Question

A company owes payments of $100 2, 4, and 6 years from now. A company can purchase zero coupon bonds with terms of 1 year and 5 years both with an annual effective rate of 10%. How much of each type of bond should be purchased to achieve Reddington Immunization? Verify that you have achieved Reddington Immunization.

Answer 1

In order to achieve Reddington Immunization, the following three conditions needs to be met:-

1. Net Present Value (NPV) of asset-side cashflows (bonds) and liabilities (payment obligations) are matched

2. Volatility of the asset side cashflows and liabilities are equal. This is measured by the first derivative of NPV calculated in Step 1

3. Convexity of the assets is higher than that of liabilities. Convexity is calculated by the second derivative of NPV calculated in Step 1

Let us assume the Company purchases 'x' units and 'y' units of 1 year and 5 year zero coupon bonds respectively.

Equation for Step 1 is as below:-

NPV (Assets) = NPV (Liabilities);

xe^-1i +ye^-5i = 100e^-2i +100e^-4i +100e^-6i; where i is annual effective rate of 10%;

Therefore, 0.90x + 0.61y = 203.79

Equation for Step 2 is as below:-

d/ di (NPV (Assets)) = d/ di (NPV (Liabilities)) ; i.e.

-1xe^-1i -5ye^-5i = -200e^-2i -400e^-4i -600e^-6i ; where i is annual effective rate of 10%;

Therefore, 0.90x + 3.03y = 761.16.

Solving equations 1 & 2, x=71.22 and y=229.74

Verification of Reddington Immunization having been achieved can be checked with the third condition which states that "Convexity of the assets is higher than that of liabilities".

Equation for Step 3 is as below:-

d²/ di² (NPV (Assets)) > d²/ di² (NPV (Liabilities)) ; i.e.

xe^-1i + 25ye^-5i > 400e^-2i + 1600e^-4i + 3600e^-6i ; where i is annual effective rate of 10%;

Therefore, 0.90x + 15.16y > 3375.73 . Putting the values of x and y, this equation holds true as shown below:-

0.90x + 15.16 y = 0.90 X 71.22 + 15.16 X 229.74 = 3548.04 which is greater than 3375.73.