Question

A fund manager is managing a bond portfolio against her client's liabilities. The liability has a duration of 5 years and a market value of $100,000. The fund manager can immunize this liability using the following assets:

1) A perpetuity that pays $100 per annum.

2) A zero coupon bond with two years until maturity and a face value of $1000.

The market yield for bonds of all maturities is 10% p.a. Calculate how many of the zero coupon bonds (rounded to the nearest whole number) the investor will buy.

Answer 1

Step 5: Find the weights of investment in Perpetuity and zero coupon bond to equal portfolio duration of 5 years

Let the Investment in Zero Coupon Bond be X.

Thus, Investment in Perpetuity will be (1-X)

Thus, Duration of portfolio = (Duration of Perpetuity * Investment in Perpetuity)+(Duration of Zero Coupon Bond * Investment in Zero Coupon Bond)

5 = 2*X+11*(1-X)

5=2X+11-11X

11-5 = 11X-2X

6=9X

X=6/9 = 0.667 or 67%

Thus, the Investment in Zero Coupon Bond = 67%

Investment in Perpetuity = 33% (1-67%)

Step 6: Compute the number of Zero Coupon Bonds to be bought by investor:

Total Portfolio Value = $100,000

Investment in Zero Coupon Bond = 67% = $100,000*67% = $67,000

Price of the Zero Coupon Bond (from step 3) = $826.45

Number of Zero Coupon Bonds to be bought by investor = $67,000/$826.45 = 81.07 or 81 bonds