Question

A company must make payments of $10 annually in the form of a 10- year annuity- immediate. It plans to buy two zero coupon bonds to fund these payments. The first bond matures in 2 years and the second bond matures in 9 years, and both are purchased to yield 10% effective. What face amount of each bond should the company buy in order to be immunized from small changes in the interest rate (redington immunization)?

Answer 1

In order to immunize our portfolio we need to calculate the duration of the payments with the bond.

Formula of duration =

= Summation(n x PVn) / Summation(PVn)

= (0 x 10 + 1 x 10/1.1 + 2 x 10/1.1^2 + 3x10/1.1^3 +.........+9x10/1.1^9)/(10 + 10/1.1 + 10/1.1^2 + 10/1.1^3 +.......+10/1.1^9)

= 3.725 days.

Now we need to calculate the weights of the two zero coupan bond and since it is zero coupan bond their duration is equal to their period of maturity.

= Weights of Bond A* Duration of A + Weights of Bond B *Duration of Bond B

A * 2 + (1-A) * 9 = 3.725

Solving this Weight of Bond A = 75%(Approx) and Bond B = 25%

Now the present value of annuity payments

= (10 + 10/1.1 + 10/1.1^2 + 10/1.1^3 +........+ 10/1.1^9)

= 67.59

Hence the face amount of each bond is

Bond A(2 Year Bond) = 0.75 x 67.59 = 50.69

Bond B(9 Year Bond) = 0.25 x 67.59 = 16.90