Question

Please upload your calculations (e.g., an Excel spreadsheet, a Word document, a PDF file, or a picture of your work) for Q21.

Q21. Consider a 2-year bond. The coupon rate of the bond is 10%, and the bond pays coupons semiannually. The bond is selling at a yield to maturity of 8.0% annually, or 4.0% semi-annually.

(a) What is the duration of the bond measured in half-years and in years? (10 points)

(b) If the semi-annually yield changes from 4.0% to 5.0%, what is the predicted change in the price of the bond (a dollar amount) using duration? (7 points)

(c) Suppose you are the company that is issuing this coupon bond. To immunize your liability, you would like to invest in a portfolio consists of one-year zero-coupon bonds and three-year zero-coupon bonds. What are the durations of the one-year zero and the three-year zero, respectively? What weight of the one-year zero will you need to hold for immunization? (5 points)

Answer 1

a. The formula for duration is: Duration = Summation(nxPVn)/Summation(PVn). So, according to the cash flows we calculate the duration. We will have 4 cashflows each in 6 months time.

Duration = (10/1.04 x 0.5 + 10/1.04^2 x 1 + 10/1.04^3 x 1.5 + 110/1.04^4 x 2)/(10/1.04 + 10/1.04^2 +10/1.04^3 + 110/1.04^4) = 1.76914 years.

Measured in half years it will be: Duration = (10/1.04 x 1 + 10/1.04^2 x 2 + 10/1.04^3 x 3 + 110/1.04^4 x 4)/(10/1.04 + 10/1.04^2 +10/1.04^3 + 110/1.04^4) = 3.538

b. TO calculate this, we need to know the effective duration which is = Duration/(1+yield) = 1.76914/1.04= 1.7010. So, an increase in 1% yield will cause a decrease in the bond pricee by 1.7010%. The present value of the bond = (10/1.04 + 10/1.04^2 +10/1.04^3 + 110/1.04^4) = 121.779. So, the dollar amount change for a $100 face value bond will be = 121.779 x 0.01701 = $2.0714.

c. Duratio of a one year zero and a 3-yar zero is 1 and 3 respectively because there are no intermediate cash flows.

Let the one-year zero have the weight A. So, the three-year zero will have 1-A.

A + 3x(1-A) = 1.76914.

Solving, we have:

A = 61.543%.