Investments, tenth edition

  Excel Questions 

    1.  Experiment with different values for both income yield and 

interest rate. What happens to the size of the time spread 

(the difference in futures prices for the long versus short 

maturity contracts) if the interest rate increases by 2%?  

   2.  What happens to the time spread if the income yield 

increases by 2%?  

   3.  What happens to the spread if the income yield equals the 

interest rate?    

Spot Futures Parity and Time Spreads

Spot price

Income yield (%)

Interest rate (%)

Today’s date

Maturity date 1

Maturity date 2

Maturity date 3

Time to maturity 1

Time to maturity 2

Time to maturity 3

100

100.00

101.26

101.58

102.66

2

4.5

5/14/09

11/17/09

1/2/10

6/7/10

0.51

0.63

1.06

Futures prices versus maturity

Spot price

Futures 1

Futures 2

Futures 3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A

B

C

D

E

1500

1520

1540

1560

1580

1600

1620

1640

1660

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

Date in July 2012

F

u

tu

re

s P

ri

ce

, $

/o

u

n

ce

Dec ‘12  delivery

June ‘13  delivery

Dec ‘13  delivery

 Figure 22.6 

Gold futures prices 

bod61671_ch22_770-798.indd   790

bod61671_ch22_770-798.indd   790

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 

791

  Forward versus Futures Pricing 

 Until now we have paid little attention to the differing time profile of returns of futures and 

forward contracts. Instead, we have taken the sum of daily mark-to-market proceeds to the 

long position as  P  

 T 

   2   F  

0

  and assumed for convenience that the entire profit accrues on the 

delivery date. Our parity theorems apply only to forward pricing because they assume that 

contract proceeds are in fact realized only on delivery. In contrast, the actual timing of cash 

flows conceivably might affect the futures price. 

 Futures prices will deviate from parity when marking to market gives a systematic 

advantage to either the long or short position. If marking to market tends to favor the long 

position, for example, the futures price should exceed the forward price, because the long 

position will be willing to pay a premium for the advantage of marking to market. 

 When will marking to market favor either a long or short trader? A trader will benefit 

if daily settlements are received (and can be invested) when the interest rate is high and 

are paid (and can be financed) when the interest rate is low. Because long positions will 

benefit if futures prices tend to rise when interest rates are high, they will be willing to 

accept a higher futures price. Therefore, a positive correlation between interest rates and 

changes in futures prices implies that the “fair” futures price will exceed the forward price. 

Conversely, a negative correlation means that marking to market favors the short position 

and implies that the equilibrium futures price should be below the forward price. 

 For most contracts, the covariance between futures prices and interest rates is so low 

that the difference between futures and forward prices will be negligible. However, con-

tracts on long-term fixed-income securities are an important exception to this rule. In this 

case, because prices have a high correlation with interest rates, the covariance can be large 

enough to generate a meaningful spread between forward and future prices.    

  So far we have considered the relationship between futures prices and the  current   spot 

price. What about the relationship between the futures price and the  expected value   of 

the spot price? In other words, how well does the futures price forecast the ultimate spot 

price? Three traditional theories have been put forth: the expectations hypothesis, normal 

backwardation, and contango. Today’s consensus is that all of these traditional hypotheses 

are subsumed by modern portfolio theory.  Figure 22.7  shows the expected path of futures 

under the three traditional hypotheses.