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### Subject 3. Pricing and valuation of equity forward contracts

Equity forward contracts are priced and valued much like the generic contract described in los a, with one important additional feature: many stocks pay dividends, and the effects of these dividends must be incorporated into the pricing and valuation process.

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:

PV(D, 0, T) = ∑[Di/(1 + r)ti] (i is from 1 to n).

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.

F(0, T) = [S0 - PV(D, 0, T)] (1 + r)T

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.

F(0, T) = S0 (1 + r)T - FV(D, 0, T)

Example 1

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

• F(0, T) = F(0, 0.25) = [50 - 4/(1.055)60/365] 1.0550.25 = \$46.66, or
• F(0, T) = 50 (1 + 0.055)0.25 - 4 x 1.05530/365 = \$46.66.

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:

F(0, T) = (S0 e-δT) erT

• The term in parentheses, the stock price discounted at the dividend yield rate, is equivalent to the stock price minus the present value of the dividends.
• The value is then compounded at the risk-free rate over the life of the contract.

Example 2

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.

Vt(0, T) = St - PV(D, t, T) - F(0, T)/(1 + r)(T-t)

or if we are using continuous compounding:

Vt(0, T) = St e-δ(T-t) - F(0, T) e-r(T - t)

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.

#### Practice Question 1

Consider a stock priced at \$65, which will pay a dividend of \$0.75 in 50 days and another \$0.75 in 100 days. The risk-free rate is 6.4%. If an investor decides to commit to a future purchase of the stock by going long a forward contract (which expires in 150 days) on the stock, at what price would the investor commit to purchase the stock in 150 days through a forward contract?

Correct Answer: First find the present value of the dividends: PV(D, 0, T) = PV(D, 0, 150/365) = 0.75/(1.064)50/365 + 0.75/(1.064)100/365 = \$1.48.

Then find the forward price: F(0, T) = F(0, 150/365) = (65 - 1.48) (1.064)150/365 = \$65.16.

#### Practice Question 2

Continue with question 1. Consider a stock priced at \$65, which will pay a dividend of \$0.75 in 50 days and another \$0.75 in 100 days. The risk-free rate is 6.4%. Suppose the investor enters into the contract which expires in 150 days at \$65.16. Now, 55 days later, the stock price is \$60. What's the value of the forward contract at this point?

Correct Answer: At this point the first dividend has been paid so it is not relevant. The second dividend will be paid in 45 days. The present value of this dividend is 0.75/(1.064)45/365 = 0.74. The value of the contract is Vt(0, T) = V95/365(0, 150/365) = (60 - 0.74) - 65.16/(1.064)95/365 = -\$4.86. The contract has a negative value.

#### Practice Question 3

Continue with question 1. Consider a stock priced at \$65, which will pay a dividend of \$0.75 in 50 days and another \$0.75 in 100 days. The risk-free rate is 4.6%. Suppose the investor enters into the contract which expires in 150 days at \$65.16. At expiration date the stock price is \$61. What's value of the forward contract at this time?

Correct Answer: The value of the contract is VT(0, T) = V150/365(0, 150/365) = 61 - 65.16 = -\$4.16. The negative value indicates a loss for the investor: he has to pay \$65.16 for a stock which is worth \$61 only.

#### Practice Question 4

A portfolio manager expects to purchase a portfolio of stocks in 180 days. To hedge against the market he decides to take a long position on a 180-day forward contract on the S&P 500 stock index. The index is currently at 1245. The continuously compounded dividend yield is 1.45%. The discrete risk-free rate is 4.6%. What is the no-arbitrage forward price on this contract?

Correct Answer: r = ln(1 + r) = ln(1 + 0.046) = 0.045.

F(0, T) = (S0 e-δT) erT = (1245 x e -0.0145 x (180/365)) (e 0.045 x (180/365)) = \$1263.87.

#### Practice Question 5

Consider a stock priced at \$150, which will pay a dividend of \$1.25 in 30 days, \$1.25 in 120 days, and another \$1.25 in 210 days. The risk-free rate is 5.25%. If you take a short position in a forward contract that expires in 250 days, what is the forward price of a contract established today and expiring in 250 days?

A. \$150.
B. \$149.74.
C. \$151.53.

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.

#### Practice Question 6

An investor took a long position in a forward contract on a stock 100 days ago. At the time of the contract initiation:
• The stock was selling at \$100.
• The stock would pay a dividend of \$1.5 in 60 days, \$1.5 in 120 days, and another \$1.5 in 180 days.
• The no-arbitrage forward price was \$98.98.
• The contract would expire in 250 days.
Suppose the risk-free rate is 5.25%. The stock price now is \$110. What's the value of the forward contract at this point?

A. \$0.
B. \$10.1.
C. \$11.02.

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.

#### Practice Question 7

An investor took a long position in a forward contract on a stock. At the time of the contract initiation:
• The stock was selling at \$100.
• The stock would pay a dividend of \$1.5 in 60 days, \$1.5 in 120 days, and another \$1.5 in 180 days.
• The no-arbitrage forward price was \$98.98.
• The contract would expire in 250 days.
Suppose the risk-free rate is 5.25%. At expiration the stock price is \$80. What's the value of the forward contract at expiration?

A. \$18.98.
B. -\$20.
C. -\$18.98.

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.

#### Practice Question 8

A portfolio manager took a long position on a 180-day forward contract on the S&P 500 stock index 30 days ago. The no-arbitrage forward price was 1263.87. The index was at 1245. Now (30 days later) the index is still at 1245. Suppose that the continuously compounded dividend yield is 1.45%, and the discrete risk-free rate is 4.6%. Both of these two rates have not been changed since the contract initiation date. What is the current value of the forward contract?

A. \$0, since the index price and two rates are still the same.
B. -\$3.11.
C. \$3.72.

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.

#### Practice Question 9

A portfolio manager took a long position on a 180-day forward contract on the S&P 500 stock index 30 days ago. At the time of the contract initiation:
• The no-arbitrage forward price was 1263.87.
• The index was at 1245.
• The constant continuously compounded dividend yield is 1.45%.
• The constant discrete risk-free rate is 4.6%.
At expiration the index value is still 1245. What is the value of the forward contract at expiration?

A. \$0.
B. -\$18.87.
C. -\$12.34.