# Solutions for Mcdonald Chapter 6 Essays

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

F0,T

⇔ 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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Suppose gold cannot be loaned. Then our cash and carry “arbitrage” is: Transaction Short forward Buy gold Borrow @ 0.05 Total Time 0 Time T = 1 0 310.686 − ST −$300 ST $300 −$315.38 0 −4.6953

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Chapter 6 Commodity Forwards and Futures

The forward price bears an implicit lease rate. Therefore, if we try to engage in a cash and carry arbitrage, but if we do not have access to the gold loan market, and thus to the additional revenue on our long gold position, it is not possible for us to replicate the forward price. We incur a loss. c) If gold can be loaned, we engage in the following cash and carry arbitrage: Transaction Time 0 Time T = 1 Short forward 0 310.686 − ST Buy tailed gold −$295.5336 ST position, lend