Simple explanation of Currency Roll Yield?

CFA Level III Derivatives
#1

Hi,

I know this is a topic that has been widely discussed over the years. Can it be broken down to a simple explanation? This is my understanding – is it that simple or could someone help point out where I go wrong in my reasoning?

t = 0

We are a US investor investing in a EUR asset and want to hedge our currency exposure.

  • · USD/EUR = 2 – We pay 2 dollar per EURO.

  • 1y Forward USD/EUR = 1.90

  • · 1y USD interest = 10%

  • · 1y EUR interest = 5%

  • · Based on Uncovered Interest Parity it tells us the USD is expected to depreciate by 5% against the EUR over the next year (in other words, the EUR is expected to appreciate).

  • · This expectation is reflected in the futures price: 1y forward USD/EUR = 1.90 (5% USD depreciation). We enter into a short 1y forward to sell EUR to hedge our position in the EUR asset.

  • · We are here expecting a negative roll yield from hedging our position, since we are selling the EUR to a lower price in the future compared to today, but we need to do it to hedge our position. (?)

  • o Expecting a negative roll yield means we expect to settle our forward to a lower price compared to todays spot value(?). In other words:

  • If we are long the forward: F > S --> Negative roll yield.

  • If we are short the forward S > F --> Negative roll yield.

t =1

One year has gone and we now need to settle the Forward and we can now asses if we actually have had a negative or positive roll yield (?)

· We expected to have a negative roll yield. Since Forward Price < Spot Price initially.

    1. We settle our forward contract and buy back the EUR at 1.90 as agreed.
    • a. To see if the roll yield was positive or negative we compare the price we sold EUR for with what we could have done in the spot market had we not entered a forward in the first place (?)
      • i. Let’s say the exchange rate still is USD/EUR = 2. We have had a negative roll yield of 5% as expected.
      • ii. Let’s say the exchange rate in the spot market now is USD/EUR = 1.71. In this case the forward has benefited us and the moves in the exchange rate has been different to what was initially priced into the future. Here we have had a positive roll yield of 10% since we made 10% on our forward?
#2

In your example, you don’t have any roll yield. (Another way to look at it is that your roll yield is zero.)

To have roll yield, you have to roll one (expiring) forward contract into another (new) forward contract. If you simply settle the first contract, you haven’t rolled it, so you don’t have any roll yield.

#4

Thanks for the reply.

This based on this reasoning:

  • If the price currency has a higher IR than the base currency then the forward would be trading at a premium( Spot x (1xPrice IR / 1x Spot Ir)).
  • When Fo > So the forward is trading at a premium.
  • If we sell the forward at a premium we make a positive return. This is what I called the roll yield.

Is this reasoning correct?
What is I am calculating here if it is not roll yield?

I also watched this YouTube video that described in this way:

#5

Let me try again.

t = 0
USD/EUR = 2.
USD Ir = 10 %
EUR ir = 5%
1y forward: 2 x (1.10/1.05) = 2.31
Forward > Spot. It is trading at a forward premium.

We are a US investor who buys a EU asset.

We hedge this position by selling the EU short forward, agreeding to sell EUR for 2.31 USD per Euro, 1 year from now.

Question 1: since we are selling at a premium we expect a positive roll yield?

t = 1
The current spot rate USD/EUR = 2.40
Now the 1y forward (t =2) is trading at 2.50

I want to keep My hedge for one more year. I do that by entering an FX Swap, which consists of the following transaction:

  1. I buy EUR to settle My initial forward contract. I buy at 2.40 and sell at 2.31 as agreed in My initial forward. I lose 0.09 here.
  2. I sell the EUR forward 1y for 2.50

How do I calculate the roll yield in this example?

Thanks a lot for the help here, I spent a couple of days in this topic but can’t seem to grasp it. I find it easier to understand in terms of examples.

#6

How did you get the initial forward rate of $2.31/€?

#7

I listened to the video. What he calls roll yield isn’t roll yield.

Roll yield is the change in yield from rolling one contract into another contract. It’s generally defined (for the short position) as the difference in the forward rates less the difference in the spot rates:

Roll\ yield_{short}=\left(F_1-F_0\right)-\left(S_1-S_0\right)

While this formula is correct mathematically, it doesn’t help understanding what roll yield measures. A little algebra, however, improves it immensely:

Roll\ yield_{short}=\left(F_1-F_0\right)-\left(S_1-S_0\right)
=F_1-F_0-S_1+S_0
=F_1-S_1-F_0+S_0
=\left(F_1-S_1\right)-\left(F_0-S_0\right)

So . . . the roll yield is nothing more nor less than the change in the forward premium when you roll over a forward contract.

#8

I was thinking about this some more and thought, “What if you don’t roll over the last contract? How would you compute roll yield then?”

In that case, you can think of it as rolling the old contract over into a new contract that expires immediately. That would mean that F1 = S1, so:

Roll\ yield_{short}=\left(F_1-S_1\right)-\left(F_0-S_0\right)=0-\left(F_0-S_0\right)=S_0-F_0

This agrees with the reading and the video. So their definition of roll yield is the special case where you don’t roll the contract forward.

It’s stupid, but there you have it.

By the way, in all cases,

Roll\ yield_{long}=-Roll\ yield_{short}
#9

Thank you very much for the explanation - it makes sense ( I think, I have explained my understanding below).

In in my example: The roll yield would be:

(2.50 - 2.40) - (2.31 - 2) = - 0.21 ?

Is this the correct way oy thinking about?

  • I went from having a positive forward premium of 0.31, then I rolled over and now I have a forward premium of 0.10, this is 0.21 lower, I lost 0.21(in terms of forward premium) from rolling over, hence the roll yield is -0.21)

Pretend we don’t toll the contract forward at time t =1, would it then be correct to say the “roll yield” is (2.31-2) = 0.31?

To answer your earlier question:

This is how I calculated the initial forward rate.

F = So x (1+ IRp)/(1+IRb) - but I can see this should be 2.09 and not 2.31 based on that calculation. My mistake.

#10

That’s all correct.

Note that the roll yield of −$0.21/€ is the roll yield to the short; the roll yield to the long is $0.21/€.

If you don’t roll over the contract, then it would be $0.31/€ for the short, −$0.31/€ for the long.

Yes: to get 2.31 you multiplied by 1.05 instead of dividing by 1.05.

#11

Thanks a lot for your help.

When we are saying being short (long) a forward, are we then always referring to selling (buying) the base currency in the P/B currency quote?

#12

Yup.

:racehorse:

#13

Thank you very much for all the help.

#14

My pleasure.

#15

Hi, I am trying to wrap my head around this and I am not sure why the profit and loss on the roll of the fx swap isn’t just S1-F0 =2.4-2.095= $0.305 /€. Where, 2.095= 2*(1.1/1.05).

The investor purchased at t=0 one euro spot at S0 for $2 and sold forward the euro at F0. So at t=1 when the investor rolls the fx swap (my understanding is that the actions performed when an investor rolls the fx swap is that they close out the far leg i.e buy the eur that was sold forward at t=0, and they do so by buying spot), they buy €1 spot at S1=2.4.

So they had agreed to buy eur at 2.095 , i.e pay $2.095 and sell the eur forward (hence roll) at F1=2.5 USD per EUR. So they pay $2.4, give the €1 purchased spot to the forward counterparty and receive F0 =2.095.(counterparty is the the same as it’s one transaction fx swap near leg spot /far leg forward, so their account is debited $0.305 and that would be the PnL on the day? (excl. the unrealised fx gain on the €1 that was purchased by the investor , for simplicity assuming it has not been invested or local value of investment is still par).
This means F1 doesn’t enter into it. As at t=1 forward maturing 1 year out (t=2) is valued at 0. Value = price at inception. I am sure that I am missing something…