Question

a) You are considering investing in bonds and have collected the following information about the prices of a 1-year zero-coupon bond and a 2-year coupon bond.

- The 1-year discount bond pays $1,000 in one year and sells for a current price of $950.

- The 2-year coupon bond has a face value of $1,000 and an annual coupon of $60. The bond currently sells for a price of $1,050.

i) What are the implied yields to maturity on one- and two-year discount bonds?

ii) What is the implied forward rate between years 1 and 2?

iii) Consider a 2-year annuity with annual coupon payments of $800. What is the most that you would be willing to pay for this annuity?

b) A 5%, $1,000 bond makes coupon payments on June 15 and December 15 and is trading with a YTM of 4% (APR). The bond is purchased and will settle on August 21 when there will be four coupons remaining until maturity. Calculate the full price of the bond using actual days.

Answer 1

a)

i) Let the rate today for one year and 2 year be y1 and y2

y1 is the implied YTM of the one year discount bond and y2 is the YTM of the 2 year discount bond

950=  1000/(1+y1)

=> y1 = 0.052632 or 5.26% which is the implied YTM of the one year bond

From 2 year coupon bond

60/(1+y1) +1060/(1+y2)^2 = 1050

.=> 57+1060/(1+y2)^2 = 1050

=> (1+y2)^2 =1.067472

=> y2 =0.033186 or 3.32%

So, implied ytm of the 2 year discount bond is 3.32%

ii) implied forward rate between year 1 and year 2= (1+y2)^2/(1+y1) -1 = 0.014099 or 1.41%

iii) Price of annuity = 800/(1+y1) +800/(1+y2)^2 = 800/1.052632+800/1.067472 = $1509.43

For the 2 year annuity , I will pay an amount of $1509.43 at the most.

iv)

Coupon amount = $1000*5%/2 = $25

Semiannual YTM = 4%/2 =2%

So, clean price of the bond with 4 payments remaining

= 25/1.02+25/1.02^2+25/1.02^3+25/1.02^4+1000/1.02^4

=$1019.04

So full price of the bond = dirty price = Clean price + accrued interest

Accrued Interest (from June15 to August 21 i.e. 67 days) = 67/365*$1000*5% = $9.18

So, Full price = $1019.04+$9.18 = $1028.22