- Email: [email protected]
A simple test of optimal hedging policy
Statistics and Probability Letters 83 (2013) 1062–1070
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A simple test of optimal hedging policy Wan-Yi Chiu National United University, Department of Finance, Taiwan, ROC
article
info
Article history: Received 20 August 2012 Received in revised form 15 November 2012 Accepted 13 December 2012 Available online 26 December 2012 Keywords: Optimal hedge ratio Risk-return measure Sharpe ratio Hedging effectiveness Mean–variance analysis
abstract This paper investigates the equivalence between the optimal hedge ratio derived in a risk-return simplification and the optimal hedge ratio using mean–variance analysis. In accordance with this relationship, we develop a simple regression-based test for evaluating the hedging effectiveness of the risk-return hedging. As a result, a t-test and an F -test are designed to examine the hedge ratio and hedging effectiveness, respectively. An example of hedging is also provided to illustrate this process. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Futures markets supply vital hedge instruments that investors require to make ‘‘a reasoned choice in conduct of risk reduction’’. One of the major research interests in futures markets is to solve the optimal hedging policy for a position to be hedged with futures. Two general methods of determining the optimal hedge ratio are the minimum-variance model and the risk-return approach. The fundamental difference between the risk-return approach and the minimum-variance model is that the risk-return approach highlights the trade-off between return and risk compared to the minimum-variance model, which focuses solely on risk reduction and requires fewer estimates (variances and covariance). Traditionally, the determination of the optimal hedge ratio of a hedged portfolio has been considered as a solution to a risk reduction problem since Ederington’s (1979) proposal of minimum-variance hedging. Researchers have presented ample statistical inferences on this issue based on minimizing portfolio variance (Stulz, 1984; Mcnew and Fackler, 1994; Jong et al., 1997; Brooks et al., 2002; Harris and Shen, 2003; Terry, 2005). Starting with Howard and D’Antonio’s (1984) proposal of the relative risk-return measure to determine the optimal hedge ratio, both practitioners and researchers have advanced a variety of risk-return measures to evaluate hedging effectiveness (Chang and Shanker, 1987; Kuo and Chen, 1995; Satyanarayan, 1998). In particular, Kuo and Chen adapt and simplify the Howard–D’Antonio approach for practical settings. However, these studies do not suggest an analytical test-statistic to evaluate hedging effectiveness. Two main results are obtained in this paper. First, we link the relationship between the revised Howard–D’Antonio model derived by Kuo–Chen and mean–variance analysis according to Heifner (1972) and Kahl (1983). Second, we integrate the mean–variance approach and the Britten-Jones regression approach (1999) into a simple test that is used to estimate the optimal hedge ratio. As a consequence, a t-test and an F -test are analytically designed to test the hedge ratio and the hedging effectiveness, respectively. This article is organized as follows. In Section 2, we briefly review Howard–D’Antonio risk-return optimization as well as the relative measure of hedging effectiveness, the Kuo–Chen risk-return simplification, and the Heifner–Kahl mean–variance
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.12.014
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1063
framework. According to this relative measure, we investigate the equivalence between the optimal hedge ratio derived in the Kuo–Chen framework and the optimal hedge ratio using mean–variance analysis. In Section 3, we employ Britten-Jones’ regression approach to develop a statistical test that is used to evaluate the Howard–D’Antonio hedging effectiveness. In Section 4, we give an example of how to construct a hedged portfolio using stock index futures. Section 5 concludes. 2. Basic features of optimal hedge ratio 2.1. Howard–D’Antonio risk-return measure A riskless asset with a rate of return i is assumed to be available in each period. Howard and D’Antonio consider an investor who wants to maximize the Sharpe ratio problem of a spot position hedged with a futures position. The Sharpe ratio captures both risk and return in a single measure, which is defined as the ratio of a portfolio’s excess return per unit of risk associated with the excess return (see Sharpe, 1994). A higher Sharpe ratio of the hedged portfolio indicates higher profitability. A simplification is suggested by Kuo and Chen in response to criticism of Howard–D’Antonio model regarding the difficulty of estimating futures excess return. Our work is related to the revised Howard–D’Antonio model. Given a portfolio comprising one spot position hedged with a futures position b, the Kuo–Chen simplification may be summarized as follows:
µs + b µa µp (1) = max 2 b σp σs + 2bσsa + b2 σa2 where µp = µs + bµa and σp = σs2 + 2bσsa + b2 σa2 are the expected excess return and the risk of a hedged portfolio, max b
respectively. In addition, we employ the following notation for further derivations. Pts = the spot price at period t , f
Pt = the futures price at period t ,
µs = the expected excess return of spot, E
Pts+1 − Pts Pts
µa = the expected excess return of futures, E
f
−i , f
P t +1 − P t Pts
,
µ = the vector of expected excess returns, [µa µs ]′ , and 2 σa σsa Σ = the covariance matrix, , σsa σs2 σsa ρsa = the coefficient of correlation, . σs σa Note that there are different ways suggested to calculate the return of futures positions. In this paper, we compute the futures excess return using the Kuo–Chen definition, which is based on the initial margin. Because the futures position not only aligns with the spot position but also does not have to tie cash up with a margin requirement, it appears reasonable for us to apply the spot price Pts as a basis to calculate the futures return. Therefore, we consider µa as a proxy for the ‘‘expected futures excess return’’. Remark 1. Some available results related to the solution of Kuo–Chen optimization (1) are listed as follows: 1. The optimal hedge ratio of optimization (1) is b∗ =
σs (Γ − ρsa ) , σa (1 − Γ ρsa )
(2)
θa µa = θs σa
(3)
where
Γ =
µ
µs . σs µ
We see that θs = σ s and θa = σ a are the Sharpe ratios of spot and futures, respectively. Moreover, Γ can be expressed s a as Sharpe ratio of futures expected excess return
Γ =
Sharpe ratio of spot expected excess return
.
Note that Γ represents a hedging multiplier of the Sharpe ratio of futures against the Sharpe ratio of the spot position. In fact, Γ is an intuitive relative because the futures position provides less (more) return for a unit of risk than the spot position if Γ < 1 (λ > 1). Of course, if Γ = 1, the spot position and futures position are equally attractive.
1064
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
2. The corresponding maximal Sharpe ratio of optimization (1) is
θ∗ =
µ∗p σ∗
=
µ′ Σ −1 µ.
(4)
p
3. Given a hedged portfolio p, Howard and D’Antonio define the relative measure of hedging effectiveness as HE =
θp µp = θs σp
µs . σs
(5)
A higher relative measure indicates higher hedging potential. With respect to the optimal hedge ratio b∗ in Eq. (2), the corresponding hedging effectiveness may be represented as ∗
HE =
θp∗ θs
=
µ∗p
σ∗ p
µs = σs
Γ 2 − 2ρsa Γ + 1 . 2 1 − ρsa
(6)
Excluding the trivial cases where ρsa = ±1, it is easy for us to show that HE ∗ ≥ 1. 2.2. Heifner–Kahl mean–variance analysis Some available results note that the Sharpe ratio is closely related to quadratic utility (see e.g. Černý’s book (2009), p. 69). Consequently, there exists a general one-to-one relationship between the maximum quadratic utility attainable in a market and the market Sharpe ratio. Instead of citing these well-known results, we outline the following theorem, which builds up the equivalence related to the optimal hedge ratio between optimization (1) and mean–variance analysis. By doing so, our discussion may be a self-contained exposition of the determination and estimation of the optimal hedge ratio. Let an investor be risk averse and hold a mean–variance profit-maximizing utility. The investor forms a hedged portfolio (Qa , Qs ), where Qa is the size of the futures position and Qs is the size of the spot position. When plugging futures excess return (µa ) and spot excess return (µs ) into the general mean–variance model, the investor’s objective is to solve the optimization as
max Qa µa + Qs µs − Qa ,Qs
λ 2
(Qa2 σa2 + 2Qa Qs σsa + Qs2 σs2 − σ02 )
(7)
where µp = Qa µa + Qs µs and σp2 = Qa2 σa2 + 2Qa Qs σsa + Qs2 σs2 are the portfolio’s expected excess return and variance.
Moreover, σ02 is an arbitrarily chosen variance of the portfolio return and λ > 0 is the risk parameter. With respect to the optimal hedge ratio, the following theorem shows that there exists an equivalence between optimizations (1) and (7). Theorem 1 (Heifner and Kahl). The optimal hedge ratio derived by mean–variance analysis is given as b˜ =
µa σs2 − µs σsa = b∗ . µs σa2 − µa σsa
(8)
Proof. The first-order conditions by quadratic programming show that the optimal futures and spot positions are proportional to the well-known tangent portfolio
2 1 Σ −1 µ Q˜ a σs = = Q˜ s λˆ p λˆ p |Σ | −σsa
−σsa σa2
−1
µa , µs
and the corresponding risk parameter is
λ˜ p =
µ′ Σ − 1 µ 1 = σ0 σ0
′ 2 µa σa µs σsa
σsa σs2
−1 µa . µs
Following the tangent portfolio (Q˜ a , Q˜ s ), which leads the optimal hedge ratio to
µa σs2 − µs σsa b˜ = = = µs σa2 − µa σsa Q˜ s Q˜ a
This completes the proof.
σa σs
µa σa
µs σs
− ρsa µs (Γ − ρsa ) = b∗ . = µ a (1 − Γ ρsa ) µa µs 1− σ ρsa σs a
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1065
Remark 2. The properties of the optimal positions, Q˜ a and Q˜ s . 1. Given Q˜ a , Q˜ s , and σ02 , we can rearrange the portfolio’s expected excess return as
µ′ Σ − 1 µ µ ˜ p = Q˜ a µa + Q˜ s µs = = σ0 µ′ Σ −1 µ, ˜λp and thus Eq. (4) implies that the maximal Sharpe ratio of this hedged portfolio is
θ˜p =
µ ˜p = µ′ Σ − 1 µ = θ ∗ . σ0
2. The corresponding hedging effectiveness of the optimal hedge ratio b˜ is
˜ = θp = σs HE θs σ0
Γ 2 − 2ρsa Γ + 1 . 2 1 − ρsa
is a function of σ0 , the mean–variance analysis allows the investor to impose her (or his) choice on risk Because HE settlement. From a risk-reduction perspective, the risk of the investor holding a hedged portfolio should not be greater than the risk of holding a spot position. Therefore, a meaningful consideration is that the investor will set her allowable maximum risk equal to the risk of holding only the spot position. In this particular case, we choose that σ0 = σs and then lead to ˜ = θp = HE θs
Γ 2 − 2ρsa Γ + 1 = HE ∗ . 2 1 − ρsa √ ′ −1 ˜ p = σ0 µ 2Σ µ = 3. Note that the risk parameter, λ σ 0
µ ˜p , σ02
may be interpreted as a relative measure of (Qa , Qs ) in the
mean–variance space. 4. Note that the risk aversion affects the size of Qa and Qs but does not account for the determination of the optimal hedge ratio and hedging effectiveness. Therefore, optimization (7) is flexible because the model allows the investor to consider both the spot and futures as determinable weights in mean–variance analysis compared to the optimization (1), for which the investor is endowed only with a sure spot position and then must choose her optimal futures position in solving the Sharpe ratio problem. 3. Main results The relationship between optimizations (1) and (7) enables us to quickly proceed with the statistical inference of hedging effectiveness based on mean–variance techniques (see e.g., Korkie and Jobson, 1982, Roll, 1985, Gibbons et al., 1989, and Britten-Jones, 1999). Researchers also present the Bayesian approach to test mean–variance efficiency (see e.g., Shanken, 1987, Black and Litterman, 1992). Usually, special attention of these mean–variance approaches is for portfolio selection purposes and mean–variance efficiency tests. Our work is related to Britten-Jones procedures which integrate the mean–variance framework and the regression approach into an elegant method to implement both the estimation of the optimal hedge ratio and the evaluation of the effectiveness of Howard–D’Antonio hedging. 3.1. Regression approach to the optimal hedge ratio To solve our problem, we transform the excess returns in optimization (1) and (7) into an arbitrage expression as 1 = wa µa + ws µs ,
(9)
which meets the data structure of the Britten-Jones regression, where there is no intercept, the dependent variable is nonstochastic, and the residual is correlated with the excess returns. Remark 3. Model (9) has an important interpretation: one unit arbitrage excess return can be achieved by holding wa units’ futures and ws units’ spot position. Moreover, in the optimization (1), we have wa = µb and ws = µ1 . Similarly, for the p
p
Qa optimization (7) of Heifner–Kahl model, we have wa = µ and ws = µQs . p p
Because the market parameters µ and Σ are unknown, the estimator bˆ must be determined based on the sample ˆ and used to solve the optimization (1). There is a large supply of techniques developed to treat counterpart µ ˆ as well as Σ this problem in finance theory (see e.g., Britten-Jones, 1999 and Černý, 2009, p. 30).
1066
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
In this section, the least squares method is used to obtain the best hedge in model (9). When using the historical data sample we illustrate how the Britten-Jones procedures are applied to determine the optimal hedge ratio. First, the sample data are plugged into the Britten-Jones unrestricted OLS regression: 1 1
ra1 ra2
rs1 rs2
. . . = . . . 1
.. wa . ws
ran
rsn
(R)
(ℓ)
u1
u2 + ... ,
(10)
un
(u)
(w)
where w = [wa ws ]′ , wa and ws represent the futures position and spot position, respectively. ℓ = (1, . . . , 1)′ is the n × 1 vector of ones, the n observations of the hedged residuals are contained in n × 1 vector u, and the observations of Kuo–Chen futures excess returns and spot excess returns are contained in the n × 2 matrix R = [ra rs ], [rat rst ] is the vector of futures excess return and spot excess return at period t. a Definition 1. The hedge ratio of model (10) is defined as b = w . ws
In model (10), we solve the equation for the hedge ratio only, removing the need to sum the weights to one. Britten-Jones shows that the unscaled OLS estimator and the unrestricted sum of squared residuals are
ˆ + r¯ r¯ ′ )−1 r¯ w ˆ = (Σ ˆ −1 r¯ r¯ ′ Σ ˆ −1 Σ −1 ˆ r¯ = Σ − ˆ −1 r¯ 1 + r¯ ′ Σ =
ˆ −1 r¯ Σ , ˆ −1 r¯ 1 + r¯ ′ Σ
(11)
and SSRu = (ℓ − Rw) ˆ ′ (ℓ − Rw) ˆ = n(1 − w ˆ ′ r¯ ) =
r¯
ˆ = where µ ˆ = r¯ = r¯as and Σ
s2a
ssa
ssa s2s
n
ˆ −1 r¯ 1 + r¯ ′ Σ
,
(12)
are the vector of sample means of excess returns and the sample covariance matrix
ˆ are the maximum likelihood estimators of µ between excess returns. In addition, under multivariate normality, r¯ and Σ and Σ . ˆ into Eqs. (11) and (12) will lead to the following result when we determine the Substituting the components of r¯ and Σ optimal hedging policy. Theorem 2 (Sample Errors in Estimating The Optimal Hedge Ratio). The optimal hedge ratio and the sum of squared residuals from this hedged regression are bˆ =
r¯a s2s − r¯s ssa
(13)
−¯ra ssa + r¯s s2a
and SSRu = where θˆ =
n
ˆ −1 r¯ 1 + r¯ ′ Σ
=
n 1 + θˆ 2
,
(14)
ˆ −1 r¯ is the corresponding maximal sample Sharpe ratio. r¯ ′ Σ
Proof. The unscaled OLS estimator of [wa ws ]′ follows from Eq. (11)
2 ˆ −1 r¯ Σ 1 w ˆa ss = = w ˆs ˆ | −ssa 1 + θˆ 2 (1 + θˆ 2 )|Σ
−ssa s2a
r¯a , r¯s
ˆ | can be regarded as a multiplier reflecting investor’s risk aversion. It affects the positions but does not where (1 + θˆ 2 )|Σ affect the optimal hedge ratio. The estimated optimal hedge ratio follows immediately from bˆ =
r¯a s2s − r¯s ssa w ˆa = . w ˆs −¯ra ssa + r¯s s2a
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1067
ˆ the resulting portfolio’s excess return and risk are rp = r¯ ′ Σ ˆ −1 r¯ /(1 + θˆ 2 ) and Relative to the optimal hedge ratio b,
ˆ −1 r¯ /(1 + θˆ 2 ), respectively. Consequently, the sample Sharpe ratio and the sum of squared residuals from this sp = r¯ ′ Σ hedged regression may be reduced to θˆ =
rp sp
=
ˆ −1 r¯ r¯ ′ Σ
This completes the proof.
SSRu =
and
n
ˆ −1 r¯ 1 + r¯ ′ Σ
=
n 1 + θˆ 2
.
Remark 4. Theorem 2 has two intuitive implications: 1. A larger sample Sharpe ratio of a hedged portfolio implies a lower sum of squared residuals from this hedged regression. In other words, in terms of SSR, Eq. (14) sets up the one-to-one relationship between the mean–variance efficiency test and the tangent portfolio (sample Sharpe ratio). The result is consistent with Černý’s conclusion. 2. Under the normality assumption, the optimal hedge ratio bˆ is fully expected if we plug directly the maximum likelihood ˆ into Eq. (8). estimators r¯ and Σ 3.2. Testing hedging effectiveness Investors may wish to assess whether a target portfolio relative to a specific hedging policy is effective. For example, investors wish to test the hedging effectiveness with a given hedge ratio b0 . The null hypothesis is stated as follows: or H0 : wa = b0 ws .
H 0 : b = b0
(15)
Under the null hypothesis, the positions vector becomes w ′ = [b0 ws ws ]. We note that this hedged portfolio can be duplicated by a risky spot asset alone. The restricted regression of unhedged portfolio is therefore reduced into a simple regression as b0 ra1 − rs1 b0 ra2 − rs2
1 1
. . = . 1
(ℓ)
u1
u2 ws + .. .
.. . b0 ran − rsn (r0 ) (ws )
(16)
un ( u)
where the n observations of pre-determined excess returns contained in the n × 1 matrix r0 = b0 ra − rs . ˆ 0 = s20 = b20 s2a − 2b0 ssa + s2s and Similarly, in terms of the sample mean r¯0 = b0 r¯a − r¯s as well as the sample variance Σ the sample Sharpe ratio θˆ0 = this unhedged portfolio is SSRr =
n 1 + θˆ02
r¯0 , s0
n
=
1+
r¯02 s20
some algebraic rearrangements show that the restricted sum of squared residuals from n
= 1+
(b0 r¯a −¯rs )2 b20 s2a −2b0 ssa +s2s
.
(17)
Under the normality assumption, it is widely known that the F (OLS)-statistic used to test the hypothesis (15) is given by F =
(SSRr − SSRu ) , SSRu /(n − 2)
(18)
and the test statistic F is distributed as a central F (1, n − 2) with 1 and n − 1 degrees of freedom. The following main result is based on Eq. (18), and we present a statistic for testing a specific hedging policy. Theorem 3 (Sample Errors in Testing a Specific Hedge Ratio). Given the null hypothesis H0 : b = b0 , under multivariate normality, the t (OLS)-statistic for testing a specific hedge ratio b0 based on the optimal hedge ratio bˆ can be obtained as bˆ − b0 s.e.(bˆ )
∼ t (n − 2),
and the standard error of bˆ in testing H0 : b = b0 can be approximated by
s.e.(bˆ ) =
(s2a s2s − s2sa )(s20 + r¯02 ) . (−¯ra ssa + r¯s s2a )2 (n − 2)
(19)
1068
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
Proof. Substituting Eqs. (14) and (17) into Eq. (18) leads to
(SSRr − SSRu ) θˆ 2 − θˆ02 = ∼ F (1, n − 2). SSRu /(n − 2) (1 + θˆ02 )/(n − 2)
(20)
ˆ −1 r¯ and θˆ02 = r¯0′ Σ ˆ 0−1 r¯0 , direct calculation yields Given θ 2 = r¯ ′ Σ (SSRr − SSRu ) = SSRu /(n − 2)
=
=
r¯a2 s2s −2r¯a r¯s ssa +¯rs2 s2a s2a s2s −s2sa
1+
−
(b0 r¯a −¯rs )2 b20 s2a −2b0 ssa +s2s
(b0 r¯a −¯rs )2 b20 s2a −2b0 ssa +s2s
×
1
(n−2)
[(¯ra s2s − r¯s ssa ) − (−¯ra ssa + r¯s s2a )b0 ]2 (s2a s2s − s2sa )(s20 + r¯02 )/(n − 2) 2 2 r¯a ss −¯rs ssa − b0 2 −¯r s +¯r s a sa
s a
(s2a s2s −s2sa )(s20 +¯r02 ) (−¯ra ssa +¯rs s2a )2 (n−2)
=
(bˆ − b0 )2 (s2a s2s −s2sa )(s20 +¯r02 ) (−¯ra ssa +¯rs s2a )2 (n−2)
.
The t (OLS)-statistic for testing the hedge ratio is derived.
Remark 5. Theorem 3 results in the following conclusions: 1. Eq. (18) establishes the relationship between the hedge ratio and the square of the Sharpe ratio to be tested. 2. Although Theorem 3 presents a closed form of the t-statistic for testing H0 : b = b0 . However, Eq. (18) provides a more efficient method of accessing the F -value in computation aspects. That is, the F (OLS)-test for testing the optimal hedge ratio bˆ may be implemented by the standard OLS associated with a specific hedged regression, which may be accomplished by several available statistical software programs. As a direct consequence of Theorem 3, we now turn our attention to assessing the hedging effectiveness with respect to the hedge ratio b0 . Consider the special case b0 = 0: this case is trivial, but the model does have an important implication. Because this restricted portfolio holds only the spot position, this test is one of the unhedged portfolio against a hedged portfolio with futures. The following theorem shows that an F (OLS)-test for testing the hedging effectiveness is constructed. Theorem 4 (Sample Errors in Testing The Hedging Effectiveness). Under multivariate normality, the test statistic of the hedging effectiveness with the optimal hedge ratio bˆ against the unhedged portfolio regression of H0 : b0 = 0 is distributed as 2
−1 HE (s2s + r¯s2 )/(n − 2)¯rs2
∼ F (1, n − 2),
(21)
= θˆ /θˆs . The sample mean and the sample variance of spot return are r¯s and s2s , respectively. where the hedging effectiveness is HE Proof. This is a direct result from Theorem 3 because
(SSRr − SSRu ) θˆ 2 − θˆs2 = SSRu /(n − 2) (1 + θˆs )/(n − 2) 2 − 1 HE = 2 2 ∼ F (1, n − 2) (ss +¯rs ) (n−2)¯rs2
where θˆs =
r¯s ss
is the sample Sharpe ratio of spot excess return.
≥ 1, a large enough F -value indicates a significant hedging Remark 6. There is an intuition about Eq. (21). Because HE effectiveness compared to the spot portfolio, where HE = 1.
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1069
Table 1 Summary statistics and regression analysis. Mean return (in %)
Covariance matrix (in %)
r¯a −0.2124 r¯ = = r¯s −0.3321
2 ˆ = sa Σ
Unrestricted regression (9) 1 = w a µa + w s µs + u
Restricted regression (15) 1 = ws µs + u (under H0 : b = 0)
Model result: SSRu = 83.02 w ˆ a = 0.1734, w ˆ s = −0.2147 bˆ = −0.8075, (t = 1.3715)a 2 = 2.72, (F = 1.8809)b HE
Model result: SSRr = 84.88
a b
ssa
ssa s2s
=
9.422 8.561
8.561 8.410
Significantly different 0 against alternative hypothesis H1 : b < 0 at the 10% level. Significantly different 1 at the 20% level.
4. Illustration In 2011, the annual trading volume on the Taiwan Futures Exchange (TAIFEX) totaled 182 million contracts at an average daily volume of 740,871 contracts. These two figures represented increases of 30.9% and 33.02%, respectively, over 2010. In particular, a remarkable increase resulted in a transaction on single stock futures reaching 2,471,605 contracts for an increase of 241.21% over 2010. The growth in the popularity of stock index futures may be attributed to the use as stock portfolio management. Currently, the futures contract ‘‘TAIEX Futures’’ is actively traded, and the underlying investment asset is the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) issued by the Taiwan Stock Exchange Corporation (TWSE). Information source: The 2011 annual report of the Taiwan Futures Exchange (TAIFEX); refer to http://www.taifex.com.tw/eng/eng10/AnnualRep.asp. Both the optimal hedge ratio and hedging effectiveness are tested using weekly closing prices of TAIEX Futures and TAIEX. Data sources are the weekly data from the TEJ database maintained by the Taiwan Economic Journal. These weekly percentage returns are used to estimate the expected return and the corresponding variances as well as covariance. Summary statistics and regression results for the data (January 2011–October 2012, with 86 weekly observations) are reported in Table 1. Because both the futures return and the spot return are negative, the investment strategy is implemented by shorting one spot and longing 0.8075 unit futures contract. Regression analysis is based on comparing Eq. (9) against Eq. (15): F =
(SSRr − SSRu ) = 1.8809. SSRu /(n − 2)
The F -value indicates that the square of the hedging effectiveness at the optimal hedge ratio −0.8075 is significantly different from 1 at the 20% level. 5. Conclusion In this paper, an equivalence between the optimal hedge ratio derived by a revised risk-return approach and the optimal hedge ratio derived by mean–variance portfolio selection is constructed. The link between two approaches will provide researchers abundant mean–variance techniques for further study on the evaluation of hedging effectiveness. According to this finding a regression-based t-statistic to test a specific hedge ratio and an F (OLS)-statistic for evaluating hedging effectiveness are derived herein. Acknowledgment I am grateful to the editor and referee for helpful comments. The views in this paper are solely the responsibility of the author. References Black, F., Litterman, R., 1992. Global portfolio optimization. Financial Analysts Journal 48, 28–43. Britten-Jones, M., 1999. The sampling error in estimating of mean–variance efficient portfolio weights. Journal of Finance 54, 655–671. Brooks, C., Henry, Ó.T., Persand, G., 2002. The effect of asymmetries on optimal hedge ratios. Journal of Business 75, 333–352. Černý, A., 2009. Mathematical Techniques in Finance: Tools for Incomplete Markets. Princeton, New Jersey. Chang, J., Shanker, L., 1987. A risk-return measure of hedging effectiveness: a comment. Journal of Financial and Quantitative Analysis 22, 373–376. Ederington, L.H., 1979. The hedging performance of the new futures markets. Journal of Finance 34, 157–170. Gibbons, M.R., Ross, S.A., Shanken, J., 1989. A test of the efficiency of a given portfolio. Econometrica 57, 1121–1152. Harris, R.D.F., Shen, J., 2003. Robust estimation of the optimal hedge ratio. Journal of Futures Markets 23, 799–816. Heifner, , 1972. Optimal hedging levels and hedging effectiveness in cattle feeding. Agricultural Economics Research 24, 25–36. Howard, C., D’Antonio, L., 1984. A risk-return measure of hedging effectiveness. Journal of Financial and Quantitative Analysis 19, 101–112.
1070
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
Jong, A.D., Roon, F.D., Veld, C., 1997. Out-of-sample hedging effectiveness of currency futures for alternative models and hedging strategies. Journal of Futures Markets 17, 817–837. Kahl, K.H., 1983. Determination of the recommended hedging ratio. American Journal of Agricultural Economics 65, 603–605. Korkie, B., Jobson, J.D., 1982. Potential performance and tests of portfolio efficiency. Journal of Financial Economics 10, 433–466. Kuo, C.-K., Chen, K.-W., 1995. A risk-return measure of hedging effectiveness: a simplification. Journal of Futures Markets 15, 39–44. Mcnew, K.P., Fackler, P.L., 1994. Nonconstant optimal hedge ratio estimation and nested hypotheses tests. Journal of Futures Markets 14, 619–635. Roll, R., 1985. A note on Shanken’s CSR T 2 test for mean/variance efficiency. Journal of Financial Economics 14, 349–357. Satyanarayan, S., 1998. A note on a risk-return measure of hedging effectiveness. Journal of Futures Markets 18, 595–605. Shanken, J., 1987. A Bayesian approach to testing portfolio efficiency. Journal of Financial Economics 19, 195–215. Sharpe, W.F., 1994. The Sharpe ratio. Journal of Portfolio Management 21, 49–58. Stulz, R.M., 1984. Optimal hedging policies. Journal of Financial and Quantitative Analysis 19, 127–140. Terry, E., 2005. Minimum-variance futures hedging under alternative return specifications. Journal of Futures Markets 25, 537–552.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A simple test of optimal hedging policy Wan-Yi Chiu National United University, Department of Finance, Taiwan, ROC
article
info
Article history: Received 20 August 2012 Received in revised form 15 November 2012 Accepted 13 December 2012 Available online 26 December 2012 Keywords: Optimal hedge ratio Risk-return measure Sharpe ratio Hedging effectiveness Mean–variance analysis
abstract This paper investigates the equivalence between the optimal hedge ratio derived in a risk-return simplification and the optimal hedge ratio using mean–variance analysis. In accordance with this relationship, we develop a simple regression-based test for evaluating the hedging effectiveness of the risk-return hedging. As a result, a t-test and an F -test are designed to examine the hedge ratio and hedging effectiveness, respectively. An example of hedging is also provided to illustrate this process. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Futures markets supply vital hedge instruments that investors require to make ‘‘a reasoned choice in conduct of risk reduction’’. One of the major research interests in futures markets is to solve the optimal hedging policy for a position to be hedged with futures. Two general methods of determining the optimal hedge ratio are the minimum-variance model and the risk-return approach. The fundamental difference between the risk-return approach and the minimum-variance model is that the risk-return approach highlights the trade-off between return and risk compared to the minimum-variance model, which focuses solely on risk reduction and requires fewer estimates (variances and covariance). Traditionally, the determination of the optimal hedge ratio of a hedged portfolio has been considered as a solution to a risk reduction problem since Ederington’s (1979) proposal of minimum-variance hedging. Researchers have presented ample statistical inferences on this issue based on minimizing portfolio variance (Stulz, 1984; Mcnew and Fackler, 1994; Jong et al., 1997; Brooks et al., 2002; Harris and Shen, 2003; Terry, 2005). Starting with Howard and D’Antonio’s (1984) proposal of the relative risk-return measure to determine the optimal hedge ratio, both practitioners and researchers have advanced a variety of risk-return measures to evaluate hedging effectiveness (Chang and Shanker, 1987; Kuo and Chen, 1995; Satyanarayan, 1998). In particular, Kuo and Chen adapt and simplify the Howard–D’Antonio approach for practical settings. However, these studies do not suggest an analytical test-statistic to evaluate hedging effectiveness. Two main results are obtained in this paper. First, we link the relationship between the revised Howard–D’Antonio model derived by Kuo–Chen and mean–variance analysis according to Heifner (1972) and Kahl (1983). Second, we integrate the mean–variance approach and the Britten-Jones regression approach (1999) into a simple test that is used to estimate the optimal hedge ratio. As a consequence, a t-test and an F -test are analytically designed to test the hedge ratio and the hedging effectiveness, respectively. This article is organized as follows. In Section 2, we briefly review Howard–D’Antonio risk-return optimization as well as the relative measure of hedging effectiveness, the Kuo–Chen risk-return simplification, and the Heifner–Kahl mean–variance
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.12.014
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1063
framework. According to this relative measure, we investigate the equivalence between the optimal hedge ratio derived in the Kuo–Chen framework and the optimal hedge ratio using mean–variance analysis. In Section 3, we employ Britten-Jones’ regression approach to develop a statistical test that is used to evaluate the Howard–D’Antonio hedging effectiveness. In Section 4, we give an example of how to construct a hedged portfolio using stock index futures. Section 5 concludes. 2. Basic features of optimal hedge ratio 2.1. Howard–D’Antonio risk-return measure A riskless asset with a rate of return i is assumed to be available in each period. Howard and D’Antonio consider an investor who wants to maximize the Sharpe ratio problem of a spot position hedged with a futures position. The Sharpe ratio captures both risk and return in a single measure, which is defined as the ratio of a portfolio’s excess return per unit of risk associated with the excess return (see Sharpe, 1994). A higher Sharpe ratio of the hedged portfolio indicates higher profitability. A simplification is suggested by Kuo and Chen in response to criticism of Howard–D’Antonio model regarding the difficulty of estimating futures excess return. Our work is related to the revised Howard–D’Antonio model. Given a portfolio comprising one spot position hedged with a futures position b, the Kuo–Chen simplification may be summarized as follows:
µs + b µa µp (1) = max 2 b σp σs + 2bσsa + b2 σa2 where µp = µs + bµa and σp = σs2 + 2bσsa + b2 σa2 are the expected excess return and the risk of a hedged portfolio, max b
respectively. In addition, we employ the following notation for further derivations. Pts = the spot price at period t , f
Pt = the futures price at period t ,
µs = the expected excess return of spot, E
Pts+1 − Pts Pts
µa = the expected excess return of futures, E
f
−i , f
P t +1 − P t Pts
,
µ = the vector of expected excess returns, [µa µs ]′ , and 2 σa σsa Σ = the covariance matrix, , σsa σs2 σsa ρsa = the coefficient of correlation, . σs σa Note that there are different ways suggested to calculate the return of futures positions. In this paper, we compute the futures excess return using the Kuo–Chen definition, which is based on the initial margin. Because the futures position not only aligns with the spot position but also does not have to tie cash up with a margin requirement, it appears reasonable for us to apply the spot price Pts as a basis to calculate the futures return. Therefore, we consider µa as a proxy for the ‘‘expected futures excess return’’. Remark 1. Some available results related to the solution of Kuo–Chen optimization (1) are listed as follows: 1. The optimal hedge ratio of optimization (1) is b∗ =
σs (Γ − ρsa ) , σa (1 − Γ ρsa )
(2)
θa µa = θs σa
(3)
where
Γ =
µ
µs . σs µ
We see that θs = σ s and θa = σ a are the Sharpe ratios of spot and futures, respectively. Moreover, Γ can be expressed s a as Sharpe ratio of futures expected excess return
Γ =
Sharpe ratio of spot expected excess return
.
Note that Γ represents a hedging multiplier of the Sharpe ratio of futures against the Sharpe ratio of the spot position. In fact, Γ is an intuitive relative because the futures position provides less (more) return for a unit of risk than the spot position if Γ < 1 (λ > 1). Of course, if Γ = 1, the spot position and futures position are equally attractive.
1064
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
2. The corresponding maximal Sharpe ratio of optimization (1) is
θ∗ =
µ∗p σ∗
=
µ′ Σ −1 µ.
(4)
p
3. Given a hedged portfolio p, Howard and D’Antonio define the relative measure of hedging effectiveness as HE =
θp µp = θs σp
µs . σs
(5)
A higher relative measure indicates higher hedging potential. With respect to the optimal hedge ratio b∗ in Eq. (2), the corresponding hedging effectiveness may be represented as ∗
HE =
θp∗ θs
=
µ∗p
σ∗ p
µs = σs
Γ 2 − 2ρsa Γ + 1 . 2 1 − ρsa
(6)
Excluding the trivial cases where ρsa = ±1, it is easy for us to show that HE ∗ ≥ 1. 2.2. Heifner–Kahl mean–variance analysis Some available results note that the Sharpe ratio is closely related to quadratic utility (see e.g. Černý’s book (2009), p. 69). Consequently, there exists a general one-to-one relationship between the maximum quadratic utility attainable in a market and the market Sharpe ratio. Instead of citing these well-known results, we outline the following theorem, which builds up the equivalence related to the optimal hedge ratio between optimization (1) and mean–variance analysis. By doing so, our discussion may be a self-contained exposition of the determination and estimation of the optimal hedge ratio. Let an investor be risk averse and hold a mean–variance profit-maximizing utility. The investor forms a hedged portfolio (Qa , Qs ), where Qa is the size of the futures position and Qs is the size of the spot position. When plugging futures excess return (µa ) and spot excess return (µs ) into the general mean–variance model, the investor’s objective is to solve the optimization as
max Qa µa + Qs µs − Qa ,Qs
λ 2
(Qa2 σa2 + 2Qa Qs σsa + Qs2 σs2 − σ02 )
(7)
where µp = Qa µa + Qs µs and σp2 = Qa2 σa2 + 2Qa Qs σsa + Qs2 σs2 are the portfolio’s expected excess return and variance.
Moreover, σ02 is an arbitrarily chosen variance of the portfolio return and λ > 0 is the risk parameter. With respect to the optimal hedge ratio, the following theorem shows that there exists an equivalence between optimizations (1) and (7). Theorem 1 (Heifner and Kahl). The optimal hedge ratio derived by mean–variance analysis is given as b˜ =
µa σs2 − µs σsa = b∗ . µs σa2 − µa σsa
(8)
Proof. The first-order conditions by quadratic programming show that the optimal futures and spot positions are proportional to the well-known tangent portfolio
2 1 Σ −1 µ Q˜ a σs = = Q˜ s λˆ p λˆ p |Σ | −σsa
−σsa σa2
−1
µa , µs
and the corresponding risk parameter is
λ˜ p =
µ′ Σ − 1 µ 1 = σ0 σ0
′ 2 µa σa µs σsa
σsa σs2
−1 µa . µs
Following the tangent portfolio (Q˜ a , Q˜ s ), which leads the optimal hedge ratio to
µa σs2 − µs σsa b˜ = = = µs σa2 − µa σsa Q˜ s Q˜ a
This completes the proof.
σa σs
µa σa
µs σs
− ρsa µs (Γ − ρsa ) = b∗ . = µ a (1 − Γ ρsa ) µa µs 1− σ ρsa σs a
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1065
Remark 2. The properties of the optimal positions, Q˜ a and Q˜ s . 1. Given Q˜ a , Q˜ s , and σ02 , we can rearrange the portfolio’s expected excess return as
µ′ Σ − 1 µ µ ˜ p = Q˜ a µa + Q˜ s µs = = σ0 µ′ Σ −1 µ, ˜λp and thus Eq. (4) implies that the maximal Sharpe ratio of this hedged portfolio is
θ˜p =
µ ˜p = µ′ Σ − 1 µ = θ ∗ . σ0
2. The corresponding hedging effectiveness of the optimal hedge ratio b˜ is
˜ = θp = σs HE θs σ0
Γ 2 − 2ρsa Γ + 1 . 2 1 − ρsa
is a function of σ0 , the mean–variance analysis allows the investor to impose her (or his) choice on risk Because HE settlement. From a risk-reduction perspective, the risk of the investor holding a hedged portfolio should not be greater than the risk of holding a spot position. Therefore, a meaningful consideration is that the investor will set her allowable maximum risk equal to the risk of holding only the spot position. In this particular case, we choose that σ0 = σs and then lead to ˜ = θp = HE θs
Γ 2 − 2ρsa Γ + 1 = HE ∗ . 2 1 − ρsa √ ′ −1 ˜ p = σ0 µ 2Σ µ = 3. Note that the risk parameter, λ σ 0
µ ˜p , σ02
may be interpreted as a relative measure of (Qa , Qs ) in the
mean–variance space. 4. Note that the risk aversion affects the size of Qa and Qs but does not account for the determination of the optimal hedge ratio and hedging effectiveness. Therefore, optimization (7) is flexible because the model allows the investor to consider both the spot and futures as determinable weights in mean–variance analysis compared to the optimization (1), for which the investor is endowed only with a sure spot position and then must choose her optimal futures position in solving the Sharpe ratio problem. 3. Main results The relationship between optimizations (1) and (7) enables us to quickly proceed with the statistical inference of hedging effectiveness based on mean–variance techniques (see e.g., Korkie and Jobson, 1982, Roll, 1985, Gibbons et al., 1989, and Britten-Jones, 1999). Researchers also present the Bayesian approach to test mean–variance efficiency (see e.g., Shanken, 1987, Black and Litterman, 1992). Usually, special attention of these mean–variance approaches is for portfolio selection purposes and mean–variance efficiency tests. Our work is related to Britten-Jones procedures which integrate the mean–variance framework and the regression approach into an elegant method to implement both the estimation of the optimal hedge ratio and the evaluation of the effectiveness of Howard–D’Antonio hedging. 3.1. Regression approach to the optimal hedge ratio To solve our problem, we transform the excess returns in optimization (1) and (7) into an arbitrage expression as 1 = wa µa + ws µs ,
(9)
which meets the data structure of the Britten-Jones regression, where there is no intercept, the dependent variable is nonstochastic, and the residual is correlated with the excess returns. Remark 3. Model (9) has an important interpretation: one unit arbitrage excess return can be achieved by holding wa units’ futures and ws units’ spot position. Moreover, in the optimization (1), we have wa = µb and ws = µ1 . Similarly, for the p
p
Qa optimization (7) of Heifner–Kahl model, we have wa = µ and ws = µQs . p p
Because the market parameters µ and Σ are unknown, the estimator bˆ must be determined based on the sample ˆ and used to solve the optimization (1). There is a large supply of techniques developed to treat counterpart µ ˆ as well as Σ this problem in finance theory (see e.g., Britten-Jones, 1999 and Černý, 2009, p. 30).
1066
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
In this section, the least squares method is used to obtain the best hedge in model (9). When using the historical data sample we illustrate how the Britten-Jones procedures are applied to determine the optimal hedge ratio. First, the sample data are plugged into the Britten-Jones unrestricted OLS regression: 1 1
ra1 ra2
rs1 rs2
. . . = . . . 1
.. wa . ws
ran
rsn
(R)
(ℓ)
u1
u2 + ... ,
(10)
un
(u)
(w)
where w = [wa ws ]′ , wa and ws represent the futures position and spot position, respectively. ℓ = (1, . . . , 1)′ is the n × 1 vector of ones, the n observations of the hedged residuals are contained in n × 1 vector u, and the observations of Kuo–Chen futures excess returns and spot excess returns are contained in the n × 2 matrix R = [ra rs ], [rat rst ] is the vector of futures excess return and spot excess return at period t. a Definition 1. The hedge ratio of model (10) is defined as b = w . ws
In model (10), we solve the equation for the hedge ratio only, removing the need to sum the weights to one. Britten-Jones shows that the unscaled OLS estimator and the unrestricted sum of squared residuals are
ˆ + r¯ r¯ ′ )−1 r¯ w ˆ = (Σ ˆ −1 r¯ r¯ ′ Σ ˆ −1 Σ −1 ˆ r¯ = Σ − ˆ −1 r¯ 1 + r¯ ′ Σ =
ˆ −1 r¯ Σ , ˆ −1 r¯ 1 + r¯ ′ Σ
(11)
and SSRu = (ℓ − Rw) ˆ ′ (ℓ − Rw) ˆ = n(1 − w ˆ ′ r¯ ) =
r¯
ˆ = where µ ˆ = r¯ = r¯as and Σ
s2a
ssa
ssa s2s
n
ˆ −1 r¯ 1 + r¯ ′ Σ
,
(12)
are the vector of sample means of excess returns and the sample covariance matrix
ˆ are the maximum likelihood estimators of µ between excess returns. In addition, under multivariate normality, r¯ and Σ and Σ . ˆ into Eqs. (11) and (12) will lead to the following result when we determine the Substituting the components of r¯ and Σ optimal hedging policy. Theorem 2 (Sample Errors in Estimating The Optimal Hedge Ratio). The optimal hedge ratio and the sum of squared residuals from this hedged regression are bˆ =
r¯a s2s − r¯s ssa
(13)
−¯ra ssa + r¯s s2a
and SSRu = where θˆ =
n
ˆ −1 r¯ 1 + r¯ ′ Σ
=
n 1 + θˆ 2
,
(14)
ˆ −1 r¯ is the corresponding maximal sample Sharpe ratio. r¯ ′ Σ
Proof. The unscaled OLS estimator of [wa ws ]′ follows from Eq. (11)
2 ˆ −1 r¯ Σ 1 w ˆa ss = = w ˆs ˆ | −ssa 1 + θˆ 2 (1 + θˆ 2 )|Σ
−ssa s2a
r¯a , r¯s
ˆ | can be regarded as a multiplier reflecting investor’s risk aversion. It affects the positions but does not where (1 + θˆ 2 )|Σ affect the optimal hedge ratio. The estimated optimal hedge ratio follows immediately from bˆ =
r¯a s2s − r¯s ssa w ˆa = . w ˆs −¯ra ssa + r¯s s2a
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1067
ˆ the resulting portfolio’s excess return and risk are rp = r¯ ′ Σ ˆ −1 r¯ /(1 + θˆ 2 ) and Relative to the optimal hedge ratio b,
ˆ −1 r¯ /(1 + θˆ 2 ), respectively. Consequently, the sample Sharpe ratio and the sum of squared residuals from this sp = r¯ ′ Σ hedged regression may be reduced to θˆ =
rp sp
=
ˆ −1 r¯ r¯ ′ Σ
This completes the proof.
SSRu =
and
n
ˆ −1 r¯ 1 + r¯ ′ Σ
=
n 1 + θˆ 2
.
Remark 4. Theorem 2 has two intuitive implications: 1. A larger sample Sharpe ratio of a hedged portfolio implies a lower sum of squared residuals from this hedged regression. In other words, in terms of SSR, Eq. (14) sets up the one-to-one relationship between the mean–variance efficiency test and the tangent portfolio (sample Sharpe ratio). The result is consistent with Černý’s conclusion. 2. Under the normality assumption, the optimal hedge ratio bˆ is fully expected if we plug directly the maximum likelihood ˆ into Eq. (8). estimators r¯ and Σ 3.2. Testing hedging effectiveness Investors may wish to assess whether a target portfolio relative to a specific hedging policy is effective. For example, investors wish to test the hedging effectiveness with a given hedge ratio b0 . The null hypothesis is stated as follows: or H0 : wa = b0 ws .
H 0 : b = b0
(15)
Under the null hypothesis, the positions vector becomes w ′ = [b0 ws ws ]. We note that this hedged portfolio can be duplicated by a risky spot asset alone. The restricted regression of unhedged portfolio is therefore reduced into a simple regression as b0 ra1 − rs1 b0 ra2 − rs2
1 1
. . = . 1
(ℓ)
u1
u2 ws + .. .
.. . b0 ran − rsn (r0 ) (ws )
(16)
un ( u)
where the n observations of pre-determined excess returns contained in the n × 1 matrix r0 = b0 ra − rs . ˆ 0 = s20 = b20 s2a − 2b0 ssa + s2s and Similarly, in terms of the sample mean r¯0 = b0 r¯a − r¯s as well as the sample variance Σ the sample Sharpe ratio θˆ0 = this unhedged portfolio is SSRr =
n 1 + θˆ02
r¯0 , s0
n
=
1+
r¯02 s20
some algebraic rearrangements show that the restricted sum of squared residuals from n
= 1+
(b0 r¯a −¯rs )2 b20 s2a −2b0 ssa +s2s
.
(17)
Under the normality assumption, it is widely known that the F (OLS)-statistic used to test the hypothesis (15) is given by F =
(SSRr − SSRu ) , SSRu /(n − 2)
(18)
and the test statistic F is distributed as a central F (1, n − 2) with 1 and n − 1 degrees of freedom. The following main result is based on Eq. (18), and we present a statistic for testing a specific hedging policy. Theorem 3 (Sample Errors in Testing a Specific Hedge Ratio). Given the null hypothesis H0 : b = b0 , under multivariate normality, the t (OLS)-statistic for testing a specific hedge ratio b0 based on the optimal hedge ratio bˆ can be obtained as bˆ − b0 s.e.(bˆ )
∼ t (n − 2),
and the standard error of bˆ in testing H0 : b = b0 can be approximated by
s.e.(bˆ ) =
(s2a s2s − s2sa )(s20 + r¯02 ) . (−¯ra ssa + r¯s s2a )2 (n − 2)
(19)
1068
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
Proof. Substituting Eqs. (14) and (17) into Eq. (18) leads to
(SSRr − SSRu ) θˆ 2 − θˆ02 = ∼ F (1, n − 2). SSRu /(n − 2) (1 + θˆ02 )/(n − 2)
(20)
ˆ −1 r¯ and θˆ02 = r¯0′ Σ ˆ 0−1 r¯0 , direct calculation yields Given θ 2 = r¯ ′ Σ (SSRr − SSRu ) = SSRu /(n − 2)
=
=
r¯a2 s2s −2r¯a r¯s ssa +¯rs2 s2a s2a s2s −s2sa
1+
−
(b0 r¯a −¯rs )2 b20 s2a −2b0 ssa +s2s
(b0 r¯a −¯rs )2 b20 s2a −2b0 ssa +s2s
×
1
(n−2)
[(¯ra s2s − r¯s ssa ) − (−¯ra ssa + r¯s s2a )b0 ]2 (s2a s2s − s2sa )(s20 + r¯02 )/(n − 2) 2 2 r¯a ss −¯rs ssa − b0 2 −¯r s +¯r s a sa
s a
(s2a s2s −s2sa )(s20 +¯r02 ) (−¯ra ssa +¯rs s2a )2 (n−2)
=
(bˆ − b0 )2 (s2a s2s −s2sa )(s20 +¯r02 ) (−¯ra ssa +¯rs s2a )2 (n−2)
.
The t (OLS)-statistic for testing the hedge ratio is derived.
Remark 5. Theorem 3 results in the following conclusions: 1. Eq. (18) establishes the relationship between the hedge ratio and the square of the Sharpe ratio to be tested. 2. Although Theorem 3 presents a closed form of the t-statistic for testing H0 : b = b0 . However, Eq. (18) provides a more efficient method of accessing the F -value in computation aspects. That is, the F (OLS)-test for testing the optimal hedge ratio bˆ may be implemented by the standard OLS associated with a specific hedged regression, which may be accomplished by several available statistical software programs. As a direct consequence of Theorem 3, we now turn our attention to assessing the hedging effectiveness with respect to the hedge ratio b0 . Consider the special case b0 = 0: this case is trivial, but the model does have an important implication. Because this restricted portfolio holds only the spot position, this test is one of the unhedged portfolio against a hedged portfolio with futures. The following theorem shows that an F (OLS)-test for testing the hedging effectiveness is constructed. Theorem 4 (Sample Errors in Testing The Hedging Effectiveness). Under multivariate normality, the test statistic of the hedging effectiveness with the optimal hedge ratio bˆ against the unhedged portfolio regression of H0 : b0 = 0 is distributed as 2
−1 HE (s2s + r¯s2 )/(n − 2)¯rs2
∼ F (1, n − 2),
(21)
= θˆ /θˆs . The sample mean and the sample variance of spot return are r¯s and s2s , respectively. where the hedging effectiveness is HE Proof. This is a direct result from Theorem 3 because
(SSRr − SSRu ) θˆ 2 − θˆs2 = SSRu /(n − 2) (1 + θˆs )/(n − 2) 2 − 1 HE = 2 2 ∼ F (1, n − 2) (ss +¯rs ) (n−2)¯rs2
where θˆs =
r¯s ss
is the sample Sharpe ratio of spot excess return.
≥ 1, a large enough F -value indicates a significant hedging Remark 6. There is an intuition about Eq. (21). Because HE effectiveness compared to the spot portfolio, where HE = 1.
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
1069
Table 1 Summary statistics and regression analysis. Mean return (in %)
Covariance matrix (in %)
r¯a −0.2124 r¯ = = r¯s −0.3321
2 ˆ = sa Σ
Unrestricted regression (9) 1 = w a µa + w s µs + u
Restricted regression (15) 1 = ws µs + u (under H0 : b = 0)
Model result: SSRu = 83.02 w ˆ a = 0.1734, w ˆ s = −0.2147 bˆ = −0.8075, (t = 1.3715)a 2 = 2.72, (F = 1.8809)b HE
Model result: SSRr = 84.88
a b
ssa
ssa s2s
=
9.422 8.561
8.561 8.410
Significantly different 0 against alternative hypothesis H1 : b < 0 at the 10% level. Significantly different 1 at the 20% level.
4. Illustration In 2011, the annual trading volume on the Taiwan Futures Exchange (TAIFEX) totaled 182 million contracts at an average daily volume of 740,871 contracts. These two figures represented increases of 30.9% and 33.02%, respectively, over 2010. In particular, a remarkable increase resulted in a transaction on single stock futures reaching 2,471,605 contracts for an increase of 241.21% over 2010. The growth in the popularity of stock index futures may be attributed to the use as stock portfolio management. Currently, the futures contract ‘‘TAIEX Futures’’ is actively traded, and the underlying investment asset is the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) issued by the Taiwan Stock Exchange Corporation (TWSE). Information source: The 2011 annual report of the Taiwan Futures Exchange (TAIFEX); refer to http://www.taifex.com.tw/eng/eng10/AnnualRep.asp. Both the optimal hedge ratio and hedging effectiveness are tested using weekly closing prices of TAIEX Futures and TAIEX. Data sources are the weekly data from the TEJ database maintained by the Taiwan Economic Journal. These weekly percentage returns are used to estimate the expected return and the corresponding variances as well as covariance. Summary statistics and regression results for the data (January 2011–October 2012, with 86 weekly observations) are reported in Table 1. Because both the futures return and the spot return are negative, the investment strategy is implemented by shorting one spot and longing 0.8075 unit futures contract. Regression analysis is based on comparing Eq. (9) against Eq. (15): F =
(SSRr − SSRu ) = 1.8809. SSRu /(n − 2)
The F -value indicates that the square of the hedging effectiveness at the optimal hedge ratio −0.8075 is significantly different from 1 at the 20% level. 5. Conclusion In this paper, an equivalence between the optimal hedge ratio derived by a revised risk-return approach and the optimal hedge ratio derived by mean–variance portfolio selection is constructed. The link between two approaches will provide researchers abundant mean–variance techniques for further study on the evaluation of hedging effectiveness. According to this finding a regression-based t-statistic to test a specific hedge ratio and an F (OLS)-statistic for evaluating hedging effectiveness are derived herein. Acknowledgment I am grateful to the editor and referee for helpful comments. The views in this paper are solely the responsibility of the author. References Black, F., Litterman, R., 1992. Global portfolio optimization. Financial Analysts Journal 48, 28–43. Britten-Jones, M., 1999. The sampling error in estimating of mean–variance efficient portfolio weights. Journal of Finance 54, 655–671. Brooks, C., Henry, Ó.T., Persand, G., 2002. The effect of asymmetries on optimal hedge ratios. Journal of Business 75, 333–352. Černý, A., 2009. Mathematical Techniques in Finance: Tools for Incomplete Markets. Princeton, New Jersey. Chang, J., Shanker, L., 1987. A risk-return measure of hedging effectiveness: a comment. Journal of Financial and Quantitative Analysis 22, 373–376. Ederington, L.H., 1979. The hedging performance of the new futures markets. Journal of Finance 34, 157–170. Gibbons, M.R., Ross, S.A., Shanken, J., 1989. A test of the efficiency of a given portfolio. Econometrica 57, 1121–1152. Harris, R.D.F., Shen, J., 2003. Robust estimation of the optimal hedge ratio. Journal of Futures Markets 23, 799–816. Heifner, , 1972. Optimal hedging levels and hedging effectiveness in cattle feeding. Agricultural Economics Research 24, 25–36. Howard, C., D’Antonio, L., 1984. A risk-return measure of hedging effectiveness. Journal of Financial and Quantitative Analysis 19, 101–112.
1070
W.-Y. Chiu / Statistics and Probability Letters 83 (2013) 1062–1070
Jong, A.D., Roon, F.D., Veld, C., 1997. Out-of-sample hedging effectiveness of currency futures for alternative models and hedging strategies. Journal of Futures Markets 17, 817–837. Kahl, K.H., 1983. Determination of the recommended hedging ratio. American Journal of Agricultural Economics 65, 603–605. Korkie, B., Jobson, J.D., 1982. Potential performance and tests of portfolio efficiency. Journal of Financial Economics 10, 433–466. Kuo, C.-K., Chen, K.-W., 1995. A risk-return measure of hedging effectiveness: a simplification. Journal of Futures Markets 15, 39–44. Mcnew, K.P., Fackler, P.L., 1994. Nonconstant optimal hedge ratio estimation and nested hypotheses tests. Journal of Futures Markets 14, 619–635. Roll, R., 1985. A note on Shanken’s CSR T 2 test for mean/variance efficiency. Journal of Financial Economics 14, 349–357. Satyanarayan, S., 1998. A note on a risk-return measure of hedging effectiveness. Journal of Futures Markets 18, 595–605. Shanken, J., 1987. A Bayesian approach to testing portfolio efficiency. Journal of Financial Economics 19, 195–215. Sharpe, W.F., 1994. The Sharpe ratio. Journal of Portfolio Management 21, 49–58. Stulz, R.M., 1984. Optimal hedging policies. Journal of Financial and Quantitative Analysis 19, 127–140. Terry, E., 2005. Minimum-variance futures hedging under alternative return specifications. Journal of Futures Markets 25, 537–552.