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Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_h$ and $K_f$ are rates of return on home and foreign currency if it is invested over a period $T$.

(A) Assume that the forward rate of exchange $F$ satisfies $F > S(0) \frac{1+K_h}{1+K_f}$ Construct a portfolio that offers a risk-free profit.
(B) Assume that the forward rate of exchange $F$ satisfies $F < S(0) \frac{1+K_h}{1+K_f}$ Construct a portfolio that offers a risk-free profit.

The answer my textbook gives me is if $F > S(0) \frac{1+K_h}{1+K_f}$, then $(\frac{1}{1+K_f}, -\frac{1}{1+K_f}, -1)$, however I have no idea what this means?
Why does this answer give us an arbitrage (risk free profit) and how do you know this when we don't have any numers to work with? How do you derive this answer so I can try to get the answer for part B?

idknuttin
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I'll work out what the arbitrage free forward has to be (assuming a spot rate of $1$ as you didn't say which way $S(0)$ went).

There's only one type of trade that makes sense in this set up: borrow $1$ unit in currency $X$...you will then owe $1+K_X$ at the end of the period. Swap your unit into currency $Y$ at the spot rate. Let's say the spot rate is $1$...that's just a question of units. Invest it so that you'll have $1+K_Y$ at the end of the period. Then swap it back at the Forward rate $F_{Y\to X}$ and pay off your loan. This has to be a zero sum transaction since every step is risk free. The math is $$F_{Y\to X}\times (1+K_Y)=(1+K_X)\implies F_{Y\to X}=\frac {1+K_X}{1+K_Y}$$

If this equality does not hold then either this or the opposite of this transaction is guaranteed to make a riskless profit.

Is this enough for you to finish?

lulu
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  • thank you, but I am confused on the terminology of $K_f$ and $K_h$, is it possible if you can explain the problem in simple terms without using finance terms? – idknuttin Feb 15 '16 at 18:20
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    What do you mean? these are your terms, not mine. I assume that $K_h$ refers to the riskless rate at home, let's say $5%$ for illustration. thus in my trade, if you borrow $$100$ for a period then you'd owe $$105$ at the end of the period. Similarly, $K_f$ denotes the foreign riskless rate. – lulu Feb 15 '16 at 18:25