Defining Returns for a Long-Short Commodity Spread

Summary

The document asks how to calculate returns for a spread that is long aluminium and short a weighted amount of lead. It compares a change in spread value divided by a contemporaneous sum of leg values with a weighted combination of the two commodities’ price changes. The answer says neither expression, as written, gives the intended rate of return. For the spread-value approach, it points out that the denominator should be the spread value at the start of the measurement period, rather than the current period’s value.

It also cautions that the second expression does not correctly calculate each leg’s return: its terms resemble a log-return relationship without applying logarithms, and unequal position sizes further invalidate that construction. The exchange does not provide a complete alternative formula or specify how to define invested capital for a long-short spread. The appropriate return measure therefore depends on the portfolio and capital convention being evaluated.

Key ideas

• A spread’s change in value divided by its current value is not the usual rate-of-return calculation.
• For a simple spread-value return, the denominator should be the spread value at the start of the period.
• A weighted difference of price changes is not automatically a weighted difference of leg returns.
• Unequal long and short position sizes require careful treatment when defining spread performance.
• The note does not specify a complete capital or margin convention for the spread return.

Tags

# Return On a Spread


# Return On a Spread












This is a beginner level question.

I have a $spread = aluminium - 0.7*lead $

$s = a - 0.7*l$

I have two methods to calculate return on this spread:

$ return = (s_t - s_{t-1})/(a_t + 0.7*l_t) $

which is nothing but:

$return = change/value_{investment} $

or

$ return = a_t - a_{t-1}/a_{t-1} - 0.7*(l_t - l_{t-1)/l_{t-1} $

which is nothing but:

$return_{aluminium} - 0.7*return_{lead} $

Which return definition is correct assuming I am taking 2 investment along a spread that is long in $1$ unit of aluminium and correspondingly short in $0.7$ unit of lead and vice-versa.

## Answer by Iñaki Viggers (score 2)

https://quant.stackexchange.com/a/44612

> Which return definition is correct

Neither one, and it appears that you are actually interested in the rate of return. In the first definition of rate of return, the denominator should be $s_{t-1}$.

In the second definition, the ratios do not really compute the return of $a$ and $l$. Each term somewhat resembles the logarithm property, except that you are missing the logarithm function in each term. Even if you applied the logarithm function, your calculations would fail as soon as the positions in $a$ and $l$ differ in magnitude.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.