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Portfolio selection with conservative short-selling
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Finance Research Letters 0 0 0 (2016) 1–7
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Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Portfolio selection with conservative short-selling Jang Ho Kim a,∗, Woo Chang Kim b, Frank J. Fabozzi c a b c
Department of Industrial and Management Systems Engineering, Kyung Hee University, Republic of Korea Department of Industrial and Systems Engineering, KAIST, Republic of Korea EDHEC Business School, France
a r t i c l e
i n f o
Article history: Received 18 February 2016 Revised 3 May 2016 Accepted 27 May 2016 Available online xxx JEL classification: C44 C61 G11 Keywords: Mean-variance portfolio selection No short-selling constraint Conservative short positions
a b s t r a c t Mean-variance analysis is considered the foundation of portfolio selection. Among various attempts to address the limitations of the original model as formulated by Markowitz more than 60 years ago, one simple solution has been to impose constraints on weights in order to reduce efficient portfolios with extreme weights that may be caused by estimation errors in the inputs. Although no short-selling constraints are often considered, the restriction removes opportunities to gain from short-selling and short positions provide various investment opportunities such as long/short strategies. In this paper we propose a portfolio selection model that allows short positions while examining the worst case only for assets that are assigned negative weights. The proposed model constructs portfolios with conservative short positions and the conservative level can be adjusted by the investor. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Mean-variance analysis proposed by Markowitz (1952, 1959) is still considered the foundation of portfolio management and in recent years this framework for portfolio construction has become even more popular as a result of the advancement in automated investment management. When the mean-variance model is applied to asset allocation, the non-negativity constraint on portfolio weights is reasonable since the objective is often to divide the overall investment amount among several asset classes. But short positions in candidate assets can be beneficial and even necessary in portfolio selection when focusing on the construction of portfolios within an individual asset class. Fundamentally, short sales allow investors to gain from assets that are overvalued. When short sales are not restricted, long/short strategies provide opportunities to exploit both long and short positions and the advantage of such strategies over long-only strategies has been widely studied (Jacobs and Levy, 1993, and Grinold and Kahn, 20 0 0). For example, combining long and short positions allows constructing market-neutral strategies. It is also argued that restricting short sales leads to a less efficient market and higher volatility (Saffi and Sigurdsson, 2011, and Yeh and Chen, 2014). These observations illustrate the need for short sales in financial markets. More importantly, in portfolio construction, Levy (1983), Green and Hollifield (1992), and Brennan and Lo (2010) observe that mean-variance efficient portfolios contain negative weights. Furthermore, Behr et al. (2013) show that minimum-variance portfolios without constraints have lower variance and a higher Sharpe ratio than the portfolios constructed with short-sale constraints when the number of candidate assets is small. Finally, short positions in some assets allow taking larger long positions in other assets and thus become advantageous when large ∗
Corresponding author. E-mail address: [email protected] (J.H. Kim).
http://dx.doi.org/10.1016/j.frl.2016.05.015 1544-6123/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: J.H. Kim et al., Portfolio selection with conservative short-selling, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.015
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exposure is favored. In summary, short selling is an inevitable part of investing and the advantages motivate investors to consider short positions in their portfolios. Nonetheless, short positions are generally associated with risky investment behavior. In fact, Jagannathan and Ma (2003) find that imposing no-shorting constraints may help reduce estimation errors in practice. However, imposing noshort-sale constraints is not always a recommended approach because many asset weights are bounded by the constraint and given zero weight (Black and Litterman, 1992). This is unappealing to investors because the recommended portfolio only invests in a few assets among many that are available. Thus, while no-shorting constraints prevent extreme weights that may appear in unconstrained portfolios, applying the constraint may be too restrictive because some assets in no-shorting portfolios may be given zero weight and thus being determined by the lower bound set by the investor. In this paper, we propose a portfolio selection model based on the mean-variance framework that forms portfolios allocating short positions in a more conservative fashion. The model considers estimation errors in expected returns when finding the optimal short positions. By employing the proposed model, investors will be able to form portfolios that contain negative positions that are less risky than portfolios without constraints and yet more aggressive than portfolios with no-shorting constraints. The remainder of the paper is organized as follows. Section 2 summarizes the mean-variance portfolio selection framework. Our proposed approach for forming portfolio with short sales is derived in Section 3. Simulation results comparing various portfolios are presented in Section 4, and Section 5 concludes the paper. 2. Mean-variance portfolio selection The efficient portfolios in the mean-variance framework can be found using several formulations. The following formulation finds the portfolio with minimum variance among portfolios with expected return of at least r and bounds on weights imposed,
ωT ω s.t. μ ω ≥ r, ωT ι = 1, l ≤ ω ≤ h min ω
1 2 T
(1)
where ω ∈ Rn is the portfolio weight allocated to n assets, μ ∈ Rn is the expected return of assets, ∈ Rn×n is the covariance matrix of asset returns, r ∈ R is the desired level of portfolio return, ι ∈ Rn is the vector of ones, l ∈ Rn sets the lower bound on portfolio weights, and h ∈ Rn sets the upper bound on portfolio weights. We assume that l ≤ 0 and 0 ≤ h for the problems we are investigating. Unconstrained portfolios are formed if the last constraint in (1) is removed, and short-sale-restricted portfolios are constructed if l is set to zeros and h is set to ones (or when h is dismissed). For portfolios with short-selling allowed, the lower bound l will contain negative values. We should note that the problem given by (1) may be infeasible if a high expected return is required while setting a tight bound on portfolio weights. The major concerns with the basic mean-variance formulation given by (1) are the difficulty in accurately estimating the input values and the high sensitivity to changes in the input parameters (Best and Grauer, 1991). Therefore, large exposure, especially in short-selling, can be damaging when realized asset returns are different from the ex-ante estimates. We next present a revised mean-variance model that invests in short-sale positions cautiously where the conservative level can be adjusted. 3. The conservative short sale model The proposed portfolio selection model is based on the classical formulation given by (1). The main idea behind our model is to consider the worst case of asset combination given negative weights for finding the optimal portfolio instead of restricting short positions altogether. For considering the worst asset returns, we borrow the concept of uncertainty sets from robust optimization (see, for example, Fabozzi, Huang, and Zhou, 2010; Kim, Kim, and Fabozzi, 2014, 2016). If the expected return of an asset i is assumed to deviate at most δi from an estimated value μ ˆ i , the uncertainty set for expected asset returns can be written as
{μ | |μi − μˆ i | ≤ δi , i = 1, . . . , n}
(2)
where μ ˆ ∈ Rn is an estimate of the means and δ ∈ Rn sets the possible deviation from the estimated value for each asset (Fabozzi et al., 2007). Hence, the worst expected return from investing in asset i when ωi < 0 is μ ˆ i + δi because (μ ˆ i + δi ) ω i will have the smallest value among the possible outcomes from the uncertainty set. These observations lead to a formulation for forming portfolios with conservative short-selling, which is explained in the following proposition. Proposition 1. When the expected asset returns are described by the set given by (2), the following portfolio formulation finds the optimal portfolio with weights between l and h by considering the worst outcome for only the amount of short positions that exceed γ ,
min ω ω+ ,ω− 2 1
T
ω
s.t. μ ˆ T ω − δ T ω− ≥ r,
ωT ι = 1, ω = ω+ − ω− − γ , 0 ≤ ω− ≤ −l − γ , 0 ≤ ω+ ≤ h + γ
(3)
Please cite this article as: J.H. Kim et al., Portfolio selection with conservative short-selling, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.015
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3
Table 1 Average correlation and absolute deviation among portfolios. Panel A. Portfolios investing in 10 assets with −100% ≤ ω ≤ 100%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.73 1 1 1.18 1.14 0
0.81 0.94 1.25 1.36 0.93 0.31
0.83 0.93 1.32 1.40 0.86 0.40
0.90 0.88 1.54 1.53 0.64 0.66
0.94 0.83 1.69 1.61 0.45 0.85
0.97 0.80 1.77 1.66 0.32 0.95
0.98 0.78 1.83 1.70 0.23 1.01
1 0.73 1.94 1.78 0 1.14
Panel B. Portfolios investing in 100 assets with −100% ≤ ω ≤ 100%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.33 1 1 1.84 12.11 0
0.48 0.49 3.81 4.45 11.20 3.41
0.59 0.52 5.03 5.55 10.26 4.48
0.84 0.45 8.51 8.73 7.06 7.85
0.96 0.38 10.82 10.91 3.86 10.20
0.99 0.35 12.04 12.07 1.57 11.43
1.00 0.33 12.54 12.55 0.43 11.94
1 0.33 12.70 12.71 0 12.11
where, l ≤ 0, 0 ≤ h, 0 ≤ γ ≤ −l, and ω+ , ω− ∈ Rn . Proof. See the appendix. A special case of the formulation given by (3) is when γ is set to zero. In this case, the worst possible outcome for an asset given negative weight is considered when computing the overall expected portfolio return. Therefore, assets will be assigned short positions if it is attractive to short-sell even when considering their worst returns. As the value of γ increases, the portfolio becomes less conservative. For example, if γ is set to 10% and the portfolio weight given to asset i is –15%, then the –10% allocation is assumed to collect the estimated expected return of μ ˆ i and only the additional –5% allocation in asset i is assumed to observe the worst situation. Separating the expected return of a single asset into two components allows investors to set the level of desired conservativeness as further illustrated in Section 4. We note that while the formulation given by (3) focuses on cautiously allocating negative weights, taking smaller short positions will affect the magnitude of long positions as well and thus become more conservative in both long and short allocations. Moreover, the proposed formulation only models possible deviations in expected returns but it is applicable in practice because errors in expected returns affect portfolios much more than the other inputs of mean-variance analysis (Chopra and Ziemba, 1993). 4. Simulation results In this section, we perform simulation to show how the proposed model forms portfolios that have weights that are not as extreme as the mean-variance portfolios formed from (1), and a comparison is also made with mean-variance portfolios with no-shorting constraints. For simplicity, we denote portfolios from formulation (1) as MV, portfolio from formulation (1) with no-shorting constraint (i.e., l is set to zero) as MV-NS, and portfolios from formulation (3) as MV-γ for a specific level of γ . The simulation is based on the steps performed by Jorion (1992), which are outlined below. Step Step Step Step Step
1. 2. 3. 4. 5.
Compute the means and covariance matrix from historical returns of N assets. From a multivariate normal distribution with parameters from Step 1, draw T samples for N assets. From the T estimated returns, compute the means and covariance matrix. Find the optimal portfolio by using the estimates from Step 3. Repeat Steps 2 to 4 for 10 0 0 iterations.
The first step asserts that the simulated returns will represent a realistic situation, and we perform the simulation with three separate data sets: historical returns of 10 industries, 30 industries, and 100 portfolios formed on size and the ratio of book equity to market equity.1 These three data sets are chosen because they divide the U.S. stock market into 10, 30, and 100 assets, and these will provide observations when the number of candidate assets is small as well as large. Investment availability and liquidity of the candidate assets are unimportant in our analysis since we are focusing on comparing the composition of portfolios. Monthly returns from 1991 to 2015 are used in the experiment. 1 Historical data are collected from the data library of Kenneth R. French, available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_ library.html.
Please cite this article as: J.H. Kim et al., Portfolio selection with conservative short-selling, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.015
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Table 2 Average correlation and absolute deviation among portfolios with tight bounds. Panel A. Portfolios investing in 30 assets with −20% ≤ ω ≤ 20%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.55 1 1 1.44 2.33 0
0.69 0.83 1.52 1.82 1.98 0.70
0.76 0.81 1.77 2.00 1.75 0.93
0.91 0.69 2.42 2.49 1.05 1.65
0.98 0.60 2.80 2.79 0.51 2.07
1.00 0.56 2.96 2.92 0.18 2.25
1 0.55 3.03 2.97 0 2.33
1 0.55 3.03 2.97 0 2.33
Panel B. Portfolios investing in 100 assets with −20% ≤ ω ≤ 20%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.31 1 1 1.82 9.82 0
0.52 0.51 3.35 3.96 8.56 2.80
0.63 0.50 4.65 5.12 7.68 3.97
0.87 0.41 7.99 8.16 4.74 7.22
0.98 0.34 9.87 9.89 1.90 9.12
1.00 0.32 10.49 10.48 0.22 9.77
1 0.31 10.54 10.52 0 9.82
1 0.31 10.54 10.52 0 9.82
Fig. 1. Average number of assets with absolute weight above threshold.
Please cite this article as: J.H. Kim et al., Portfolio selection with conservative short-selling, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.015
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Fig. 2. Average weights of portfolios.
In each iteration of the simulation, 250 sample returns are produced (T = 250) and portfolios with expected monthly return of at least 1% are formed. For setting the value of δi for asset i, twice the standard error of the sample mean returns is used, which is equivalent to considering 95% of the outcomes. Various levels of γ are used to investigate portfolios with different conservative levels. Table 1 shows the average results for portfolios investing in 10 and 100 candidate assets when portfolio weights are restricted to be between –100% and 100%. Measures collected include correlation of portfolio weights with MV and also with MV-NS. Absolute deviations in portfolio weights relative to MV, MV-NS, and the equally-weighted portfolio are also presented, where absolute deviation is computed by taking the sum of absolute differences in asset weights between two portfolios. The values reported are average values from 10 0 0 iterations. All outcomes for every measure exhibit a very clear pattern: MV-0 is most similar to MV-NS and MV-γ becomes closer to MV as the value of γ increases. Strong evidence of this pattern is also shown in Table 2 where portfolios have weights between –20% and 20%. Note that MV-0.2 portfolios in this case are identical to MV because no assets can be given a weight less than –20%. Fig. 1 plots the average number of assets in each portfolio with absolute weight greater than 10%, 20%, 30%, 40%, and 50%. As expected MV has the largest number of assets with large exposures representing their aggressiveness while MV-NS has the smallest number of assets since restricting short-selling limits the overall exposure. More interestingly, the same patterns shown in Tables 1 and 2 also appear in Fig. 1: MV-0 is most similar to MV-NS and MV-γ becomes closer to MV as the value of γ is increased. These findings also certainly demonstrate how the proposed portfolio formulation can control the level of conservativeness through γ while examining short-sale opportunities. The results so far display a clear pattern of MV-γ having characteristics that fall between MV and MV-NS, and observing average weights allocated to each asset also display similar ordering as shown in Fig. 2. However, the average weights of MV-γ for some assets are sometimes either above both MV and MV-NS or below both MV and MV-NS. These occur because MV-γ portfolios are not computed from the weights of MV and MV-NS but rather a portfolio optimized from the proposed formulation in this paper. Thus, the optimized portfolios from our model form portfolios that are conservative as well as being optimal under the worst-case modeling of short positions. Please cite this article as: J.H. Kim et al., Portfolio selection with conservative short-selling, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.015
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5. Conclusion In this paper, imposing restrictions on short-selling in mean-variance analysis is revisited. Constraining short positions in portfolio selection addresses limitations of the unconstrained mean-variance model by reducing the sensitivity caused by estimation errors in means, variances, and covariances of asset returns, but it also introduces new concerns such as not exploiting all candidate assets and also not taking advantage of downside movements. Therefore, this paper presents a revised mean-variance formulation that allows short-selling but examines the worst case only when short positions are allocated to assets. The worst case is computed from a predefined set of possible values, which are defined by confidence intervals in our experiment. Simulation results show that the proposed portfolios have characteristics that fall between mean-variance portfolios with and without no-shorting constraints. As the conservative level of the proposed model is increased, the resulting portfolio becomes more similar to the no-shorting constrained portfolio. These findings confirm that our model is effective at forming optimal mean-variance portfolios with conservative short-selling. Acknowledgements The authors are grateful for the suggestions of the anonymous referees and for the support by a grant from Kyung Hee University in 2015 (KHU-20151538). Appendix Proof of Proposition 1. The formulation (1) can be written as the following where portfolio weight ω is split into positive and negative components,
min ω ω+ ,ω− 2 1
T
ω
s.t. μT ω ≥ r,
ωT ι = 1, ω = ω+ − ω− , 0 ≤ ω− ≤ −l, 0 ≤ ω+ ≤ h
(4)
where ω+ ∈ Rn represents the positive components of ω and ω− ∈ Rn represents the negative components of ω. Recall that we assume l ≤ 0 and 0 ≤ h. The above reformulation applies standard optimization technique for handling absolute value functions and at least one of ω+,i or ω−,i is guaranteed to be zero because the objective function can be further minimized otherwise. The worst possible expected return from investing in asset i when ωi < 0 is μ ˆ i + δi because (μ ˆ i + δi )ωi will have the smallest value among the possible outcomes from the uncertainty set defined by (2). But since we only want to apply the worst case to the values of ω− , the constraint μT ω ≥ r becomes μ ˆ T ω + δ T ( −ω − ) = μ ˆ T ω − δ T ω− ≥ r. The formulation given by (4) can be further modified to only consider the worst case for the amount of short sales that ∗ + γ where ω ∗ is the amount of short sales exceeding γ . Then, exceed γ ≥ 0 by replacing all occurrences of ω− with ω− − the last three constraints of (4) become
ω = ω+ − ω−∗ − γ , −γ ≤ ω−∗ ≤ −l − γ , 0 ≤ ω+ ≤ h. ∗ < 0 and therefore the worst case should not be considered in this case. This is resolved by But ω > −γ when −γ ≤ ω− ∗ to be non-negative but allowing ω to have a maximum of h + γ , and the three constraints are now written restricting ω− + as
ω = ω+ − ω−∗ − γ , 0 ≤ ω−∗ ≤ −l − γ , 0 ≤ ω+ ≤ h + γ . Finally, we arrive at a portfolio problem identical to (3) in Proposition 1.
References Behr, P., Guettler, A., Miebs, F., 2013. On portfolio optimization: imposing the right constraints. J. Bank. Finance 37 (4), 1232–1242. Best, M.J., Grauer, R.R., 1991. On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. Rev. Financ. Stud. 4 (2), 315–342. Black, F., Litterman, R., 1992. Global portfolio optimization. Financ. Anal. J. 48 (5), 28–43. Brennan, T.J., Lo, A.W., 2010. Impossible frontiers. Manag. Sci. 56 (6), 905–923. Chopra, V.K., Ziemba, W.T., 1993. The effect of errors in means, variances, and covariances on optimal portfolio choice. J. Portfolio Manag. 19 (2), 6–11. Fabozzi, F.J., Huang, D., Zhou, G., 2010. Robust portfolios: contributions from operations research and finance. Ann. Oper. Res. 176 (1), 191–220. Fabozzi, F.J., Kolm, P.N., Pachamanova, D., Focardi, S.M., 2007. Robust Portfolio Optimization and Management. Wiley, Hoboken, NJ. Green, R.C., Hollifield, B., 1992. When will mean-variance efficient portfolios be well diversified? J. Finance 47 (5), 1785–1809. Grinold, R.C., Kahn, R.N., 20 0 0. The efficiency gains of long-short investing. Financ. Anal. J. 56 (6), 40–53. Jacobs, B.I., Levy, K.N., 1993. Long/short equity investing. J. Portfolio Manag. 20 (1), 52–63. Jagannathan, R., Ma, T., 2003. Risk reduction in large portfolios: why imposing the wrong constraints helps. J. Finance 58 (4), 1651–1684. Jorion, P., 1992. Portfolio optimization in practice. Financ. Anal. J. 48 (1), 68–74. Kim, J.H., Kim, W.C., Fabozzi, F.J., 2014. Recent developments in robust portfolios with a worst-case approach. J. Optim. Theory Appl. 161 (1), 103–121. Kim, W.C., Kim, J.H., Fabozzi, F.J., 2016. Robust Equity Portfolio Management + Website: Formulations, Implementations, and Properties using MATLAB. Wiley, Hoboken, NJ.
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Levy, H., 1983. The capital asset pricing model: theory and empiricism. Econ. J. 93 (369), 145–165. Markowitz, H., 1952. Portfolio selection. J. Finance 7 (1), 77–91. Markowitz, H., 1959. Portfolio Selection: Efficient Diversification of Investments. Wiley, New York, NY. Saffi, P., Sigurdsson, K., 2011. Price efficiency and short selling. Rev. Financ. Stud. 24 (3), 821–852. Yeh, J.H., Chen, L.C., 2014. Stabilizing the market with short sale constraint? New evidence from price jump activities. Finance Res. Lett. 11 (3), 238–246.
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Finance Research Letters 0 0 0 (2016) 1–7
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Portfolio selection with conservative short-selling Jang Ho Kim a,∗, Woo Chang Kim b, Frank J. Fabozzi c a b c
Department of Industrial and Management Systems Engineering, Kyung Hee University, Republic of Korea Department of Industrial and Systems Engineering, KAIST, Republic of Korea EDHEC Business School, France
a r t i c l e
i n f o
Article history: Received 18 February 2016 Revised 3 May 2016 Accepted 27 May 2016 Available online xxx JEL classification: C44 C61 G11 Keywords: Mean-variance portfolio selection No short-selling constraint Conservative short positions
a b s t r a c t Mean-variance analysis is considered the foundation of portfolio selection. Among various attempts to address the limitations of the original model as formulated by Markowitz more than 60 years ago, one simple solution has been to impose constraints on weights in order to reduce efficient portfolios with extreme weights that may be caused by estimation errors in the inputs. Although no short-selling constraints are often considered, the restriction removes opportunities to gain from short-selling and short positions provide various investment opportunities such as long/short strategies. In this paper we propose a portfolio selection model that allows short positions while examining the worst case only for assets that are assigned negative weights. The proposed model constructs portfolios with conservative short positions and the conservative level can be adjusted by the investor. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Mean-variance analysis proposed by Markowitz (1952, 1959) is still considered the foundation of portfolio management and in recent years this framework for portfolio construction has become even more popular as a result of the advancement in automated investment management. When the mean-variance model is applied to asset allocation, the non-negativity constraint on portfolio weights is reasonable since the objective is often to divide the overall investment amount among several asset classes. But short positions in candidate assets can be beneficial and even necessary in portfolio selection when focusing on the construction of portfolios within an individual asset class. Fundamentally, short sales allow investors to gain from assets that are overvalued. When short sales are not restricted, long/short strategies provide opportunities to exploit both long and short positions and the advantage of such strategies over long-only strategies has been widely studied (Jacobs and Levy, 1993, and Grinold and Kahn, 20 0 0). For example, combining long and short positions allows constructing market-neutral strategies. It is also argued that restricting short sales leads to a less efficient market and higher volatility (Saffi and Sigurdsson, 2011, and Yeh and Chen, 2014). These observations illustrate the need for short sales in financial markets. More importantly, in portfolio construction, Levy (1983), Green and Hollifield (1992), and Brennan and Lo (2010) observe that mean-variance efficient portfolios contain negative weights. Furthermore, Behr et al. (2013) show that minimum-variance portfolios without constraints have lower variance and a higher Sharpe ratio than the portfolios constructed with short-sale constraints when the number of candidate assets is small. Finally, short positions in some assets allow taking larger long positions in other assets and thus become advantageous when large ∗
Corresponding author. E-mail address: [email protected] (J.H. Kim).
http://dx.doi.org/10.1016/j.frl.2016.05.015 1544-6123/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: J.H. Kim et al., Portfolio selection with conservative short-selling, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.015
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exposure is favored. In summary, short selling is an inevitable part of investing and the advantages motivate investors to consider short positions in their portfolios. Nonetheless, short positions are generally associated with risky investment behavior. In fact, Jagannathan and Ma (2003) find that imposing no-shorting constraints may help reduce estimation errors in practice. However, imposing noshort-sale constraints is not always a recommended approach because many asset weights are bounded by the constraint and given zero weight (Black and Litterman, 1992). This is unappealing to investors because the recommended portfolio only invests in a few assets among many that are available. Thus, while no-shorting constraints prevent extreme weights that may appear in unconstrained portfolios, applying the constraint may be too restrictive because some assets in no-shorting portfolios may be given zero weight and thus being determined by the lower bound set by the investor. In this paper, we propose a portfolio selection model based on the mean-variance framework that forms portfolios allocating short positions in a more conservative fashion. The model considers estimation errors in expected returns when finding the optimal short positions. By employing the proposed model, investors will be able to form portfolios that contain negative positions that are less risky than portfolios without constraints and yet more aggressive than portfolios with no-shorting constraints. The remainder of the paper is organized as follows. Section 2 summarizes the mean-variance portfolio selection framework. Our proposed approach for forming portfolio with short sales is derived in Section 3. Simulation results comparing various portfolios are presented in Section 4, and Section 5 concludes the paper. 2. Mean-variance portfolio selection The efficient portfolios in the mean-variance framework can be found using several formulations. The following formulation finds the portfolio with minimum variance among portfolios with expected return of at least r and bounds on weights imposed,
ωT ω s.t. μ ω ≥ r, ωT ι = 1, l ≤ ω ≤ h min ω
1 2 T
(1)
where ω ∈ Rn is the portfolio weight allocated to n assets, μ ∈ Rn is the expected return of assets, ∈ Rn×n is the covariance matrix of asset returns, r ∈ R is the desired level of portfolio return, ι ∈ Rn is the vector of ones, l ∈ Rn sets the lower bound on portfolio weights, and h ∈ Rn sets the upper bound on portfolio weights. We assume that l ≤ 0 and 0 ≤ h for the problems we are investigating. Unconstrained portfolios are formed if the last constraint in (1) is removed, and short-sale-restricted portfolios are constructed if l is set to zeros and h is set to ones (or when h is dismissed). For portfolios with short-selling allowed, the lower bound l will contain negative values. We should note that the problem given by (1) may be infeasible if a high expected return is required while setting a tight bound on portfolio weights. The major concerns with the basic mean-variance formulation given by (1) are the difficulty in accurately estimating the input values and the high sensitivity to changes in the input parameters (Best and Grauer, 1991). Therefore, large exposure, especially in short-selling, can be damaging when realized asset returns are different from the ex-ante estimates. We next present a revised mean-variance model that invests in short-sale positions cautiously where the conservative level can be adjusted. 3. The conservative short sale model The proposed portfolio selection model is based on the classical formulation given by (1). The main idea behind our model is to consider the worst case of asset combination given negative weights for finding the optimal portfolio instead of restricting short positions altogether. For considering the worst asset returns, we borrow the concept of uncertainty sets from robust optimization (see, for example, Fabozzi, Huang, and Zhou, 2010; Kim, Kim, and Fabozzi, 2014, 2016). If the expected return of an asset i is assumed to deviate at most δi from an estimated value μ ˆ i , the uncertainty set for expected asset returns can be written as
{μ | |μi − μˆ i | ≤ δi , i = 1, . . . , n}
(2)
where μ ˆ ∈ Rn is an estimate of the means and δ ∈ Rn sets the possible deviation from the estimated value for each asset (Fabozzi et al., 2007). Hence, the worst expected return from investing in asset i when ωi < 0 is μ ˆ i + δi because (μ ˆ i + δi ) ω i will have the smallest value among the possible outcomes from the uncertainty set. These observations lead to a formulation for forming portfolios with conservative short-selling, which is explained in the following proposition. Proposition 1. When the expected asset returns are described by the set given by (2), the following portfolio formulation finds the optimal portfolio with weights between l and h by considering the worst outcome for only the amount of short positions that exceed γ ,
min ω ω+ ,ω− 2 1
T
ω
s.t. μ ˆ T ω − δ T ω− ≥ r,
ωT ι = 1, ω = ω+ − ω− − γ , 0 ≤ ω− ≤ −l − γ , 0 ≤ ω+ ≤ h + γ
(3)
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Table 1 Average correlation and absolute deviation among portfolios. Panel A. Portfolios investing in 10 assets with −100% ≤ ω ≤ 100%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.73 1 1 1.18 1.14 0
0.81 0.94 1.25 1.36 0.93 0.31
0.83 0.93 1.32 1.40 0.86 0.40
0.90 0.88 1.54 1.53 0.64 0.66
0.94 0.83 1.69 1.61 0.45 0.85
0.97 0.80 1.77 1.66 0.32 0.95
0.98 0.78 1.83 1.70 0.23 1.01
1 0.73 1.94 1.78 0 1.14
Panel B. Portfolios investing in 100 assets with −100% ≤ ω ≤ 100%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.33 1 1 1.84 12.11 0
0.48 0.49 3.81 4.45 11.20 3.41
0.59 0.52 5.03 5.55 10.26 4.48
0.84 0.45 8.51 8.73 7.06 7.85
0.96 0.38 10.82 10.91 3.86 10.20
0.99 0.35 12.04 12.07 1.57 11.43
1.00 0.33 12.54 12.55 0.43 11.94
1 0.33 12.70 12.71 0 12.11
where, l ≤ 0, 0 ≤ h, 0 ≤ γ ≤ −l, and ω+ , ω− ∈ Rn . Proof. See the appendix. A special case of the formulation given by (3) is when γ is set to zero. In this case, the worst possible outcome for an asset given negative weight is considered when computing the overall expected portfolio return. Therefore, assets will be assigned short positions if it is attractive to short-sell even when considering their worst returns. As the value of γ increases, the portfolio becomes less conservative. For example, if γ is set to 10% and the portfolio weight given to asset i is –15%, then the –10% allocation is assumed to collect the estimated expected return of μ ˆ i and only the additional –5% allocation in asset i is assumed to observe the worst situation. Separating the expected return of a single asset into two components allows investors to set the level of desired conservativeness as further illustrated in Section 4. We note that while the formulation given by (3) focuses on cautiously allocating negative weights, taking smaller short positions will affect the magnitude of long positions as well and thus become more conservative in both long and short allocations. Moreover, the proposed formulation only models possible deviations in expected returns but it is applicable in practice because errors in expected returns affect portfolios much more than the other inputs of mean-variance analysis (Chopra and Ziemba, 1993). 4. Simulation results In this section, we perform simulation to show how the proposed model forms portfolios that have weights that are not as extreme as the mean-variance portfolios formed from (1), and a comparison is also made with mean-variance portfolios with no-shorting constraints. For simplicity, we denote portfolios from formulation (1) as MV, portfolio from formulation (1) with no-shorting constraint (i.e., l is set to zero) as MV-NS, and portfolios from formulation (3) as MV-γ for a specific level of γ . The simulation is based on the steps performed by Jorion (1992), which are outlined below. Step Step Step Step Step
1. 2. 3. 4. 5.
Compute the means and covariance matrix from historical returns of N assets. From a multivariate normal distribution with parameters from Step 1, draw T samples for N assets. From the T estimated returns, compute the means and covariance matrix. Find the optimal portfolio by using the estimates from Step 3. Repeat Steps 2 to 4 for 10 0 0 iterations.
The first step asserts that the simulated returns will represent a realistic situation, and we perform the simulation with three separate data sets: historical returns of 10 industries, 30 industries, and 100 portfolios formed on size and the ratio of book equity to market equity.1 These three data sets are chosen because they divide the U.S. stock market into 10, 30, and 100 assets, and these will provide observations when the number of candidate assets is small as well as large. Investment availability and liquidity of the candidate assets are unimportant in our analysis since we are focusing on comparing the composition of portfolios. Monthly returns from 1991 to 2015 are used in the experiment. 1 Historical data are collected from the data library of Kenneth R. French, available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_ library.html.
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Table 2 Average correlation and absolute deviation among portfolios with tight bounds. Panel A. Portfolios investing in 30 assets with −20% ≤ ω ≤ 20%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.55 1 1 1.44 2.33 0
0.69 0.83 1.52 1.82 1.98 0.70
0.76 0.81 1.77 2.00 1.75 0.93
0.91 0.69 2.42 2.49 1.05 1.65
0.98 0.60 2.80 2.79 0.51 2.07
1.00 0.56 2.96 2.92 0.18 2.25
1 0.55 3.03 2.97 0 2.33
1 0.55 3.03 2.97 0 2.33
Panel B. Portfolios investing in 100 assets with −20% ≤ ω ≤ 20%
Corr. with MV Corr. with MV-NS Abs. dev. Abs. dev. from 1/N Abs. dev. from MV Abs. dev. from MV-NS
MV-NS
MV-0
MV-0.01
MV-0.05
MV-0.1
MV-0.15
MV-0.2
MV
0.31 1 1 1.82 9.82 0
0.52 0.51 3.35 3.96 8.56 2.80
0.63 0.50 4.65 5.12 7.68 3.97
0.87 0.41 7.99 8.16 4.74 7.22
0.98 0.34 9.87 9.89 1.90 9.12
1.00 0.32 10.49 10.48 0.22 9.77
1 0.31 10.54 10.52 0 9.82
1 0.31 10.54 10.52 0 9.82
Fig. 1. Average number of assets with absolute weight above threshold.
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Fig. 2. Average weights of portfolios.
In each iteration of the simulation, 250 sample returns are produced (T = 250) and portfolios with expected monthly return of at least 1% are formed. For setting the value of δi for asset i, twice the standard error of the sample mean returns is used, which is equivalent to considering 95% of the outcomes. Various levels of γ are used to investigate portfolios with different conservative levels. Table 1 shows the average results for portfolios investing in 10 and 100 candidate assets when portfolio weights are restricted to be between –100% and 100%. Measures collected include correlation of portfolio weights with MV and also with MV-NS. Absolute deviations in portfolio weights relative to MV, MV-NS, and the equally-weighted portfolio are also presented, where absolute deviation is computed by taking the sum of absolute differences in asset weights between two portfolios. The values reported are average values from 10 0 0 iterations. All outcomes for every measure exhibit a very clear pattern: MV-0 is most similar to MV-NS and MV-γ becomes closer to MV as the value of γ increases. Strong evidence of this pattern is also shown in Table 2 where portfolios have weights between –20% and 20%. Note that MV-0.2 portfolios in this case are identical to MV because no assets can be given a weight less than –20%. Fig. 1 plots the average number of assets in each portfolio with absolute weight greater than 10%, 20%, 30%, 40%, and 50%. As expected MV has the largest number of assets with large exposures representing their aggressiveness while MV-NS has the smallest number of assets since restricting short-selling limits the overall exposure. More interestingly, the same patterns shown in Tables 1 and 2 also appear in Fig. 1: MV-0 is most similar to MV-NS and MV-γ becomes closer to MV as the value of γ is increased. These findings also certainly demonstrate how the proposed portfolio formulation can control the level of conservativeness through γ while examining short-sale opportunities. The results so far display a clear pattern of MV-γ having characteristics that fall between MV and MV-NS, and observing average weights allocated to each asset also display similar ordering as shown in Fig. 2. However, the average weights of MV-γ for some assets are sometimes either above both MV and MV-NS or below both MV and MV-NS. These occur because MV-γ portfolios are not computed from the weights of MV and MV-NS but rather a portfolio optimized from the proposed formulation in this paper. Thus, the optimized portfolios from our model form portfolios that are conservative as well as being optimal under the worst-case modeling of short positions. Please cite this article as: J.H. Kim et al., Portfolio selection with conservative short-selling, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.015
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5. Conclusion In this paper, imposing restrictions on short-selling in mean-variance analysis is revisited. Constraining short positions in portfolio selection addresses limitations of the unconstrained mean-variance model by reducing the sensitivity caused by estimation errors in means, variances, and covariances of asset returns, but it also introduces new concerns such as not exploiting all candidate assets and also not taking advantage of downside movements. Therefore, this paper presents a revised mean-variance formulation that allows short-selling but examines the worst case only when short positions are allocated to assets. The worst case is computed from a predefined set of possible values, which are defined by confidence intervals in our experiment. Simulation results show that the proposed portfolios have characteristics that fall between mean-variance portfolios with and without no-shorting constraints. As the conservative level of the proposed model is increased, the resulting portfolio becomes more similar to the no-shorting constrained portfolio. These findings confirm that our model is effective at forming optimal mean-variance portfolios with conservative short-selling. Acknowledgements The authors are grateful for the suggestions of the anonymous referees and for the support by a grant from Kyung Hee University in 2015 (KHU-20151538). Appendix Proof of Proposition 1. The formulation (1) can be written as the following where portfolio weight ω is split into positive and negative components,
min ω ω+ ,ω− 2 1
T
ω
s.t. μT ω ≥ r,
ωT ι = 1, ω = ω+ − ω− , 0 ≤ ω− ≤ −l, 0 ≤ ω+ ≤ h
(4)
where ω+ ∈ Rn represents the positive components of ω and ω− ∈ Rn represents the negative components of ω. Recall that we assume l ≤ 0 and 0 ≤ h. The above reformulation applies standard optimization technique for handling absolute value functions and at least one of ω+,i or ω−,i is guaranteed to be zero because the objective function can be further minimized otherwise. The worst possible expected return from investing in asset i when ωi < 0 is μ ˆ i + δi because (μ ˆ i + δi )ωi will have the smallest value among the possible outcomes from the uncertainty set defined by (2). But since we only want to apply the worst case to the values of ω− , the constraint μT ω ≥ r becomes μ ˆ T ω + δ T ( −ω − ) = μ ˆ T ω − δ T ω− ≥ r. The formulation given by (4) can be further modified to only consider the worst case for the amount of short sales that ∗ + γ where ω ∗ is the amount of short sales exceeding γ . Then, exceed γ ≥ 0 by replacing all occurrences of ω− with ω− − the last three constraints of (4) become
ω = ω+ − ω−∗ − γ , −γ ≤ ω−∗ ≤ −l − γ , 0 ≤ ω+ ≤ h. ∗ < 0 and therefore the worst case should not be considered in this case. This is resolved by But ω > −γ when −γ ≤ ω− ∗ to be non-negative but allowing ω to have a maximum of h + γ , and the three constraints are now written restricting ω− + as
ω = ω+ − ω−∗ − γ , 0 ≤ ω−∗ ≤ −l − γ , 0 ≤ ω+ ≤ h + γ . Finally, we arrive at a portfolio problem identical to (3) in Proposition 1.
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