Question

The following bonds and liabilities are given: • Bond A: A zero-coupon bond with a face value of $100 and a time to maturity of 3 years. • Bond B: A zero-coupon bond with a face value of $100 and a time to maturity of 11 years. • Liability X: A one-time liability maturing in 4 years with the present value of $100. • Liability Y: A one-time liability maturing in 8 years with the present value of $100. Suppose you have both liabilities X and Y and want to immunize your liabilities using bonds A and B. What would be the weights of two bonds in your immunizing bond portfolio? Please round your calculation to the nearest 2nd decimal.

Select one: A. 62% in Bond A and 38% in bond B

B. 38% in Bond A and 62% in bond B

C. 50% in Bond A and 50% in bond B

D. 66% in Bond A and 34% in bond B

E. 34% in Bond A and 66% in bond B

Answer 1

Liability X: Time to maturity = 4 years | PV = 100

Liability Y: Time to maturity = 8 years | PV = 100

As both liabilities are one-time liability, therefore, their Time to maturity is their Duration.

Duration of X = 4 yrs | Duration of Y = 8 yrs

As both Liabilities have same Present Value, therefore, we can take average of both Liabilities' duration to find the Weighted Average duration of Liabilities

Weighted Average Duration of Liabilities = (Duration of X + Duration of Y) / 2

Weighted Average Duration of Liabilities = (4 + 8) / 2 = 12 / 2 = 6 years

Now we will look at the Bonds.

Bond A: Time to maturity = 3 years | Face value = 100

Bond B: Time to maturity = 11 years | Face value = 100

Both are zero-coupon bonds, therefore, their Time to maturity is their Duration.

Duration of A = 3 years | Duration of B = 11 years

For Immunization against interest rate risk, we need to match the duration of Assets and Liabilities.

As Duration of Liabilities is 6 years, therefore, Weights to Bonds should be allocated such a way that their duration matches duration of Liabilities.

Let Weight allocated to Bond A be W and Weight for Bond B be (1 - W)

Duration of Bonds = Duration of Liabilities

=> W * 3 + (1 - W) * 11 = 6

=> 3W + 11 - 11W = 6

=> 5 = 8W

=> W =5/8

Weight that should be allocated to Bond A = 62.50%

Weight that should be allocated to Bond B = 1 - 62.50% = 37.50%

The resulting weightage for immunization to Bond A of 62.5% and to Bond B of 37.5% matches answer in Option A.