Essay Solutions for Mcdonald Chapter 6

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Chapter 6
Commodity Forwards and Futures
Question 6.1. The spot price of a widget is $70.00. With a continuously compounded annual risk-free rate of 5%, we can calculate the annualized lease rates according to the formula: F0,T = S0 × e(r−δl )×T ⇔ F0,T S0 = e(r−δl )×T S0 = (r − δl ) × T F0,T 1 ln T S0

⇔ ln


⇔ δl = r −

Time to expiration Forward price Annualized lease rate 3 months $70.70 0.0101987 6 months $71.41 0.0101147 9 months $72.13 0.0100336 12 months $72.86 0.0099555 The lease rate is less than the risk-free interest rate. The forward curve is upward sloping, thus the prices of exercise 6.1. are an example of contango. Question 6.2. The spot price of oil is $32.00 per barrel. With a continuously compounded annual
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It does not make sense to store pencils in equilibrium, because even if we have an active lease market for pencils, the lease rate is smaller than the risk-free interest rate. Lending money at ten percent is more profitable than lending pencils at five percent. b) The equilibrium forward price is calculated according to our pricing formula: F0,T = S0 × e(r−δl )×T = $0.20 × e(0.10−0.05)×1 = $0.20 × 1.05127 = $0.2103, which is the price given in the exercise. c) c1) Let us first look at the different arbitrage strategies we can use in each case. Pencils can be sold short. We can engage in our usual reverse cash and carry arbitrage: Transaction Time 0 Time T = 1 Long forward 0 ST − F0,T Short-sell tailed pen- $0.19025 −ST cil position, @ 0.05 Lend short-sale −$0.19025 $0.2103 proceeds @ 0.1 Total 0 $0.2103 − F0,T For there to be no arbitrage, F0,T ≥ $0.2103 c2) Suppose pencils cannot be sold short. Then we have no ability to create the short position necessary to offset the pencil price risk from the long forward. Consequently, we are not able to find a lower boundary for the pencil forward in this case.

Part 2 Forwards, Futures, and Swaps


Pencils can be loaned. We engage in a cash and carry arbitrage: Transaction Time 0 Short forward 0 Buy tailed pencil −$0.19025 position, lend @0.05 borrow @ 0.1 $0.19025 Total 0 Time T = 1 F0,T − ST ST −$0.2103 F0,T − $0.2103

For there
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