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FACULTY WORKING PAPER NO. 1265

-w

A Note On the Arch Effects in Hedge Ratio Estimation: Stock Index Futures

Anil Bera Hun Park Edward Bubnys

College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 1265 College of Commerce and Business Administration University of Illinois at Urbana-Champaign June, 1986

A Note on the Arch Effects in Hedge Ratio Estimation: Stock Index Futures

Anil Bera, Assistant Professor Department of Economics

Hun Park, Assistant Professor Department of Finance

Edward Bubnys Memphis State University

This research is supported in part by the Investors in Business Education at the University of Illinois. The computational and programming assistance of Angela Lee and George Relyea are gratefully acknowledged.

A NOTE ON THE ARCH EFFECTS IN HEDGE RATIO ESTIMATION: STOCK INDEX FUTURES

ABSTRACT

This paper investigates the validity of the simple OLS model developed by Johnson (1960) and Stein (1961) and used by numerous sub- sequent studies to estimate the optimal hedge ratio using futures contracts. Focusing on the variance structure of the model, this paper provides some theoretical reasons for possible existence of heteroscedasticity (conditional as well as unconditional). Using data on three index futures we find significant heteroscedasticity and non- normality in the conventional model. Alternative hedge ratios are obtained using an autoregressive conditional heteroscedastic (ARCH) model. Information provided by empirical results in this paper suggests the importance of taking account of the ARCH effects in esti- mating the optimal hedge ratio.

A NOTE ON THE ARCH EFFECTS IN HEDGE RATIO ESTIMATION: STOCK INDEX FUTURES

I. INTRODUCTION

One of the important functions of futures contracts is to facilitate hedging, i.e., transferring the risk inherent in spot positions to speculators in the futures market. In this regard, stock index futures contracts are of particular interest to investors since they can provide a means to hedge the market risk exposure.

The key to any hedging strategy using futures contracts is a knowl- edge of the hedge ratio, i.e., the number of futures contracts to sell short per a long position in the cash market. Following Johnson (1960) and Stein (1961), the predominant method used in previous studies to estimate the optimal hedge ratio is the regression approach (the ordi- nary least squares regression) relating changes in cash prices to changes in futures prices. Inherent in the regression is the assump- tion that the optimal combination of cash position with futures is the one whose variance is minimized (see Ederington 1979 for a literature review on the hedging theory).

This paper investigates the validity of the simple regression model to estimate the optimal hedge ratio in stock index futures, focusing on the variance structure of the model. We provide some theoretical reasons for possible existence of heteroscedasticity in the conven- tional hedge ratio estimation model, and attempt to reestimate the hedge ratio taking heteroscedasticity into account. The most common method to correct for heteroscedasticity is to introduce some exoge- neous variables which may predict the variance. However, as pointed

-2-

out by Engle (1982), this method requires a specification of the causes of the changing variance in an ad-hoc fashion rather than recognizing that both means and variances conditional on the past information available may jointly evolve over time. We obtain alternative hedge ratios based upon the autoregressive conditional heteroscedastic (ARCH) model, introduced by Engle (1982), which is characterized by mean zero, serially uncorrelated processes with non-constant variances conditional on the past but constant unconditional variances.

In Section II, we provide some reasons for possible existence of heteroscedasticity in the conventional hedge ratio model for index futures. In Section III, we discuss econometric methodology. Section IV describes the data and presents empirical results. A brief summay is contained in Section V.

II. HETEROSCEDASTICITY IN THE CONVENTIONAL MODEL Following Johnson (1960) and Stein (1961), the commonly used ordinary least squares (OLS) technique to estimate the optimal hedge ratio can be written as:

AS = a + 8AF + e (1)

where AS and AF are the random changes in spot and futures prices, respectively, in period t. The slope coefficient 8 measures the optimal hedge ratio. Apart from assuming that the functional form of model (1) is correct, the success of the OLS procedure heavily relies on at least three assumptions on the distribution of e ; (i) homoscedasticity (both conditional and unconditional), (ii) normality and (iii) serial independence. Although we discuss issues (ii) and (iii), this paper

-3-

mainly deals with the problem of heteroscedasticity , particularly con- ditional heteroscedasticity.

Following Engle (1982), suppose we could write:

et =Vt (Yo+Yl £t-l2+ •••• +%et-p2)1/2. (2)

where the v 's are independent N(0,1). Then the conditional first two moments are (conditional on e , the past values of e ) :

E(et|e_t) = 0

net|e_t) =YQ+Y1 z\_x + ... +Yp£t_p2 (3)

P whereas unconditionally E(e ) = 0 and V(e ) = y_/(1- E y.). There-

i=l fore, the disturbances are conditionally heteroscedastic but unconditionally

homoscedastic. We can have both conditional and unconditional

heteroscedasticities just by assuming y as a time varying parameter,

say Y , and there will be only unconditional heteroscedasticity if

Y, = y„ = .. = y =0. However, there are a number of reasons to 1 2 p

suspect possible existence of conditional heteroscedasticity.

First, omitted variables in the model may cause heteroscedasticity. As pointed out earlier, the regression model in (1) is based solely on risk-minimization alone. Recall that hedging in the modern portfolio theory should be viewed as an activity that reduces total expected return in exchange for a smaller variance (see Howard and D 'Antonio (1984)). Taking account of the risk-return tradeoff, the hedge ratio can be alternatively derived as (see the Appendix for proof):

-4-

Cov(AS ,AF ) E(F )-F

6* = — - — (it)

Var(AF ) 2* Var(AF ) ' *" '

t r t

where $ represents risk, aversion parameter and t > t.

For analytical convenience, suppose that an investor holds a futures position until its maturity date, i.e., E(F ) = E(S ), where T repre- sents the maturity date. Note that on the maturity date, the futures price should be equal to the spot price to rule out the costless arbitrage (see Cox, Ingersoll and Ross (1981), and Richard and Sundaresan (1981)). Then, it becomes clear that eq. (4) narrows down to the conventional hedge ratio only if the futures price is an unbiased estimate of the expected spot price and/or a hedger is ex- tremely risk averse (<}> ■+ °°).

Whether or not the futures price is a systematically biased estimate of the expected spot price has been a long-time controversial issue in financial literature, i.e., the issue of normal-backwardation or con- tango (e.g., Keynes (1930), Hicks (1939), Houthakker (1957), Cootner (1960), Carter, Rausser and Schmitz (1983), Rockwell (1967), Richard and Sundaresan (1981) and Telser (1958)). However, the existence of normal backwardation or contango, if any, and/or changing risk aver- sion over time will be reflected in the disturbance term, e , in the

' t '

simple regression model. It is more clear by rewriting (1) using (4)

as

E(AF ) ASt =a + [8* + 2*Var(AFt)1AFt +£t

E(AF )AF -« +6*AFt + [et + ^Var(AF }] (5)

-5-

= a + 6*AF + [et+rit],

where E(AF ) = E(Ft> - Ffc. We can easily see from (5) that as long as the futures price is a systematically biased estimate of the expected spot price over time and thus E(AF ) depends on t, the variance of (e +n ) may depend on t even if we assume that e and n are independent.

Second, conditional heteroscedasticity may exist because of the basis risk (the changes in the relationship between spot and futures prices). Figlewski (1984) provides evidence that the basis risk is positively related to the market trend and decreases as the time to maturity decreases. To the extent that the basis risk is significant and related to market trend (i.e., the underlying index itself) over time, the heteroscedasticity may exist and be reflected in y . .

Intuitively, the variance of e represents the uncertainty about the agents choice of the hedge ratio in each period. As the maturity of a futures contract approaches, investors will have more information about the spot price in the future, so that the heterogeneity of information among investors will be reduced and thus the variance of e will decrease over time. Therefore, the possibility exists that V(e ) depends in part on the past information, e_ . This error-learning hedging behavior, if any, will be reflected in y. and its impact will be smaller as the maturity approaches. The reasoning is quite conceivable in light of the speculative behavior of investors due to heterogeneous information, and the ample evidence of maturity effect (e.g., Figlewski (1984) and Ederington (1979)).

-6-

III. ECONOMETRIC METHODOLOGY

Effects of unconditional heteroscedasticity are well known and one of them is the inconsistency of the standard errors if we use the usual formula. However, asymptotically valid influences can be drawn by using the White (1980) heteroscedasticity consistent covariance matrix estimator. This consistent covariance matrix is easily obtained by running an instrumental variable (IV) regression (see Messer and White (1984)). The procedure can be summarized as follows:

Suppose we have the OLS estimates a and 6 and the residuals

e = AS - a - BAF . Then define the variables: t t t

* ASt * AFt s = ■* , f = ■* and z = AF • e ,

t £ t g t t t

£t t

and run an IV regression of s on f using z as an instrument. The

t t t

resulting covariance matrix from this regression will provide the correct standard errors. White (1980) also provides a test for the presence of unconditional heteroscedasticity which does not assume any particular form of heteroscedasticity. Assuming that the e 's are homokurtic, the White test statistics can be calculated as the product

of the sample size and the coefficient of determination of the regres-

-2 2

sion of e on AF and AF . Under homoscedas ticity , this test sta- t t t J

2 tistic asymptotically follows x distribution with 2 degrees of

freedom. We also test for the normality of the disturbances and use

the following test statistic:

(/b^2 (b2-3)2 6 + 24

-7-

where n is the sample size, and /b and b are the sample skewness and

kurtosis coefficients of the OLS residuals e , respectively. Under normality,

2 this statistic also asymptotically follows x distribution with two

degrees of freedom (see Bera and Jarque (1982) for details).

Concerning conditional heteroscedasticity , Engle (1982) has shown that ARCH disturbances are uncorrelated (but dependent) and have a fatter tail than the normal distribution. Therefore, the above nor- mality test may also reveal ARCH effects and vice-versa. For simplic- ity, we consider only a first order ARCH model, i.e., p = 1 in (3).

vogs.j.) =^0 + Yiet-i •

2 2 Note that Y-. is the simple correlation coefficient between e and e

Therefore, the first order ARCH model postulates a non-linear

relationship between e and £■__■.•

The log-likelihood function for the first-order ARCH model can be

written as

1 n 2 £ = constant - — Y. log [y + y, (AS , - a - 6AF •• , ) ]

2 o 1 t-1 t-1

. n (AS - a - 8AF )

-\ z E l- . <6>

t=2 y +Y,(AS^ - a - 6AF^ ) o 1 t-1 t-1

Maximum likelihood estimates (MLEs) are obtained by maximizing (6) with respect to (a , 8 , Y , Y,) using the GRADX Subroutine of the R.E. Quandt's program GQ0PT3. The program also gives the asymptotic stand- ard errors of MLEs. A significance test on y, provides evidence of the presence of ARCH effects.

-8-

This type of conditional heteroscedastic model has particular appeal for estimating hedge ratios. First, realistic measures of hedge ratios can be estimated when the underlying variance may change over time and is predicted by past forecasting errors rather than making conventional assumptions about the disturbance. Second, it provides more efficient estimators using the maximum likelihood method. Third, it does not employ an arbitrary exogeneous variable to explain heterosce- dasticity. Lastly, by the nature of the ARCH process, the effect of possibly omitted variables from the estimated model as discussed earlier might be picked up.

IV. DATA AND EMPIRICAL RESULTS In February 1982, the Commodity Futures Trading Commission approved the trading of futures contracts on the Value Line Index in the Kansas City'Board of Trade. This action was followed by the intro- duction of futures contracts on the S&P500 Index (Chicago Mercantile Exchange) and the NYSE Index (New York Futures Exchange) in April and May of 1982, respectively. This paper uses daily data on S&P500, NYSE and Kansas City Value Line futures (KCVAL) from the first trading dates

of each contract to June 1985. All of the data (spot and futures

3 closing prices) were secured from the MJK associate computer tapes.

The OLS and ARCH model results on three stock indexes: S&P500, NYSE

and KCVAL for 36 futures contracts are presented in Table 1. First,

for the OLS results, the White test indicates the presence of strong

unconditional heteroscedasticity in two thirds of the cases. The

normality test statistics are significant for almost half of the

contracts. The D.W. test statistics, which are not presented here to

-9-

save space, do not show any presence of serial correlation. In the presence of non-normality and heteroscedasticity, the estimates of the OLS hedge ratios are inefficient and the t-statistics are unreliable.

For each of the contracts, we calculate the White consistent esti- mate of the variance and report the resulting modified t-statistics in brackets in Table 1. It is interesting to note that these t-statistics are overall lower than those calculated using the standard formula which in most cases underestimates the variance. These alternative estimates of the variances of the hedge ratios have some practical importance. They take account of unconditional heteroscedasticity and provide reliable confidence intervals which are of interest to the investors.

It is difficult to formulate an explicit model for the non-normal

disturbances. Non-normality coupled with zero autocorrelation may

imply some form of dependency among the disturbances. As we mentioned

before, the ARCH model whose distrubances allow a simple form of

dependency is a very convenient tool to capture non-normality. The

results in the first-order ARCH model are given in the last columns of

Table 1. Significance of the test H : y. =0 indicates the presence

o I

of first-order ARCH effects, i.e., conditional heteroscedasticity. Note that the test statistics are significant in all of the contracts of S&P500 and NYSE, and half of the contracts of KCVAL.

Comparing the OLS with ARCH hedge ratios, we can see that the OLS regressions overestimate (underestimate) significantly the optimal hedge ratio for some cases and thus cause the investors to sell short too many (few) futures contracts. Of particular interest is the

-10-

improved efficiency of the hedge ratio through the ARCH model. Since the ARCH model takes account of conditional heteroscedasticity and in part of dependency and non-normality, they are more efficient than the OLS regression. The results on the tests for non-normality and homoscedasticity are similar to those for the OLS model. The ARCH model incorporates non-normality, but the presence of unconditional heteroscedasticity once again make the t-statistics given in paren- theses invalid. A somewhat better estimate of the variance can be obtained by computing the White variance estimate using the ARCH residuals. The resulting t-statistics are in brackets. As before, these t-statistics are much lower, and they are more reliable in the presence of conditional and unconditional heteroscedasticity.

We also broke down the data of each index futures into four cate- gories in terms of time-to-maturity of the futures to examine the impact of the time-to-maturity on the hedge ratio estimates. Four non-overlappig time-to-maturity contracts were chosen for each index futures: contracts maturing within three months, three to six months, six to nine months, and finally over nine months.

Table 2 shows the results of the OLS and ARCH models on each cate- gory of the three index futures. Presence of strong heteroscedastic- ity (conditional as well as unconditional) and non-normality is evi- dent in all the cases. We do not observe much evidence of the rela- tionship between the degree of heteroscedasticity (or non-normality) and the time-to-maturity in both the OLS and ARCH models. Only for KCVAL, the conditional and unconditional heteroscedas ticities become stronger with the length of the time horizon. An interesting result

-11-

to note in Table 2 is a general pattern of decreasing hedge ratios as the time-to-maturity horizon is lengthened. Theoretically, this is what we should expect. For shorter horizon, in equation (4), the numerator of the first part increases because of close correspondence between spot and futures prices and the numerator of the second part becomes smaller.

V. SUMMARY

We have investigated the validity of the conventional OLS model developed by Johnson (1960) and Stein (1961) and used by numerous sub- sequent studies to estimate the optimal hedge ratio using futures contracts. Focusing on the variance structure of the model, we pro- vide some theoretical reasons for possible existence of heterosce- dasticity (conditional as well as unconditional).

Using data on spot and futures prices of three indexes: S&P 500, NYSE and K. C. Value Line indexes from the first trading date of each contract to June 1985, we find significant heteroscedasticity (conditional and unconditional) and non-normality of the disturbance term of the OLS regression. This paper provides alternative hedge ratios based on an ARCH model, introduced by Engle. Comparing the OLS and ARCH hedge ratios, we find that the conventional hedge ratio esti- mating model causes investors to sell short too many or few futures contracts. Of particular interest is the markedly improved efficiency of the hedge ratio estimates. Information provided by the empirical results in this paper suggests the importance of taking account of the ARCH effects in estimating the optimal hedge ratios for futures con- tracts.

-12-

Footnotes

See Ederington (1979) for a brief derivation of the hedge ratio.

2 Normal backwardation/contango refers to the process in which

futures prices are systematically downward/upward biased estimates of

expected spot prices over time.

3 The MJK Associate is a computer service in California specializing

in futures markets. All prices are quoted in their normal trading units

as determined by the various exchanges.

-13-

REFERENCES

Bera, A. and Jarque, C. , "Model Specification Tests: A Simultaneous Approach," Journal of Econometrics 20, 1982.

Carter, C. , Rausser, G. and Schmitz, A., "Efficient Asset Portfolios and the Theory of Normal Backwardation," Journal of Political Economy 91, April 1983.

Cootner, P., "Returns to Speculators: Telser vs. Keynes," Journal of Political Economy 68, August 1960.

Cox, J., Ingersoll, J. and Ross, R. , "The Relation Between Forward Prices and Futures Prices," Journal of Financial Economics 9, December 1981.

Ederington, L. , "The Hedging Performance of the New Futures Markets," The Journal of Finance, March 1979.

Engle, R. , "Autoregressive Conditional Heteroscedasticity with Esti- mates of the Variance of United Kingdom Inflation," Econometrica, July 1982.

Figlewski, S., "Hedging Performance and Basis Risk in Stock Index Futures,' Journal of Finance, July 1984.

Hicks, J., "Value and Capital: An Inquiry Into Some Fundamental

Principles of Economic Theory," 2nd ed. , Oxford University Press, Oxford, 1939.

Houthakker, H. , "Can Speculators Forecast Prices," Review of Economics and Statistics, May 1957.

Howard, C. and D'Antonio, L. , "A Risk-Return Measure of Hedging

Effectiveness." Journal of Financial and Quantitative Analysis, March 1984.

Johnson, L. , "The Theory of Hedging and Speculation in Commodity Futures," Review of Economic Studies 3, 1960.

Keynes, J., "A Treatise on Money," Harcourt Brace and Company, New York, 1930.

Messer, C. and White H. , "A Note on Computing Heteroskedasticity Consistent Covariance Matrix Using Instrumental Variable Technique," Oxford Bulletin of Economics and Statistics, 46, 1984.

Richard, S. and Sundaresan, M. , "A Continuous Time Equilibrium Model of Forward Prices and Futures Prices in a Multigood Economy," Journal of Financial Economics, December 1981.

-14-

Rockwell, C. , "Normal Backwardation, Forecasting and the Returns to

Commodity Futures Traders," Food Research Institute Studies 8, 1967,

Stein, J., "The Simultaneous Determination of Spot and Futures Prices," American Economic Review 5, 1961.

Telser, L. , "Futures Trading and the Storage of Cotton and Wheat," Journal of Political Economy 66, June 1958.

White, H. , "A Heteroskedasticity — Consistent Covariance Matrix Estima- tor and a Direct Test for Heteroskedasticity," Econometrica, 1980.

D/388

Appendix

An alternative hedge ratio can be derived using the basic princi- ples of the modern portfolio theory. Let us use the following notations

S : Spot price at time t today

S : Spot price at time t , where t > t and x - t represents hedging period.

F : Futures price at time t maturing at sometime later than x

F : Futures price at time x maturing at sometime later than x

N: The proportion of the portfolio held in futures

contracts with N > 0 representing a long position and N < 0 representing a short position

W : End-of-period return distribution at x x

<J> : Hedger's risk aversion parameter

U: Hedger's utility which is a function of only expected return and its variance.

Holding N futures contracts short per spot position, the hedger's

return distribution at time x will be

W = S - S - N(F - F ) (Al)

X X t X t

Then, the hedger's expected utility can be written as

EU = E(W ) - <}> Var(W ) = E(S ) - S - N(E(F ) - F ) (A2)

t X X X t X t

- d> [Var (S -S ) + N2 Var(F -F ) T x t x t

- 2N Cov(ST-St, FT-Ft)]

The first order condition of (A2) yields the optimal hedge ratio as:

Cov(S -S , F -F ) E(F )-Ft_

N = t t x t x t

Var(F -F ) 2<j> Var(F -F ) '

x t x t

which is eq. (4),

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ARCH Results

Normality

Homosce-

dastlcity

Test

8 Y0

Yl

Statistics

Table 2

Results Based on the OLS and ARCH Models for Different Maturity Periods

( t-statis tics are in parentheses and White modified t-statistics are in brackets)

OLS Results

Normality Homosce- dastlcity

Index/Maturity Period _ _ Test

(t: month) a 8 Statistics o

S&P500/t <_ 3 .040 .784 77.65a .037 .752 .281 .126 94.95a

(n=748) (2.46)a (64.06)a 38.30a (2.64)a (81.97)a (22.53)a (4.48)a 55.35a

[40.74]a [41.83]a

S&P500/3 < t _< 6 .038 .768 60.60a .034 .736 .290 .126 67.93a

(n=748) (2.26)b (62.77)a 37.85a (2.42)a (80.51)a (22.35)a (4.40)a 38.90a

[42.12]a [43.31]a

S&P500/6 < t < 9 .040 .720 117. 64a .034 .724 .290 .141 134. 88a

(n=748) (1.88)b (57.00)a 39.87a (2.38)a (80.33)a (21.65)a (4.49)a 43.38a

[40.35]a [41.65]a

S&P500/t > 9 .037 .772 35.63a .035 .733 .296 .138 36.07a

(n=647) (2.04)b (58.16)a 31.19a (2.26)a (74.99)a (20.07)a (4.01)a 21.35a

[39.93]a [40.46]a

NYSE/t < 3 .027 .660 68.00a .023 .663 .090 .183 79.61a

(n=745) (2.15)b (53.00)a 29.43a (2.87)a (75.29)a (22.01)a (5.81)a 43.96a

[37.83]a [38.65]a

NYSE/3 < t _< 6 .026 .644 47.85a .023 .645 .092 .177 52.47a

(n=745) (2.07)b (52.72)a 29.13a (2.81)a (73.95)a (22.14)a (5.73)a 32.04a

[39.36]a [39.80]a

NYSE/6 < t <_ 9 .025 .635 41.91a .023 .635 .092 .177 45.28a

(n=745) (1.98)b (52.59)a 27.34a (2.80) (73.54) (22.19) (5.79) 30.55a

[39.66]a [40.04]a

NYSE/t > 9 .023 .632 26.42a .023 .626 .094 .160 26.64a

(n=707) (1.83)b (51.45)a 19.30a (2.72) (69.99) (21.66) (5.23) 16.97a

[40.52]a [40.64]a

KCVAL/t < 3 .055 .509 19.69a .051 .504 .471 .076 16.77a

(n=79Z") (2.19)b (38.14)a 18.52a (2.88)a (50.57)a (22.81)a (2.74)a 54.06a

[28.15)a [28.89]a

KCVAL/3 < t <_ 6 .063 .480 12.96a .052 .476 .498 .112 12.18a

(n=795) (2.36)a (35.26)a 23.61a (2.79)a (46.75)a (22.69)a (3.92)a 27.83a

[30.25]3 [29.68Ja

KCVAL/6 < t £ 9 .057 .483 12.46a .047 .480 .478 .135 12. 15a

(n=767) (2.13)b (35.12)a 23.85a (1.73)b (46.60) (21.78) (4.31) 30.68a

[29.64)a [29.18]a

KCVAL/t > 9 .066 .433 85.95a .047 .420 .624 .188 12.34a

(n-514) (1.71)b (24.26)a 44.79a (1.73)b (31.32)a (15.36)a (3.35)a 37.84a

[18.01]a [16.68J3

Significant at the 1 percent level. The critical values for t-statistics and x statistics are respectively 2.57 and 9.21.

Significant at the 5 percent level. The critical values for t-statistics and x-" statistics are respectively 1.66 and 5.99

*ECKMAN

WDERy tNCm

JUN95