Pricing and Valuation of Equity Forwards

Derivatives 3 min readModule 2 of 7Save

What you'll learn

LOS 28.a
Describe how equity forwards and futures are priced, and calculate and interpret their no arbitrage value.

Equity Forward Contracts With Discrete Dividends

Recall that the no arbitrage forward price in our earlier example was calculated for an asset with no periodic payments. A stock, a stock portfolio, or an equity index may have expected dividend payments over the life of the contract. In order to price such a contract, we must either adjust the spot price for the present value of the expected dividends (PVD) over the life of the contract, or adjust the forward price for the future value of the dividends (FVD) over the life of the contract.

The no arbitrage price of an equity forward contract in either case is:

F P_{equity} = (S_{0} - P V D) (1 + R_{f})^{T} = S_{0} (1 + R_{f})^{T} - F V D

For equity contracts, use a 365 day basis for calculating T. For example, if it is a 60 day contract, T = 60 / 365.

Worked example: Calculating the price of a forward contract on a stock

Calculate the no arbitrage forward price for a 100 day forward on a stock that is currently priced at $30.00 and is expected to pay a dividend of $0.40 in 15 days, $0.40 in 85 days, and $0.50 in 175 days. The annual risk free rate is 5%, and the yield curve is flat.

Solution

Ignore the dividend in 175 days because it occurs after the maturity of the forward contract.

P V D F P = \frac{$0.40}{1.0 5 ^{15/365}} + \frac{$0.40}{1.0 5 ^{85/365}} = $0.7946 = ($30.00 - $0.7946) 1.0 5^{100/365} = $29.60

The timeline of cash flows is shown in the following figure.

Pricing a 100 Day Forward Contract on Dividend Paying Stock

To calculate the value of the long position in a forward contract on a dividend paying stock, we make the adjustment for the present value of the remaining expected discrete dividends at time t ( $P V D_{t}$ ) to get:

V_{t} (long) = (S_{t} - P V D_{t}) - \frac{F P}{( 1 + R _{f} ) ^{T - t}}

If given the current forward price ( $F P_{t}$ ) on the same underlying and with the same maturity:

V_{t} (long) = \frac{F P _{t} - F P}{( 1 + R _{f} ) ^{T - t}}

Worked example: Calculating the value of an equity forward contract on a stock

After 60 days, the value of the stock in the previous example is $36.00. Calculate the value of the equity forward contract on the stock to the long position, assuming the risk free rate is still 5% and the yield curve is flat.

Solution

There’s only one dividend remaining (in 25 days) before the contract matures (in 40 days) as shown here, so:

P V D_{60} V_{60} (long) = \frac{$0.40}{1.0 5 ^{25/365}} = $0.3987 = $36.00 - $0.3987 - \frac{$29.60}{1.0 5 ^{40/365}} = $6.16

Valuing a 100 Day Forward Contract After 60 Days

Equity Forward Contracts With Continuous Dividends

To calculate the price of an equity index forward contract, rather than take the present value of each dividend on (possibly) hundreds of stocks, we can make the calculation as if the dividends are paid continuously (rather than at discrete times) at the dividend yield rate on the index. Using continuous time discounting, we can calculate the no arbitrage forward price as:

F P_{equity index} = S_{0} e^{(R_{f}^{c} - δ^{c}) T} = (S_{0} e^{- δ^{c} T}) e^{R_{f}^{c} T}

$R_{f}^{c}$ is the continuously compounded risk free rate. $δ^{c}$ is the continuously compounded dividend yield.

Worked example: Calculating the price of a forward contract on an equity index

The value of the S&P 500 Index is 1,140. The continuously compounded risk free rate is 4.6% and the continuous dividend yield is 2.1%. Calculate the no arbitrage price of a 140 day forward contract on the index.

Solution

F P = 1, 140 e^{(0.046 - 0.021) (140/365)} = 1, 151

On a TI BA II PLUS calculator, use the following keystrokes:

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