A firm is expected to pay quarterly dividend of $1 per share at the end of March, June, Sept and December. Suppose that today is the beginning of May and the stock is currently priced at $20 per share. You short-sell the stock today and intend to settle the short-sale at the end of October. Assume a constant 5% p.a. continuously compounded rate of interest. Calculate the minimum value that the share price must drop at the end of October before your short-selling strategy yields a positive payoff.
Question
A firm is expected to pay quarterly dividend of 20 per share. You short-sell the stock today and intend to settle the short-sale at the end of October. Assume a constant 5% p.a. continuously compounded rate of interest. Calculate the minimum value that the share price must drop at the end of October before your short-selling strategy yields a positive payoff.
Solution
To calculate the minimum value that the share price must drop at the end of October before your short-selling strategy yields a positive payoff, we need to consider the dividends you would need to pay as a short seller and the interest you would earn on the proceeds of the short sale.
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Calculate the present value of dividends: As a short seller, you would need to pay the dividends to the lender of the shares. The dividends to be paid in June, September, and December are $1 each. However, since we are settling the short-sale at the end of October, we only need to consider the dividends to be paid in June and September. The present value of these dividends can be calculated using the formula for the present value of a cash flow, which is CF / e^(rt), where CF is the cash flow, r is the interest rate, and t is the time in years.
For the June dividend, t = 1/12 (since it's one month away), so the present value is 1 / e^(0.05 * 1/12) = $0.958.
For the September dividend, t = 4/12 (since it's four months away), so the present value is 1 / e^(0.05 * 4/12) = $0.902.
The total present value of the dividends is 0.902 = $1.86.
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Calculate the present value of the short sale: When you short sell, you receive the proceeds of the sale immediately. The present value of this is simply the price at which you sold the stock, which is $20.
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Calculate the future value of the short sale: This is the amount you would have at the end of October if you invested the proceeds of the short sale at the given interest rate. The formula for future value is PV * e^(rt), where PV is the present value, r is the interest rate, and t is the time in years. In this case, t = 6/12 (since it's six months from May to October), so the future value is 20 * e^(0.05 * 6/12) = $20.51.
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Calculate the minimum value that the share price must drop to: This is the future value of the short sale minus the present value of the dividends. So, the minimum value that the share price must drop to is 1.86 = $18.65.
Therefore, the share price must drop to at least $18.65 at the end of October for your short-selling strategy to yield a positive payoff.
Similar Questions
The share price must drop below $X for the short-sale to be profitable. The objective is to calculate $X. Opportunity cost forgone must be equal to the future values of dividends paid on 30 June and 30 Sept. The future values are calculated on 31 Oct. As such, the sum of the future values is: FV = 1e(0.05×4/12) + 1e(0.05×1/12) = 2.02 Thus, $X = 20 − 2.02 = $17.98 So, as long as the share price drops below $17.98 (say it is priced at $17 ) on 31 Oct, the short sale will be profitable. Comment: Usually we should also consider the time value of money associated with the $20. Assume that after you receive the $20, you will reinvest it at the risk-free rate, then by the end of October, you will have $20e0.05×1/2 = $20.51. In this case, your share price can drop to $18.49 and still make a profit. If we do not consider the time value of money, then the assumption you will make is that you will hold the $20 cash on hand till the end of October.
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