Question

(20pts) 5. The term structure of interest rates for zero-coupon bonds with $100 face value is shown below: Maturity 1 year 2 years 3 years YTM Price 4.60% 2 4.80% 2 5.00% 2 (5pts) (a) Find the current price of the zero-coupon bonds. (15pts) (b) Consider a three-year coupon bond with a $2000 face value that pays 10% annual coupons. Show that the price of this three-year bond must be equal to a portfolio of the above zero-coupon bonds. What must be the composition of this portfolio? Draw the necessary timelines and show your work. (See the Appendix to Chapter 6).

Answer 1

a)

Price of zero coupon bond with 1 year maturity= Face Value/(1+YTM)^year to maturity=100/(1+4.6%)^1=$95.60

Price of zero coupon bond with 2 year maturity= Face Value/(1+YTM)^year to maturity=100/(1+4.8%)^2=$91.05

Price of zero coupon bond with 3 year maturity= Face Value/(1+YTM)^year to maturity=100/(1+5%)^3=$86.38

b) Three year coupon bond have 1 initial price and 3 cash flow at the end of 1, 2 and 3 year. Cash flow related to this coupon paying bond is given below:

Price of Bond= Present Value year 1 cash flow+Present Value of year 2 cash flow+Present Value of year 3 cash flow

or, price of bond= 2000*10%/(1+4.6%)^1+ 2000*10%/(1+4.8%)^2+2000*(1+10%)/(1+5%)^3

or, price of bond=200/(1+4.6%)+200/(4.8%)^2+2200/(1+5%)^3

or price of bond= 2* 1 year zero coupon bond+2*2 year zero coupon bond+22*3 year zero coupon bond

Hence, composition of the 1 year, 2 year and 3 year zero coupon bond 2, 2 and 22 respectively.