‘Roll yield’ and ‘roll return’ are one and the same as far as the curriculum is concerned.

This being said, the first of the two equations you have shown above:

Roll return = (change in futures price) - (change in spot price)

results in an absolute number (e.g. roll yield - or roll return - is $0.55).

The second equation gives the ’ roll yield as a % of the spot price (i.e. its magnitude).

Under certain assumptions which are discussed below, the equations are identical.

The reason why the two terms look quite different is to be found in footnote 26 of section 6.1.2 (Reading 19) where the authors explain that in the equation

%roll yield = |(F-S)/S|

they assume _ **“all else equal”** _, or, in other words, they assume that the spot rate at settlement date is equal to the spot rate at contract inception.

This (pretty strong) assumption that Spot rate (S) at time 0 is equal to S at time *t* allows them to write the formula as

%roll yield =|(F-S)/S|)

Removing the (/S) to obtain the absolute roll yield value ( |roll yield| ) we can say that, *under the assumption in footnote 26*,

roll yield = |(F-S)| = (change in futures price) - (change in spot price)

To prove it let’s zoom in on the equation:

roll yield = (change in futures price) - (change in spot price)

under the assumption that **S(at time 0) =** **S(at time t)** this equation becomes

roll yield = (change in futures price) - 0

The assumption

S(at time t) = S(at time 0)

also implies that

F(at time t) = S(at time 0) (because by definition F(at time t) = S(at time t))

which implies that the gain/loss on the futures contract, **all else equal** , is |F-S|

hence, **roll yield =** **(change in futures price) - 0** (i.e. the first equation) can be re-written as **roll yield = |F(at time 0) - S(at time 0)| =** **|F-S|** (i.e. the second equation)

This allows us to conclude that, yes, under the ‘all else equal’ assumption and ignoring the fact that one gives absolute values and the other a %, the two equations are the same.

If it is of any consolation, I did not find this part easy either.

The trick for me was to start from the very basic concept of roll yield which (in commodities space) can be thought as “I want to buy a cow. I can buy it for delivery today at a spot price of £100 or I can buy it for delivery in 3-month via a futures contract priced at £50 [we are in spring so it makes sense for the cow market to be in backwardation*]. If I choose the delivery in 3-month time I have a roll yield of |50 - 100| = positive £50 or |(50 - 100)/100| = 50%”.

There is no more than that…

Good luck, Carlo

* reason being the cow can mow your lawn, keep a family of 5 in milk and cheese and provide plenty of rich manure for the vegetable patch,… so I would rather have it now thank you very much…