Question

Your friend David has just bought a futures contract on a stock index, and the contract specifies one year to expiration. The current share price is $80, and the annually compounded interest rate is 10%. The stock will pay quarterly dividends of $2 during the next year, with dividends payments on the following dates:

• January 25
• April 25
• July 25
• October 25

Assume that this is a non-leap year.

1. What is the futures price on this contract on January 3?

Answer 1

First we will calculate present value of dividends

Interest rate = r = 10%

No of days to dividend on Jan 25 = t1 = No of days from Jan 3 to Jan 25 = 22 days

No of days to dividend on April 25 = t2 = No of days from Jan 3 to Jan 31 + days in february + Days in march + no of days from 1st April to 25th April = 28 + 28 + 31 + 25 = 112 days

No of days to dividend on July 25 = t3 = No of days from Jan 3 to Jan 31 + days in february + Days in march + Days in April + Days in May + Days in June + no of days from 1st July to 25th July = 28 + 28 + 31 + 30 + 31 + 30 + 25 = 203

No of days to dividend on 25 Oct = t4 = No of days from Jan 3 to Jan 31 + days in february + Days in march + Days in April + Days in May + Days in June + Days in July + Days in Aug + Days in Sep + no of days from 1st Oct to 25th Oct = 28 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 25 = 295

Present value of dividends = PVD = Dividend on Jan 25 / (1+r)^t1/365 + Dividend on April 25 / (1+r)^t2/365 + Dividend on July / (1+r)^t3/365 + Dividend on Oct / (1+r)^t4/365 = 2 / (1+10%)^22/365 + 2 / (1+10%)^112/365 + 2 / (1+10%)^203/365 + 2 / (1+10%)^295/365 = 2/ 1.005761 + 2/1.029677 + 2 / 1.054438 + 2 / 1.080076 = 1.9885 + 1.9423 + 1.8967 + 1.8517 = 7.6792

Time to expiration of future contract = t = 1 year, Current share price = S₀ = $80

Future price of contract = (S₀ - PVD) (1+r)^t = (80 - 7.6792)(1+10%)¹ = 72.3208 x 1.10 = 79.5528 = 79.55 (rounded to two decimals)

Hence Future price of contract on Jan 3 = $79.55