# Return on investment in spreads

I have a hard time getting my head around this.
Let's say you have a strategy that consists in buying one future spread, for instance CL Z7-Z8 (crude oil dec17 minus dec18). It's easy to calculate the PnL of that strategy:

PnL = quantity * lot_size * (spread(t) - spread(0))


where spread(t) is the spread at time t and spread(0) is the level at which you entered.

Now, to calculate the return of that strategy, you'd have to assume that this corresponds to a certain capital that's at risk.

Return = PnL / InvestedCapital


Given that entering the future spread will demand an initial margin call, and subsequent maintenance margins (potentially infinite), how does one "assign" a capital to that strategy?

Is it just a matter of choice? Can tell myself: I'm committing 100k to that strategy, and with that I can buy the amount of spreads that I think will not make me go bust, which I'm assessing corresponds to 10k of initial margin calls?

But if that's the case then I could just as well commit 200k, and buy the same 10k of margin calls, which means I would be half as leveraged.

Is there a usual way of doing this? For instance where we say that the capital that's at risk is the amount you would loose if the spreads goes 7 stddevs out, or something like that?

Sorry if I'm not being very clear, it's kind of messy in my head right now...

• Note that ideally the expected return on a futures trade should just be the risk-free rate. Also, how can you invest twice as much but have the same margin call amount? – D Stanley Jan 27 '17 at 15:21

What is the return on a strategy which has no up-front cost to implement? I argue that it doesn't really make sense, and that the most sensible approach is to define a 'trading capital' that you are comfortable with, and measure returns against that.

In fact, to come up against this problem you don't even need to think about spread strategies. You might see the return on a futures contract priced at $F_t$ defined as

$$R_{t+1} = \frac{F_{t+1} - F_t}{F_t}$$

but this is misleading, since it is zero cost to enter a futures contract (if you ignore initial margin). The quantity $F_t$ that you are measuring return against is the notional contract size, but it has no relation to the amount of capital required to run the strategy.

Compare it to the return on a cash equities position with price $P_t$ and dividend $D_t$,

$$R_{t+1} = \frac{P_{t+1} + D_{t+1} - P_t}{P_t}$$

Here it makes sense that the denominator is $P_t$ since this is the capital outlay required to buy the stock at time $t$.

If you want to think in terms of return rather than profit and loss, I would define a nominal 'trading capital' $X_t$ for the strategy, and measure returns against that,

$$R_{t+1} = \frac{\textrm{PnL}_{t+1}}{X_t}$$

You can choose whatever value of $X_t$ you are comfortable with. A common approach is to choose $X_t$ so that the annualized standard deviation of returns is some acceptable number (5%, 10%, 15% etc) or to choose it so that a certain sized drawdown (say \$10k) would result in a loss of a specific percentage of your capital (e.g. if you wanted a$10k drawdown to correspond to a 10% loss of capital, you choose your initial capital to be \$100k). As you say, you can choose your initial capital to be \$200k and size your positions exactly the same way as if you had committed \\$100k, in which case you have half the risk (i.e. volatility is halved, drawdowns are halved etc).

As a general piece of advice, it is probably sensible risk management to actually hold your trading capital, whatever amount it is, in a money market account (or somewhere else safe) so that you earn a risk-free return on it, and you can use it to meet margin calls when they become necessary.

• Thanks a lot, great answer. That's what I thought, but wasn't able to articulate it fully. – d--b Jan 29 '17 at 10:22