Definition of Return of A Long/short Portfolio

This can either be a silly question or a question with no sure rigorous answer but defined with some convention. Any way, here it is.

What is the (industrial recognized) definition of the return of a long-short portfolio? Normally, return is defined as profit/initial investment. The initial portfolio may be predominantly short or even neutral (net zero dollar). Should we take the initial investment as the sum of the absolute value of the investment, because the short most likely requires collateral? If so, should one take the leverage rate into account?

Answer

The initial investment is the capital in the account used to support the portfolio, not the cost of the assets in the portfolio. For example, when you sell a stock or bond short, your account doesn't actually accrue any cash. Instead you start receiving a regular cash flow.

There isn't necessarily a difference between these quantities in a long-only portfolio but for a long-short portfolio of any kind you automatically must make some assumption about the leverage -- the amount of cash required to support a given position size.

If your portfolio is a set of positions with notional values $A_i$ (possibly including cash $A_0$) and it is supported by capital $K$ then the gross leverage is

$$ L_G =\frac1K \sum_{i>0} |A_i| $$

After a month, if your positions are now worth $\tilde{A}_i$ then your one-month return on capital is

$$ r =\frac1K \sum_i (\tilde{A}_i-{A}_i) $$.

This obviously depends on the capital $K$, so return is dependent on leverage assumptions.

Depending on the riskiness of a strategy and how well it can be measured within their existing risk control software, prime brokers will typically allow leverage between 2:1 and 20:1.

If your position exceeds the agreed leverage ratio at any time, you will receive a margin call, where you will either have to come up with some further long positions to assign to the portfolio, or liquidate some existing positions to cash.