Vol. 17 No. 2 Journal of Systems Science and Complexity Apr., 2004 THE OPTIMAL HEDGE RATIO FOR UNCERTAIN MULTI-FOREIGN CURRENCY CASH FLOW YANG Ming LI Chulin (Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China) Abstract. The paper extends the Adle and Dumas s simple regression approach of foreign currency hedging to the case of exposure to multiple foreign currencies and provides extension methodology. Key words. Hedge, exchange rate risk. 1 Introduction Consider the case where a parent corporation will receive foreign currency cash flow at the end of the year from its overseas subsidiary. The parent corporation faces the foreign exchange rate risk in its foreign currency translation. If the corporation knows the amount of the cash flow at the beginning of the year, they can avoid the risk of foreign exchange rate by signing a forward contract that matures at the end of the year. This is the financial instrument used for hedging in practice. But the parent corporation is not sure how many profits they will receive from its overseas subsidiary at the end of the year due to the uncertainty of the markets and the competition. So the scale of the foreign currency cash flows would be random amounts and cannot be determined in advance in a forward contract. In this case, determining a hedge ratio to minimize the risk of exchange rate is what the corporation is concerned about. Adler and Dumas [1] gave a definition and measurement of the exposure to currency risk in the research of the problem. They also developed a regression approach to determining the optimal hedge ratio (short for AD procedure). By taking only one foreign currency as a factor of the AD procedure showed that the slop coefficient of the regression model is the measure of exposure to foreign currency risk. This slop coefficient is also the optimal hedge ratio where optimal means the hedge ratio minimizes the variance of translating foreign currency to domestic currency. AD procedure also shows that the currency flow isn t sensitive to exchange rate variance after hedging. The coefficient of AD procedure solves the problem of determining optimal hedge ratio. It is a method of being extensively accepted in recent years. In the trend of globalization, an international parent corporation would set up more than one subsidiary in foreign countries. Thus when it transforms its revenue to the domestic currency, its revenue will be affected by more than one exchange rate. It is limited to use AD procedure that deals with one exchange rate to such a case. Schnabel [2,3] extended the AD procedure to the case of two exchange rates. But the Schnabel s methodology may not be suitable to the cases involving more than three foreign currencies. The work extending the AD procedure to such cases and proving the properties analogous to AD procedure would be useful. The object of this paper is to extend the AD procedure to the case of the multiple foreign revenue cash flows. We will see that the method we develop in this paper can be easily used for the multi-foreign currencies. Received July 10, 2001. Revised April 4, 2003.
No. 2 HEDGE RATIO FOR UNCERTAIN FOREIGN CURRENCY FLOW 165 2 Extension of the AD Procedure 2.1 The Extension model for three foreign currencies Without loss of generality, we consider a Chinese parent corporation. In the beginning of the year, its subsidiary corporation at the American, Canada and England implements the projects separately; the parent corporation will receive cash flows in three foreign countries at the end of the year; but their figures cannot be determined in advance. Assume the exchange rates of those foreign currencies to the domestic currency are S 1, S 2 and S 3 respectively. Denote the total revenue in domestic currency by P. Applying the AD s concept, the risk of P to exchange rate S i is defined as the expectation of the partial sensitivities, with the effects of all other elements of the state variations held constant, i.e. Sensitivity of P [ P ] to S i = E, i = 1, 2, 3. S i For extension AD progress, we write the regression model for three foreign currencies as follows: p = α + β 1 S 1 + β 2 S 2 + β 3 S 3 + ε, (1) where ε is an error term. The following are the usual regression assumptions: From Eqn (??), we have so, cov(ε, S 1 ) = cov(ε, S 2 ) = cov(ε, S 3 ) = 0. (2) α + β 1 S 1 + ε = P β 2 S 2 β 3 S 3, cov(α + β 1 S 1 + ε, S 1 ) = cov(p β 2 S 2 β 3 S 3, S 1 ). (3) If we denote cov(s i, S j ) = σ ij, cov(s i, S i ) = σi 2, then σ ij is the covariance of exchange rate S i to S j and σi 2 is the variance of the exchange rate S i. Substituting the above items into Eqn (??) yields β 1 σ1 2 = cov(p, S 1) β 2 σ 21 β 3 σ 31 or Meanwhile, we can write Eqn (??) as β 1 σ 2 1 + β 2 σ 21 + β 3 σ 31 = cov(p, S 1 ). (4) α + β 2 S 2 + ε = P β 1 S 1 β 3 S 3. Using the same procedure, we get the following equation β 1 σ 12 + β 2 σ 2 2 + β 3 σ 32 = cov(p, S 2 ). (5) Similarly, we have β 1 σ 13 + β 2 σ 23 + β 3 σ 2 3 = cov(p, S 3). (6) Combining Eqn (??), (??), (??), we can find the regression coefficients of the model: β 1, β 2 and β 3. 2.2 The optimal hedging ratio In this section we consider the problem how the parent corporation determines the optimal hedging ratio for their cash flows in the three foreign currencies. Define h 1 as the dollar hedge ratio or the number of U.S. dollar to sell forward at the U.S. dollar forward rate of F 01. Similarly,
166 YANG MING LI CHULIN Vol. 17 define h 2 as number of pound to sell forward at the forward rate F 02 and h 3 as the number of Canada dollar to sell forward at the forward rate F 03. Thus the value of the Chinese Parent s hedged cash flow (in Chinese yuan) is given by HC = P + h 1 (F 01 S 1 ) + h 2 (F 02 S 2 ) + h 3 (F 03 S 3 ). (7) The optimal hedging ratios are h 1, h 2 and h 3, which minimize the variance of HC. The variance of HC can be counted as σ 2 (HC) = σ 2 (P ) + h 2 1σ 2 1 + h 2 2σ 2 2 + h 2 3σ 2 3 2h 1 cov(p, S 1 ) 2h 2 cov(p, S 2 ) 2h 3 cov(p, S 3 ) +2h 1 h 2 σ 12 + 2h 1 h 3 σ 13 + 2h 2 h 3 σ 23. Differentiating σ 2 (HC) with respect to h 1 yield the result: h 1 = 2h 1 σ 2 1 2cov(P, S 1) + 2h 2 σ 12 + 2h 3 σ 13. Setting it equal to zero, we get the first-order condition as follows: h 1 σ 2 1 + h 2σ 12 + h 3 σ 13 = cov(p, S 1 ). By the same process, differentiating σ 2 (HC) with respect to h 2 and h 3 respectively yield the first-order conditions that the optimal hedging ratio is contented with h 1 σ1 2 + h 2σ 21 + h 3 σ 31 = cov(p, S 1 ), h 1 σ 12 + h 2 σ2 2 + h 3 σ 32 = cov(p, S 2 ), (8) h 1 σ 13 + h 2 σ 23 + h 3 σ3 2 = cov(p, S 3). We can get the optimal hedging ratios by solving the linear system of equations (??). Comparing the above linear system with the equation group (??), (??) and (??), we get the conclusion of h i = β i, i = 1, 2, 3. Thus we complete the proof that the optimal hedging ratios are the regression coefficients in three foreign currencies. 2.3 The extension model for n foreign currencies The model of three foreign currencies has been given in 2.1 and 2.2. The method can be extended easily to the cases that encompass more than three foreign currencies. Generally, if the parent corporation has n foreign currencies cash flow, we take the multi-regression model as follows: P = α + β 1 S 1 + β 2 S 2 + + β n S n + ε (9) where ε is an error term and the usual regression assumptions are cov(ε, S i ) = 0, i = 1, 2,, n. The exchange rate S i is independent exchange rate S j, i j. For equation (??) we can write α + β i S i + ε = P β j S j. i j We have the equation, ( cov(α + β i S i + ε, S i ) = cov P β j S j, S i ), i j
No. 2 HEDGE RATIO FOR UNCERTAIN FOREIGN CURRENCY FLOW 167 that is, β i σ 2 i = cov(p, S i) i j β i σ ij, i = 1, 2,, n. Writing the above equations as a linear system of n equations: β 1 σ1 2 + β 2σ 21 + + β n σ n1 = cov(p, S 1 ), β 1 σ 12 + β 2 σ2 2 + + β n σ n2 = cov(p, S 2 ), β 1 σ 1n + β 2 σ 2n + + β n σn 2 = cov(p, S n ); the coefficient matrix, A, of the linear system (??) is σ 2 1 σ 21 σ n1 σ 12 σ2 2 σ n2..... σ 1n σ 2n σn 2 Matrix A is the variance-covariance matrix of exchange rates S 1, S 2,, S n, and A is an n n nonsingular matrix. Using Cramer s rule, linear system (??) has a unique solution (β 1, β 2,, β n ). Now, take such β i as the foreign currency hedge ratio or the number of foreign currency i to sell forward at the foreign currency i forward rate of F 0i, i = 1, 2,, n. Thus the value of the parent s hedged cash flow(in domestic currency) is given as (10) HC = P + β 1 (F 01 S 1 ) + β 2 (F 02 S 2 ) + + β n (F 0n S n ). (11) From (??), the variance of HC can be written as n n σ 2 (HC) = σ 2 (P ) + β i β j σ ij 2 β i cov(p, S i ). i=1 β 2 i σ 2 i + 2 i j Differentiating σ 2 (HC) with respect to β i yields β i = β 1 σ 1i + β 2 σ 2i + + β i 1 σ i 1i i=1 +β i σ 2 i + β i+1σ i+1i + + β n σ ni cov(p, S i ). For (β 1, β 2,, β n ) is the solution of system (??), we get the results: β j = 0, i = 1, 2,, n. This means (β 1, β 2,, β n ) minimize the variance of HC. That is to say that (β 1, β 2,, β n ) is the optimal hedging ratios and the model can be used to determine the optimal hedging ratio of n foreign currencies. 3 The Optimal Hedge Ratio And the Risk of the Exchange Rate Here we consider the foreign exchange exposure of the cash flows hedged using the hedge ratios computed above. That is the sensitive cash flows HC to exchange rate S i. The hedged cash flow is given by HC = P + β 1 (F 01 S 1 ) + β 2 (F 02 S 2 ) + + β n (F 0n S n ). (12)
168 YANG MING LI CHULIN Vol. 17 Substituting Eqn (??) for P in Eqn (??) results in HC = α + β 1 S 1 + β 2 S 2 + + β n S n + ε +β 1 (F 01 S 1 ) + β 2 (F 02 S 2 ) + + β n (F 0n S n ) = α + β 1 F 01 + β 2 F 02 + + β n F 0n + ε. Combining conditions (??), we have cov(hc, S 1 ) = cov(ε, S 1 ) = 0, cov(hc, S 2 ) = cov(ε, S 2 ) = 0, cov(hc, S n ) = cov(ε, S n ) = 0. These results mean that the cash flow is independent of the exchange rate. The risk correlated with the exchange rate is removed by hedging, what remains after hedging whose variance is minimal and which is statistically unrelated to the exchange rate is demonstrated. So we can believe that the above hedge is effective to prevent the risk caused by exchange rate. 4 Conclusion This paper extends the AD procedure to the case of multi-foreign currencies. extended multiple-regression model: Use the P = α + β 1 S 1 + β 2 S 2 + + β n S n + ε. Taking three foreign currencies as an example, and applying a method different from Schnabel s, we have proved that regression coefficient β i is the optimal hedging ratios which minimize the variance of remains after hedging. We also have demonstrated that the cash flow is not sensitive to exchange rate after hedging using such ratio. The methodology we develop here is applicable for n foreign currencies, thus we complete the extension of AD procedure and solve the problem of how to determine the optimal ratio of preventing the risk of exchange rate. It is convenient to obtain the regression coefficient β i by multi-regression procedure, so we provide an effective method which can be used to determine the optimal hedge ratio for random multi-foreign currencies cash flows. References [1] M. Adler and B. Dumas, Exposure to currency risk: Definition and measurement, Finance Management, 1984, 13: 41 50. [2] J.A. Schnabel, Exposure to foreign exchange risk: A multi-currency extension, Management and Decision Economics, 1998, 10: 331 333. [3] J. A. Schnable, Real Exposure to foreign currency risk, Management Finance, 1994, 20(4): 69 77.