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

To immunize our portfolio, we must match the duration of the payments with the bonds. Hence, the duration will be calculated as :

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 + 4x10/1.1^4 + 5 x10/1.1^5 + 6x10/1.1^6 + 7x10/1.1^7 + 8x10/1.1^8 + 9x10/1.1^9)/(10 + 10/1.1 + 10/1.1^2 + 10/1.1^3 + 10/1.1^4 + 10/1.1^5 + 10/1.1^6 + 10/1.1^7 + 10/1.1^8 + 10/1.1^9) = 3.7254.

Hence, since the two bonds are zero coupon bonds, their duration will be equal to their maturity. So, their weights should be in the proportion that the net duration comes out to be 3.7254. Hence,

2 x A + 9 x (1 - A) = 3.7254

A = 0.7535. Hence, the 2 year bond should be around 75% and the 9 year bond should be around 25%.

The present value of the payment is = (10 + 10/1.1 + 10/1.1^2 + 10/1.1^3 + 10/1.1^4 + 10/1.1^5 + 10/1.1^6 + 10/1.1^7 + 10/1.1^8 + 10/1.1^9) = 67.59.

Hence, the face value of the 2 year bond is 0.75 x 67.59 = 50.69 and the face value of the 9 year bond is 0.25 x 67.59 = 16.897