# Coupon bond pricing problem with reinvestment

The three year bond has face value USD 100, and pays USD 5 coupons annually, the last one at maturity. Assume that the continuously compounding rate is 7%.
(a) Find the price of this bond.
(b) Consider the investor who invests 1000 in these bonds. Each year after the coupon payments are issued, the investor buys the bonds from that money. What is the amount of money that the investor receives at the maturity of the bonds?

Is this correct for part a? in class I learned bond price = $\frac{C}{1+r}+\frac{C}{(1+r)^2}+...+\frac{C+FaceValue}{(1+r)^2}$
where c = coupon payment and r = interest rate

$$\frac{5}{1+0.07}+\frac{5}{(1+0.07)^2}+\frac{105}{(1+0.07)^3} = 94.75$$
Is the correct price of the bond $\$94.75$? I am having trouble with part b, from what I understand and investor is able to buy 10 bonds at 100 USD face value with his 1000 USD, then after one year he wants to buy more bonds from his coupon payments. After one year he will get$\frac{5}{1+0.07}=4.67$from each bond he bought, since he bought 10 bonds he will have 46.73, this is not enough to reinvest for another bond since hte face value is 100 USD, even if he waits after year 2 it still won't be enough. What am I doing wrong? • In part (a) you say the rate is 7% but you discount at 7.25% – Alex C Feb 15 '16 at 18:11 • I thought I needed to change it to a continous rate with$e^{0.7}-1$, If I don't need to do that then all I have to do is change the 0.0725 to 0.07 in my equation, would that give me the correct answer? And I am still confused on part b – idknuttin Feb 15 '16 at 18:14 • OK, I understand what you did – Alex C Feb 15 '16 at 19:01 ## 1 Answer In part (a) use discount rate$e^.07 -1 = .072508181$to get the right answer. For part (b) I am just giving you hint: 1. Calculate bond price at the end of 1st year and 2nd year in the same way as you did in part (a). 2. Use the above calculated price to buy bond from the dividend at the end of first and second year. You may assume bond can be purchased in fraction. 3. At the end of 2nd year donot forget to consider the dividend received on the bond purchased in the first year. 4. Calculate the maturity value at the end of 3rd year. • two questions, 1. if the question is telling me the continuous compounding rate is$7\%$, what would you call 0.0725? 2. Is$\frac{5}{1+0.0725}=4.66$how much money you will receive for one bond after year 1 (the coupon payment)? Is the value of the bond after one year$PV=\frac{FV}{1+r}$or$100=\frac{FV}{1+.0725}$where the future value equals 107.25 USD? – idknuttin Feb 15 '16 at 18:38 • @idknuttin .0725 is annually compounding interest rate which is equivalent to .07 continuous compounding rate. – Neeraj Feb 15 '16 at 18:47 • Further, You will receive$5$for one bond at the end of one year.$4.66\$ is present value of this coupon payment. – Neeraj Feb 15 '16 at 18:48
• To compute the value of bond at the end of one year, just assumed one year has elapsed, and now discount only future cash flow only. – Neeraj Feb 15 '16 at 18:49
• @idknuttin if your calculation is accurate, then you are right. But it is better to say that investor would get .536 units of bond instead of saying 53.6\%. So after the end of one year he would have 10+.536=10.536 units of bonds. So, at the end of second year he will get 10.536*5 coupon payment. Now use this coupon payment to buy further bond at prevailing price (ie price at the end of 2nd year). And finally calculate your maturity value. – Neeraj Feb 16 '16 at 12:31