THE ECONOMICS OF MONEY,BANKING, AND FINANCIAL MARKETS 155

CHAPTER 6

The Risk and Term Structure of Interest Rates

123

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
it

today s (time t) interest rate on a one-period bond

i te 1

interest rate on a one-period bond expected for next period (time t

i2t

today s (time t) interest rate on the two-period bond

1)

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
(1

i2t )(1

i2t )

1

1

(i2t )2

2i2t

1

2i2t

(i2t )2

After the second period, the $1 investment is worth (1
i2t )(1
i2t ). Subtracting
the $1 initial investment from this amount and dividing by the initial $1 investment
gives the rate of return calculated in the above equation. Because (i2t )2 is extremely
small if i2t 10% 0.10, then (i2t )2 0.01 we can simplify the expected return
for holding the two-period bond for the two periods to
2i2t
With the other strategy, in which one-period bonds are bought, the expected
return on the $1 investment over the two periods is
(1

it )(1

i te+ 1) - 1

1 + it + i te+ 1

it (i te+ 1) - 1 = it + i te+ 1 + it (i te+ 1)

This calculation is derived by recognizing that after the first period, the $1 investment
becomes 1 it , and this is reinvested in the one-period bond for the next period,
e
yielding an amount (1 it )(1 i t+1
). Then, subtracting the $1 initial investment
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
e
e
it (i et+1) is also extremely small if i t i t+1
0.10, then it (i t+1
)
0.01 we can
simplify this to
it +i et+1
Both bonds will be held only if these expected returns are equal, that is, when
2i2t

it

i et+1

Solving for i2t in terms of the one-period rates, we have
i2t

it + i et

2

1

(1)

which tells us that the two-period rate must equal the average of the two oneperiod rates. Graphically, this can be shown as:
Today
0

Year
1

it
i 2t

it

ite
2

ite 1
1

Year
2