3) Forward prices and convergence

a) Assume that 1 and 2 are the futures prices of two contracts on the same investment asset, with times to maturity 1and 2, 2 > 1. Prove that 2 = 1 (21) .

b) We observe from market data that, for a given non-dividend paying stock, 0 0 . What might explain the weak inequality in this relationship (i.e. why dont we observe 0 = 0 ?) Hint: in McDonald Ch. 5, see if you can identify a model for a friction that might cause cases where 0 < 0 .

c) At time = 0, a speculator takes a long position in a futures contract on stock i that will expire at time . Rearranging McDonald equation (6.5), the present value of this contract to the speculator is given by: 0 + 0 [ ]

Where is the risk-free rate, and the discount rate for the expected price of stock i at time T, 0[], is given by = = + ([] ), which is the CAPM required rate of return on the stock. Assume no-arbitrage pricing. Show analytically that if the return from stock i is positively correlated with the overall return on the stock market, the market risk premium is positive, and stock i is risky, then the futures market must be in backwardation.

d) Download the following data series:

(1) Monthly NYMEX futures prices for crude oil (light-sweet, Cushing, OK, Contract 1), 1983-2018, from www.eia.gov

(2) Monthly spot prices for crude oil (WTI Cushing, OK), 1986-2019, from www.eia.gov

(3) 1-month treasury bill rates (1-Month Treasury Constant Maturity Rate), monthly average, Jan. 2010 to Dec. 2018, from www.fred.stlouisfed.org

Combine the data series in a spreadsheet (make sure dates are aligned) for the years 2010 to 2018. Assume that futures prices are determined under no-arbitrage, and that storage costs are 12% annually. Assume the risk premium for crude oil is 6% annually when calculating expected spot prices ( = + 0.06). Derive an estimate for the monthly convenience yield, and generate a scatter plot showing these estimates over time. (Label your axes and include a title.) Pay close attention to units here.

Based on this scatter plot, what can you say about oil producers perceptions of the likelihood of a negative supply shock from 2016 to 2019?