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
