Question

Andi is considering investing $1000 in Canada, where she expects an interest rate of 3 percent, or in the U.K., where the expected interest rate would be 5 percent. The current exchange rate is 0.6 £/$, which could take by the end of the year any value between 0.4 and 0.7£/$ with equal probability. Daisy Cortez

a) What should Andi do?

b) How does your answer change if there is a currency conversion fee of 3 percent?

Answer 1

I was not sure how mathematically accurate this question requires to be and you might know better,so I have included both approaches that I thought could be expected,one like a typical question in economics using rougha and sensible estimates and one more mathematically rigorous. You can completely ignore the notes if you think they weren't needed at the level.

a) If she invests in Canada, her $1000 will become $1030 in a year at 3% interest rate

However if she invests in UK, she will first get 0.6*1000=600 pounds which will become 600*1.05= 630 pounds at the end of the year with 5 percent interest rate

Let the exchange rate at the end be X pounds/$

So in the end she will have 630/X $

As a rough assumption Since X can have an equal probability of anything between 0.4 and 0.7, we can calculate expected value by finding the expected value of the exchange rate=(0.4+0.7)/2=0.55

630/0.55=1145.4545

Since this expected value is higher than 1030, she should invest in UK

Note: (I say an assumption because I have assumed for simplicity that Expected value of a variable (1/X) = 1/(Expected value of of variable X) where I directly divided 630 by 0.55 which is the expected variable of X, however it is mathematically incorrect. I am solving it correctly here but it is a bit tedious, however it doesnt affect the decisions of where she should invest so you can skip,but it will affect the values

We have to find the Expected value of (630/X) given X has an equal probability of being between anywhere 0.4 to 0.7

Expected value of (630/X)= 630*Expected value(1/X)

Expected value of (1/X) over equal probability between 0.4 and 0.7will be =1.86538

70.7 1/0.3 (1/0)dr

so 630 pounds will become 630*Expected value(1/X) i.e. 630*Expected value of (1/x)= 630*1.86538=1175.193

As opposed to the 630*(1/Expected value of X)=630*(1/0.55) we did earlier)

b) If there is a currency conversion rate of 3%, all conversions will be multiplied by 1-0.03=0.97

If she invests 1000 in UK, she will first have 1000*0.6*0.97=582 pounds

which will become 611.1 after 5% interest rate after a year.

After converting it back with the exchange rate X or 0.55 as shown above, we will have

611.1/0.55*0.97=1077.758 which is the expected value of the return

(Or more accurately 611.1*1.865386*0.97=1105.739 as we did in the Note above)

which is still higher than the previous 1030$ if she invested in canada, so she should invest in UK in both cases with or without currency conversion fee.

Hope it helps. Do ask for any clarifications required.

Note: To understand better why I added the note I can say that the question says the probability of the exchange rate in pounds/$ is equal between 0.4 to 0.7,so the expected value of the exchange rate can be written as 0.55 if you were converting it by multiplying the dollar amount to this exchange rate, however when you convert pounds to dollar, you're dividing this exchange rate X, at X=0.4, the exchange rate in dollars/pounds= 1/0.4=2.5 and at X=0.7 it is 1/0.7=1.428,

If X is distributed evenly between 0.4 and 0.7 such that each has equal probability, it doesn't mean 1/x would be evenly distributed between 1.428 dollars/pounds and 2.5 dollars/pounds. You can see this at the expected value of X i.e. 0.55, the exchange rate will be 1.8181 which is not the mid value of 1.428 and 2.5, so it is not infact evenly distributed between 1.428 and 2.5 while converting pounds to dollars.