Question

Q4 a) On January 1st 2019 one share of BBK plc is priced at $35 and is expected to pay a dividend of $1.25 in 6 months and a further dividend of $1.00 in 1 year. The relevant risk-free rate of interest is 3% per annum with continuous compounding. What should be the price of a forward contract, written on a BBK share, which matures immediately after the second dividend is paid and what is the initial value of the forward contract?

b) Explain and discuss your answer to part a.

c) BBK shares are included in the RNB500 stock index which is trading at 4,000 index points with a contract multiplier of $10 per full index point. If the annual dividend yield is 2% and the risk-free rate of interest is 3%, what is the value of a futures contract written on the RNB500 that matures in 6 months?

Answer 1

a) Spot Price = $35
Risk-Free Rate (Continuous Compounding) - 3% p.a.
Dividend1 (D1) = $1.25 (time = 6 months)
  Dividend2 (D2) = $1.00 (time = 12 months)

Present Value of Dividend = Dividend * e-r*t
where r = risk free rate
t = time period when dividend is paid

Present Value of Dividend = D1 * e-r*t + D2 * e-r*t
= 1.25 * e-0.03*(6/12) + 1.00 * e-0.03*(12/12)
= 1.25 * 0.98512 + 1.00 * 0.97045
= 1.23139 + 0.97045
= 2.20184

Forward Contract = (Spot Price - Present Value of Div) * er*t
= (35 - 2.20184) * e0.03*(12/12)
= 32.79816 * 1.0305
Forward Contract = $33.7970

Initial Value of Contract is $0 as there was no exchange of money as they do not require any down or early payment.

b) Since the dividend is paid on a future date i.e. 6 months and 1 year from today. Therefore, we have to calculate the present value of the dividend and subtract it from the spot price as the buyer of the forward contract does not hold physical shares when the dividend was given out and also, the price of the stock after ex-date reduces causing a notional loss to the buyer of the forward contract (if the spot price was not adjusted for the dividends). Hence, the present value of the dividend is subtracted from the spot price.

c) Spot Price = 4000
Maturity = 6 months
Continuous Risk Free Rate = 3% p.a.
Continuous Dividend Yield = 2% p.a.

Futures Price = Spot Price * e[(Risk Free Rate - Dividend Yield) * maturity]
= 4000 * e[(0.03-0.02)*0.5]
= 4000 * 1.005
Futures Price = 4,020.05