# The three year bond has face value $100, and pays \$5 coupons annually, the last one at maturity. Assume that the continuously compounding rate is 7%.

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 $\$94.75$? For part b I don't understand what I have to do to solve the question? ## 1 Answer You are confusing compounding with continuously compounding. The former uses the 7% you have used. The latter would require you to calculate a compounded rate using (e^annual_rate)-1 Once you have that rate, you can use it in the formula you used. On the second part: At the end of each year, as the investor you'll get a lump sum of money - a coupon payment from each bond you bought at the beginning of the period. Assuming you buy new bonds with the same features as your t0 bonds, at the end of year 2, you'll get money from t0 bonds held over the 2 year period AND the bonds you purchased at the end of year 1 etc. So every year you hold the bonds you get coupons based on the compounded interest using the continuously compounded rate. You then buy more bonds with that money to add to your holding. At the end of the holding period, you get the last set of coupons (which you will not reinvest) and you also get the face value. So: At the end of year 1 you hold n-bonds. At the end of year 1 you get n-bonds*coupon. You then buy m-bonds At the end of year 2 you hold n-bonds+m-bonds. At the end of year 2 you get the coupon of year 2 on n-bonds + coupon of year 1 on n-bonds.... Coupon calculation as I described above. Hope it helps. Last point: when you buy a 3-year bond a year after issuance, you get a 2-year bond...etc. • so$e^{0.07}-1 = 0.0725$I would just replace this rate with the rate I used in the my question to get the correct answer? The formula (e^annual_rate)-1, what is it called because I want to look it up and see how it's derived, it will make remembering it easier. Feb 15 '16 at 0:18 • You're looking for continuously compounded interest rate. Plenty of examples on Google :) a good one is mathsisfun.com/money/compound-interest-periodic.html Feb 15 '16 at 10:12 • If I understood correctly, for part b, the investor has 1000 and with that money he is able to buy 10 bonds at a face value of 100 each. At the end of the first year he will reveive$\frac{5}{(1+0.0725)}=4.66$from each bond, since he bought 10 bonds he will make 46.62 dollars, that is not enough to reinvest in anither bond since the face value is 100 dollars? Feb 15 '16 at 17:25 • Ok. I hope this will help. If the bond is trading at a discount (ie not 100), then you'll have some change left. If I'm not mistaken, you'll have about$60 USD. At the end of year 1 you'll get another \$50. Hence, assuming bond price < 110 you could buy another bond... Feb 15 '16 at 17:45
• the question doesn't say anything about the bond trading at a discount? It says the bond has a face value of 100 USD, where did you get 60 from? Feb 15 '16 at 17:49