Futures III

# Futures III - Econ 174 FINANCIAL RISK MANAGEMENT LECTURE...

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Econ 174 – FINANCIAL RISK MANAGEMENT LECTURE NOTES Foster, UCSD October 16, 2011 FUTURES III – PRICING THEORY A. The Determination of Forward and Futures Prices 1. A Perfect 1-Year Hedge: [Table 1] a) Sharon owns Q a units of asset A with a current value V(a) 0 = S 0 Q a , where S 0 is the current spot price of A. 1) She receives cash flow C per unit of A per year, or ΣC = C×Q a /yr. 2) She wants to hedge against changes in V(a) over the next year (T = 1). 3) There are futures contracts in exactly the same asset A with one-year deliveries. Each contract is for q a units of A. The current futures price is \$F 0 per unit. 4) To fix ideas, use the values in Table 1. b) Sharon shorts 4 futures contracts in A for one-year deliveries at F 0 = \$2.97. c) Sharon keeps her position open and delivers 500 × 4 = 2000 units of A on the contracts in exactly one year, receiving F 0 per unit delivered, or \$2.97 × 500 × 4 = \$5,940. She also collected ΣC = \$300 in cash flows during the year. 1) She ends up with total final value Y T = F 0 × Q a + ΣC = 5,940 + 300 = \$6,240. Note that this can be written as: Y T = [S T + (F 0 − S T )] × 2000 + ΣC = V(a) T + π short + ΣC 2) She started with an investment value of V(a) 0 = \$3 × 2000 = \$6,000. 3) Table 2 shows ending investment value Y T for various S T . d) It is a perfect hedge. No matter what hap-pens to S T , Sharon makes overall rate of return r y = (6240-6000)/6000 = 4.0%. 2. The Spot-Futures Parity Theorem: a) From the above example, we see the following for a perfect one-year hedge: Table 1. Perfect Hedge Data Q a = 2000 Units of A owned S 0 = \$3.00 Spot price of A V(a) 0 = \$6,00 0 = S 0 × Q a C = \$0.15 Cash flow/unit of A/yr c (or d) = 5% Current Yield = C/S 0 ΣC = \$300 = C × Q a (\$/yr) F 0 (T) = \$2.97 Futures price of A q a = 500 Units per contract Table 2. Perfect Hedge Results S T = \$2.90 \$3.00 \$3.10 V(a) T = S T × 2000= \$580 0 \$600 0 \$620 0 π sh = (F 0 −S T ) × 2000 = \$140 −\$60 \$260 ΣC = \$300 \$300 \$300 Ending value Y T = \$624 0 \$624 0 \$624 0 [ ] [ ] 0 0 0 0 0 0 0 0 ) ( ) ( ) ( S S C F Q S Q S Q S F C Q S a V a V Y r a a a T a T T y - + = - - + Σ + = - =

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Ec 174 FUTURES III p. 2 of 17 b) Because r y is certain, it is a risk-free return. 1) Therefore, r y = r f . 2) This establishes the Spot-Futures Parity Theorem, which asserts that arbitrage will drive the futures price into line with spot prices according to the equation at right: c) Using the spot-futures parity equation. 1) C is the cash flow received by the owner of asset A, in \$/unit per year. 2) c = C/S 0 = “current yield,” the cash flow per dollar of asset value, in %/year. For stock or stock portfolio, C = dividends per share. For corporate or govt bond, C = annual coupon interest per bond. For a commodity like corn, C = 0.
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Futures III - Econ 174 FINANCIAL RISK MANAGEMENT LECTURE...

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