Rolling the Hedge Forward: An ExtensionAuthor(s): Dwight GrantSource: Financial Management, Vol. 13, No. 4 (Winter, 1984), pp. 26-28Published by: Wiley on behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665298 .

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Rolling the Hedge Forward: An Extension

Dwight Grant

Dwight Grant is an Associate Professor of Finance in the Department of Economics and Business at North Carolina State Universit, in

Raleigh.

* In a recent article McCabe and Franckle [13] identify an interesting question: How can futures markets be used to hedge risky cash positions which mature at later dates than those covered by existing futures con- tracts? The technique they present -"rolling the

hedge forward" -is a valid and potentially useful one.

McCabe and Franckle motivate this multiperiod problem by assuming that the trader has a cash position that matures at a time later than the maturity of any currently available futures contract. This creates a

multiperiod problem because a hedger must trade in futures at more than one date. The purpose of this article is to extend McCabe and Franckle's analysis by incorporating basis risk and deriving a general multi-

period hedging rule which has not previously been

recognized in the literature.'

This paper has been improved by the suggestions of Mark Eaker and two reviewers.

'Baesel and Grant [1] examined a similar problem using a mean/vari- ance framework. They derived the same result but did not recognize that it was a general rule for multiperiod hedging.

The Problem The problem McCabe and Franckle pose arises

whenever an individual or firm wants to hedge a spot position which has a maturity date later than that of the latest maturing futures contract. Such a position can be

hedged for its entire life only by trading in an existing contract that matures before the spot position and sub-

sequently switching to a contract that matures at the same time or later than the spot position. This switch-

ing is referred to as "rolling the hedge forward." McCabe and Franckle consider an example in which the hedge is rolled forward just once, but there is no theoretical or practical limit to the process. Rolling the hedge forward creates a multiperiod decision problem which can be solved using the backward recursive technique of dynamic programming. (See Mossin [4].) This technique requires that the optimal last period decision be determined first. Then the optimal second- to-last period decision is determined, taking into con- sideration the optimal decision rule which will be im- plemented for the last period. This process is repeated until the optimal decision for each period has been determined.

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GRANT/ROLLING THE HEDGE FORWARD

The Solution McCabe and Franckle neither explicitly discuss the

dynamic aspect of this problem, nor explicitly employ the backward recursive solution method.2 Their ap- proach is feasible because they assume away basis risk and restrict themselves to rolling the hedge forward just once. A solution incorporating basis risk is devel- oped below. The hedge is rolled forward twice because this is sufficient to demonstrate the general rule.

Several simplifying assumptions McCabe and Fran- ckle make are retained in this note. They are as fol- lows: (1) There are no costs to hedging; (2) All hedging gains and losses occur at the time the cash position is closed; and (3) The objective is variance minimiza- tion.3 The following symbols are used:

W is the value of the hedged position. Ai is the value of accumulated gains and losses on

futures positions. R3 is the value of the cash position at time 3. fi is the price at which a futures position is opened. Fi is the price at which a futures position is closed. bi is the quantity of a futures contract bought (+)

or sold (-) at time i. Tildes - denote random variables. var and cov are the symbols for variance and covar-

iance. In the problem being considered an individual holds

a cash position at time 0 which matures at time 3. At time 0 there is one futures contract which matures at time 1.4 At time 1 another futures contract is opened which matures at time 2; and at time 2 another contract opens which matures at time 3 or later. At time 0 only the price of the current futures contract is known. The wealth position is

W = bo(F, - f0) + b,(F2 -fl) + b2(F3 -f2) + R,. (1)

The objective is to choose b0, b, and b2 to minimize var(W). Following the backward recursive technique

2In the absence of basis risk the optimal hedge at the final decision point is a perfect match of the futures position to the spot. Thus the price changes in the last period do not alter wealth and the goal becomes one of minimizing the variance caused by price changes up to the last period. McCabe and Franckle focus on this element of the problem and do not emphasize the recursive aspect of their analysis.

3This solution can be extended directly to the case of a mean/variance objective function. See Baesel and Grant [1].

4It is not necessary that only one futures contract trade at one time. The problem depends only on one or more contracts maturing before the cash position matures.

the first step is to choose the risk-minimizing b2 from the perspective of time 2, then the optimal b, from the perspective of time 1, and so on. At time 2 the out- comes on the first 2 hedges, A2, and the opening price of the third futures contract, f2, are known. Therefore, at time 2,

W = A2 + b2(F3 - f2) + R3, and var(W) = bj var(F3) + b2 cov(F3,R3)

+ var(R3).

(2)

(3)

Ederington [2] and others have derived the value of b2 which minimizes var(W). It is

b* = -cov(F3,R3)/var(F3) (4)

Given this decision rule, the next step is to determine the optimal hedge at time 1. At time 1 the returns from the initial futures contract and the opening price of the second are known:

W = Al + bl(F2 - f) + b2(F3 -f2) + R3. (5)

The risk-minimizing value of b, can be identified quickly if the expression [b*(F3 - f) + R3] is consid- ered one term and identified as X. Then (5) can be written as

W = A, + b,(F2 - f,) + X, (5.1)

which has the same structure as (2). Therefore, the risk-minimizing hedge at time 1 is identical in structure to that at time 2:

b* = - cov(F2,X)/var(F2). (6)

Thus the optimal hedge at time 1 is expressed in the usual form, as a ratio of the covariance between the price of the hedging instrument and a cash position, divided by the variance of the hedging instrument. The only difference is in the definition of the cash position. In a multiperiod problem the cash position at a decision point is defined as the original cash position plus the returns from optimal futures positions which will be taken at later decision points.

This multiperiod hedging rule applies regardless of how many times the hedge is rolled forward. Solving the current problem for the optimal hedge at time 0 illustrates this. At time 0:

W = bo(F1 - fo) + b*(F2 -f,) + b2*(3 -f2) + R3. (7)

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FINANCIAL MANAGEMENT/WINTER 1984

When the cash position is redefined as:

V = b0(F, - f,) + Y,

where

Y =b , -) b(F: -ff) + ( + ,.

Conclusion The risk-minimizing hedge in a multiperiod prob-

lem is analogous to that in a single-period problem. (7.1) They differ only with respect to the definition of the

cash position. In a multiperiod setting the cash position is defined as the original cash position plus the returns from futures positions which will be taken at subse- quent trading dates. This result provides a general rule for managing long-term risk exposure by rolling the

(8) hedge forward.

Y is the cash position at time 0 and includes the origi- nal cash position, R3, plus the returns from futures

positions which will be taken at times 1 and 2. The risk

minimizing value of b, assumes the usual form:

b* = -cov(F,,Y)/var(F,), (9)

differing only with respect to the definition of the cash

position.

References 1. J. Baesel and D. Grant, "Optimal Sequential Futures Trad-

ing," Journal of Financial and Quantitative Analysis (De- cember 1982), pp. 683-695.

2. L. Ederington, "The Hedging Performance of the New Fu- tures Markets," Journal of Finance (March 1979), pp. 157- 170.

3. G. McCabe and C. Franckle, "The Effectiveness of Rolling the Hedge Forward in the Treasury Bill Futures Market," Financial Management (Summer 1983), pp. 21-29.

4. J. Mossin, "Optimal Multiperiod Portfolio Decisions," Journal of Business (April 1968), pp. 215-229.

EUROPEAN FINANCE ASSOCIATION CALL FOR PARTICIPATION AT THE TWELFTH ANNUAL MEETING

UniversiOt of Bern, Switzerland, August 29-August 31, 1985

The European Finance Association is a professional society for academicians and practitioners with interest in financial management, financial theory and applications. It serves as a center of communication for its members residing in Europe or abroad. It also provides a framework for the better dissemination of information and exchange at the international level. Its Twelfth Annual Meeting is scheduled for August 29-

August 31, 1985 at the University of Bern, Bern, Switzerland. Those who wish to present a paper at the meeting should send a copy of the paper (or a detailed abstract) by

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