# Where am I making a mistake in my calculation of profit on a short-sale?

I am studying financial math and here is a problem and the solution from the author: Here are my calculations:

The short sale is $200\cdot24.82 = 4964$. Now half of this amount will be taken for a haircut and $0.3$% will be taken for a commission. Then we will have (0.5)(0.997)4964 = 2474.55 at t = 0. At t = 3 months, we will have $2474.55e^{0.06 \cdot 0.25} = 2511.95.$

Covering the short position at t = 3 requires 200(21.64) = 4328. The close commission is 200(21.64)(0.003) = 12.98.

We will receive the haircut and rebate which will be $4964(0.50)e^{0.04 \cdot 0.25} = 2506.94.$

The profit is then 2511.95 - 4328 - 12.98 + 2506.94 = 677.91.

## 1 Answer

You assume that you receive the 6% interest rate on the receivings of the short sale (2474.55) and you assume that you will receive the 4% interest rate on the haircut (2474.55). This is not stated in the question and the only thing you should calculate based on the information in the question is the loss from not being able to receive the 6% interest on the haircut, which is:

$$2474.55*e^{.06/4}-2474.55e^{.04/4} = 12.53$$

By adding the interest proceedings to both the sale proceedings and the haircut you arrive at your answer. The correct answer is indeed A, as given by the author:

\begin{align} \text{Proceedings: } & 200*24.82*.997 &= 4949.11 \\ \text{- Interest Loss: } & 2474.55*e^{.06/4}-2474.55e^{.04/4} &= 12.53 \\ \text{- Close: } & 200*21.64*1.003 &= 4340.98 \\ \text{Profit: } & 595.60 \end{align}

• I think my problem is that I don't see why we wouldn't take the time value of money into account here for the proceeds. For instance, when calculating the profit on a call option, we have to look at the time value of the call premium. I know it's not explicitly stated in the problem that the proceeds are invested at the market rate, but it's still difficult to wrap my head around. – user2521987 Sep 21 '15 at 14:18
• I think it is because the question only asks for the profits related to the short sale. If you include the time value of money you would calculate the profits of the position you take by short selling. However, I agree that this is confusing. Perhaps someone else can comment on this. – mkdevoogd Sep 23 '15 at 15:14