## Forward Contracts On A Security That Provides A Known Dividend Yield

As will be explained in later sections, both currencies and stock indices can be regarded as securities that provide known dividend yields. In this section, we provide a general analysis of forward contracts on such securities.

A known dividend yield means that the income when expressed as a percentage of the security price is known. We will assume that the dividend yield is paid continuously at an annual rate q. To illustrate what this means, suppose that q = 0.05 so that the dividend yield is 5 percent per annum. When the security price is \$10, dividends in the next small interval of time are paid at the rate of 50 cents per annum; when the security price is \$100, dividends in the next small interval of time are paid at the rate of \$5 per annum; and so on.

To value the forward contract, portfolio B in Section 3.2 can be replaced by:

Portfolio B: e~q(T~!> of the security with all income being reinvested in the security

The security holding in portfolio B grows as a result of the dividends which are paid, so that at time T exactly one unit of the security is held. Portfolios A and B are therefore worth the same at time T. From equating their values at time, i, we obtain

and the forward price, F, is given by the value of K that makes / zero:

Note that if the dividend yield rate varies during the life of the forward contract, Equation (3.10) is still correct with q equal to the average dividend yield rate.

### Example 3.7

Consider a 6-month forward contract on a security that is expected to provide a continuous dividend yield of 4% per annum. The risk-free rate of interest (with continuous compounding) is 10% per annum. The stock price is \$25 and the delivery price is \$27. In this case S = 25, K = 27, r = 0.10, q = 0.04, and T - t = 0.5. From Equation (3.9) the value of a long position, /, is given by

From Equation (3.10) the forward price, F, is given by