Question

Bill buys a 10-year $1000 par value 6% bond with semi- annual coupons. The price assumes a nominal yield of 6%, compounded semi-annually. As Bill receives each coupon payment, he immediately puts the money into an account earning interest at an annual effective rate of i. At the end of 10 years, immediately after Bill receives the final coupon payment and the redemption value of the bond, Bill has earned an annual effective yield of 7% on his investment in the bond. Calculate i.

Answer 1

To determine the annual effective interest rate (i) that reinvested bond coupon payments need to earn for Bill to achieve a 7% effective annual yield over 10 years, we utilise the future value annuity formula. As there is no direct algebraic way to solve for i, we resort to iterative methods or financial calculators to find an approximate rate that matches the desired future value.

Step-by-step explanation:

The student's question involves calculating the effective annual interest rate (i) that a bond investor, Bill, would need to earn on reinvested coupon payments to achieve an effective annual yield of 7% over 10 years, given that the bond has a 6% nominal yield, compounded semi-annually. Our goal is to determine the rate i when Bill reinvests the semi-annual coupon payments of a $1000 par value bond at this unknown rate.

To find this rate, we must use the future value of an ordinary annuity formula. Each coupon payment is $30 (6% of $1000 divided by 2 because of semi-annual payments), and there are 20 payments over 10 years.

The future value of these reinvested payments at rate i, along with the $1000 redemption value, must equal the future value of the entire investment at a 7% yield.

We set up the equation considering the future value of an annuity (FV₀₁ = (Pmt * ((1 + i)n - 1) / i) where Pmt is the payment per period, n is the number of periods, i is the interest rate per period, and FV₀₁ is the future value of the ordinary annuity:

FV₀₁ (reinvested coupons at rate i) + $1000 = Lump sum equivalent at 7% yield

The lump sum equivalent can be calculated using the formula for compound interest: P * (1 + r)n

However, finding i requires iterative methods or financial calculators because we're solving for the interest rate in an annuity equation, which does not have a direct algebraic solution.

The approximate value of i can be obtained by trying different rates until the left-hand side of the equation comes close enough to the future value of Bill's investment compounded at an annual effective rate of 7% over 10 years.