Given a series of dividends of D1, D2, ...Dn, whose values are known, that occur at times t1, t2, ... tn, the present value is defined as PV(D, 0, T) and computed as:
An equity forward contract is priced by taking the stock price, subtracting the present value of the dividends over the life of the contract, and then compounding this amount at the risk-free rate to the expiration date of the contract.
Note that the dividends reduce the forward price, a reflection of the fact that holders of long positions in forward contracts do not benefit from dividends in comparison to holders of long positions in the underlying stock.
Another approach to incorporating the dividends is to use the future value of the dividends. An equity forward can be priced by compounding the stock price to the expiration date and then subtracting the future value of the dividends at the expiration date.
Let's consider a stock priced at $50, which pays a dividend of $4 in 60 days. The risk-free rate is 5.5%. A forward contract expiring in three months (T = 0.25) would have a price of
Not surprisingly, the two formulas give the same answer.
If the stock has more than one dividend, we would simply subtract the present (or future) value of all dividends over the life of the contract from the stock price (or future value of the stock price).
We can alternatively express dividends as a fixed percentage of the stock price. If we assume that the stock, portfolio, or index pays dividends continuously at a rate of δ, we then allow the dividends to be uncertain and completely determined by the stock price at the time the dividends are being paid.
Because we pay dividends continuously, for consistency we must also compound the interest continuously. The continuously compounded equivalent of the discrete risk-free rate r is denoted as r , and is found as r = ln(1 + r). The future value of $1 at time T is exp(rT). The forward price is then:
The France's CAC 40 index is at 5475. The continuously compounded dividend yield is 1.5%. The continuously compounded risk-free interest rate is 4.625%. The contract life is two years. With T = 2, the contract price is: F(0, T) = F(0, 2) = (5475 x e-0.015(2)) e0.04625(2) = 5828.11.
Regardless of how the dividend is specified or even whether the underlying stock, portfolio, or index pays dividends, the value of an equity forward contract is the stock price minus the present value of the dividends minus the present value of the forward price that will be paid at expiration.
or if we are using continuous compounding:
Note that the forward price should not be interpreted as a forecast of the future price of the underlying. If the forward price is higher than the spot price, it merely indicates that the effect of the risk-free rate is greater than the effect of the dividends. In fact, such is usually the case with equity forwards: interest rates are usually greater than dividend yields.
Then find the forward price: F(0, T) = F(0, 150/365) = (65 - 1.48) (1.064)150/365 = $65.16.
F(0, T) = (S0 e-δT) erT = (1245 x e -0.0145 x (180/365)) (e 0.045 x (180/365)) = $1263.87.
First find the present value of the dividends: PV(D, 0, T) = PV(D, 0, 250/365) = 1.25/(1.0525)30/365 + 1.25/(1.0525)120/365 + 1.25/(1.0525)210/365 = $3.69.
Then find the forward price: F(0, T) = F(0, 250/365) = (150 - 3.69) (1.0525) 250/365 = $151.53.
At this time point two dividends remain: the first one in 20 days, and the second one in 80 days.
PV(D, t, T) = 1.5/1.052520/365 + 1.5/1.052580/365 = $2.98.
Vt(0, T) = 110 - 2.98 - 98.98/1.0525150/365 = $10.1.
A positive value is the gain to the long.
ST = $80.
F(0, T) = $98.98.
VT(0, T) = 80 - 98.98 = -$18.98.
The contract expires with a value of negative $18.98, a gain to the short.
A. $0, since the index price and two rates are still the same.
r = ln(1 + r) = ln(1 + 0.046) = 0.045.
Vt(0, T) = Vt(0, T) = St e-δ(T-t) - F(0, T) e-r(T - t) = 1245 x e -0.0145 x (150/365) - 1263.87 e -0.045 x (150/365) ) = -$3.11.
This is a loss to the long position.
ST = 1245.
F(0, T) = 1263.87.
VT(0, T) = 1245 - 1263.87 = -$18.87.
This is a loss to the long position.