Forward Contracts on Investment Assets with Known Yield

Example 2.7. (Assets with Known Yield):

The spot price of ABC stock is $100 today. It will pay 4% dividend in two months. What should be the equilibrium forward price, F₀, maturing in 8 months?

S₀= $100 [Stock price at t = 0]
T = 8 months: Time to maturity of the forward
Dividend Dates        Dividend Rates
t = 2 mo                     q = 4%
Risk-free rate = 6%

Keep in mind that when you close the forward position in 8 months, you will have one share of stock, but you will have missed the dividend to be paid in 2 months. Furthermore, the amount of dividend is not known at t = 0. All that is known is that it will be 4% of the market price in 2 months, S_2month, whatever it will be.

Analysis:

1. Create a synthetic forward:
   
   t = 0:       Borrow $100 and buy one share
   t = 8 mo: Pay off the loan, $100 · (1 + 6%)^8/12
                  Receive one share (S_8mo + FV_8mo(Div_2mo))
   
   Difficulty: This will not help because the amount of dividend is not known - it is 4% of the stock price at t = 2 mo, which is unknown at t = 0

2. Create an alternative synthetic forward:
   
   t = 0:       Borrow $100 · (1 – 4%) and buy 96% of one share
   t = 2 mo: Buy additional stock with the dividend:
                  Dividend received: 4% on 0.96 shares = 4% · (0.96 · S_2mo)
                  Ex div stock price = S_2mo – 4% div = 0.96 · S_2mo
                  Buy additional stock with the dividend:
                  => Additional share = [4% · (0.96*S_2mo) /[0.96 · S_2mo] = 4% of a share

   Now you have (96% + 4%) =100% of a share

   T = 8 mo: Pay off the loan, $100(1 - 4%) · (1 + 6%)^8/12
                   Have one share worth S_8mo
   → F(0,8mo) = $100(1 - 4%) · (1 + 6%)^8/12 = 99.80 (= FV{S₀(1 - q)})

Note: If you need further assistance in solving this example, view the Forward Contract on Assets With One-time Known Yield video by clicking on the Instructional Videos link in the left menu.