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.
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
=> 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.
Coupon amount = $1000*5%/2 = $25
Semiannual YTM = 4%/2 =2%
So, clean price of the bond with 4 payments remaining
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
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