okay my main problem is I have to work out a $3$ month interest rate for the us dollar. The question I'm stuck on is At the end of trading on $1$ January $2012$ the dollar/pound spot exchange rate was $\$ 1.5700$ per pound and the three month forward rate was $\$1.5825$ per pound. answer the following:
estimate the us interest rate for the next quarter if the interest rate on risk free pound deposits in the uk for the next quarter is $0.125\%$ ($90$ day rate).
my solution given by the lecturer just says
$F = S \frac{1+r}{1+r£}$ where $F$ is the forward rate, $S$ is the spot exchange rate and $r$ stands for the interest rate $$1.5825 = 1.5700 (1+r)/1.00125$$ (re arranging from this line is where i become confused ) $r= 0.9222\%$
any help would be appreciated, cheers :)
You want to solve the equation $$1.5825=\frac{1.5700(1+r)}{1.00125}.$$ We want to "isolate" $r$. First multiply both sides by $1.00125$. We get $$(1.5825)(1.00125)=1.5700(1+r).$$ Better, the $r$ is less buried. Now divide both sides by $1.5700$. We get $$\frac{(1.5825)(1.00125)}{1.5700}=1+r.$$ Now it is time to use the calculator. Mine gives that the left side is equal to $1.0092217$. Of course this is not exact, but it is pretty close. Your instructor seems to have rounded this up very slightly.
Subtract $1$ to get $r$. We get $r=0.0092217$. Now as usual to express this in percent, multiply by $100$. We get $0.92217\%$.