Investments, tenth edition

  Modern Portfolio Theory 

 The three traditional hypotheses all envision a mass of speculators willing to enter either 

side of the futures market if they are sufficiently compensated for the risk they incur. 

Future Prices

Expectations Hypothesis

Contango

Normal Backwardation

Time

E(P

T

)

Delivery

Date

 Figure 22.7 

Futures price over time, in the special case that the 

expected spot price remains unchanged 

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  C H A P T E R  

2 2

 Futures 

Markets 

793

Modern portfolio theory fine-tunes this approach by refining the notion of risk used in the 

determination of risk premiums. Simply put, if commodity prices pose positive systematic 

risk, futures prices must be lower than expected spot prices. 

 To illustrate this approach, consider once again a stock paying no dividends. If  E ( P  

 T 

 ) 

denotes the expected time- T  stock price and  k  denotes the required rate of return on the 

stock, then the price of the stock today must equal the present value of its expected future 

payoff as follows:   

 

P

0

5

E(P

T

)

(1 1 k)

T

 

 (22.4)   

 We also know from the spot-futures parity relationship that   

 

P

0

5

F

0

(1 1 r

f

)

T

 

 (22.5)   

 Therefore, the right-hand sides of Equations 22.4 and 22.5 must be equal. Equating these 

terms allows us to solve for  F  

0

 :   

 

F

0

5 E(P

T

)

¢

1 1 r

f

1 1 k

≤

T

 

 (22.6)   

 You can see immediately from Equation 22.6 that  F  

0

  will be less than the expectation of  P  

 T 

  

whenever  k  is greater than  r  

 f 

 , which will be the case for any positive-beta asset. This means 

that the long side of the contract will make an expected profit [ F  

0

  will be lower than  E ( P  

 T 

 )] 

when the commodity exhibits positive systematic risk ( k  is greater than  r  

 f 

 ). 

 Why should this be? A long futures position will provide a profit (or loss) of  P  

 T 

   2   F  

0

 .  If 

the ultimate value of  P  

 T 

  entails positive systematic risk, so will the profit to the long posi-

tion. Speculators with well-diversified portfolios will be willing to enter long futures posi-

tions only if they receive compensation for bearing that risk in the form of positive expected 

profits. Their expected profits will be positive only if  E ( P  

 T 

 ) is greater than  F  

0

 .  The  short 

position’s profit is the negative of the long’s and will have 

negative systematic risk. Diversified investors in the short 

position will be willing to suffer that expected loss to 

lower portfolio risk and will be willing to enter the con-

tract even when  F  

0

  is less than  E ( P  

 T 

 ). Therefore, if  P  

 T 

     has 

positive beta,  F  

0

  must be less than the expectation of  P  

 T 

 . 

The analysis is reversed for negative-beta commodities.      

 What must be true of the risk of the spot price of 

an asset if the futures price is an unbiased esti-

mate of the ultimate spot price? 

 CONCEPT CHECK