Question

Problem 2.5.6. (Piecewise constant dividend yield) It is now January 1, 3018. You are given: (i) The current price of the stock is 1,000. (ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield changes throughout the year. In March, June, September, and December, the dividend yield is 3%. In other months, the dividend yield is 2% (iii) The continuously compounded risk-free interest rate is 9%. Calculate the price of a 1-year forward contract. (Hint: How many shares should you buy at time 0 to end up with exactly one share in ono year?)

Answer 1

ANSWER:

Given,

Current Stock Price = $1000 (S)

Dividend yield for march , June, September, December = 3%

i.e.., Dividend for these moths (D) = 3% * $1000 = 30 at t = 4 months

Dividend yield for remaining months (D) = 2%

i.e.., Dividend for these moths (D) = 2% * $1000 = 20 at t = 8 months

Risk-free rate (r) = 9% = 0.09

Formula for forward Price is:

F = (S-D) * e^rt (where e= 2.7183 approx)

now,

Calculating Forward Price for 4 months:

F = (1000-30) * (2.7183)^(0.09*1/4)

=970 * (2.7183) ^ (0.0225)

= 992.07 ( round to 2 decimals)

now,

Calculating Forward Price for 8 months:

F = (1000-20) * (2.7183)*(0.09*1/8)

= 980* (2.7183)^(0.01125)

= 991.09 ( round to 2 decimals)

SO, Forward Price for year is by adding F for 4 months and F for 8 months

= 992.07 + 991.09

F = 1983.16

Therefore, Forward Price for 1 year is 1983.16