# Question on duration matching and reddinggton's immunisation

An insurance company has liabilities of 6 million due in 8 years’ time and 11 million due in 15 years’ time. The assets consist of two zero-coupon bonds, one paying X in 5 years’ time and the other paying Y in 20 years’ time. The current interest rate is flat and 8% per annum effective. The insurance company wishes to ensure that it is immunized against small changes in the rate of interest.

(a) Determine the values of X and Y such that the two conditions for duration matching immunization strategy are satisfied.

(b) Demonstrate that the third condition for Redington’s immunization is also satisfied

a) I know we must equate value of assets and liabilities, nd also its duration.

I get, by disounting all the cashflows VL = 6.709272062 (in millions)

To get VA = X (1.08^-5) + Y (1.08^-20) = 6.709272062

then DL = (8.6(1.08)^-8 + 15.11(1.08)^-15) /6.709272062 = 11.61792025

The DA = 5X(1.08^-5) + 20Y(1.08^-20)/6.709272062 = 11.61792025

Then by solving, I get X = 5.508770881 (in millions) = $5 508 770.88 I get Y =$13 796 876.70

Can somebody comment on whether my method and answers are right? Thank you!

For b) I'm not entirely sure abt what reddington's immunisation is. What do i need to show?

The third condition for Redington immunization is that $P_A''(.08)>P_L''(.08)$, where $P_A''(i)$ is the second derivative of the pricing function for assets and $P_L''(i)$ is for liabilities.
We need to show that $\frac{d^2}{di^2}[\frac{X}{(1+i)^5}+\frac{Y}{(1+i)^{20}}]>\frac{d^2}{di^2}[\frac{6}{(1+i)^8}+\frac{11}{(1+i)^{15}}]$ when evaluated at $i=.08$.