Financial Markets and Institutions (2-downloads)

1. Purchase a one-year bond, and when it matures in one year, purchase another

one-year bond.

2. Purchase a two-year bond and hold it until maturity.

Because both strategies must have the same expected return if people are hold-

ing both one- and two-year bonds, the interest rate on the two-year bond must equal

the average of the two one-year interest rates.

The current interest rate on a one-year bond is 9%, and you expect the interest rate on

the one-year bond next year to be 11%. What is the expected return over the two years?

What interest rate must a two-year bond have to equal the two one-year bonds?

Solution

The expected return over the two years will average 10% per year ([9% + 11%]/2 = 10%).

The bondholder will be willing to hold both the one- and two-year bonds only if the expected

return per year of the two-year bond equals 10%. Therefore, the interest rate on the two-

year bond must equal 10%, the average interest rate on the two one-year bonds.

Graphically, we have:

E X A M P L E   5 . 2 Expectations Theory

We can make this argument more general. For an investment of $1, consider the

choice of holding, for two periods, a two-period bond or two one-period bonds. Using

the definitions

= today’s (time t) interest rate on a one-period bond

= interest rate on a one-period bond expected for next period (time t + 1) 

= today’s (time t) interest rate on the two-period bond 

the expected return over the two periods from investing $1 in the two-period bond

and holding it for the two periods can be calculated as

After the second period, the $1 investment is worth 

Subtracting the $1 initial investment from this amount and dividing by the initial

$1 investment gives the rate of return calculated in the previous equation. Because

11 ⫹ i

2t

2 11 ⫹ i

2t

2.

11 ⫹ i

2t

2 11 ⫹ i

2t

2 ⫺ 1 ⫽ 1 ⫹ 2i

2t

⫹ 1i

2t

2

2

⫺ 1 ⫽ 2i

2t

⫹ 1i

2t

2

2

i

2t

i

e

t

⫹ 1

i

t

Today

0

9%

10%

Year

1

Year

2

11%

---

100

Part 2 Fundamentals of Financial Markets

is extremely small—if 

, then 

—we can sim-

plify the expected return for holding the two-period bond for the two periods to

2i

2t

With the other strategy, in which one-period bonds are bought, the expected

return on the $1 investment over the two periods is

This calculation is derived by recognizing that after the first period, the $1 invest-

ment becomes 1 + i

t

, and this is reinvested in the one-period bond for the next

period, yielding an amount (1 + i

t

) (1 + 

). Then subtracting the $1 initial invest-

ment from this amount and dividing by the initial investment of $1 gives the

expected return for the strategy of holding one-period bonds for the two periods.

Because 

is also extremely small—if i

t

= 

= 0.10, then i

t

(

) = 0.01—

we can simplify this to

Both bonds will be held only if these expected returns are equal—that is, when

Solving for i

2t

in terms of the one-period rates, we have

(1)

which tells us that the two-period rate must equal the average of the two one-period

rates. Graphically, this can be shown as