Question

Consider an equity index with a continuously compounded dividend yield that is known to vary
throughout the year: 4% per annum in February, May, August, and November, and 1% per annum
during other months. The continuously compounded risk-free rate is fixed at 5% per annum.
a) (5%) Suppose it is now 31 January and that the current index level is 1,000. Calculate the
current forward price on the index for delivery on 31 December.
b) (5%) Suppose that the index level is 900 on 30 June. Calculate the value of a long position in a
forward contract on the index entered on 31 January with delivery on 31 December, and with a
contract size of $1 times the index.
c) (5%) Suppose that, on 31 July, the forward price on the index for delivery on 31 December is
1,010. Furthermore, on 30 September, the forward price on the index for delivery on 31
December is 910. Calculate the return on the index between 31 July and 31 September

Answer 1

For all the parts r will be risk free rate - dividend yield

a.

The futures contract lasts for five months. The dividend yield is 4% for first 4 months and 1% for two of the months. Total number of months between31 Jan to 31 December are 11.

Weighted Dividend Yield for 11 months = (4*5 + 1*6)/11 = 2.26%

Current forward price of the contract = 1000*e^rt (Where r = dividend rate, t = time period i.e. 11 months)

= 1000*e^(0.05-0.026)*11/12 = 1000*e^0.022

= 1000*1.0222437844704 = $1022.24

b.

Value of contract on 31 january will be index value on 30th june discount back to January 31.

2 months i.e. February and May have dividend yield of 4%. Total months are 5.

Weighted Dividend Yield = (4*2 + 1*3)/5 = 11/5 = 2.2%

Hence, the value of index on 31st Jan is 889.50

Future contract value will be value at Dec 31. The dividend yield for this will be 2.26%

Futrue Contract value of the index = 889.50*e^(0.05-0.026)*11/12 = 1000*e^0.022

= 889.50*1.0222437844704 = $1022.24

= 909.286.

Hence the long positive would be valued at 909.286 on January 31st for the forward contract delievering on 31st December.

c.

For part c. please refer to the image.

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