 # Ex-post return

A U.S. investor purchased a U.K. bond one year ago. The exchange rate at the time was 1.5 to 1 (dollars to pounds) and the beginning-of-period ratio of the price levels of the consumption baskets was 2 to 1 (dollars to pounds). Beginning of the year interest rates were 7% in the U.S. and 12% in the U.K. Inflation during the year was expected to be 5% in the U.S. and 10% in the U.K. Today the exchange rate is 1.6. What is the ex-post domestic currency return on the U.K. bond to the U.S. investor?

A) 19.67%. B) 11.67%. C) 18.67%.

Explain why.

This is a standard question, but I’m not getting the right answer!!

Correct answer: 18.67%. The dollar was expected to appreciate (lower inflation), but depreciated by 6.67 percent (= 1.6/1.5 = 1.0667 or 6.67%) instead. Hence, the return to U.S. investor is the foreign interest rate plus currency appreciation, or 18.67 percent (= 12% foreign rate + 6.67% appreciation). Ex-post domestic currency return is the return, in term of domestic currency, for holding the foreign investment denominated in foreign currency. Inflation plays no role here. The inflation information is only useful when you calculate the real exchange rate.

You’re right.

I wanted to think of it like this:

1. You start with \$1.50, which you convert to 1 pound at the current exchange rate of \$1.50/Pound.

2. You then deposit the 1 pound for one year and earn 12%, so you get 1.12 pounds at year end.

3. Now the current exchange rate is \$1.60/pound. So, you convert back and you get 1.12 pounds * \$1.60 = \$1.79.

4. You started with \$1.50 and now you have \$1.79, that’s a return of 19.46%.

I get what you are saying and that’s how the book does it, but I don’t see what’s wrong with the logic above!

You can also get it in another way.

Foreign currency risk premium = currency appreciation - forward premium = (1.6/1.5-1) - (-5) = 11.67%

Forward premium is obtained by interest rate parity. Since the UK intreset rate is more, the forward premium will be negative and is simply the interest rate differenctial, i.e. 7 - 12 = -5%

So the ex-post return = 7 + 11.67 = 18.67%

This is more lengthy (but you should atleast be aware of it). I prefer the (forerign return + foreign currency appreciation), as Krisztina mentioned.

How about the simple real life logic above…where am I wrong?

test

this site keeps kicking me out

Yours is an accurate version. There is also an approx version using interest rate differentials.

Accurate solution to Kris is 1.12 x (1.6/1.5) -1 = 19.46.

You should know both, as it is sure shot question on the exam.

I sure hope so! Note though that answers A and B are mine, changed from real version.

OMG! 2011 Level II Mock Exam: Morning Session/ Result of Q21 seems to support your version of 19.46%. Not to mention logic! Schweser is a different story, isn’t it?

I knew there was something wrong here, because my numbers were always off!.

The good news is that I figured out the mistake in their calculations. You see the 6.67% appreciation of the pound, in their formula, affected only the initial exchange rate (the \$1.5/pound), but in reality the appreciation of the pound impacts the final amount, the 1.12 pound after interest. So, you should do it like this:

6.67% + 12% + (0.0667 * 12%) = 19.47%

remember in all this the delta-r+delta-s+delta-r*delta-s -> they ignore the 3rd component -> delta_r*delta_s saying if the numbers are small - they can ignore it.

Not small here, cpk. That’s about 0.80%!! If you did this with \$100 milion, the difference is about \$800,000. You should not ignore it.

should not… I know. But they usually do.

and they mention that usually the numbers are small - and can be ignored. That is all I am saying.

I agree that in the exam the difference may not be important because the answer choices will be quite apart from each other, but it’s good to know the actual logic behind these things, and not rely completely on formulas.

Dreary your logic is perfect. It is the most precise way of solving the problem.

Btw elan has given both of these approximate and precise methods in their example.

I disagree with the book and agree with Dreary’s logic. The logic is right, however, the formula given in the CFA only works if there is a small difference in interest rate. If there is a large difference like in this example, the formula doesn’t work. Some econ textbook use the same formula for this to find nominal interest rate, but if there is a big difference in rates you have to use the multiple method. I always use the multiple method because it’s always right, but you might be safer to use the CFA formula in its exams.