# Roll Yield Formula

Hi all, on page 257 of SS15, the formula for Roll yield is provided as roll yield = (F - S)/S

Also, it is said that roll yield is positive when buying a futures contract that is trading at a discount. Suppose then, that a base currency is trading for 2.00 P (ie price currency) on the spot market, and 1.50 on the forward market. my roll yield should then be calculated as:

roll yield = (1.50 - 2)/2 = -.25 or -25%

Clearly the forward contract is trading at a discount, so why does the calculation give a negative roll yield? what am I doing wrong?

Your base currency is going to appreciate against the currency you are trying to buy. By locking in today’s rate to buy the forward, you are going to lose money.

I’m not sure I agree with naren…

(F - S) / S is the formula for the implied roll yield in a situation where the current forward price is an unbiased estimate of the future spot price (i.e. spot drops from 2.00 to 1.50). In this case, the implied roll yield being negative at (-25%) implies that if you are long EUR assets and decide to fully hedge by selling EUR forward , it will cost you because under no arbitrage, your EUR assets will depreciate as the spot drifts down towards the unbiased forward estimate.

Does this make sense? Can someone elaborate? I got confused.

Now that I am done the SS, I have a better grasp on this concept. iossif is correct, naren_ has got it backwards. Here is what I’ve learned in my studies:

If a currency is trading at a forward discount (which I will call B for Base currency), and you go LONG on that forward contract, that means F P/B < S P/B, which is consistent with my example. I had F = 1.50, and S = 2.00. In the book, it says price movements tend to not appreciate/depreciate up/down to the forward prices - and I say TEND to because it’s a skewed distribution, meaning that when shit gets out of hand, it gets really out of hand (ie market stress). For that reason, if you buy a forward contract at a discount, you should have POSITIVE roll yield, because the expectation is that the price depreciation from the current spot rate will not be as extreme as the forward price. So if F = 1.50 and S = 2.00, and I go LONG on S, that means I commit to buying this foreign currency for 1.50 units of my own domestic currency. If the spot price stays the same until maturity, that means my profit will be .50 P because I will pay 1.50 P and receive 1.00 B, which I will then be able to immediately sell back on the spot market at a price of 2.00 P.

The simple roll yield formula here ignores interest rates, however, which is wrong, but it gets the basic idea of the abvoe right. Also, VERY IMPORTANT: The formula is if you go LONG on the S, and SHORT on the F, and I incorrectly assumed it was for the other way around.

So I still used this overly simple formula correctly when I did this:

roll yield = (1.50 - 2)/2 = -.25 or -25%

The roll yield is indeed -25% for this scenario, except that this is the roll yield if you SHORTL the F and LONG the S. The roll yield for LONG the F and SHORT the S is +25% (you just use the same formula and reverse the sign). This is consistent with the book’s repeated concept that if you Buy a Futures contract that is selling at a discount, you have a positive expected roll yield.

Remember, it is an EXPECTED roll yield, not a guarantee. The text describes chasing roll yield as picking up quarters off a railway track, or something like that. I think the analogy is more valid if Helen Keller is the one picking up the quarters, but whatever.

Another thing: this formula, (F - S)/S, is an over simplified one that ignores interest rates and assumes the spot price will stay the same, so if you are provided with interest rates, here is the formula you should use:

unannualized Roll yield = [(ip - ib)*t]/(1+ib*t)]

where ip and ib are the interest rates for the price and base currencies, respectively, and it is the roll yield if you SELL the forward contract. Just switch the sign for if you’re going long on it instead.

Also, in the formula (F - S)/S, I think the S in the bracket is the EXPECTED spot price, and the S in the denominator is the current one. Can someone confirm that for me please?

Thanks

both are the expected spot. neither of them is the current spot.

at the time of the contract expiration - the future is supposed to “converge” towards the “expected spot” - and this causes the roll yield.

^ sorry dood, but that is soooo incorrect, it says so right in the text on page 243:

"We can simplify this equation by assuming a one-year time horizon (360 days) and rearranging it to isolate the forward premium or discount (expressed as a percentage of the spot rate): FPB−SPB = [(ip - ib)*t]/(1+ib*t)]

(Institute 243) Institute, CFA. CFA Institute Level III 2014 Volume 5 Alternative Investments, Risk Management, and the Application of Derivatives. John Wiley & Sons P&T, 2013-07-12. VitalBook file. “expressed as a percentage of the spot rate”, NOT the expected spot rate. Expected spot rate wouldn’t make any sense cuz the roll yield is based off of what you’ve gotta invest today to get the roll yield in the first place. It’s basically a holding period return with the current spot rate as your cost. You generally have some good posts CPK, but step up your game a bit and check your facts before answering stuff with such certainty.

I’m pretty sure the expected spot goes in the numerator, but I can’t find that explicitly said in the text anywhere.

If you are calculating the implied roll yield today , (F - S)/S gives you the discount/premium as a percentage of current spot. S is the current spot rate in the denominator, no doubt.

Again, today, S is also the current spot in the numerator because if F is an unbiased estimate of the future spot, replacing S in the numerator with F would turn the implied roll yield fomula into (F - F)/S which would always give you zero and it would make no sense to use it.

What the roll yield turns out to be at expiration is an entirely different question because the future spot might not end up being equal to the F that you calculated at t = 0.

Are you arguing for no reason or am I missing something?

not for no reason - if you get a question like “the research department expects the spot rate to be X at the time of the contract’s maturity. Here’s F and here’s S, calculate the expected roll yield”, then the roll yield would obviously need to use X in the numerator, and thus, the expected roll yield would then be (F-X)/S.

Also, F is not an unbiased estimator of expected S, the text pretty much says F is biased to be too extreme, and that prices tend to not appreciate/depreciate to F. The thing is though, sometimes they do, and even go much further, and that is why people will accept a negative expected roll yield - it’s like a fee to protect yourself from extreme movements and lock in a rate that you know with certainty, and can plan around.

Check your facts. The forward rate applicable to covered interest rate parity is an unbiased estimate of the future spot rate, by definition, assuming that interest rate parity holds. Now, it may, or it may not, but this is why it’s called the implied roll yield. Under the uncovered interest rate parity, the actual future exchange rate may very well deviate from our initial calculation (i.e. F calculated at t = 0) in which case we get a different final roll yield, the actual roll yield. And finally, sure, you can also have an expected roll yield that you calculate at t = 0 coming from the research department’s estimate of their expected forward rate.

Do you agree with all that? Let’s call it a day?

The equation quoted above is only correct if you are short the forward contract. If you are long then you want to reverse the numerator to be written as (S-F).

If you are LONG the base currency, the converge of the base currency from 1.5 to 2 (appreciation of the base) will result in a gain/positive roll yield.

If you were SHORT the forward, then the convergence of the forward to the spot would result in a loss and thus negative roll yield.

Agreed.

To Swhip, ya, I also wrote the same later on after I spent some time reading.

To iossif in regards to fact checking, I would refer you to page 242 of the CFA text, which confirms that the forward rate is, in fact, a biased predictor (Kritzman, 1999). I have bolded it below. the preceding paragraphs provide context.

" Recall that uncovered interest rate parity asserts that, on a longer-term average, the return on an unhedged foreign-currency asset investment will be the same as a domestic-currency investment. Assuming that the base currency in the P/B quote is the low-yield currency, stated algebraically uncovered interest rate parity asserts that %ΔSH L ≈iH −iL where %ΔSH/L is the percentage change in the SH/L spot exchange rate (the low-yield currency is the base currency), iH is the interest rate on the high-yield currency and iL is the interest rate on the low-yield currency. If uncovered interest rate parity holds, the yield spread advantage for the high-yielding currency (the right side of the equation) will, on average, be matched by the depreciation of the high-yield currency (the left side of the equation; the low-yield currency is the base currency and hence a positive value for %ΔSH/L means a depreciation of the high-yield currency). According to the uncovered interest rate parity theorem, it is this offset between (1) the yield advantage and (2) the currency depreciation that equates, on average, the unhedged currency returns. But in reality, the historical data show that there are persistent deviations from uncovered interest rate parity in FX markets, at least in the short to medium term. Indeed, high-yield countries often see their currencies appreciate, not depreciate, for extended periods of time. The positive returns from a combination of a favorable yield differential plus an appreciating currency can remain in place long enough to present attractive investment opportunities. This persistent violation of uncovered interest rate parity described by the carry trade is often referred to as the forward rate bias. An implication of uncovered interest rate parity is that the forward rate should be an unbiased predictor of future spot rates. The historical data, however, show that the forward rate is not the center of the distribution for future spot rates; in fact, it is a biased predictor (for example, see Kritzman 1999)." (Institute 242) Institute, CFA. CFA Institute Level III 2014 Volume 5 Alternative Investments, Risk Management, and the Application of Derivatives. John Wiley & Sons P&T, 2013-07-12. VitalBook file.

^ Also, to iossif, I want you to picture me singing “I believe I can fly” by R Kelly, sort of like this, when Jim Carry does it

https://www.youtube.com/watch?v=JZzFtQmsHfI

You’re killing me! Did you even read or understood what I wrote? I basically said the same thing. I said that F is biased if you assume uncovered interest rate parity, but unbiased if you assume covered interest rate parity. Everybody knows this since Level 1.

So, enough aruguing. Keep flying.

lol then what facts are you asking me to check? what are you talking about? I’ll listen from up here, where I am currently flying