Short selling
1. The idea
A short sale sells shares that we do not own. A broker lends us the shares from another client’s account; we sell them now and buy them back later to return them. We gain if the price falls and lose if it rises, and since a price can rise without limit, so can the loss.
The lender of the shares must not lose anything by lending them, so any dividend paid while the shares are borrowed is paid by the short seller. The profit on \(N\) shares sold at \(P_0\) and bought back at \(P_1\), with a dividend \(D\) per share paid meanwhile, is
\[ \text{profit} = N\,(P_0 - P_1 - D), \]
leaving out interest and the broker’s fees.
An example. We sell 200 shares short at $40. The company pays a dividend of $0.50 a share, and we buy the shares back at $35. Our profit is \(200 \times (40 - 35 - 0.50) = \$900\): the $5 fall on each share, less the $0.50 dividend we pay the lender.
Short selling is how arbitrage with a mispriced forward works when a forward is quoted too low: sell the share short, invest the money, and buy the share back through the forward.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The profit on a chart
First, a short sale without a dividend.
Now move the price at which the shares are bought back. Each $1 rise in that price lowers the profit by $1 a share, $500 on the 500 shares, and the dividend of $1.20 a share moves the whole line down by $600.
4. One worked example, then your turn
In the worked example, we find the buying-back price behind a loss.
Now the profit or loss of a short sale, with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a profit and one for a buying-back price. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: