CHAPTER 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 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 (i2t )2 2i2t 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 + it + i te+ it (i te+ 1) - = it + i te+ + it (i te+ 1) This calculation is derived by recognizing that after the first period, the $1 investment becomes 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 (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 Year it i 2t it ite ite 1 Year