# 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?

• You should be able to find the money weighted return if you get the inflows / outflows correctly. – user1869020 Mar 14 '14 at 23:52
• @user1869020: Could you be more specific with a formula? – Hans Mar 15 '14 at 0:13

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.

• I suppose for each $A_i$ there is a support capital $K_i$? It is always positive (nonnegative), correct? Is this the collateral to the bank for loans for either long or short? If I define leverage $L_i=\frac{A_i}{K_i}$, is it reasonable to assume the positive leverages are mostly of similar absolute magnitude, and the negative ones are as well but quite distinct from the former? – Hans Mar 15 '14 at 18:35
• It really depends on the type of assets in the portfolio. For example, futures hedges on futures options will reduce capital charges. – Brian B Mar 15 '14 at 19:44

* For a given period t and a set of securities and cash denoted with index i which individually have returns r and weights w in a portfolio the portfolio return could be computed as

$$R = \sum_i w^s _i r^s _i + w_i^l r_i^l$$ where the sups l and s mean short and and long respectively. Note that the weights need to sum up to unity

$$\sum_i (w^s_i + w^l_i) =1$$

Also note that the weights of the shorts are negative

hope this helps.

*adding an example. time = 0, sec1 = 0, sec2 = 0, margin cash = 0, available cash = 100.

time = 1, sec1 = 80, sec2 = 0, margin cash = 0, available cash = 20.

time = 2, sec1 = 80, sec2 = -40, margin cash = 60, available cash = 0.

weights thus can be read as: sec1 = 80% sec2 = -40% margin cash = 60% sum= 100%

This sort of normalization makes attribution with short positions possible, so it may help you as well.

*Source: 'Performance Attribution with Short Positions' Dr. Jose Menchero, The Journal of Performance Measurement

• This is not valid. For a long-short portfolio, you can not impose extraneously the total portfolio value be always positive. Not only positive would be a problem, even positive net weight but close to zero would wreak havoc. – Hans Mar 20 '14 at 20:26
• Also, a purely short portfolio would real the absurdity of this formulation whether one allows $w_i$'s to take on arbitrary signs or not. What you have written here is the definition of return on a net long portfolio. I would not have posed the question, had that definition been fit for extending to general long-short portfolio. – Hans Mar 20 '14 at 21:05
• check out the example for clarity – Sason Torosean Apr 1 '14 at 22:50
• Are your examples supposed to resolve the two objections I have raised in my two comments? I do not see that. Please explain in detail if you do think they do resolve them. – Hans Apr 2 '14 at 3:47
• "This is not valid. For a long-short portfolio, you can not impose extraneously the total portfolio value be always positive" Ans: They are normalized weights, so they sum up to 1 or 100% – Sason Torosean Apr 3 '14 at 0:26