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On the weight sign of the global minimum variance portfolio
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Finance Research Letters 0 0 0 (2016) 1–6
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On the weight sign of the global minimum variance portfolio Wan-Yi Chiu∗, Ching-Hai Jiang Department of Finance, National United University, Miaoli 36003, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 15 June 2016 Revised 25 July 2016 Accepted 10 August 2016 Available online xxx JEL codes: G11 C58
a b s t r a c t We investigate the one-to-one mapping between the global minimum variance portfolio and regression hedge coefficients. The analysis demonstrates that assets with a superior (inferior) regression hedged effect in terms of marginal return create a negative (positive) weight. The asset has a weight of zero when both the asset and regression hedge enjoy the same marginal return. In addition, we develop a modified information ratio to compare the magnitudes of two arbitrary weights of the global minimum variance portfolio. From the perspective of hedging, we determine that the asset with a higher modified information ratio yields a larger weight.
Keywords: Global minimum variance portfolio Inverse covariance matrix Regression hedge Modified information ratio
© 2016 Elsevier Inc. All rights reserved.
1. Introduction There is ample evidence to suggest that certain intuitive portfolios may perform better than the market portfolio. The particular weighting scheme includes the naive portfolio with equal weights (e.g., Haugen and Baker, 1991; Duchin and Levy, 2009; DeMiguel et al., 2009) and the global minimum variance portfolio (e.g., Clarke et al., 2006; Frahm, 2010). Specifically, the global minimum variance portfolio (GMVP) strategy has triggered an attraction for seeking less volatile portfolios as a result of swings in the financial markets in recent years. The approach has been discussed by a number of researchers (Kempf and Memmel, 2006; Clarke et al., 2011; Scherer, 2011; Kim et al., 2013; Fu et al., 2015; Maillet et al., 2015, and Yanushevsky and Yanushevsky, 2015). Among these studies, (Clarke et al., 2011) provide empirical evidence that the GMVP is superior to the market portfolio. The authors indicate that the GMVP’s performance may be improved by incorporating long constraints into the minimum variance strategy. Kim et al. (2016) show that the portfolios with short positions are less risky than portfolios without constraints and more aggressive than portfolios without short sales. Scherer (2011) conjectures a low beta (e.g., below one) to create a positive GMVP weight. As a result of these classifications, the relationship of the beta value and GMVP weight is depicted in an approximate equation of residual risk. Recently, Yanushevsky and Yanushevsky (2015) disprove Scherer’s conjecture. However, whether there exists an analytic method that can accurately identify the GMVP weight sign has not been determined. In this paper, we investigate the GMVP weight sign from the viewpoint of hedging and compare the magnitudes of two arbitrary weights of the GMVP. Our study has obtained two main results.
∗
Corresponding author. Fax: +886 037 381545. E-mail addresses: [email protected] (W.-Y. Chiu), [email protected] (C.-H. Jiang).
http://dx.doi.org/10.1016/j.frl.2016.08.008 1544-6123/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008
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W.-Y. Chiu, C.-H. Jiang / Finance Research Letters 000 (2016) 1–6
First, building on prior research that shows that the inverse covariance matrix is closely related to a regression hedge among assets (e.g., Anderson and Danthine, 1981; Stevens, 1998), we consider an alternative to analyzing the one-to-one relationship between the GMVP weight sign and regression hedge coefficients. Consequently, we present a simple way to identify the weight sign of the GMVP. We show that assets with a superior (inferior) regression hedged effect create a negative (positive) GMVP weight. The asset has a zero weight of when both the asset and hedged portfolio enjoy the same marginal return. Our approach is computationally easy to implement in any statistical software packages. Second, we investigate an important economic implication of the GMVP weight. A modified information ratio (MIR) is developed as the standardized metric for interpreting the risk-adjusted return of assets. It is defined as the active return divided by the squared tracking error, where the active return is the difference between the asset’s return and the return of the regression hedge. The squared tracking error is the variance of residual risk relative to the regression hedge. From the MIR perspective, we find that the asset with a higher MIR yields a larger weight in the GMVP strategy. 2. Global minimum variance portfolio In modern portfolio theory, return and risk are measured as the expectation and standard deviation of the investment’s payoff, respectively. We consider the typical universe of k risky assets with return r˜ = (r1 r2 · · · rk ) that has the k × 1 expectation μ ˜ = (μ1 μ2 · · · μk ) and a k × k positive-definite covariance matrix . Subject to being fully invested, ω ˜ ˜ = 1, the expected return and variance are denoted as μ p = ω ˜ μ ˜ and σ p2 = ω ˜ ω ˜, ˜ where ω ˜ = (ω1 ω2 · · · ωk ) is the portfolio weight and = (1 1 · · · 1 ) is the k × 1 unity vector. One version to trace out the efficient frontier is the mean-variance utility optimization (Das et al., 2010; Bodnar and Okhrin, 2013):
max E (U (ω ˜ )) = max
ω˜ ˜=1
ω˜ ˜=1
ω˜ μ˜ −
γ 2
ω˜ ω˜ .
(1)
where γ is the risk aversion coefficient, a measure of the additional return required to bear additional risk. That is, γ is defined as the marginal portfolio return when risk increases by one unit. In the mean-variance utility optimization, the investor aims to maximize expected utility. The optimal mean-variance weight is given by
−1 ˜ 1 (˜ −1 ˜) −1 μ ˜ − (˜ −1 μ ˜ ) −1 ˜ ω˜ p = −1 + . ˜ ˜ γ ˜ −1 ˜
(2)
The return and variance of returns with respect to the optimal portfolio ω ˜ p are thus obtained as
μp
˜ −1 μ ˜ 1 = + γ ˜ −1 ˜
and
σ = 2 p
1 ˜ −1 ˜
+
1
γ2
(˜ −1 ˜)(μ˜ −1 μ˜ ) − (˜ −1 μ˜ )2 ˜ −1 ˜
(˜ −1 ˜)(μ˜ −1 μ˜ ) − (˜ −1 μ˜ )2 . ˜ −1 ˜
(3)
(4)
Substituting different positive values for γ into Eqs. (3) and (4), we can trace out the efficient frontier on the (σ p , μ p ) plane. In particular, if the risk-aversion coefficient γ approaches infinity, the GMVP that is the vertex of the efficient frontier has weight
ω¯ =
−1 ˜ . ˜ −1 ˜
(5)
The variance corresponding to the GMVP weight is
σ¯ 2 =
1 . ˜ −1 ˜
(6)
As such, the inverse covariance matrix of returns is the key factor in determining the GMVP weight. 3. Inverse covariance matrix of a three-asset portfolio Both Scherer (2011) and Yanushevsky and Yanushevsky (2015) apply an approximate equation of residual risk to analyze the relationship of the beta value and GMVP weight. In this study, we link up the regression hedge coefficients with the GMVP weights. Our regression hedge contains the intercept (the active return) that is crucially needed for the ensuing derivations. Note that the intercept term not only helps to solve the other regression coefficients, and will not affect the determination of the GMVP weight sign. The proposed regression hedge can accurately identify the GMVP weight sign. Results from previous studies show that the inverse covariance matrix is closely related to a regression hedge among assets. For example, Anderson and Danthine (1981, page 1188) present that the hedge proportions of the multiple futures contracts are given by the multiple regression coefficients of spot prices on the futures prices. Specifically, (Stevens, 1998) Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008
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derives an explicit form to the inverse covariance matrix expressed in terms of the regression coefficients and residual variances. To explore the properties of the GMVP weight and the regression hedge, we outline some results such that our discussion may be a self-contained exposition of the GMVP weight. To do so, we provide a simple case with three assets to establish the link between the inverse covariance matrix and a regression hedge among assets. Example Without loss of generality, we illustrate here a simple case of investment opportunities consisting of asset 1, asset 2, and asset 3 with returns r1 , r2 , and r3 , respectively. In addition, the expectations of r1 , r2 , and r3 are E (r1 ) = μ1 , E (r2 ) = μ2 , and E (r3 ) = μ3 . It is well-known that the inverse covariance matrix (σi j )3×3 is given as
⎡ σ 2σ 2 − σ 2 2 3 23 1 ⎢ −1 2 ⎢σ σ − σ3 σ12 = | | ⎣ 13 23
σ13 σ23 − σ32 σ12
σ12 σ23 − σ22 σ13
σ12 σ13 − σ12 σ23
σ12 σ23 − σ22 σ13 ⎤ ⎥ σ12 σ13 − σ12 σ23 ⎥ ⎦,
2 σ12 σ32 − σ13
(7)
2 σ12 σ22 − σ12
2 σ 2 − σ 2 σ 2 − σ 2 σ 2 denotes the determinant of . where | | = σ12 σ22 σ32 + 2σ12 σ13 σ23 − σ13 2 23 1 12 3 Alternatively, if we first regress the returns from asset 1 on the remaining assets’ returns, then the conditional expectation is given as
E (r1 |r2 , r3 ) = β11 + β12 r2 + β13 r3
(8)
and the residual variance using (r2 , r3 ) to hedge r1 is denoted as
τ12 = E (V ar (r1 |r2 , r3 )) =
| | . 2 σ22 σ32 − σ23
(9)
Stevens (1998) refers to Eq. (8) as the “regression hedge”. In the regression hedge setting, the conditional expectation is designed to replicate the return of asset 1 using the remaining assets. Note that the asset holdings are β12 for asset 2 and β13 for asset 3. Eq. (9) measures the residual variance of the regression hedge. To solve the parameters (β11 , β12 , β13 , and τ12 ) in Eqs. (8) and (9), the least squares method leads to the intercept β11 = μ1 − β12 μ2 − β13 μ3 and the optimal holdings of asset 2 and asset 3 as follows:
β12 β13
⎡
⎤ σ12 σ32 − σ13 σ23 2 ⎥
⎢ σ22 σ32 − σ23 ⎥ = Cov(r1 , (r2 , r3 ))[V ar (r2 , r3 )]−1 = ⎢ ⎣ σ13 σ22 − σ12 σ23 ⎦. 2 σ22 σ32 − σ23
(10)
Consequently, the hedged variance due to the regressors r2 and r3 is given by
V ar (E (r1 |r2 , r3 )) = V ar (β11 + β12 r2 + β13 r3 ) = =
β12 β13
σ22 σ23
σ23 σ32
β12 β13
2 2 σ12 σ32 − 2σ12 σ13 σ23 + σ13 σ22 . 2 2 2 σ2 σ3 − σ23
(11)
Using the law of total variance, V ar (r1 ) = E (V ar (r1 |r2 , r3 )) + V ar (E (r1 |r2 , r3 )), the residual variance τ12 is actually the difference between the unhedged variance of return V ar (r1 ) and the hedged variance of return V ar (E (r1 |r2 , r3 )). Therefore, the residual’s variance can be reduced to
τ12 = E (V ar (r1 |r2 , r3 )) = V ar (r1 ) − V ar (E (r1 |r2 , r3 )) =
| | . 2 σ22 σ32 − σ23
(12)
Similar computations are the same for the regression hedges: E (r2 |r1 , r3 ) = β22 + β21 r1 + β23 r3 with the residual variance τ22 and E (r3 |r1 , r2 ) = β33 + β31 r1 + β32 r2 with the residual variance τ32 . The corresponding results are summarized as follows:
⎤ σ12 σ32 − σ13 σ23 2 ⎢ σ12 σ32 − σ13 ⎥ 2 | | β21 ⎥, τ = E (V ar (r2 |r1 , r3 )) = =⎢ , 2 2 ⎣ β23 σ12 σ32 − σ13 σ23 σ1 − σ12 σ13 ⎦ 2 2 σ 2 σ 2 − σ13 ⎡ 1 23 ⎤ σ13 σ2 − σ12 σ23 2 ⎢ σ12 σ22 − σ12 ⎥ 2 | | β31 ⎥ =⎢ 2 ⎣ σ23 σ1 − σ12 σ13 ⎦, τ3 = E (V ar (r3 |r1 , r2 )) = σ 2 σ 2 − σ 2 . β32 1 2 12 2 σ12 σ22 − σ12
⎡
(13)
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Combining Eqs. (7), (10), (12), and (13), we rewrite the inverse covariance matrix as
⎡ 1 ⎢ τ12 ⎢ −β ⎢ 21 −1 = ⎢ 2 ⎢ τ2 ⎣ −β 31 τ32
−β12
τ
2 1
1
τ22 −β32 τ32
−β13 ⎤
τ12 ⎥ ⎥ −β23 ⎥ . 2 τ2 ⎥ ⎥ ⎦ 1
(14)
τ32
With this formulation, Eqs. (5), (6), and (14) produce the GMVP weight
⎡
⎤ σ¯ 2 (1 − β12 − β13 ) ⎢ ⎥ τ12 ⎢ 2 ⎥ ω1 −1 ˜ ⎢ σ¯ (1 − β21 − β23 ) ⎥ ⎢ ⎥. ω ω¯ = = = 2 ⎥ τ22 ˜ −1 ˜ ⎢ ⎢ ⎥ ω3 ⎣ σ¯ 2 (1 − β31 − β32 ) ⎦ τ32
(15)
Because the discussions of all assets are similar, we use the asset 1 as our example through the following sections. The value of ω1 is the product of (1 − β12 − β13 ) and the ratio of the GMVP’s variance to the residual variance of the regression hedge. Note that σ¯ 2 /τ12 is always positive and the value (1 − β12 − β13 ) determines the sign of ω1 . Therefore, the sign of ω1 has three possibilities:
+
ω1 = 0
−
if 1 > β12 + β13 , if 1 = β12 + β13 , if 1 < β12 + β13 .
(16)
There are two ways to explain the position of the asset 1, ω1 , in the GMVP. First, we can compute the marginal contribution of the regression hedge (8) using the total differential rule
dE (r1 |r2 , r3 ) = β12 dr2 + β13 dr3 .
(17)
Note that Eq. (17) may be considered as the marginal return of the hedged portfolio with respect to the increments of dr2 and dr3 . For the purposes of comparison, we standardize the marginal returns for all assets, which is defined as dr1 = dr2 = dr3 = 1. Compare the marginal return dr1 of the asset 1 with the marginal return (β12 dr2 + β13 dr3 ) of the regression hedge, we can easily see the link between the GMVP weight sign and the regression hedge coefficients. The most obvious situation for constructing a hedged portfolio is full replication, β12 + β13 = 1. The asset 1 can be completely replicated by the other assets. With this technique, the asset 2 and asset 3 are purchased in proportion to their regression hedge coefficients. Note that the asset 1 and the hedged portfolio both exhibit the same marginal return. Because buying the asset 1 increases transaction costs and detracts from performance, a zero position on asset 1 helps ensure optimal tracking. It follows that ω1 = 0. In the case β12 + β13 > 1, the regression hedge would have earned a higher marginal return than asset 1. The return of the GMVP increases if the investor is able to take a long position on the hedged portfolio and therefore increase its marginal return. Under the fully invested restriction where all the weights sum to one, the holding of the asset 1 is negative. Clearly, in the case of β12 + β13 < 1, the asset 1 would have earned a higher marginal return than the hedged portfolio. The condition states that a long position of the hedged portfolio can never earn more the GMVP return. As a result, the optimal holding for the asset 1 should be positive. Second, we can also identify the GMVP weight sign using the risk aversion coefficient. For example, we assume that the investor applies a long (or a short) position to asset 1 as β12 + β13 > 1 (or β12 + β13 < 1) in Eq. (8). Such a decision not only changes an amount of the GMVP return but also results in an increase of the portfolio variance. Due to the GMVP’s uniqueness on the efficient frontier, the investor’s portfolio selection is forced to deviate from the vertex of the efficient frontier. Consequently, the investor will select a new portfolio with a finite risk aversion coefficient. This contradicts the GMVP’s infinite risk aversion coefficient assumption. Therefore, the investor should assign a negative (positive) weight to the asset 1 as β12 + β13 > 1 (or β12 + β13 < 1). The discussion for the case β12 + β13 = 1 is similar. If an investor applies a long (or short) position to the asset 1, the GMVP’s infinite risk aversion coefficient assumption will be distorted. Thus, the GMVP weight of the asset 1 should be zero as β12 + β13 = 1. 4. Generalization of k assets portfolio In the k assets portfolio, the conditional expectation is used by analogy with a regression hedge. The k conditional expectations are given by
E (ri |r1 , . . . , ri−1 , ri+1 , . . . , rk ) = βii +
k
j=i
βi j r j ,
i = 1, 2, . . . , k.
(18)
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Note that the regression coefficient βi j denotes the marginal contribution when using r j to hedge ri . As in the case of the conditional expectation, the k residual variances using (r1 , . . . , ri−1 , ri+1 , . . . , rk ) to hedge ri are denoted as
E (V ar (ri |r1 , . . . , ri−1 , ri+1 , . . . , rk )) = τi2 ,
i = 1, 2, . . . , k.
(19)
The generalization of −1 can refer to the direct characterization of the inverse covariance matrix derived by Stevens (1998). In the Stevens’s framework, the general form of the inverse covariance matrix for k assets may be rewritten as
⎡
−β12
1
⎢ τ12 ⎢ ⎢ −β21 ⎢ 2 ⎢ τ2 −1 =⎢ ⎢ .. ⎢ . ⎢ ⎢ ⎣ −βk1
−β13
τ12 τ22
τ12 −β23 τ22
.. .
.. .
−βk2
−βk3
1
τk2
τk2
τk2
···
−β1k
⎤
τ12 ⎥ ⎥ −β2k ⎥ ⎥ ··· τ22 ⎥ ⎥. .. ⎥ .. . . ⎥ ⎥ ⎥ 1 ⎦ ···
(20)
τk2
In response to the regression hedge to the asset i, Eqs. (5), (6), (18), (19), and (20) then lead to the optimal holding of the asset i in the GMVP.
σ¯ 2 (1 − ωi = τi2
j=i
βi j )
,
i = 1, 2, . . . , k.
(21)
As explained in the previous example, Eq. (21) links a one-to-one relationship between the GMVP weight and its corre sponding regression hedge coefficients. The sign of ωi depends on the sign of (1 − j=i βi j ). An alternative commonly put forward to explain Eq. (21) concerns comparative advantages. In terms of a standardized marginal return, the marginal re turn between the asset i and corresponding regression hedge is (1 − j=i βi j ). Because σ¯ 2 /τi2 is positive, the sign of ωi in the GMVP weight is critically dependent on whether the asset i outperforms or underperforms the regression hedge portfolio. In the first case of the linear restriction j=i βi j > 1, the hedged portfolio outperforms the asset i by an amount reflecting the marginal return. This finding implies a short position of the asset i and a negative value of ωi in the GMVP weight. In the second case of the restriction j=i βi j < 1, the hedged portfolio underperforms the asset i by an effect of the marginal return, which implies a long position of the asset i and a positive value of ωi in the GMVP weight. In the third case of the full replication j=i βi j = 1, both the asset i and the hedged portfolios enjoy the same marginal returns. Because buying the asset i increases transaction costs and detracts from performance, only a zero position on the asset i will ensure the optimality of the GMVP. In fact, Eq. (21) has an important economic implication for comparing the magnitudes of two arbitrary weights ωi and ω j based on the GMVP strategy. For this purpose, Eq. (21) can be rewritten as
ωi = σ¯ 2 × MIR(i ),
i = 1, 2, . . . , k.
(22)
where the modified information ratio MIR(i), is a standardized metric of the risk-adjusted return of the asset i. It is defined as the active return divided by the squared tracking error, where the active return is the difference between the return of asset i and the return of its regression hedge. The squared tracking error is the variance of residual risk relative to the regression hedge. Note that the MIR(i) is similar to the information ratio. Whereas the information ratio is the active return of a portfolio over the return of a benchmark divided by the standard deviation of active return, the MIR(i) is the active return of asset i over the return of its regression hedge divided by the variance of residual risk. The MIR captures the trade-off between the active return of the hedged portfolio and the variance of residual risk. Therefore, the MIR provides a simple way to compare the magnitudes of two arbitrary weights ωi and ω j based on the GMVP strategy. Using the case ωi > ω j as an example, we find that it is equivalent to the expression MIR(i) > MIR(j). In this case, the higher the MIR(i), the higher the active return of asset i, given the amount of residual risk of the regression hedge taken, and the larger the weight of asset i in the GMVP weight. 5. Conclusion Building on prior research that indicates that the inverse covariance matrix is closely related to a regression hedge among assets, we reform the Stevens inverse covariance matrix to analyze the one-to-one relationship between the GMVP weight sign and regression hedge coefficients. In terms of the standardized marginal return, we demonstrate that a superior (inferior) regression hedged effect leads to a negative (positive) weight for the GMVP constituents. Of course, the asset has a weight of zero when both the asset and its corresponding regression hedge enjoy the same marginal return. In addition, we use a modified information ratio (MIR) as the standardized measure for interpreting the risk-adjusted return of assets. The MIR is developed to judge the economic implication of comparing the magnitudes of two arbitrary Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008
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weights of the GMVP. From the perspective of hedging, we determine that the asset with a higher MIR yields a larger weight in the GMVP strategy. In practice, the approach is computationally easy to implement in any statistical software packages. Acknowledgments We are grateful to the anonymous referees for helpful comments and suggestions. The views in this paper are solely the responsibility of the authors. References Anderson, R.W., Danthine, J.-P., 1981. Cross hedging. J. Polit. Econ. 89 (6), 1182–1196. Bodnar, T., Okhrin, Y., 2013. Boundaries of the risk aversion coefficient: Should we invest in the GMV portfolio? Appl. Math. Comput. 219 (10), 5440–5448. Clarke, R., de Silva, H., Thorley, S., 2006. Minimum-variance portfolios in the U.S. equity market. J. Portf. Manage. 33 (1), 10–24. Clarke, R., de Silva, H., Thorley, S., 2011. Minimum-variance portfolio composition. J. Portf. Manage. 37 (2), 31–45. Das, S., Markowitz, H., Scheid, J., Statman, M., 2010. Portfolio optimization with mental accounts. J. Finan. Quant. Anal. 45 (2), 311–334. DeMiguel, V., Garlappi, L., Uppal, R., 2009. Optimal versus naive diversification: how inefficient is the 1/n portfolio strategy? Rev. Finan. Stud. 22 (5), 1915–1953. Duchin, R., Levy, H., 2009. Markowitz versus the talmud portfolio diversification strategies. J. Portf. Manag. 35 (2), 71–74. Frahm, G., 2010. Linear statistical inference for global and local minimum variance portfolios. Stat. Pap. 51 (4), 789–812. Fu, C., Jacoby, G., Wang, Y., 2015. Investor sentiment and portfolio selection. Finance Res. Lett. 15, 266–273. Haugen, R.A., Baker, N.L., 1991. The efficient market inefficiency of capitalization-weighted stock portfolios. J. Portf. Manag. 17 (3), 35–40. Kempf, A., Memmel, C., 2006. Estimating the global minimum variance portfolio. Schmalenbach Bus. Rev. 58 (4), 332–348. Kim, J.H., Kim, W.C., Fabozzi, F.J., 2013. Composition of robust equity portfolios. Finance Res. Lett. 10, 72–81. Kim, J.H., Kim, W.C., Fabozzi, F.J., 2016. Portfolio selection with conservative short-selling. Finance Res. Lett In press http://dx.doi.org/10.1016/j.frl.2016.05.015. Maillet, B., Tokpavi, S., Vaucher, B., 2015. Global minimum variance portfolio optimisation under some model risk: a robust regression-based approach. Eur. J. Oper. Res. 244 (1), 289–299. Scherer, B., 2011. A note on the returns from minimum variance investing. J. Empir. Finance 18 (4), 652–660. Stevens, G.V.G., 1998. On the inverse of the covariance matrix in portfolio analysis. J. Finance 53 (5), 1821–1827. Yanushevsky, R., Yanushevsky, D., 2015. Comment on a note on the returns from minimum variance investing. J. Empir. Finance 31, 109–110.
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Finance Research Letters 0 0 0 (2016) 1–6
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
On the weight sign of the global minimum variance portfolio Wan-Yi Chiu∗, Ching-Hai Jiang Department of Finance, National United University, Miaoli 36003, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 15 June 2016 Revised 25 July 2016 Accepted 10 August 2016 Available online xxx JEL codes: G11 C58
a b s t r a c t We investigate the one-to-one mapping between the global minimum variance portfolio and regression hedge coefficients. The analysis demonstrates that assets with a superior (inferior) regression hedged effect in terms of marginal return create a negative (positive) weight. The asset has a weight of zero when both the asset and regression hedge enjoy the same marginal return. In addition, we develop a modified information ratio to compare the magnitudes of two arbitrary weights of the global minimum variance portfolio. From the perspective of hedging, we determine that the asset with a higher modified information ratio yields a larger weight.
Keywords: Global minimum variance portfolio Inverse covariance matrix Regression hedge Modified information ratio
© 2016 Elsevier Inc. All rights reserved.
1. Introduction There is ample evidence to suggest that certain intuitive portfolios may perform better than the market portfolio. The particular weighting scheme includes the naive portfolio with equal weights (e.g., Haugen and Baker, 1991; Duchin and Levy, 2009; DeMiguel et al., 2009) and the global minimum variance portfolio (e.g., Clarke et al., 2006; Frahm, 2010). Specifically, the global minimum variance portfolio (GMVP) strategy has triggered an attraction for seeking less volatile portfolios as a result of swings in the financial markets in recent years. The approach has been discussed by a number of researchers (Kempf and Memmel, 2006; Clarke et al., 2011; Scherer, 2011; Kim et al., 2013; Fu et al., 2015; Maillet et al., 2015, and Yanushevsky and Yanushevsky, 2015). Among these studies, (Clarke et al., 2011) provide empirical evidence that the GMVP is superior to the market portfolio. The authors indicate that the GMVP’s performance may be improved by incorporating long constraints into the minimum variance strategy. Kim et al. (2016) show that the portfolios with short positions are less risky than portfolios without constraints and more aggressive than portfolios without short sales. Scherer (2011) conjectures a low beta (e.g., below one) to create a positive GMVP weight. As a result of these classifications, the relationship of the beta value and GMVP weight is depicted in an approximate equation of residual risk. Recently, Yanushevsky and Yanushevsky (2015) disprove Scherer’s conjecture. However, whether there exists an analytic method that can accurately identify the GMVP weight sign has not been determined. In this paper, we investigate the GMVP weight sign from the viewpoint of hedging and compare the magnitudes of two arbitrary weights of the GMVP. Our study has obtained two main results.
∗
Corresponding author. Fax: +886 037 381545. E-mail addresses: [email protected] (W.-Y. Chiu), [email protected] (C.-H. Jiang).
http://dx.doi.org/10.1016/j.frl.2016.08.008 1544-6123/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008
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First, building on prior research that shows that the inverse covariance matrix is closely related to a regression hedge among assets (e.g., Anderson and Danthine, 1981; Stevens, 1998), we consider an alternative to analyzing the one-to-one relationship between the GMVP weight sign and regression hedge coefficients. Consequently, we present a simple way to identify the weight sign of the GMVP. We show that assets with a superior (inferior) regression hedged effect create a negative (positive) GMVP weight. The asset has a zero weight of when both the asset and hedged portfolio enjoy the same marginal return. Our approach is computationally easy to implement in any statistical software packages. Second, we investigate an important economic implication of the GMVP weight. A modified information ratio (MIR) is developed as the standardized metric for interpreting the risk-adjusted return of assets. It is defined as the active return divided by the squared tracking error, where the active return is the difference between the asset’s return and the return of the regression hedge. The squared tracking error is the variance of residual risk relative to the regression hedge. From the MIR perspective, we find that the asset with a higher MIR yields a larger weight in the GMVP strategy. 2. Global minimum variance portfolio In modern portfolio theory, return and risk are measured as the expectation and standard deviation of the investment’s payoff, respectively. We consider the typical universe of k risky assets with return r˜ = (r1 r2 · · · rk ) that has the k × 1 expectation μ ˜ = (μ1 μ2 · · · μk ) and a k × k positive-definite covariance matrix . Subject to being fully invested, ω ˜ ˜ = 1, the expected return and variance are denoted as μ p = ω ˜ μ ˜ and σ p2 = ω ˜ ω ˜, ˜ where ω ˜ = (ω1 ω2 · · · ωk ) is the portfolio weight and = (1 1 · · · 1 ) is the k × 1 unity vector. One version to trace out the efficient frontier is the mean-variance utility optimization (Das et al., 2010; Bodnar and Okhrin, 2013):
max E (U (ω ˜ )) = max
ω˜ ˜=1
ω˜ ˜=1
ω˜ μ˜ −
γ 2
ω˜ ω˜ .
(1)
where γ is the risk aversion coefficient, a measure of the additional return required to bear additional risk. That is, γ is defined as the marginal portfolio return when risk increases by one unit. In the mean-variance utility optimization, the investor aims to maximize expected utility. The optimal mean-variance weight is given by
−1 ˜ 1 (˜ −1 ˜) −1 μ ˜ − (˜ −1 μ ˜ ) −1 ˜ ω˜ p = −1 + . ˜ ˜ γ ˜ −1 ˜
(2)
The return and variance of returns with respect to the optimal portfolio ω ˜ p are thus obtained as
μp
˜ −1 μ ˜ 1 = + γ ˜ −1 ˜
and
σ = 2 p
1 ˜ −1 ˜
+
1
γ2
(˜ −1 ˜)(μ˜ −1 μ˜ ) − (˜ −1 μ˜ )2 ˜ −1 ˜
(˜ −1 ˜)(μ˜ −1 μ˜ ) − (˜ −1 μ˜ )2 . ˜ −1 ˜
(3)
(4)
Substituting different positive values for γ into Eqs. (3) and (4), we can trace out the efficient frontier on the (σ p , μ p ) plane. In particular, if the risk-aversion coefficient γ approaches infinity, the GMVP that is the vertex of the efficient frontier has weight
ω¯ =
−1 ˜ . ˜ −1 ˜
(5)
The variance corresponding to the GMVP weight is
σ¯ 2 =
1 . ˜ −1 ˜
(6)
As such, the inverse covariance matrix of returns is the key factor in determining the GMVP weight. 3. Inverse covariance matrix of a three-asset portfolio Both Scherer (2011) and Yanushevsky and Yanushevsky (2015) apply an approximate equation of residual risk to analyze the relationship of the beta value and GMVP weight. In this study, we link up the regression hedge coefficients with the GMVP weights. Our regression hedge contains the intercept (the active return) that is crucially needed for the ensuing derivations. Note that the intercept term not only helps to solve the other regression coefficients, and will not affect the determination of the GMVP weight sign. The proposed regression hedge can accurately identify the GMVP weight sign. Results from previous studies show that the inverse covariance matrix is closely related to a regression hedge among assets. For example, Anderson and Danthine (1981, page 1188) present that the hedge proportions of the multiple futures contracts are given by the multiple regression coefficients of spot prices on the futures prices. Specifically, (Stevens, 1998) Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008
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derives an explicit form to the inverse covariance matrix expressed in terms of the regression coefficients and residual variances. To explore the properties of the GMVP weight and the regression hedge, we outline some results such that our discussion may be a self-contained exposition of the GMVP weight. To do so, we provide a simple case with three assets to establish the link between the inverse covariance matrix and a regression hedge among assets. Example Without loss of generality, we illustrate here a simple case of investment opportunities consisting of asset 1, asset 2, and asset 3 with returns r1 , r2 , and r3 , respectively. In addition, the expectations of r1 , r2 , and r3 are E (r1 ) = μ1 , E (r2 ) = μ2 , and E (r3 ) = μ3 . It is well-known that the inverse covariance matrix (σi j )3×3 is given as
⎡ σ 2σ 2 − σ 2 2 3 23 1 ⎢ −1 2 ⎢σ σ − σ3 σ12 = | | ⎣ 13 23
σ13 σ23 − σ32 σ12
σ12 σ23 − σ22 σ13
σ12 σ13 − σ12 σ23
σ12 σ23 − σ22 σ13 ⎤ ⎥ σ12 σ13 − σ12 σ23 ⎥ ⎦,
2 σ12 σ32 − σ13
(7)
2 σ12 σ22 − σ12
2 σ 2 − σ 2 σ 2 − σ 2 σ 2 denotes the determinant of . where | | = σ12 σ22 σ32 + 2σ12 σ13 σ23 − σ13 2 23 1 12 3 Alternatively, if we first regress the returns from asset 1 on the remaining assets’ returns, then the conditional expectation is given as
E (r1 |r2 , r3 ) = β11 + β12 r2 + β13 r3
(8)
and the residual variance using (r2 , r3 ) to hedge r1 is denoted as
τ12 = E (V ar (r1 |r2 , r3 )) =
| | . 2 σ22 σ32 − σ23
(9)
Stevens (1998) refers to Eq. (8) as the “regression hedge”. In the regression hedge setting, the conditional expectation is designed to replicate the return of asset 1 using the remaining assets. Note that the asset holdings are β12 for asset 2 and β13 for asset 3. Eq. (9) measures the residual variance of the regression hedge. To solve the parameters (β11 , β12 , β13 , and τ12 ) in Eqs. (8) and (9), the least squares method leads to the intercept β11 = μ1 − β12 μ2 − β13 μ3 and the optimal holdings of asset 2 and asset 3 as follows:
β12 β13
⎡
⎤ σ12 σ32 − σ13 σ23 2 ⎥
⎢ σ22 σ32 − σ23 ⎥ = Cov(r1 , (r2 , r3 ))[V ar (r2 , r3 )]−1 = ⎢ ⎣ σ13 σ22 − σ12 σ23 ⎦. 2 σ22 σ32 − σ23
(10)
Consequently, the hedged variance due to the regressors r2 and r3 is given by
V ar (E (r1 |r2 , r3 )) = V ar (β11 + β12 r2 + β13 r3 ) = =
β12 β13
σ22 σ23
σ23 σ32
β12 β13
2 2 σ12 σ32 − 2σ12 σ13 σ23 + σ13 σ22 . 2 2 2 σ2 σ3 − σ23
(11)
Using the law of total variance, V ar (r1 ) = E (V ar (r1 |r2 , r3 )) + V ar (E (r1 |r2 , r3 )), the residual variance τ12 is actually the difference between the unhedged variance of return V ar (r1 ) and the hedged variance of return V ar (E (r1 |r2 , r3 )). Therefore, the residual’s variance can be reduced to
τ12 = E (V ar (r1 |r2 , r3 )) = V ar (r1 ) − V ar (E (r1 |r2 , r3 )) =
| | . 2 σ22 σ32 − σ23
(12)
Similar computations are the same for the regression hedges: E (r2 |r1 , r3 ) = β22 + β21 r1 + β23 r3 with the residual variance τ22 and E (r3 |r1 , r2 ) = β33 + β31 r1 + β32 r2 with the residual variance τ32 . The corresponding results are summarized as follows:
⎤ σ12 σ32 − σ13 σ23 2 ⎢ σ12 σ32 − σ13 ⎥ 2 | | β21 ⎥, τ = E (V ar (r2 |r1 , r3 )) = =⎢ , 2 2 ⎣ β23 σ12 σ32 − σ13 σ23 σ1 − σ12 σ13 ⎦ 2 2 σ 2 σ 2 − σ13 ⎡ 1 23 ⎤ σ13 σ2 − σ12 σ23 2 ⎢ σ12 σ22 − σ12 ⎥ 2 | | β31 ⎥ =⎢ 2 ⎣ σ23 σ1 − σ12 σ13 ⎦, τ3 = E (V ar (r3 |r1 , r2 )) = σ 2 σ 2 − σ 2 . β32 1 2 12 2 σ12 σ22 − σ12
⎡
(13)
Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008
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W.-Y. Chiu, C.-H. Jiang / Finance Research Letters 000 (2016) 1–6
Combining Eqs. (7), (10), (12), and (13), we rewrite the inverse covariance matrix as
⎡ 1 ⎢ τ12 ⎢ −β ⎢ 21 −1 = ⎢ 2 ⎢ τ2 ⎣ −β 31 τ32
−β12
τ
2 1
1
τ22 −β32 τ32
−β13 ⎤
τ12 ⎥ ⎥ −β23 ⎥ . 2 τ2 ⎥ ⎥ ⎦ 1
(14)
τ32
With this formulation, Eqs. (5), (6), and (14) produce the GMVP weight
⎡
⎤ σ¯ 2 (1 − β12 − β13 ) ⎢ ⎥ τ12 ⎢ 2 ⎥ ω1 −1 ˜ ⎢ σ¯ (1 − β21 − β23 ) ⎥ ⎢ ⎥. ω ω¯ = = = 2 ⎥ τ22 ˜ −1 ˜ ⎢ ⎢ ⎥ ω3 ⎣ σ¯ 2 (1 − β31 − β32 ) ⎦ τ32
(15)
Because the discussions of all assets are similar, we use the asset 1 as our example through the following sections. The value of ω1 is the product of (1 − β12 − β13 ) and the ratio of the GMVP’s variance to the residual variance of the regression hedge. Note that σ¯ 2 /τ12 is always positive and the value (1 − β12 − β13 ) determines the sign of ω1 . Therefore, the sign of ω1 has three possibilities:
+
ω1 = 0
−
if 1 > β12 + β13 , if 1 = β12 + β13 , if 1 < β12 + β13 .
(16)
There are two ways to explain the position of the asset 1, ω1 , in the GMVP. First, we can compute the marginal contribution of the regression hedge (8) using the total differential rule
dE (r1 |r2 , r3 ) = β12 dr2 + β13 dr3 .
(17)
Note that Eq. (17) may be considered as the marginal return of the hedged portfolio with respect to the increments of dr2 and dr3 . For the purposes of comparison, we standardize the marginal returns for all assets, which is defined as dr1 = dr2 = dr3 = 1. Compare the marginal return dr1 of the asset 1 with the marginal return (β12 dr2 + β13 dr3 ) of the regression hedge, we can easily see the link between the GMVP weight sign and the regression hedge coefficients. The most obvious situation for constructing a hedged portfolio is full replication, β12 + β13 = 1. The asset 1 can be completely replicated by the other assets. With this technique, the asset 2 and asset 3 are purchased in proportion to their regression hedge coefficients. Note that the asset 1 and the hedged portfolio both exhibit the same marginal return. Because buying the asset 1 increases transaction costs and detracts from performance, a zero position on asset 1 helps ensure optimal tracking. It follows that ω1 = 0. In the case β12 + β13 > 1, the regression hedge would have earned a higher marginal return than asset 1. The return of the GMVP increases if the investor is able to take a long position on the hedged portfolio and therefore increase its marginal return. Under the fully invested restriction where all the weights sum to one, the holding of the asset 1 is negative. Clearly, in the case of β12 + β13 < 1, the asset 1 would have earned a higher marginal return than the hedged portfolio. The condition states that a long position of the hedged portfolio can never earn more the GMVP return. As a result, the optimal holding for the asset 1 should be positive. Second, we can also identify the GMVP weight sign using the risk aversion coefficient. For example, we assume that the investor applies a long (or a short) position to asset 1 as β12 + β13 > 1 (or β12 + β13 < 1) in Eq. (8). Such a decision not only changes an amount of the GMVP return but also results in an increase of the portfolio variance. Due to the GMVP’s uniqueness on the efficient frontier, the investor’s portfolio selection is forced to deviate from the vertex of the efficient frontier. Consequently, the investor will select a new portfolio with a finite risk aversion coefficient. This contradicts the GMVP’s infinite risk aversion coefficient assumption. Therefore, the investor should assign a negative (positive) weight to the asset 1 as β12 + β13 > 1 (or β12 + β13 < 1). The discussion for the case β12 + β13 = 1 is similar. If an investor applies a long (or short) position to the asset 1, the GMVP’s infinite risk aversion coefficient assumption will be distorted. Thus, the GMVP weight of the asset 1 should be zero as β12 + β13 = 1. 4. Generalization of k assets portfolio In the k assets portfolio, the conditional expectation is used by analogy with a regression hedge. The k conditional expectations are given by
E (ri |r1 , . . . , ri−1 , ri+1 , . . . , rk ) = βii +
k
j=i
βi j r j ,
i = 1, 2, . . . , k.
(18)
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Note that the regression coefficient βi j denotes the marginal contribution when using r j to hedge ri . As in the case of the conditional expectation, the k residual variances using (r1 , . . . , ri−1 , ri+1 , . . . , rk ) to hedge ri are denoted as
E (V ar (ri |r1 , . . . , ri−1 , ri+1 , . . . , rk )) = τi2 ,
i = 1, 2, . . . , k.
(19)
The generalization of −1 can refer to the direct characterization of the inverse covariance matrix derived by Stevens (1998). In the Stevens’s framework, the general form of the inverse covariance matrix for k assets may be rewritten as
⎡
−β12
1
⎢ τ12 ⎢ ⎢ −β21 ⎢ 2 ⎢ τ2 −1 =⎢ ⎢ .. ⎢ . ⎢ ⎢ ⎣ −βk1
−β13
τ12 τ22
τ12 −β23 τ22
.. .
.. .
−βk2
−βk3
1
τk2
τk2
τk2
···
−β1k
⎤
τ12 ⎥ ⎥ −β2k ⎥ ⎥ ··· τ22 ⎥ ⎥. .. ⎥ .. . . ⎥ ⎥ ⎥ 1 ⎦ ···
(20)
τk2
In response to the regression hedge to the asset i, Eqs. (5), (6), (18), (19), and (20) then lead to the optimal holding of the asset i in the GMVP.
σ¯ 2 (1 − ωi = τi2
j=i
βi j )
,
i = 1, 2, . . . , k.
(21)
As explained in the previous example, Eq. (21) links a one-to-one relationship between the GMVP weight and its corre sponding regression hedge coefficients. The sign of ωi depends on the sign of (1 − j=i βi j ). An alternative commonly put forward to explain Eq. (21) concerns comparative advantages. In terms of a standardized marginal return, the marginal re turn between the asset i and corresponding regression hedge is (1 − j=i βi j ). Because σ¯ 2 /τi2 is positive, the sign of ωi in the GMVP weight is critically dependent on whether the asset i outperforms or underperforms the regression hedge portfolio. In the first case of the linear restriction j=i βi j > 1, the hedged portfolio outperforms the asset i by an amount reflecting the marginal return. This finding implies a short position of the asset i and a negative value of ωi in the GMVP weight. In the second case of the restriction j=i βi j < 1, the hedged portfolio underperforms the asset i by an effect of the marginal return, which implies a long position of the asset i and a positive value of ωi in the GMVP weight. In the third case of the full replication j=i βi j = 1, both the asset i and the hedged portfolios enjoy the same marginal returns. Because buying the asset i increases transaction costs and detracts from performance, only a zero position on the asset i will ensure the optimality of the GMVP. In fact, Eq. (21) has an important economic implication for comparing the magnitudes of two arbitrary weights ωi and ω j based on the GMVP strategy. For this purpose, Eq. (21) can be rewritten as
ωi = σ¯ 2 × MIR(i ),
i = 1, 2, . . . , k.
(22)
where the modified information ratio MIR(i), is a standardized metric of the risk-adjusted return of the asset i. It is defined as the active return divided by the squared tracking error, where the active return is the difference between the return of asset i and the return of its regression hedge. The squared tracking error is the variance of residual risk relative to the regression hedge. Note that the MIR(i) is similar to the information ratio. Whereas the information ratio is the active return of a portfolio over the return of a benchmark divided by the standard deviation of active return, the MIR(i) is the active return of asset i over the return of its regression hedge divided by the variance of residual risk. The MIR captures the trade-off between the active return of the hedged portfolio and the variance of residual risk. Therefore, the MIR provides a simple way to compare the magnitudes of two arbitrary weights ωi and ω j based on the GMVP strategy. Using the case ωi > ω j as an example, we find that it is equivalent to the expression MIR(i) > MIR(j). In this case, the higher the MIR(i), the higher the active return of asset i, given the amount of residual risk of the regression hedge taken, and the larger the weight of asset i in the GMVP weight. 5. Conclusion Building on prior research that indicates that the inverse covariance matrix is closely related to a regression hedge among assets, we reform the Stevens inverse covariance matrix to analyze the one-to-one relationship between the GMVP weight sign and regression hedge coefficients. In terms of the standardized marginal return, we demonstrate that a superior (inferior) regression hedged effect leads to a negative (positive) weight for the GMVP constituents. Of course, the asset has a weight of zero when both the asset and its corresponding regression hedge enjoy the same marginal return. In addition, we use a modified information ratio (MIR) as the standardized measure for interpreting the risk-adjusted return of assets. The MIR is developed to judge the economic implication of comparing the magnitudes of two arbitrary Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008
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weights of the GMVP. From the perspective of hedging, we determine that the asset with a higher MIR yields a larger weight in the GMVP strategy. In practice, the approach is computationally easy to implement in any statistical software packages. Acknowledgments We are grateful to the anonymous referees for helpful comments and suggestions. The views in this paper are solely the responsibility of the authors. References Anderson, R.W., Danthine, J.-P., 1981. Cross hedging. J. Polit. Econ. 89 (6), 1182–1196. Bodnar, T., Okhrin, Y., 2013. Boundaries of the risk aversion coefficient: Should we invest in the GMV portfolio? Appl. Math. Comput. 219 (10), 5440–5448. Clarke, R., de Silva, H., Thorley, S., 2006. Minimum-variance portfolios in the U.S. equity market. J. Portf. Manage. 33 (1), 10–24. Clarke, R., de Silva, H., Thorley, S., 2011. Minimum-variance portfolio composition. J. Portf. Manage. 37 (2), 31–45. Das, S., Markowitz, H., Scheid, J., Statman, M., 2010. Portfolio optimization with mental accounts. J. Finan. Quant. Anal. 45 (2), 311–334. DeMiguel, V., Garlappi, L., Uppal, R., 2009. Optimal versus naive diversification: how inefficient is the 1/n portfolio strategy? Rev. Finan. Stud. 22 (5), 1915–1953. Duchin, R., Levy, H., 2009. Markowitz versus the talmud portfolio diversification strategies. J. Portf. Manag. 35 (2), 71–74. Frahm, G., 2010. Linear statistical inference for global and local minimum variance portfolios. Stat. Pap. 51 (4), 789–812. Fu, C., Jacoby, G., Wang, Y., 2015. Investor sentiment and portfolio selection. Finance Res. Lett. 15, 266–273. Haugen, R.A., Baker, N.L., 1991. The efficient market inefficiency of capitalization-weighted stock portfolios. J. Portf. Manag. 17 (3), 35–40. Kempf, A., Memmel, C., 2006. Estimating the global minimum variance portfolio. Schmalenbach Bus. Rev. 58 (4), 332–348. Kim, J.H., Kim, W.C., Fabozzi, F.J., 2013. Composition of robust equity portfolios. Finance Res. Lett. 10, 72–81. Kim, J.H., Kim, W.C., Fabozzi, F.J., 2016. Portfolio selection with conservative short-selling. Finance Res. Lett In press http://dx.doi.org/10.1016/j.frl.2016.05.015. Maillet, B., Tokpavi, S., Vaucher, B., 2015. Global minimum variance portfolio optimisation under some model risk: a robust regression-based approach. Eur. J. Oper. Res. 244 (1), 289–299. Scherer, B., 2011. A note on the returns from minimum variance investing. J. Empir. Finance 18 (4), 652–660. Stevens, G.V.G., 1998. On the inverse of the covariance matrix in portfolio analysis. J. Finance 53 (5), 1821–1827. Yanushevsky, R., Yanushevsky, D., 2015. Comment on a note on the returns from minimum variance investing. J. Empir. Finance 31, 109–110.
Please cite this article as: W.-Y. Chiu, C.-H. Jiang, On the weight sign of the global minimum variance portfolio, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.008