Question

An investor owned a 9 percent annual payment coupon bond for six years that was originally purchased at a 9 percent required return. She did not reinvest any coupons (she kept the money under her mattress). She redeemed the bond at par. What was her annual realized rate of return? What if she did reinvest the coupons but only earned 5 percent on each coupon? Why are your answers not equal to 9 percent?

Answer 1

Computation of Annualized rate of return

Let the Face Value of the Bond = $ 1000

Since the Coupon rate and required rate of return is same so the Bond will trade at Par . So We can purchase the Bond at face value.

Given Coupon rate = 9%

Given that Coupons are not reinvested every Year, so we Cannot gain the interest on Coupon amount

Time period = 6 Years

Coupon amount = Face value * Coupon rate

= $ 1000*9%

= $ 90

Cash flow from Bond = 6* Coupon amount + Maturity amount

= 6*$ 90+$ 1000

= $ 540+$ 1000

=$ 1540

We know that Future Value = Present Value ( 1+i)^n

Here i = Rate of interest

n = No.of Years

$ 1540= $ 1000( 1+i)^6

$ 1540/$ 1000 = ( 1+i)^6

1.54 = ( 1+i)^6

1.54^1/6 = 1+i

1.074616= 1+i

1.074616-1 = i

0.0746=i

Hence Annualized rate of interest is 7.46%

When the Reinvestment rate is 5%

Cash flow derived from the bond = Interest amount * FVIFA( 5%,6) + Maturity amount

= $ 90* 6.801913+$ 1000

= $ 612.1722+$ 1000

= $1612.17

So the Future Value is $ 1612.17

We know that Future Value = Present Value ( 1+i)^n

Here I = Rate of interest

n = No.of Years

$ 1612.17= $ 1000( 1+i)^6

$ 1612.17/$ 1000= ( 1+i)^6

$ 1.61217= (1+i)^6

$ 1.61217^1/6 = 1+i

1.08285= 1+i

1.08285-1= i

0.08285= i

Hence the Annualized rate of return is 8.285%

The Annualized rate of return is less than required rate of return i.e 9% because investors did not reinvest the Coupons at the required rate of return. In order to earn the 9% required return we should reinvest Coupons at 9% rate.