How to solve ALM — Redington immunization: asset present value and matching asset amounts questions on Exam FM?

Question

A liability of $30000$ is due in $6$ years. It is to be immunized (matching present value and duration) using two zero-coupon assets: asset 1 pays $A_1$ at time $t=2$ and asset 2 pays $A_2$ at time $t=8$. At an effective annual interest rate of $i = 0.0456$, calculate the required amount $A_{2}$ (the payment at time $t=8$).

Accepted Answer

Formula
For cash flows, $PV=\sum_t CF_t v^t$. At the immunization yield, require $PV_A=PV_L$, $D_A=D_L$, and asset convexity greater than liability convexity.

Why this formula applies
Match asset and liability present values and duration exposure at the current yield. Solve the resulting simultaneous linear equations before checking any convexity or surplus condition.

Substitution and calculation
$P = L v^n = 30000\cdot0.95639^{6} = 22957.6699$. Matching PV and duration, $A_2 v^{t_2} = P\dfrac{n-t_1}{t_2-t_1} = P\dfrac{6-2}{8-2}$, so $A_2 = P\dfrac{n-t_1}{t_2-t_1}(1+i)^{t_2} = 21865.5872$.

Answer
The answer is 21865.5872, option (A).

How to Solve ALM — Redington immunization: asset present value and matching asset amounts Questions on Exam FM

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