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Sour«: Bodnar, Hayt, Marston, and Smithson, "Wharton Survey ol Derivatives Usage by U.S. Nonfinanctal Firms'; Fmaniiot Management, Sumniw, 1995, pp. 104 114.

derivatives for hedging. Thirty-five percent of the responding companies used derivatives for hedging, and given that they hedged, 47 percent used over-the-counter forward contracts. The prevalence of transaction hedging is shown in Exhibit 17.11. The common practice is to identify a specific source of risk in a foreign currency, then take an offsetting position in forward contracts or options. The problem with this approach is that it does not take all cash flows into account (and may actually unhedge the company). Even if it does consider all cash flows, these cash flows, usually payables and receivables, cannot be forecasted with a high degree of accuracy. Therefore, the hedge may be ineffective.

If we look at pure risk (or variance) reduction, the efficiency of the hedge is the percentage reduction in the variance of the hedged variable (usually the operating cash flows of a company). The math for this type of problem is important. Suppose that we define the hedged position of a company as the change in operating cash flows minus a hedge ratio, h, times the change in the value of a forward position:

The expected hedged operating cash flows and their variance can therefore be written as: VAK(Qperiling profit) = VAR(F) - 2rbop<sf + JfI\&R(F)

Where Px - P0 = The change in operating profit.

F - F0 = The change in the price of a forward or futures contract.

h = The variance minimizing hedge ratio.

VAR(P) = The variance of operating cash flows.

VAR(F = The variance of dollar returns of a forward or futures contract.

r = The correlation between operating cash flows and dollar returns on a forward or futures contact.

Taking the derivative of the variance of operating profit with respect to the hedging ratio h, and setting the result equal to zero gives the optimal hedge ratio for the minimum variance portfolio:

Note that this is the slope of a linear regression of the operating profits against the gains (or losses) on the forward contracts. By substituting the optimal hedge ratio into the variance equation, we see that the standard deviation of the minimum variance hedge portfolio depends on the correlation between the operating profit and the gains or losses on the forward contracts:

For example, the r-squared must be 60 percent in order to reduce the variance of the unhedged operating profits by 37 percent. An r-squared of 90 percent will reduce the variance roughly 68 percent. This demonstrates that regardless of the optimal hedge ratio, one must have a very high correlation between the operating profits and the forward contracts in order to have an efficient hedge. In practice, the inability to accurately forecast receivables and payables makes it difficult to have highly efficient hedges.

Even though your company hedges at the optimal hedge ratio, the cost of hedging may exceed the benefits (Exhibit 17.12). In the exhibit, the firm's operating cash flows are characterized as a random walk (a Gauss Wiener process) through time, starting at level P0, with a positive drift to reflect expected growth. If this line should happen to touch a lower boundary, illustrated by the upward sloping straight line starting at h0, the firm experiences business disruption costs such as the loss of customers and talented management, missed investment opportunities, or even a trip to the bankruptcy courts.

Hedging has two effects on operating cash flows. The desirable effect is to reduce the volatility of cash flows. The line representing hedged cash flows has lower volatility. The undesirable effect is that the cost of hedging reduces the slope of the line. Thus, a value-maximizing approach to hedging examines the expected benefit versus the expected cost. An article by

Exhibit 17.12 Cash Flows and a Boundary Condition Over Time

Exhibit 17.12 Cash Flows and a Boundary Condition Over Time

Copeland and Copeland discusses this approach in greater depth.5 The implication for a value-maximizing firm is that the expected benefit is the present value of the change in the probability of crossing the lower boundary multiplied by the cost of touching it—the expected business disruption costs (e.g., 15 percent of the value of the firm). The cost is the present value of the expected total cost of the hedge (e.g., 50 basis points per year) from now until the expected time to touching the boundary condition, given the hedge. A possible, and even common, outcome is that the expected time before the unhedged cash flows will touch the boundary is so long that even though the cost of hedging is low (50 basis points per year), the expected cost of hedging exceeds the expected benefit. The decision to hedge depends on the volatility and rate of growth in unhedged cash flows; the initial coverage ratio measured as the distance between the starting cash flows and the starting boundary condition; the efficiency of the hedge (how much it reduces volatility); the cost of business disruption, and the cost of hedging.

Once it has been determined that the firm should hedge, Copeland and Copeland provide the optimal hedge ratio. This is the original hedge ratio, h*, based on the slope of a linear regression between the unhedged operating profit and the gains or losses on the forward position, minus an adjustment equal to the cost of the hedge per year, |x, divided by the variance of the forward contract value, Gx.

5 T. Copeland and M. Copeland, ''Managing Corporate FX Risk: A Value-Maximizing Approach," Financial Management, vol. 28, no. 3 (autumn 1999), pp. 68-75.

This value-maximizing approach predicts that very profitable firms, with adequate liquidity, are less likely to hedge than other firms in the same industry that have lower coverage ratios, lower growth in operating profits, or higher business disruption costs. It reaches the conclusion that hedging is not necessarily the best thing to do.

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