# Calculate theoretical forward price of a stock

The current price of a stock is USD400 per share and it pays no dividends. Assuming a constant interest rate of $8%$ compounded quarterly, what is the stock's theoretical forward price for delivery in $9$ months ?

I am taking the Financial Engineering and risk management course on Coursera. The above question was in the quiz and I got a wrong answer on it.

Shouldn't the answer be: $$400\times\left(1+\frac{0.08}{4}\right)^3 = 424.48\, ?$$

• What was the given answer? At a quick glance, your approach looks like. – AfterWorkGuinness Dec 8 '15 at 13:30
• The answer is 400*(1+0.08/4)^3 = 424.48. – Gordon Dec 8 '15 at 14:08

## 5 Answers

Let's use a no-arbitrage argument. Assume that the (continuously compounding) dividend yield is $q$ while the interest rate is $r$.

For portfolio 1, we go long 1 forward contract with maturity $T$ and delivery price $K$. The payoff at time $T$ is $S_T - K$.

For portfolio 2, we go long $e^{-qT}$ unit of a stock (while reinvest all dividends) and short $K e^{-rT}$ unit of a bond. The payoff at time $T$ is also $S_T - K$.

At time $t = 0$, the present value (PV) of portfolio 1 is 0, because we just entered the trade. The PV of portfolio 2 at time $t = 0$ is $S_0 e^{-qT} - K e^{-rT}$. Assuming that there is no arbitrage, we conclude that the PV at time $t = 0$ of portfolios 1 and 2 must be the same: $S_0 e^{-qT} - K e^{-rT} = 0$. Hence $\boxed{K = S_0 e^{(r-q)T}}$. Your answer of $400 (1+0.08/4)^3=424.48$ is correct.

• Is it possible to create a portfolio like portfolio 2 in reality? In other words, could one buy or sell a fraction of the stock or a fraction of a bond? – noob-mathematician May 21 at 15:41

It should be just $400*(1+0.02)^3$ Where $0.02$ is quarterly compounded rate for a quarter, and $n$ is 3 quarters, hence the exponent 3

It is correct. J.Hull's book explains it clearly in Chapter 3, paragraph 3.5 "Forward price for an investment asset". It is also shows the arbitrage strategy if the price does not match.

• Hi LorenzoQF, welcome to Quant.SE! Could you please give a more specific reference? All of Hull is quite a read. – Bob Jansen Feb 15 '16 at 16:22
• Chapter 3. Paragraph 3.5 "Forward price for an investment asset". It is also shown the arbitrage strategy if the price does not match. – LorenzQF Feb 16 '16 at 8:11

Based on No Arbitrage equation for Forward Asset combination: $$S_0 e^{−qT} − K e^{−rT} = 0$$

i.e. $$K = S_0 e^{(r-q)T} \\ = 400 * e^{(0.02 - 0) * 3} \\ = 400 * 1.061837 \\ = 424.7346$$

Assuming 8% is the quartely interest rate

I think it's: $400*(1+0.08)^3 = 503.8$

$0.08$ is the quarter interest rate and you compound for 3 quarters

Or

$r_{year} = (1+0.08)^4 - 1 = 0.36$

$400*(1+0.36)^(9/12) = 503.8$

Because there are 9 months in a 12-month year.

• At a quick glance, this doesn't look correct. The quarterly rate would be annualized when it's quoted so a quarterly rate of .08 is .08/4 – AfterWorkGuinness Dec 8 '15 at 13:31
• Yes, but how do you know if it's annualized or quarterly? It the question is not expressed that it's annualized – sparkle Dec 8 '15 at 14:18
• It is a standard convention to annualized rates, such that they would have to specify if the rate were not annualized. – AfterWorkGuinness Dec 8 '15 at 14:28