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Explain what short-selling means and determine the net profit when a dividend-paying stock is sold short.

FRM® Part I · Financial Markets and Products · Pricing Financial Forwards and Futures · FRM1.FM.10.b

The idea

Short selling means selling an asset you do not own. The seller borrows the shares from a lender, sells them in the market, and later buys the same number of shares back to return to the lender. The seller profits if the price falls and loses if it rises, and because a price can rise without limit, so can the potential loss. If the stock pays a dividend while the position is open, the short seller must pay the lender an amount equal to the dividend, because the lender must be left as well off as if the shares had never been lent. Ignoring borrowing fees and interest on margin, the result on each share comes down to one line. Net profit = sale price − repurchase price − dividends paid. Setting the profit to zero gives the break-even point, the repurchase price at which the seller neither gains nor loses. Break-even price = sale price − dividends paid. The usual slip is to ignore the dividend, or to add it as though the short seller received it.

Formulas

Net profit per share on a short sale

profit = sale price − repurchase price − dividends

sale price
the price the borrowed shares were sold at
repurchase price
the price paid to buy them back
dividends
dividends paid to the lender while the short was open

Ignores borrowing fees and margin interest. The short seller pays the dividends, never receives them; setting profit to zero gives the break-even repurchase price, sale price less dividends.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement best describes short selling a stock?

  1. Selling a futures contract on the stock without trading any sharesWhy not: Describes a short futures position; short selling involves selling borrowed shares themselves.
  2. Selling shares you own while holding a call option on themWhy not: Describes a covered or protected position, not selling borrowed shares you do not own.
  3. Buying shares on margin and selling them when the price risesWhy not: Describes a leveraged long position; a short seller sells first and buys later.
  4. Borrowing shares, selling them, and later buying the same number of shares in the market to return to the lender - correct

A short seller sells borrowed shares and profits if the price falls before the shares are repurchased and returned.

A harder one

A trader sells short 1,000 shares at USD 50.00. During the holding period the stock pays a dividend of USD 1.20 per share, and the trader buys the shares back at USD 46.50. Ignoring fees and interest, the net profit is closest to:

  1. USD 3,500Why not: Ignores the dividend that the short seller must pay to the lender.
  2. USD -2,300Why not: Reverses the sign of the whole position, treating it as a long position that lost value.
  3. USD 4,700Why not: Adds the dividend as if the short seller had received it.
  4. USD 2,300 - correct

Profit per share = 50.00 - 46.50 - 1.20 = 2.30; times 1,000 shares = USD 2,300.

The hardest tier

An investor sells short 2,000 shares at USD 62.40 and puts up an initial margin of 50% of the sale proceeds from own funds. Two dividends of USD 0.55 per share are paid during the holding period, and the shares are bought back at USD 57.10. Ignoring interest and fees, the return on the investor's initial margin is closest to:

  1. 13.5% - correct
  2. 17.0%Why not: Ignores the two dividends the short seller owed to the lender.
  3. 6.7%Why not: Divides by the full proceeds instead of the 50% margin actually posted.
  4. 20.5%Why not: Adds the dividends to the gain instead of paying them.

Net profit per share = 62.40 - 57.10 - 1.10 = 4.20, times 2,000 = 8,400. Margin = 0.5 x 62.40 x 2,000 = 62,400. Return = 8,400 / 62,400 = 13.5%.

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