnl) are denoted in order of sophistication.
Five factor model + BAB Regression re∗
p,t= αp+ βprem,t+ γpSMBt+ δpHMLt+ κpRMWt+ θpCMAt+ φpBABt+ p,t
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
ˆ α - −0.01 0.02 0.02 0.00 0.01 0.01 ˆ β (MKT) - −0.22∗∗∗ −0.43∗∗∗ −0.49∗∗∗ −0.25∗∗∗ −0.47∗∗∗ −0.42∗∗∗ ˆ γ (SMB) - −0.15∗∗∗ 0.25∗∗∗ 0.30∗∗∗ 0.32∗∗∗ 0.19∗∗∗ 0.26∗∗∗ ˆ δ (HML) - −0.03 −0.12∗∗∗ −0.13∗∗ 0.15∗∗∗ −0.09∗ −0.13∗∗∗ ˆ κ (RMW) - 0.37∗∗∗ −0.11∗ −0.11 0.13∗ −0.05 −0.10 ˆ θ (CMA) - 0.42∗∗∗ 0.18∗∗∗ 0.19∗∗ −0.17∗∗ 0.18∗∗∗ 0.25∗∗∗ ˆ φ (BAB) - 0.23∗∗∗ 0.32∗∗∗ 0.40∗∗∗ 0.28∗∗∗ 0.34∗∗∗ 0.32∗∗∗ Adj. R2 - 0.61 0.71 0.67 0.47 0.74 0.71
Five factor model + BAB Regression re∗∗
p,t = αp+ βprem,t+ γpSMBt+ δpHMLt+ κpRMWt+ θpCMAt+ φpBABt+ p,t
Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
ˆ α - - 0.04∗ 0.03 0.02 0.02 0.02 ˆ β (MKT) - - −0.21∗∗∗ −0.27∗∗∗ −0.03 −0.25∗∗∗ −0.20∗∗∗ ˆ γ (SMB) - - 0.40∗∗∗ 0.45∗∗∗ 0.46∗∗∗ 0.34∗∗∗ 0.41∗∗∗ ˆ δ (HML) - - −0.09∗ −0.10∗ 0.18∗∗∗ −0.06 −0.10∗∗ ˆ κ (RMW) - - −0.49∗∗∗ −0.48∗∗∗ −0.24∗∗∗ −0.43∗∗∗ −0.47∗∗∗ ˆ θ (CMA) - - −0.24∗∗∗ −0.23∗∗∗ −0.59∗∗∗ −0.24∗∗∗ −0.17∗∗ ˆ φ (BAB) - - 0.09∗∗ 0.17∗∗∗ 0.05 0.11∗∗∗ 0.10∗∗ Adj. R2 - - 0.37 0.38 0.25 0.41 0.38
*,* and *** denote significance at the 10%, 5% and 1% levels, respectively.
comes from accessing the size, investment and betting against beta factors against the FTSE developed.
5.4
Sensitivity analysis
To further evaluate the shrinkage methods a shorter estimation period and relaxation of the long-only constraint are considered. The results are presented in Table 7. Theoretically, relaxing the long-only constraint should improve the minimum variance portfolios. The portfolios are not forced to use the short option, and should only do so if it is beneficial in terms of reducing risk. However, the results indicate that the standard deviation increases when the constraint is relaxed. This somewhat counter-intuitive result can be explained by Jagannathan and Ma (2003). They show that minimum variance optimizers implicitly apply some form of shrinkage to the sample covariance matrix when short sales are ruled out. This is generally beneficial in terms of improving weight stability
The only viable choices in the portfolio selection problem faced by pension funds are to select an estimation period with the long-only constraint. The literature in this area holds that recent returns are more representative for stocks. However, all but the linear shrunk risk-minimizing portfolios obtain higher Sharpe ratios when the estimation period is set to five years instead of one. One possible explanation may be that the risk-minimizing portfolios benefit from the reduced focus on stocks which have relatively low volatility in the short term past.
Table 7: Risk-adjusted return of the FTSE developed and risk-minimizing portfolios us-ing an one or five year estimation periods with or without the long-only con-straint
This table presents the average return, standard deviation and related Sharpe ratio of the FTSE developed and risk-minimizing portfolios using an one or five year estimation periods with or without the long-only constraint. The number are based on 952 realized weekly returns over the investment period from April 1999 to July 2017. The average return and standard deviation are annualized after calculation. The average return represents the compounded annual growth rate. The related Sharpe ratio is based on the annualized numbers. The FTSE developed (FTSE), low volatility portfolio (Low Vol), corrected minimum variance portfolio (MVPc), and
mini-mum variance portfolio based on the five factor model by Fama and French (2006) (MVP5f), linear shrinkage
(SMVPl), constant correlation model shrinkage (SMVPc) and nonlinear shrinkage (SMPVnl) are denoted in
order of sophistication.
Estimation period of 5 years, with long-only constraint Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Average Return (%) 4.6 8.3 10.2 10.5 9.1 9.1 9.1 Standard Deviation (%) 16.8 12.8 9.5 10.1 13.1 9.0 9.6
Sharpe ratio (%) 17 52 89 87 57 82 77
Estimation period of 5 years, without long-only constraint Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Average Return (%) 4.6 8.3 12.0 11.0 10.8 13.0 11.4 Standard Deviation (%) 16.8 12.8 14.3 11.1 15.3 13.4 13.4
Sharpe ratio (%) 17 52 72 84 59 85 72
Estimation period of 1 year, with long-only constraint Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Average Return (%) 4.6 6.6 7.6 8.7 8.2 6.5 8.2 Standard Deviation (%) 16.8 12.1 10.7 9.6 10.5 9.6 10.2
Sharpe ratio (%) 17 40 55 73 61 50 63
Estimation period of 1 year, without long-only constraint Portfolio FTSE Low Vol MVPc MVP5f SMVPl SMVPc SMVPnl
Average Return (%) 4.6 6.6 10.0 10.4 9.7 11.7 9.9 Standard Deviation (%) 16.8 12.1 12.0 10.4 12.1 10.7 11.9
When the estimation period is set to five years and the long-only constrained is imposed, the corrected minimum variance portfolio does not underperform the other minimum variance portfolios. This result is counter-intuitive as the underlying shrinkage method does not aim to reduce estimation error. The portfolio uses the nearest positive definite matrix algorithm by Higham (2002). The method has a relatively small impact on the sample covariance matrix, see Appendix D. The results imply that in the out-of-sample setting under consideration it is not necessarily beneficial to use more sophisticated shrinkage methods.
When the long-only constraint is relaxed, the minimum variance portfolio based on the constant correlation model shows a higher Sharpe ratio than the corrected minimum variance portfolio. The difference is significant at the 5% level during both estimation periods. This is not sur-prising as Ledoit and Wolf (2004b) construct this portfolio based on the constant correlation model with a specific focus on the portfolio selection problem. When the estimation period is set to one year and the long-only constraint is relaxed, the minimum variance portfolio based on the five factor model also shows significantly higher Sharpe ratio at the 5% level. These results are in line with the literature which holds that sophisticated shrinkage methods have greater improvement potential when the sample size is reduced.
6
Conclusion
This paper has tested the out-of-sample performance of risk-minimizing portfolios using a va-riety of shrinkage methods, namely the low volatility portfolio, corrected minimum variance portfolio using Higman’s algorithm, and minimum variance portfolios based on the five factor model, linear shrinkage, constant correlation model shrinkage and nonlinear shrinkage. In line with the existing literature, the risk-minimizing portfolios achieve higher returns at lower risk relative to the capitalization-weighted FTSE developed index during the period under obser-vation. The Sharpe ratios of the risk-minimizing portfolios are higher than the Sharpe ratio of the FTSE developed at the 5% level. Moreover, the corrected minimum variance portfo-lio and minimum variance portfoportfo-lios based on the five factor model and constant correlation model shrinkage earn higher risk-adjusted returns compared to the low volatility portfolio. The difference in related Sharpe ratios are significant at the 5% level. The results indicate that the minimum variance approach is an improvement on the simple approach based on histori-cal volatilities, due to the ability of the minimum variance approach to capture diversification benefits.
The returns are decomposed using factor models to understand the various return profiles of the risk-minimizing portfolios. The reduced volatility of the risk-minimizing portfolios is explained by their relatively low market beta. Moreover, the excess return of the low volatility portfolio may be explained by the profitability, investment and betting against beta factors, while the excess return of all but the linear shrunk minimum variance portfolios is attributable to the size, investment and betting against beta factors. In related literature, the outperformance of risk-minimizing portfolios is partially attributable to the value factor. However, this study shows that if the five factor is considered then the part originally explained by the value factor is in fact captured by the investment and profitability factors. The results indicate that these factors can provide an improved explanation of the excess return of risk-minimizing portfolios.
excess returns of the portfolio. As a suggestion for future research, a relatively simple long-only portfolio containing low-beta stocks can be constructed. This low-beta portfolio may achieve excess return and lower risk against the low volatility portfolio because of higher loadings on the betting against beta factor and lower exposure to the market factor, respectively. In addi-tion, the low-beta portfolio may have comparable turnover rates relative to the low volatility portfolio.
Risk-minimizing portfolios typically have two disadvantageous characteristics. First, the port-folios can have highly concentrated positions in their construction. Large weights are assigned to stocks with relatively low volatility and/or good diversification properties in the recent past. The performance of these stocks can be closely tied to that of a single sector or national econ-omy. It is therefore possible that a portfolio puts too much emphasis on a specific sector or country. In line with the literature, the observed effective N of the risk-minimizing portfolios are relatively low. More country and sector diversification might be useful for improving the construction of risk-minimizing portfolios.
The second problem is that risk-minimizing portfolios generally have high turnover rates; partic-ularly those constructed according to the minimum variance approach. High turnover rates are associated with high implementation and controlling costs, since at each rebalancing moment a large volume of stock trades must take place to realign with the updated weights. If the associ-ated costs are higher than the excess returns, it is more profitable to invest in a capitalization-weighted index. As expected, the observed turnover rates for the risk-minimizing portfolios are relatively high, particularly for the minimum variance portfolios. The minimum variance portfolio based on the five factor model shows significantly lower turnover rate compared to the other minimum variance portfolios, suggesting that the covariance matrix construction using the five factor model imposes more stable weights. However, more refined portfolio engineering techniques would likely be required to construct more efficient portfolios.
While the investment universe of this research has been based on the FTSE developed in-dex, the out-of-sample test can be readily performed on other investment universes and over different periods to see if the results are consistent across universes and timeframes. Moreover, the covariance matrix is based on an estimation period of one or five years. Historically, covari-ance estimates appear to be more stable over time in comparison to individual stock varicovari-ance estimates. To increase estimator performance, the diagonals and non-diagonals could be based on different optimal estimation periods.
Finally, risk-minimizing portfolios are of interest to pension fund managers and researchers alike mainly due to their historically high returns. The economy is always changing, and it will remain up to the investor or fund manager to decide whether historical performance is to be considered a sufficiently reliable indicator of future performance. Moreover, the low volatility anomaly arises from behavioural patterns or leverage limitations, leading to overvaluation of high volatility stocks and a corresponding undervaluation of low volatility stocks. As such, investment strategies unaware of the valuation level of stocks may earn less return than the out-of-sample test suggests.
