Dealer $A$ in Chicago will buy pounds a year from now at a rate of $\$1.58$ to a pound. Dealer $B$ in London will sell pounds immediately at a rate of $\$1.60$ a pound. Furthermore, dollars can be borrowed at an annual rate of $4\%$ while pounds can be invested with an annual interest of $6\%$.
How do I find the arbitrage in this scenario?
Suppose you borrow $1.60 \,\rm{USD}$. You own $1.60 \,\rm{USD}$ in cash and owe $1.60 \,\rm{USD}$. The liability travels into the future and becomes $1.60 \exp(0.04) \,\rm{USD}$ one year from now. The asset can be immediately converted to $1 \,\rm{GBP}$, lent at a rate of $6 \%$ for one year and converted back to U.S. dollars a year from now. Hence, a year from now, the profit is given by
$$\left(1.58 \exp(0.06) - 1.60 \exp(0.04) \right) \,\rm{USD}$$
If this profit happens to be negative, try borrowing pounds and lending U.S. dollars instead.
You have two choices. The wording, which talks of investing pounds and borrowing dollars, suggests you should buy pounds now, hold them a year, and sell them to pay back the dollars you borrowed to buy the pounds. It doesn't matter how many pounds you buy, so assume one. How many dollars does that take? In one year, how many pounds do you have? When you sell them, how many dollars will you receive? How many dollars do you owe at the end of the year? If you have more dollars than needed to pay off the debt, that is your profit.
If you don't have enough dollars to pay off the debt, you just need to go the other way around. Sell pounds now, invest the dollars, sell the dollars in a year and pay off the pound loan. Since pounds are declining compared to dollars you don't want to own them
The arbitrage is the simultaneous buying and selling to take advantage of differences in prices for the same assets.
Arbitrage opportunities do not exist in the fictitious efficient market, however, somewhat ironically, we can use the notion of an efficient market to find an equivalent current rate for Sterling using the supplied numbers.
To evaluate this, we need to look at the prices for the same asset (£1) referred to the same time.
The current value from Dealer B's perspective is $1.60.
To compute the equivalent rate current $r$ (Dollars per Sterling) from Dealer A's perspective we borrow enough to buy £$x$, buy £$x$, invest the £$x$, 'uninvest' the investment, buy the \$ equivalent and repay the loan. This should result in a net gain of zero. In particular, we should have $$ {x \over r} (1.58)(1.06)-(1.04)x = 0$$ or $r= {1.58 (1.06) \over 1.04} \approx 1.6104$.
Hence Dealer A is buying at a higher current equivalent price than Dealer B is selling, so the borrow, buy, etc, strategy should be followed with as large an $x$ as possible.
