Question 7.8: Suppose an investor purchases a 5-year €10,000 bond when it ......
Suppose an investor purchases a 5-year €10,000 bond when it is issued on January 1, 2020 at a coupon rate of 6% with interest paid bi-annually, and the market rate of interest is 4%. Determine the present value of the bond on January 1, 2020.
The present value of the bond is determined from the present value of the interest payments and the present value of the bond maturity amount with respect to the market rate of interest.
Value = Pres Val Int . Payments + Pres Val. Maturity
We discussed the present value of annuities in section 7.4, and so the present value of the interest payments is determined from the following formula:
P=\frac{A}{i}\biggl[1-\frac{1}{(1+i)^{n}}\biggr]The bond is for 5 years with two payments of interest each year, and so n = 5 ∗ 2 = 10. The market rate is 4% and so the interest rate i per interest period is 4%/2 = 2% = 0.02. The interest amount for each interest period is 10,000 ∗ 0.06 ∗ 0.5 = €300.
Therefore, the present value of the interest payments is given by
300/0.02(1-1/(1.02)^{10})=15000(1-0.8203)=€2694.77The present value of the maturity amount is given by the present value formula where n = 5 ∗ 2 = 10, and the interest rate i per interest period is 4%/2 = 2% = 0.02.
PV\ =\ M V\left(1\,+\,i\right)^{-n}= 10000(1.02)^{-10}
= €8202.48
Therefore, the present value of the bond on the issue date is given by the sum of the present value of the interest payments and the present value of the maturity amount:
= 2694.77 + 8202.48
= €10,903.25
Note
The reader will note that the present value of the bond (€10,903.25) is greater than the amount of the bond on the issue date. The market rate of interest (4%) is lower than the coupon rate (6%) and so it takes a greater investment to achieve the same return at 4% as the yield from the corporate bond at 6%. If the market rate of interest is higher than the coupon rate then the present value of the bond would be less than the value on the issue date.