Measuring Returns on Leveraged Long-Short Portfolios
Summary
The document explains why a long-short portfolio’s return depends on the capital assigned to support its positions. With a short sale, the amount invested in securities does not directly represent the investor’s capital, so return calculations need a denominator tied to account capital rather than simply the sum of position values. Gross leverage relates the total absolute notional exposure to that supporting capital, and the return over a period is the portfolio’s change in value divided by that capital.
A second approach normalizes signed long and short weights, including margin cash, to support attribution. The example shows that short exposure can be represented by negative weights while cash offsets it. These are conventions for different analytical purposes; the resulting return depends on the leverage or normalization assumptions. The note also describes margin calls when exposure breaches an agreed leverage limit, but does not prescribe one universally accepted industry convention.
Key ideas
- Long-short returns require a denominator based on the capital supporting the positions.
- Gross leverage measures total absolute position value relative to supporting capital.
- Return on capital changes when the assumed capital or leverage changes.
- Signed weights can include short positions and cash for portfolio attribution.
- Exceeding a broker’s leverage limit can trigger a margin call.
Tags
Full text
# Definition of Return of A Long/short Portfolio
# 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 by Brian B (score 10, accepted)
https://quant.stackexchange.com/a/10590
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.
## Answer by Sason Torosean (score -1)
https://quant.stackexchange.com/a/10627
* 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 MeasurementShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.