Question

1. One important property of a special floater (when coupon rate = discount rate) is that its Macaulay duration is the same as the Macaulay duration of a one-year zero. We use this question to illustrate why.

For all parts of this question, we assume that coupon payment is annual and the principal is $1,000. Also assume that LIBOR0 (LIBOR rate at t=0) is 3%. The discount rate of fixed cash flows is 5% at t=0.

a. A floater has a coupon rate of LIBOR + 2%. Its discount rate is also LIBOR + 2%. Assume that its time to maturity is two years. What is the price of this bond at t=0? (Hint: write down the pricing equation using the discounted cash flow method and simply it.)

b. At t=Δ (you can think of this as one second after t=0), LIBORΔ becomes 4% (increases by 1% from 3%). What is the price of this floater at t=Δ?

c. What is the price of a one-year zero-coupon bond? This bond pays $1,050 at t=1.

d. What is the price of the one-year zero of part c if the discount rate increases from 5% to 6%?

e. Recall that definition of modified duration. Modified duration ≈ − ∆? ? . Using this∆? ?

equation, what is the modified duration of the floater of parts a and b? What is the modified duration of the one-year zero of parts c and d? (Based on the properties, the Macaulay duration is 1 for both bonds. The above equation gives you an approximated modified duration, which should be roughly equal to 1/(1+y).)

f. For any new discount rate yΔ at t=Δ, the price of the one-year zero is ????. What is the?+??

pricing equation of the floater for any given LIBORΔ?

g. The discount rate of the floater from t= Δ to t=1 is LIBORΔ+2%. The discount rate of the one-year zero for the same period is yΔ. Do you see any similarity between the two pricing equations of part f?

Need answers to all but g in particular

Answer 1

(a) Given coupon rate is same as the discount rate, the bond is valued at par. The price is $1,000.

(b) In case, the LIBOR changes to 4%, the discount rate remains the same as the coupon rate. The price of the bond doesn't change, but remains the same at $1,000.

(c) In case LIBOR is 3%, discount rate is 5%. The price of the one year zero-coupon bond is $1,000: -

(d) The new price of the one year zero coupon bond is $990.566: -