Cumulative Return on Futures

Question

In my current backtesting, I am using log returns as a proxy for simple returns via the relationship $\ln(1 + r) \approx r$ for small enough r. This gives me wonderful properties like time additivity, so that calculating rolling returns is as simple as applying a cumulative sum of the historical log returns.

This works for stocks, where the value of the stock is the dollar value of the security. However this is not true for futures. Take corn for example: corn trades at $12.5 per point.

In my thinking then, the close-to-close log return would not accurately represent the profit and loss of a portfolio. Instead, I think that calculating the delta in point value between two close values and applying that to your starting capital is a better method. Here is an example:

Given two days of data (for corn):

1. 764.3600814
2. 754.7100857

The log return is simple -

$ln(754.7100857) - ln(764.3600814) = -0.012705306$

So we lost money. But we are in futures, so this return is misleading. On a starting capital of \$10,000 our account is now worth \$9872.94694.

But if we calculate the point delta ($-9.64999577$) we see our starting capital has been reduced to \$9879.375 calculated with ($10000 + (12.5 * -9.64999577))$.

The error between the two is significant in the sense that over the course of thousands of rows of data the error could accumulate to quite a large sum.

Which of these is the preferred method for calculating returns on futures? I feel like the point value is the most accurate, but it complicates backtesting in that you need to base all returns on "capital returns" rather than just mindlessly summing your log returns.

Answer 1

When you do backtesting it is much better to keep track of your daily $ pnl, not the daily returns. Then all your problems disappear. All you need to do in the case of futures is keep track properly of your costs in addition to mark-to-market eg financing costs for your margin account, rolls etc..

Measuring everything in actual $ ensures you are indeed self-financing (eg: making sure rolls are treated correctly in the case of futures but also that corporate actions are properly accounted for in the case of stocks say) and you calculate ROE only by comparing your overall pnl with your initial capital at risk.

Answer 2

Performance in futures is typically determined based on price per point/tick. Obviously, tick size varies by contract, so it tends to be easier to calculate PnL for individual positions and calculate any performance stats based on PnL per some reference capital allocation (eg, PnL of \$4,000 for \$100,000 book = 4% return).

Log (natural log) returns are only really used in equities.