Question

An investor has two bonds in his portfolio that have a face value of $1,000 and pay a 9% annual coupon. Bond L matures in 15 years, while Bond S matures in 1 year. a. What will the value of the Bond L be if the going interest rate is 5%, 7%, and 10%? Assume that only one more interest payment is to be made on Bond S at its maturity and that 15 more payments are to be made on Bond L. Round your answers to the nearest cent. 596 10% $ Bond L Bond S b. Why does the longer-term bond's price vary more than the price of the shorter-term bond when interest rates change? 1. Long-term bonds have greater interest rate risk than do short-term bonds. II. The change in price due to a change in the required rate of return decreases as a bond's maturity increases. III. Long-term bonds have lower interest rate risk than do short-term bonds. IV. Long-term bonds have lower reinvestment rate risk than do short-term bonds. V. The change in price due to a change in the required rate of return increases as a bond's maturity decreases -Select

Answer 1

Answer:

Price of L bond at 5 percent required rate of return = $1,415.16

Price of L bond at 7 percent required rate of return = $1,182.16

Price of L bond at 10 percent required rate of return = $923.94

The price of the long term bonds change more with a change in interest rate because the long term bonds have a greater interest rate risk as compared to the short term bonds

Explanation:

L bond has a coupon rate of 9 percent, a face value of $1,000 and matures in 15 years. The coupon payments are made on annual basis. At the time of maturity the bondholder gets the face value.

We can find the present value of the coupon payments using the present value of annuity formula and the present value of the face value to be received after fifteen years using the present value formula. Sum of the present value of annuity of coupon payments and present value of the face value should equal the fair value (price) of the bond.

If the required rate of return is 5 percent, the price of the bond can be computed as under

Price = PMT [[(1+i)^n] -1]/[ix(1+i)^n] + FV/(1+i)^n

where PMT = 1,000 x 9% = $90

n = 15 years, i = 5% and FV = $1,000

Plugging the values in the formula we get

Price = 90[{(1+0.05)^15} - 1]/ [0.05 x (1+0.05)^15] + 1,000/(1+0.05)^15

Price = 90[{(1.05)^15} - 1]/ [0.05 x (1.05)^15] + 1,000/(1.05)^15

Price = 90[2.07893 - 1]/ [0.05 x 2.07893] + 1,000/2.07893

Price = 90[1.07893]/ [0.10395] + 1,000/2.07893

Price = 934.14 + 481.02 = 1,415.16

If the required rate of return increases to 7 percent, the price is computed as under

Price = 90[{(1+0.07)^15} - 1]/ [0.07 x (1+0.07)^15] + 1,000/(1+0.07)^15

Price = 90[{(1.07)^15} - 1]/ [0.07 x (1.07)^15] + 1,000/(1.07)^15

Price = 90[2.759 - 1]/ [0.07 x 2.759] + 1,000/2.759

Price = 90[1.759]/ [0.19313] + 1,000/2.759

Price = 819.71+ 362.45 = 1,182.16

If the required rate of return increases to 10 percent, the price is computed as under

Price = 90[{(1+0.1)^15} - 1]/ [0.1 x (1+0.1)^15] + 1,000/(1+0.1)^15

Price = 90[{(1.1)^15} - 1]/ [0.1 x (1.1)^15] + 1,000/(1.1)^15

Price = 90[4.1772 - 1]/ [0.1 x 4.1772] + 1,000/4.1772

Price = 90[3.1772]/ [0.41772] + 1,000/4.1772

Price = 684.55+ 239.39 = 923.94

The price of the long term bonds change more with a change in interest rate because the long term bonds have a greater interest rate risk as compared to the short term bonds