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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: $$400\times\left(1+\frac{0.08}{4}\right)^3 = 424.48\, ?$$

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    $\begingroup$ What was the given answer? At a quick glance, your approach looks like. $\endgroup$ Commented Dec 8, 2015 at 13:30
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    $\begingroup$ The answer is 400*(1+0.08/4)^3 = 424.48. $\endgroup$
    – Gordon
    Commented Dec 8, 2015 at 14:08

5 Answers 5

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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.

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  • $\begingroup$ 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? $\endgroup$ Commented May 21, 2020 at 15:41
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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

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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.

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    $\begingroup$ Hi LorenzoQF, welcome to Quant.SE! Could you please give a more specific reference? All of Hull is quite a read. $\endgroup$
    – Bob Jansen
    Commented Feb 15, 2016 at 16:22
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    $\begingroup$ Chapter 3. Paragraph 3.5 "Forward price for an investment asset". It is also shown the arbitrage strategy if the price does not match. $\endgroup$
    – LorenzQF
    Commented Feb 16, 2016 at 8:11
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Based on No Arbitrage equation for Forward Asset combination: $S_0 e^{−qT} − K e^{−rT} = 0$

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

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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.

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  • $\begingroup$ 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 $\endgroup$ Commented Dec 8, 2015 at 13:31
  • $\begingroup$ Yes, but how do you know if it's annualized or quarterly? It the question is not expressed that it's annualized $\endgroup$
    – sparkle
    Commented Dec 8, 2015 at 14:18
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    $\begingroup$ It is a standard convention to annualized rates, such that they would have to specify if the rate were not annualized. $\endgroup$ Commented Dec 8, 2015 at 14:28

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