Question

3a. Assume that you pay $1,020 for a one-year coupon bond with a face value of $1,000 and a coupon payment of $50. Calculate the coupon rate, the yield to maturity and the current yield.

b. Assume that you pay $1,250 for a consol that has a coupon payment of $50. What is the yield to maturity? What is the current yield?

c. Assume that you hold the consol in part b for one year. When you sell it, the interest rate is equal to 4.1%. What is your rate of return?

d. Assume that you have a one-year coupon bond with a face value of $1,000 and a coupon payment of $50. What is the price of the bond if the yield to maturity is 6%?

e. Assume that you have the same bond is in part d, except instead of paying one annual payment of $50, the bond pays two semi-annual payments of $25 (one six months from now and another payment in twelve months). What is the price of this bond if the yield to maturity is 6%?

f. Use the approximation formula to calculate the yield to maturity for a ten year bond that has a coupon payment of $20, a face value of $1,000 and a price of $1,040.

g. Calculate the yield to maturity of a six-month zero coupon bond that has a price of $990 and a face value of $1,000.

Answer 1

a) Coupon rate is always calculated on the face value (or par value) of the bond. It doesn't depend on the price of the bond.

In this question, coupon = $50 and face value = $1000, so coupon rate = coupon / face value = 50/1000 = 0.05 = 5%

Yield to maturity is the annual return we get on a bond. So we purchased the bond for $1020 and we at the end of 1 year we received 50 (coupon) + 1000(face value) = 1050 .

Thus return on the bond or we can say YTM = (1050/1020) - 1 = 1.029-1 = 0.029 = 2.9%

Current Yield on a bond is equal to coupon rate / current market price = 50/1020 = 0.049 = 4.9%

b) A consol is a bond that has no maturity. It only pays regular coupon payments.

Current yield for a consol = coupon/current market price = 50/1250= 0.04 = 4%

Now as a consol doesn't have any maturity, so it doesn't have any yield to maturity.

c) We know the price at which we bought the console which is $1250. After 1 year, Price of Consol = Coupon/ Interest rate = 50/0.041 = 1219.5

So after 1 year, we get a coupon of 50 and a selling price of 1219.5 i.e. a total of 50+1219.5 = 1269.5

Rate of return = (1269.5/1250 ) -1 = 0.0156 = 1.56%

d) Price of a bond is the present value of all future payments discounted at yield to maturity. In this question, Maturity of bond is of 1 year, so after 1 year we get coupon as well as face value i.e. a total of 50+1000 =1050. We will discount this 1050 at 6%.

So value of bond = 1050/1.06 = 990.57