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A note on minimum riskiness hedge ratio Sina Ehsani a, Donald Lien b,⇑ a b

Graham School of Management, Saint Xavier University, United States The University of Texas at San Antonio, United States

a r t i c l e

i n f o

Article history: Received 7 February 2015 Accepted 2 May 2015 Available online xxxx JEL classiﬁcation: G10 G11 G13 G32

a b s t r a c t This note incorporates the riskiness indexes of Aumann and Serrano (2008) and Foster and Hart (2009) into the futures hedging framework. It is shown that the minimum FH riskiness hedge strategy does not exist whereas the minimum AS riskiness hedge ratio tends to be smaller than the conventional minimum variance hedge ratio. Empirical results using daily spot and futures prices of S&P500 and FTSE100 indices over the 2009 to 2013 period support the theoretical prediction. Ó 2015 Elsevier Inc. All rights reserved.

Keywords: Economic index of riskiness Operational measure of riskiness Hedge ratio

1. Introduction While it is agreed that the futures market provides an important function for risk management, there is no universally agreed upon measure of risk. Possible choices include variance, lower partial moment, mean-Gini coefﬁcient, Value at Risk (VaR), and Conditional Value at Risk (CVaR). Aumann and Serrano (2008, AS hereafter) propose a new risk index, namely economic index of riskiness, which is strictly monotone with respect to the ﬁrst and second order stochastic dominance and is objective, in the sense that it allows a risk-averse expected utility decision maker to compare alternative prospects only based on the gambles own distributional characteristics. Later, Foster and Hart (2009, FH hereafter) suggest an alternative riskiness index. Both indexes have been applied to ﬁnancial research. ⇑ Corresponding author. Tel.: +1 210 458 8070. E-mail addresses: [email protected] (S. Ehsani), [email protected] (D. Lien). http://dx.doi.org/10.1016/j.frl.2015.05.002 1544-6123/Ó 2015 Elsevier Inc. All rights reserved.

Please cite this article in press as: Ehsani, S., Lien, D. A note on minimum riskiness hedge ratio. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.05.002

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This note ﬁrst adopts the AS riskiness index to examine the optimal futures hedge ratio. It shows that the minimum AS riskiness hedge ratio is most likely smaller than the conventional minimum variance hedge ratio. When the AS riskiness is replaced by the FH measure, the optimal strategy does not exist.

2. Riskiness measures The AS riskiness of a gamble is deﬁned as the reciprocal of the absolute risk aversion for an individual with constant absolute risk aversion (CARA) who is indifferent toward the gamble. AS show that for each gamble with some negative values and a positive expected value, there exists a unique real positive number RðgÞ which is the riskiness of gamble g:

E½eg=RðgÞ ¼ 1:

ð1Þ

To be speciﬁc, RðgÞ is characterized by the duality axiom which asserts that if the decision maker accepts a gamble at some wealth, then a less risk-averse agent will accept any gamble with smaller riskiness at that wealth. Bali et al. (2011), Bakshi et al. (2011), Homm and Pigorsch (2012a,b), Schreiber (2013, 2014), and Kadan and Liu (2014) extend and apply the riskiness index to a number of topics in ﬁnance. Foster and Hart (2009, FH hereafter) develop an alternative risk measure (the operational measure of riskiness) that ranks gambles regardless of decision makers preference and respects stochastic dominance. The FH riskiness of a gamble is deﬁned as the critical wealth below which it becomes risky to accept the gamble. FH show that for every gamble with possible losses and positive expectation, there exists a unique positive number WðgÞ, determined by:

E½logð1 þ g=WðgÞÞ ¼ 0:

ð2Þ

WðgÞ (the operational measure of riskiness) is the minimal wealth level at which gamble g can be accepted; g is deﬁnitely rejected for all wealth levels below WðgÞ. AS and FH measures share some common properties; for example, both are measured in wealth units (dollars), homogeneous of degree one and sub-additive. Nonetheless, they are different in other aspects. The AS index is based on comparing the gambles riskiness and the FH is a measure of riskiness. In other words, AS index is based on the level of risk aversion regardless of wealth whereas the FH measure searches for the critical level of wealth irrespective of preference. In contrast to conventional risk measures, both AS and FH are speciﬁed by incorporating the information of all higher moments of the gamble distribution. For a normally distributed gamble g Nðl; r2 Þ, the AS riskiness index has an analytic solution, RðgÞ ¼ r2 =2l. As expected, a higher variance increases riskiness and a higher mean reduces the index. Homm and Pigorsch (2010) derive the analytic form for AS index for the class of normal inverse Gaussian distributions. They show that, in line with investor aversion to heavy tails and tendency towards skewed payoffs (see Brunnermeier et al., 2007), higher kurtosis or smaller positive skewness always increase the AS index. However, a reduction in the skewness of a negatively skewed gamble (i.e., becoming more negatively skewed) does not necessarily lead to greater riskiness. To illustrate that the riskiness index ranks risky prospects better than other conventional criteria, consider the following two gambles. Gamble 1 (g 1 ) yields $1 with probability 0.1 and $1 with probability 0.9. Denote the mean and standard deviation with l and r, respectively. Then r1 ¼ 0:6 and l1 =r1 ¼ 1:33. Gamble 2 yields $1 with probability 0.1, $1 with probability 0.45, and $10 with probability 0.45. The corresponding mean and standard deviation are r2 ¼ 4:69 and l2 =r2 ¼ 1:03. Thus, according to the Sharpe ratio or standard deviation, g 1 is prefered to g 2 , although g 2 stochastically dominates g 1 . Furthermore, the %5 or %10 Value at Risk (or Conditional Value at Risk) for the two gambles are equal since the loss side of densities are identical. However, solving for AS riskiness index in (1) gives Rg1 ¼ 0:455 and Rg2 ¼ 0:443; thus g 2 is preferred, consistent with stochastic dominance criteria. Please cite this article in press as: Ehsani, S., Lien, D. A note on minimum riskiness hedge ratio. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.05.002

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2.1. Minimum AS riskiness hedge ratio Let x ¼ s hf denote the portfolio return where s and f are, respectively, spot and futures returns, and h is the hedge ratio. The AS index of riskiness for x is denoted by R such that:

E½ex=R ¼ 1:

ð1t Þ

It is assumed that s; f , and x are random variables with some negative values but positive expectations, thus, EðsÞ; Eðf Þ and EðxÞ are strictly positive to ensure that AS riskiness for the portfolio is well deﬁned. Since x is a function of h; R is also a function of h : R ¼ RðhÞ. Minimum riskiness hedge ratio is the hedge ratio, h , which attains the minimum of R. Note that Eq. (1t) is deﬁned for all h, upon taking derivative with respect to h,

ðshf Þ f ðs hf Þ @R @h ¼ 0; E e R þ 2 R R for every h. Suppose that the minimum riskiness hedge ratio exists.1 Then,

ð3Þ

@R @h h¼h

¼ 0. As a result,

ðsh f Þ f ¼ 0; E e R R

ð4Þ

where R ¼ Rðh Þ. Use the property EðyzÞ ¼ EðyÞEðzÞ þ Cov ðy; zÞ and note that R is non-random,

sh f sh f E e R Eðf Þ þ Cov e R ; f ¼ 0:

ð5Þ

h sh f i From Eq. (1t), E e R ¼ 1 and from the requirement of riskiness deﬁnition, Eðf Þ > 0. Hence, Eq. (5) sh f implies Cov e R ; f < 0. By projection, s can be written as follows:

s ¼ a þ bf þ u;

ð6Þ

where a and b are constants and u is a random variable such that Cov ðf ; uÞ ¼ 0. It is well known that b ¼ hv , the minimum variance hedge ratio. Consequently, sh f

a

ðbh Þf R

e R ¼ eR e

u

eR :

ð7Þ

Using the formula from Bohrnstedt and Goldberger (1969),2

sh f Cov e R ; f ¼ A1 þ A2 þ A3 ;

ð8Þ

bh f a u A1 ¼ eR E½eR Cov e R ; f ;

ð9Þ

h ðbh Þf i u

a A2 ¼ eR E e R Cov eR ; f ;

ð10Þ

h i ðbh Þf a u A3 ¼ eR E DeR De R Df :

ð11Þ

where

For a given random variable z; Dz ¼ z EðzÞ. 1 Chen et al. (2014) show that, under the assumptions that ensure the existence of the AS riskiness index, the solution to the ﬁrst-order condition in (3) is the minimum AS hedge ratio if the distribution of the gamble is not exceedingly concentrated on the gain side. 2 Let x; y, and v be jointly distributed random variables, and Dx ¼ x E½x; Dy ¼ y E½y and Dz ¼ z E½z. Bohrnstedt and Goldberger (1969) show that the covariance of products follows the following formula: Cov ðxy; v Þ ¼ EðxÞCov ðy; v Þ þ EðyÞCov ðx; v Þ þ E½ðDxÞðDyÞðDv Þ.

Please cite this article in press as: Ehsani, S., Lien, D. A note on minimum riskiness hedge ratio. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.05.002

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Table 1 ^ hedged ^ 2 hedged portfolio (r ðShv FÞ ), and minimum R Summary statistics of daily returns for spot (r S ), futures (r F ), minimum r portfolio (r ðShR FÞ ). In panels (a) and (b), we use the data from 2009 to 2013 to estimate ex-post hedge ratios. Hedge ratios are then used to form minimum variance and minimum riskiness portfolios over the same period. The ﬁrst two columns report summary statistics for returns (in percentage) on the spot and futures prices. The third and fourth columns report the statistics for the resulting returns on hedged portfolios. In panels (c) and (d), every day, we use daily returns between t ¼ 755 and t ¼ 0 to estimate hedge ratios. We use hedge ratios to form portfolios at time t ¼ 0, and track the portfolio return for the following day (t ¼ 1). The procedure is repeated every day to generate a time-series of ex-ante hedge ratios and subsequent portfolio returns. ^ R ^ represent mean, variance, skewness, kurtosis, and riskiness, respectively. hv and hR represent minimum variance and l^ ; r^ 2 ; ^S; K; minimum riskiness hedge ratios, respectively. Spot

Futures

Panel a. In-sample S&P500 (2009–2013) r Spot r Futures 0.06446 l^ l^ 1.50372 r^ 2 r^ 2 ^ ^ 0.15597 S S ^ ^ 7.10239 K K 23.00108

^ R

^ R

Panel b. In-sample FTSE100 (2009–2013) rSpot r Futures 0.039851 l^ l^ 1.306873 r^ 2 r^ 2 ^ ^ 0.09971 S S ^ ^ 5.244899 K K 32.62758

^ R

^ R

Panel c. Predictive S&P500 (2012–2013) r Futures rSpot 0.07663 l^ l^ 0.56252 r^ 2 r^ 2 ^ ^ 0.10734 S S ^ ^ 4.09347 k k 7.3754

^ R

^ R

Panel d. Predictive FTSE100 (2012–2013) r Futures rSpot 0.041426 l^ l^ 0.681677 r^ 2 r^ 2 ^ ^ 0.08659 S S ^ ^ 3.933025 k k 16.4615

^ R

^ R

0.06434 1.48708 0.15687 6.96938

Minimum variance

Minimum riskiness

r ðShv FÞ ; hv ¼ 0:979 0.00144 l^ 0.04779 r^ 2 ^ 0.23906 S

r ðShR FÞ ; hR ¼ 0:733 0.01730 l^ 0.14554 r^ 2 ^ 0.18285 S

7.55224

22.80786

^ K ^ R

0.04002 1.28598 0.10748

r ðShv FÞ ; hv ¼ 0:982 0.00055 l^ 0.04250 r^ 2 ^ 0.92697 S

5.61982 31.99506

^ K ^ R

0.07686 0.60882 0.20076

r ðShv FÞ l^ r^ 2 ^ S

4.56325 7.98607

0.04146 0.68883 0.07967 3.83214 16.6169

^ k ^ R r ðShv FÞ l^ r^ 2 ^ S ^ k ^ R

29.91495

6.89897 51.14350

0.00190 0.02496 0.16027 5.97583 13.08577

0.00001 0.03361 1.18221 9.39782 49.31772

^ K ^ R

7.16888 8.418339

r ðShR FÞ ; hR ¼ 0:792 0.00817 l^ 0.09382 r^ 2 ^ 0.07744 S 4.49389

^ K ^ R

11.45772

r ðShR FÞ l^ r^ 2 ^ S

0.02656 0.09725 0.08900 3.66233

^ k ^ R

3.66352

r ðShR FÞ l^ r^ 2 ^ S

0.00328 0.04936 0.28798

^ k ^ R

5.14202 14.7634

h i u

ðbh Þf u Since u and f are uncorrelated, we expect both Cov eR ; f ¼ 0 and E DeR De R Df to be close to sh f bh f ðbh Þf zero. The sign of Cov e R ; f is, therefore, the same as that of Cov e R ; f . Note that e R is strictly sh f increasing (decreasing) in f if b h < 0ð> 0Þ. Thus, for Cov e R ; f to be negative, it is required that

b h > 0. We have the following proposition. Proposition 1. The minimum variance hedge ratio is expected to be greater than the minimum Aumann– Serrano riskiness hedge ratio. Chen et al. (2014) analytically prove that the minimum AS riskiness hedge ratio is smaller than the conventional minimum variance hedge ratio when spot and future returns follow a bivariate normal distribution. Interestingly, despite the lack of normality in their data, their empirical estimates for hedge ratios minimizing the AS riskiness are always smaller than those minimizing variance. Proposition 1 provides a theoretical support for their empirical ﬁndings given that it holds regardless of the underlying price process. Please cite this article in press as: Ehsani, S., Lien, D. A note on minimum riskiness hedge ratio. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.05.002

S. Ehsani, D. Lien / Finance Research Letters xxx (2015) xxx–xxx

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Fig. 1. Time-series of minimum variance and minimum riskiness hedge ratios for S&P500 and FTSE100. Graphs display the daily minimum variance and minimum riskiness hedge ratios between 2012 and 2013. Starting from 01/04/2012, we use three-years of prior data (T ¼ 756) to estimate hedge ratios. We repeat the procedure everyday by rolling the estimation window by one day.

2.2. Minimum FH riskiness hedge ratio The FH riskiness (W) satisﬁes the following equation:

E½logð1 þ x=WÞ ¼ 0:

ð20 Þ

Once again, W is a function of h. Following a similar approach to the AS case, we take the derivative with respect to h to obtain

f þ ðs hf ÞðW1 @W Þ @h ¼ 0; E W þ s hf

ð12Þ

^ Then for every h. Suppose that the minimum FH riskiness hedge ratio exists and denote it by h. ¼ 0 and Eq. (12) is reduced to

@W ^ @h h¼h

Please cite this article in press as: Ehsani, S., Lien, D. A note on minimum riskiness hedge ratio. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.05.002

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S. Ehsani, D. Lien / Finance Research Letters xxx (2015) xxx–xxx

" E

#

f ¼ 0; ^ ^ þ s hf W

ð13Þ

^ This condition is reduced to Eðf Þ ¼ 0. ^ ¼ WðhÞ. where W Proposition 2. The minimum Foster–Hart riskiness hedge ratio does not exist. This ﬁnding is consistent with Riedel and Hellmann (2013) who demonstrate that FH riskiness index does not exist for many common distributions. 3. Empirical application In this section, we test our main ﬁnding by utilizing real data on daily spot and futures prices for Standard & Poors 500 and FTSE 100. Data is drawn from Datastream and covers the period 2009 to 2013. Denote the sample variance of futures returns (measured by the daily change in the natural log^ 2f , and the sample covariance between the futures and spot returns arithms of futures prices) by r ^ s;f . The minimum variance (measured by the daily change in the natural logarithms of spot prices) by r r^ s;f ^ hedge ratio is given by hv ¼ . On the other hand, generally, neither the riskiness index, nor the minr^ 2f

^R ) have closed-form analytical solutions. Therefore, we estimate the riskimum riskiness hedge ratio (h P 2 r t iness of a series by minimizing the objective function 1T Tt¼1 e R 1 , where rt is the return on the 2 ^ R , we minimize the objective function 1 PT eðst hR R f t Þ 1 . asset or portfolio on date t. To estimate h t¼1 T ^ R ), we need to compute hedge ratio and riskiness Obviously, to ﬁnd minimum riskiness hedge ratio (h index simultaneously. The optimization is implemented using R 3.1.2 with the quasi-Newton methodology of Broyden–Fletcher–Goldfarb–Shanno (BFGS). Panels (a) and (b) of Table 1 show that returns on spot and futures of both indices imply negative skewness and excess kurtosis. Therefore, using the riskiness index in hedging framework is relevant given that it incorporates all higher moments. S&P500 (FTSE) spot and futures returns have riskiness indices of 23.00 (32.63) and 22.81 (31.99), respectively. Consistent with prior evidence, the minimum ^ v ) is very close to one. However, among all portfolios, the minimum variance variance hedge ratio (h portfolio has the highest riskiness, 29.91 for S&P500 and 51.14 for FTSE100, suggesting that minimizing variance alone leads to higher riskiness as it worsens return characteristics such as mean and kurtosis. ^ R for S&P500 and FTSE100 are 0.733 and 0.792, supporting our theoretical prediction that AS The h ^ v . Further, using h ^ R is signiﬁcantly effective in reducing riskiness riskiness hedge ratio is smaller than h of spot returns from 23.00 (32.63) to 8.41 (11.46) for S&P500 (FTSE100). The results in panels (a) and (b) are based on static hedge ratios estimated using the whole sample. Next set of results utilize a rolling procedure. Speciﬁcally, on day t, we use data from t ¼ 755 to t to estimate hedge ratios. Then, we form a portfolio of spot and future prices, and compute the return on this portfolio for day t þ 1 after which the estimation and formation procedure are repeated everyday until the sample is exhausted. The procedure provides a time-series of ex-ante hedge ratios and subsequent portfolio returns. Since the estimation procedure requires three years or 755 days of data, the ^R is results presented in this section span the period 2012 to 2013. Fig. 1 shows that for both indices, h ^v . For S&P500 (FTSE100), h ^ R varies between 0.40 to 0.80 (0.65 to 0.95) although always smaller than h ^ ^ v and h ^ R is 0.55 (0.20) for hv is always close to 1. Nevertheless, the correlation coefﬁcient between h S&P500 (FTSE100), which is driven by the fact that variance is positively related to riskiness. Panels (c) and (d) of Table 1 present summary statistics for returns on spot, futures, and hedged portfolios over the period 2012 to 2013. Out-of-sample estimations conﬁrm our previous ﬁnding that minimum variance portfolios have higher riskiness, while minimum riskiness portfolios provide a more desirable risk-return proﬁle. Please cite this article in press as: Ehsani, S., Lien, D. A note on minimum riskiness hedge ratio. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.05.002

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4. Conclusion This note considers a one-period futures hedging framework in which the hedger attempts to choose the optimal strategy to minimize the Aumann–Serrano or Foster–Hart riskiness measures. It is shown that such a strategy does not exist for the FH riskiness. In addition, the minimum AS riskiness hedge ratio is generally smaller than the conventional minimum variance hedge ratio. References Aumann, R.J., Serrano, R., 2008. An economic index of riskiness. J. Pol. Econ. 116, 810–836. Bakshi, G., Chabi-Yob, F., Gaoc, X., 2011. Riskier times and asset returns. Working paper, University of Maryland. Bali, T.G., Cakici, N., Chabi-Yo, F., 2011. A generalized measure of riskiness. Manage. Sci. 57, 1406–1423. Bohrnstedt, G., Goldberger, W., 1969. On the exact covariance of products of random variables. J. Am. Stat. Assoc. 64, 1439– 1442. Brunnermeier, M.K., Gollier, C., Parker, A.J., 2007. Optimal beliefs, asset prices, and the preference for skewed returns. Am. Econ. Rev. 97, 159–165. Chen, Y.T., Ko, K.Y., Tzeng, L.Y., 2014. Riskiness-minimizing spot-futures hedge ratio. J. Bank. Finance 40, 154–164. Foster, D.P., Hart, S., 2009. An operational measure of riskiness. J. Pol. Econ. 117, 785–814. Homm, U., Pigorsch, C., 2010. The Aumann–Serrano index of riskiness for normal inverse Gaussian distributed gambles. Working paper, University of Bonn. Homm, U., Pigorsch, C., 2012a. An operational interpretation and existence of the Aumann–Serrano index of riskiness. Econ. Lett. 114, 265–267. Homm, U., Pigorsch, C., 2012b. Beyond the Sharpe ratio: an application of the Aumann–Serrano index to performance measurement. J. Bank. Finance 36, 2274–2284. Kadan, O., Liu, F., 2014. Performance evaluation with high moments and disaster risk. J. Financ. Econ. 113, 131–155. Riedel, F., Hellmann, T., 2013. The foster-hart measure of riskiness for general gambles. Working paper, Institute of Mathematical Economics (IMW), Bielefeld University. Schreiber, A., 2013. Comparing local risks by acceptance and rejection. SSRN Working paper. Schreiber, A., 2014. Economic indices of absolute and relative riskiness. Econ. Theor. 256, 309–331.

Please cite this article in press as: Ehsani, S., Lien, D. A note on minimum riskiness hedge ratio. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.05.002

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