Investments, tenth edition

  Spreads 

 Just as we can predict the relationship between spot and futures prices, there are similar 

ways to determine the proper relationships among futures prices for contracts of different 

  

6

 The very high dividend yield in November 2004 was due to Microsoft’s special one-time dividend of $3 per share. 

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D

ivi

de

nd Y

ie

ld (

A

nnua

li

z

e

d

, %

)

 Figure 22.5 

S&P 500 monthly dividend yield 

bod61671_ch22_770-798.indd   788

bod61671_ch22_770-798.indd   788

7/27/13   1:48 AM

7/27/13   1:48 AM

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  C H A P T E R  

2 2

 Futures 

Markets 

789

maturity dates. Equation 22.2 shows that the futures price is in part determined by time 

to maturity. If the risk-free rate is greater than the dividend yield (i.e.,  r  

 f 

  >  d ), then the 

futures price will be higher on longer-maturity contracts and if  r  

 f 

    <    d,   longer-maturity 

futures prices will be lower. You can confirm from  Figure  22.1  that in 2013, when the 

risk-free rate was below the dividend yield, the longer-maturity S&P 500 contract did 

have a lower futures price than the shorter term contract. For futures on assets like gold, 

which pay no “dividend yield,” we can set  d   5  0 and conclude that  F must  increase as 

time to maturity increases. 

 To be more precise about spread pricing, call  F ( T  

1

 ) the current futures price for delivery 

at date  T  

1

 , and  F ( T  

2

 ) the futures price for delivery at  T  

2

 . Let  d  be the dividend yield of the 

stock. We know from the parity Equation 22.2 that 

   F(T

1

) 5 S

0

(1 1 r

f

2 d)

T

1

 F(T

2

) 5 S

0

(1 1 r

f

2 d)

T

2

  

 As a result, 

  F(T

2

)/F(T

1

) 5 (1 1 r

f

2 d)

(T

2

2T

1

)

  

 Therefore, the basic parity relationship for spreads is 

 

  F(T

2

) 5 F(T

1

)(1 1 r

f

2 d)

(T

2

2T

1

)

  

(22.3)   

 Equation 22.3 should remind you of the spot-futures parity relationship. The major difference 

is in the substitution of  F ( T  

1

 ) for the current spot price. The intuition is also similar. Delaying 

delivery from  T  

1

   to   T  

2

  assures the long position that the stock will be purchased for  F ( T  

2

 )  dol-

lars at  T  

2

  but does not require that money be tied up in the stock until  T  

2

 . The savings realized 

are the net cost of carry between  T  

1

   and   T  

2

 . Delaying delivery from  T  

1

  until  T  

2

  frees up  F ( T  

1

 ) 

dollars, which earn risk-free interest at  r  

 f 

 . The delayed delivery of the stock also results in the 

lost dividend yield between  T  

1

  and  T  

2

 . The net cost of carry saved by delaying the delivery is 

thus  r  

 f 

   2   d.  This gives the proportional increase in the futures price that is required to com-

pensate market participants for the delayed delivery of the stock and postponement of the pay-

ment of the futures price. If the parity condition for spreads is violated, arbitrage opportunities 

will arise. (Problem 19 at the end of the chapter explores this possibility.)  

 To see how to use Equation 22.3, consider the following data for a hypothetical contract: