Learning Module 1 Pricing and Valuation of Forward

Learning Module 1 Pricing and Valuation of Forward Commitments EXAMPLE 1 Forward Contract Price An Australian stock paying no dividends is trading in Australian dollars for A563.31, and the annual Australian interest rate is 2.75% with annual compounding. 1. Based on the current stock price and the no-arbitrage approach, which of the following values is closest to the equilibrium three-month forward price? A. A$63.31 B. A$63.74 CA$65.05 Solution to 1: B is correct. Based on the information given, So = A563.31, r2.75% (annu- al compounding), and T 0.25. Therefore, FFV(S) 63.31(1+0.0275)025-AS63.7408, 2. If the interest rate immediately falls 50 bps to 2.25%, the three-month for ward price will: A. decrease. B. increase. C be unchanged. Solution to 2: A is correct, because the forward price is directly related to the interest rate. Specifically, FoFV(S) 63.31(1+0.0225)025-AS63.6632. Therefore, we see in this case that a decrease in interest rates resulted in a decrease in the forward price. This relationship between forward prices and interest rates will generally hold so long as the underlying is not also influ enced by interest rates. EXAMPLE 2 Forward Contract Value 1. Assume that at Time 0 we entered into a one-year long forward contract with price Fo=105. Nine months later, at Time t = 0.75, the observed price of the underlying stock is S0.75 = 110 and the interest rate is 5%. The value of the existing forward contract expiring in three months will be closest to: A. -6.34. B. 6.27. C. 6.34. Solution: B is correct. Note that, $0.75 = 110, r = 5%, and T-t=0.25. Therefore, the three-month forward price at Time t is equal to F₁ = FV(S) = 110(1+0.05)0.25= 111.3499. Based on Fo 105, we find that the value of the existing forward entered at Time 0 and valued at Time t using the difference method (Equation 2) is: VPV[F, Fo](111.3499-105)/(1+0.05)0.25 -6.2729. Alternatively, using Equation 3 we have, V₁SPV[F]110-[105/(1+0.05)0.25]= 6.2729. EXAMPLE 3 Forward Contract Price with Underlying Cash Flows A US stock paying a $10 dividend in two months is trading at $1,000. Assume the US interest rate is 5% with annual compounding. 1. Based on the current stock price and the no-arbitrage approach, which of the following values is closest to the equilibrium three-month forward price? A. $1,002.23 B. $1,022.40 C. $1,025.31 Solution to 1: A is correct. Based on the information given, we know So $1,000, r 5% (annual compounding), and T = 0.25. After 2 months, we will receive the benefit of a $10 dividend, which earns interest for 1 month. Therefore, FoFV(S)-FV(D)-1,000(1+0.05)3/12-10(1+0.05)1/12 $1,012.2722-$10.0407-$1,002.2315. Using Equation 4, we could have arrived at the same result. Here CC = 0, and CB, is the PV of the dividend at time 010/(1+0.05)2/12 = $9.919. Then, FoFV(SCCCB) FV (1,000+0-9.919) (990.081)×(1+0.05)3/12=$1,002.23. 2. If the dividend is instead paid in one month, the three-month forward price will: A. decrease. B. increase. C. be unchanged. Solution to 2: A is correct. The benefit of the dividend occurs one month earlier, so we can collect interest for one additional month. The future value of the dividend would be slightly higher. So, the forward price would decrease slightly, FoFV(S)-FV(D) 1,000(1+0.05)3/12-10(1+0.05)2/12 $1,012.2722-$10.0816-$1,002,1906. EXAMPLE 4 Forward Contract Price with Carry Costs and Benefits 1. A long one-year forward contract on a productive asset was entered at a forward price of €1,000. Now, seven months later, the underlying asset is selling for $1,050. The PV of the cost to store, insure, and maintain the asset for the next 5 months is €4.00, and the asset will generate income over the next 5 months with a PV of €28.00. Assume annual compounding for all costs and benefits and a risk-free rate of 2%. Based on the current spot price and the no-arbitrage approach, which of the following values is closest to the equilibrium five-month forward value? A. €34.22 B. 633.50 C €35.94 Solution to 1: A is correct. Based on the information given, we know the following: F = 1,000, S, 1,050, CC, 4, CB, 28, 7 months. T-t-5 months, and r=2%. The new forward price is F, FV(S,CCCB). So, with annual compounding, we have: F(1,050+4-28)(1+0.02)5/12€1,034.50 and V-PV[F,F)-[C1,034.50 €1000]/(1+0.02)3/12-€34.22. EXAMPLE 5 Equity Futures Contract Price with Continuously Compounded Interest Rates 1. The continuously compounded dividend yield on the EURO STOXX 50 is 3%, and the current stock index level is 3,500. The continuously compound- ed annual interest rate is 0.15%. Based on the carry arbitrage model, the three-month futures price will be closest to A. 3,473.85. B. 3,475.15. C. 3,525.03. Solution: B is correct. Based on the carry arbitrage model (see Equation 4), the futures price is SpexpCC-Cвіт We assume the carry costs (CC) are 0 for a financial asset, such as a stock index. The carry benefit (CB), in this case a 3% continuous dividend yield, is greater than the financing cost r (0.15%), so the futures price will be below the spot price. The futures price, the future value of the underlying adjusted for carry (Le., the dividend payments, over the next 3-months) is: fa-3,500exp 00150-001(3/12)-3,475.15. EXAMPLE 6 Equity Forward Pricing and Forward Valuation with Discrete Dividends 1. Suppose Nestlé common stock is trading for CHF70 and pays a CHF2.20 dividend in one month. Further, assume the Swiss one-month risk-free rate is 1.0%, quoted on an annual compounding basis. Assume that the stock goes ex-dividend the same day the single stock forward contract expires. Thus, the single stock forward contract expires in one month. The one-month forward price for Nestlé common stock will be closest to: A. CHF66.80. B. CHF67.86. C. CHF69.94. Solution: B is correct. In this case, we have Sp=70, r = 1.0%, T= 1/12, and FV(CB) = 2.20CB. Therefore, FoFV(S+CCCB)FV(S) + FV (CC)-FV(CB) 70(1+0.01)1/12+0-2.20CHF67.86.