# Calculation of carry trade returns

Hi all,

Hope yall study progress is doing fine!

I have a burning question I hope somebody can answer with regards to calculation of carry trade returns. An example of this calculation in the CFAI book and the EOC question seems to contradict.

Pg 546 of CFAI example:

A Tokyo-based asset manager enters into a carry trade position based on borrowing in yen and investing in one-year Australian Libor.

Today’s One-Year Libor Currency Pair Spot Rate Today Spot Rate One Year Later JPY 0.10% JPY/USD 81.30 80.00 AUD 4.50% USD/AUD 1.0750 1.0803

1. After one year, the all-in return to this trade, measured in JPY terms, would be closest to:
2. +1.84 percent.
3. +3.23 percent.
4. +5.02 percent.

B is correct. To calculate the all-in return to a Japanese investor in a one-year AUD Libor deposit, we must first calculate the current and one year later JPY/AUD cross rates. Because one USD buys JPY 81.30 today, and one AUD buys USD 1.0750 today, today’s JPY/AUD cross rate is the product of these two numbers: 81.30 × 1.0750 = 87.40 (rounding to two decimal places). Similarly, one year later the observed cross rate is 80.00 × 1.0803 = 86.42 (rounded to two decimal places). Accordingly, measured in yen, the investment return for the unhedged Australian Libor deposit is closest to:

187.40(1+4.50 %)86.42=(1.0 333) 187.40(1+4.50%)86.42 (1/87.40)=(1.0333)

Against this 3.33 percent gross return, however, the manager must charge the borrowing costs to fund the carry trade investment (one-year yen Libor was 0.10 percent). Hence, the net return on the carry trade is closest to 3.33% – 0.10% = 3.23%.

EOC Qn 2:

Today’s One-Year Libor Currency Pair Spot Rate Today Spot Rate One Year Later USD 0.80% CAD/USD 1.0055 1.0006 CAD 1.71% EUR/CAD 0.7218 0.7279 EUR | 2.20% |

Based on Exhibit 3, the potential all-in USD return on the carry trade is closest to:

1. 1.04%.
2. 1.40%.
3. 1.84%.

A is correct. The carry trade involves borrowing in a lower yielding currency to invest in a higher yielding one and netting any profit after allowing for borrowing costs and exchange rate movements. The relevant trade is to borrow USD and lend in Euros. To calculate the all-in USD return from a one-year EUR Libor deposit, first determine the current and one-year later USD/EUR exchange rates. Because one USD buys CAD 1.0055 today, and one CAD buys EUR 0.7218 today, today’s EUR/USD rate is the product of these two numbers: 1.0055 × 0.7218 = 0.7258. The projected rate one year later is: 1.0006 × 0.7279 = 0.7283. Accordingly, measured in dollars, the investment return for the unhedged EUR Libor deposit is equal to:

(1.0055 × 0.7218) × (1 + 0.022) × [1/(1.0006 × 0.7279)] –1 = 0.7258 × (1.022)(1/0.7283) –1 = 1.0184 – 1 = 1.84%

However, the borrowing costs must be charged against this gross return to fund the carry trade investment (one-year USD Libor was 0.80%). The net return on the carry trade is thereby closest to: 1.84% – 0.80% = 1.04%.

MY QUESTION:

It seems like for the CFAI example, it has the spot rate one year later on the numerator multiplied by the interest rate of the investment currency, divided by the current spot rate.

For the EOC question, it has the one-year rate as the numurator and divided by the current spot rate!

I couldn’t make sense (as of now, its 1.30am) why this is different? For the example in the text, why take the later spot rate in the numerator to multiply by the interest? And why is it that in the EOC qn, they took current spot rates as the numerator?

For the EOC answer, why take the future spot rate to multiply by current interest return? Shouldn’t it be the CURRENT spot rate (how much you can get, and then invested at the one year interest rate) divided by the

I figure it this way… With an FX rate, your numerator is the price (from the perspective of a person in the numerator’s country), and the denominator is an asset, just like a stock index, and one unit of that asset costs you the numerator amount.

This particular question wants you to compare what happens naturally in the FX market, based on the one-year quotes, which what you get when you do your conversions and investing. If the real FX change isn’t what you got by converting, investing, and re-converting, then somebody won and somebody lost. In reality, you’d never know the one-year spot rate while you did this maneuver. So, in other words, figure out what you need to do to position yourself to get the high yielding currency (like the stock index) and invest in it. Then once it’s over convert the price currency back so you can spend your money, etc.

Like any other future, you’ve got to reduce the spot of this asset by the present value of future cash flows, which here is the risk-free rate. The cross rate concept makes this one a bit more confusing and probably realistic. Since CAD shows up in both quotes, it should be intuitive that you’re just using that to get an EUR/USD quote now and for one year later. Also, another way of seeing that is that USD is the low yielding (cheaper to borrow) and EUR is the highest rate (ideally invest here). (We know, though, that interest rate parity should cause the higher yielding currency to depreciate; otherwise this trade would never fail. And there’s no such thing as a free lunch.)

So, after finding these new EUR/USD quotes from the cross rate I approach this as a European* buying some USD, and divide EUR/USD by (1+USDrate) and multiply by (1+USDrate). You could have taken the American approach from this point and looked at in terms of USD/EUR—i.e., how many dollars do I need for a euro. The problem is going to want an American perspective in the end, and you’ll flip it eventually either way. This is where the second problem is different. They want you to find a cross rate AND then take it a step further and switch the price and asset currency.

Sticking** with the European perspective, you compare what you got with your division and multiplication in terms of EUR per USD with what the trade is currently. .7258 was t=0 spot EUR/USD. .7359 was your actual locked in FX rate at t=T, based on what the forward market would have quoted you in the future. And .7283 is what turned out to be the true, final spot at t=T.

So the European dollar buyer signed up for .7359 – .7283 = .0075 more euros per dollar than economic principles would have had us expect. Should have just waited. As a percentage of the original starting amount of .7258 EUR/USD, that’s about 1.04%.

At * or ** you could have switched the perspective to American euro buyer.

I guess you could also look at this as whether EUR/USD increased (or maybe decreased) more as a percentage of the spot .7258 via the (implied, assuming we got a quote and contract based on parity) forward contract of .7359 or if there was a larger increase naturally based on the new spot of .7283. By now you can see the contract obviously overpaid for dollars using euros. Then flip it for the American (here winner’s) perspective. The difference, I believe, should be profit.

Answer B. looks like it’s the increase only in EUR per USD of the forward contract.

Answer C. appears to add both the increase in the forward EUR per USD with the increase of the spot for EUR per USD?

I’m not the best at these, but working on it, and I saw nobody had given this a shot. I really recommend s2000magician’s series on derivatives. He’s been thoughtful enough to share a really keen understanding of the subject matter.