Simple Return for Short Position

It is trivial to compute a simple return (or arithmetic return) for a long position. For example, a share of stock A is bought at $10 and sold at $12. The profit is $2 (=$12 – $10) and the simple return of the long position R_L is computed by

R_L = 12/10 – 1 = 0.2

When a share of stock A is shorted at $10 and the short position is closed at $12, what is the return for this trade? Some people may say that the return is the negative of that of a long position, i.e., -0.2 in this example. This is not true, however. For a short position, the orders of executions are swapped. To short sell a share, you first borrow the share, sell it, and hope the price drop, then you buy back a share some time later to close the position. So, the return of the short position should be computed by

R_S = 10/12 – 1 = -0.1667

because the share is bought at $12 and sold at $10 (but the sequence is just reversed in reality).

In general, R_S can be expressed in terms of R_L as

R_S = -R_L / (R_L + 1)

Note that R_S ≠ -R_L.

In fact, this equation can be deduced simply from the relation that

(1 + R_L)(1 + R_S) = 1

This relation tells that holding 1 long position and 1 short position of the same stock for the same time period will give you zero return (in a friction less market).

If you are still not convinced, we can also deduce the formula step by step. Suppose the price of a share goes from $v₁ at time t₁ to $v₂ at time t₂. If I simultaneously enter a long position and a short position at t₁, and close both positions at t₂, the simple returns of these positions are

R_L = v₂/v₁ – 1

R_S = v₁/v₂ – 1

From these 2 equations, we arrive at the same formula.