PS06-solution

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Finance

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Jan 9, 2024

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Problem Set 6: Solutions UCLA MQE Core Finance Prof. Pierre-Olivier Weill 1. Go to Yahoo Finance to check out the page of Procter and Gamble. (a) Using information from the statistics and historical data tabs, find or calculate: the beta, the price, the price earning ratio, the earning per share, the dividend per share per year, the distribution rate. Answer: On November 6th, 2023, the price of P&G was P = $151 . 08 . The forward price earning ratio in the statistic tab was P E = 23 . 58 . Implying that the earning per share is P P/E = 151 . 08 23 . 58 = 6 . 41 . Using the historical data tab, we find that over the last year the dividend per share was 4 × 0 . 94 = $3 . 67 . Finally, in the statistic tab we find that β = 0 . 46. (b) Assuming that the risk free rate is 5% and the risk premium is 6%, what should be the discount rate for Procter and Gamble cash flow? Answer: This discount rate is given by the CAPM formula ¯ r = r f + β r M r f ) = 0 . 05 + 0 . 46(0 . 11 0 . 05) = 0 . 0776 , that is, 7.76%.
(c) Assuming that the price of Procter and Gamble can be calculated using the Gordon Growth formula, what is the average growth rate that the market expects for Procter and Gamble in the future? Answer: The Gordon Growth model states that the price earning ratio is equal to: P E = (1 + ¯ g ) δ ¯ r ¯ g . Solving for ¯ g we obtain: ¯ g = P/E ¯ r δ P/E + δ = 5 . 14% . 2. Calculate the forward price in the following examples. Assume that the one year rate is r 01 = 5 . 33%, the two year rate is r 02 = 4 . 94%, and the one year rate a year from now is r 11 = (1 + r 02 ) 2 / (1 + r 01 ) = 4 . 55% (a) A forward contract on corn, with delivery in one year. Assume that the spot price is $ 4.79 per Bushel, and there is no storage cost. Answer: The forward formula implies that F 01 = S 0 (1 + r 01 ) = 4 . 79(1 + 0 . 0533) = 5 . 05. (b) A forward contract on crude oil, with delivery in two years. Assume that the spot price is $ 90 per Barrel, and that the storage cost per year is $ 5.4. Answer: The forward formula implies that F 02 = S 0 (1 + r 02 ) 2 + c 1 (1 + r 02 ) 2 + c 2 (1 + r 11 ) = 110 . 7. (c) A forward contract on a two year annual coupon paying bond, with coupon rate of 4%, face value of $ 1,000. The forward is for delivery in one year, after the coupon has been paid. For this calculation, you will first need to calculate the spot price. Answer: We first calculate the price of the bond using the provided rates r 01 and r 02 : P = 40 (1 + r 01 ) + 1040 (1 + r 02 ) 2 = 982 . 37 . The forward price is therefore equal to (1 + r 01 ) P c = 994 . 73.
3. The one year rate is r 01 = 5 . 33%. Suppose the spot price of corn is $ 4.79 per Bushel and that there are no storage cost. Suppose that the one year corn forward sells at $ 4.9. Show that there is an arbitrage. Describe the trade at time t = 0 and time t = 1 that would allow to take advantage of this arbitrage. Answer: According to the previous question, the no-arbitrage price of the corn forward is $ 5.05. Hence, to take advantage of the arbitrage you need to buy a corn forward, short sell corn at the spot price of $ 4.79, and invest the proceed at a rate of r 01 = 5 . 33%. At the end of one year, you have $4 . 79 × (1 + 0 . 0533) = $5 . 05 on your bank account. You use $ 4.9 to buy corn with the forward, deliver the corn that you shorted, and keep $ 0.15 in profits.
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