Aakash has a liability of 6000 due in four years. This liability will be met with payments of A in two years and B in six years. Aakash is employing a full immunization strategy using an annual effective interest rate of 5%.

Calculate [math]|A-B|[/math].

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Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Step 1: Understanding Full Immunization and Given Parameters

Aakash is employing a full immunization strategy to meet a liability. Full immunization aims to protect a portfolio from small changes in interest rates by satisfying three main conditions:

• Present Value Match: The present value of assets must equal the present value of liabilities.
• Duration Match: The Macaulay duration of assets must equal the Macaulay duration of liabilities.
• Convexity Match: The convexity of assets must be greater than or equal to the convexity of liabilities. (Often naturally met or implied when two assets are used to match a single liability.)

In this problem, we are given:

• Liability: $6,000 due in 4 years ([math]t=4[/math]).
• Assets: A payment of [math]A[/math] in 2 years ([math]t=2[/math]) and a payment of [math]B[/math] in 6 years ([math]t=6[/math]).
• Interest Rate: An annual effective interest rate of [math]i = 5\%[/math] ([math]1+i = 1.05[/math]).

Our goal is to calculate the absolute difference between [math]A[/math] and [math]B[/math], i.e., [math]|A-B|[/math].

Step 2: Formulating the Immunization Equations

We will set up a system of two equations based on the full immunization conditions, specifically focusing on the present value and duration matching. We can formulate these equations at the liability's maturity date ([math]t=4[/math]).

 Equation 1: Present Value Match (Accumulated Value at [math]t=4[/math])

The accumulated value of the assets at [math]t=4[/math] must equal the accumulated value of the liability at [math]t=4[/math]. Since the liability of $6,000 is due at [math]t=4[/math], its accumulated value at [math]t=4[/math] is simply $6,000.

• The payment [math]A[/math] received at [math]t=2[/math] accumulates to [math]A(1+0.05)^{4-2} = A(1.05)^2[/math] at [math]t=4[/math].
• The payment [math]B[/math] received at [math]t=6[/math] discounts to [math]B(1+0.05)^{4-6} = B(1.05)^{-2}[/math] at [math]t=4[/math].

Thus, the first immunization equation is:

[[math]]A(1.05)^2 + B(1.05)^{-2} = 6000 \quad \text{(Equation 1)}[[/math]]

 Equation 2: Duration Match

The Macaulay duration of the assets must equal the Macaulay duration of the liability. For a single liability at time [math]t_L[/math] (here [math]t_L=4[/math]) being immunized by two asset cash flows at [math]t_A[/math] (here [math]t_A=2[/math]) and [math]t_B[/math] (here [math]t_B=6[/math]), this condition simplifies significantly. The original problem provides a pre-simplified form of this condition:

[[math]]2A(1.05)^{1} - 2B(1.05)^{-3} = 0[[/math]]

We can further simplify this equation algebraically:

[[math]]2A(1.05) = 2B(1.05)^{-3}[[/math]]

Divide both sides by 2:

[[math]]A(1.05) = B(1.05)^{-3}[[/math]]

To isolate [math]A[/math], divide both sides by [math](1.05)[/math]:

[[math]]A = B(1.05)^{-3}(1.05)^{-1}[[/math]]

[[math]]A = B(1.05)^{-4} \quad \text{(Equation 2 Simplified)}[[/math]]

This simplified form highlights a key relationship for immunization when the time differences from the liability are equal ([math]t_L - t_A = 4-2=2[/math] and [math]t_B - t_L = 6-4=2[/math]). In such cases, the present values of the two asset cash flows must be equal: [math]A(1.05)^{-t_A} = B(1.05)^{-t_B}[/math], which means [math]A(1.05)^{-2} = B(1.05)^{-6}[/math], leading directly to [math]A = B(1.05)^{-4}[/math].

Step 3: Solving the System of Equations

Now we solve the system of two equations simultaneously to find the values of [math]A[/math] and [math]B[/math]: 1. [math]A(1.05)^2 + B(1.05)^{-2} = 6000[/math] 2. [math]A = B(1.05)^{-4}[/math] First, let's determine the numerical values of the factors at [math]i = 0.05[/math]:

Factors at [math]i = 0.05[/math]

Factor	Value (approx.)
[math](1.05)^2[/math]	[math]1.1025[/math]
[math](1.05)^{-2}[/math]	[math]0.907029478[/math]
[math](1.05)^{-4}[/math]	[math]0.822702476[/math]

Substitute Equation 2 (the simplified duration relationship) into Equation 1:

[[math]]B(1.05)^{-4}(1.05)^2 + B(1.05)^{-2} = 6000[[/math]]

Simplify the exponents:

[[math]]B(1.05)^{-2} + B(1.05)^{-2} = 6000[[/math]]

Combine like terms:

[[math]]2B(1.05)^{-2} = 6000[[/math]]

Solve for [math]B[/math]:

[[math]]B = \frac{6000}{2(1.05)^{-2}}[[/math]]

Using the pre-calculated value for [math](1.05)^{-2}[/math]:

[[math]]B = \frac{6000}{2 \times 0.907029478}[[/math]]

[[math]]B = \frac{6000}{1.814058956}[[/math]]

[[math]]B \approx 3307.5028[[/math]]

Rounding to two decimal places, [math]B \approx 3307.50[/math]. Now, substitute the value of [math]B[/math] back into Equation 2 to find [math]A[/math]:

[[math]]A = B(1.05)^{-4}[[/math]]

[[math]]A = 3307.5028 \times 0.822702476[[/math]]

[[math]]A \approx 2721.0934[[/math]]

Rounding to two decimal places, [math]A \approx 2721.09[/math].

Step 4: Calculate [math]|A-B|[/math]

Finally, we calculate the absolute difference between [math]A[/math] and [math]B[/math]:

[[math]]|A-B| = |2721.09 - 3307.50|[[/math]]

[[math]]|A-B| = |-586.41|[[/math]]

[[math]]|A-B| = 586.41[[/math]]

The closest answer choice is 586.

Key Insights

• Full immunization is a strategy to manage interest rate risk by matching the present value, duration, and convexity of assets to liabilities.
• For a single liability at time [math]t_L[/math] to be immunized by two asset cash flows, one at [math]t_A[/math] and one at [math]t_B[/math] (with [math]t_A \lt t_L \lt t_B[/math]), the two critical conditions are present value matching and duration matching.
• When the time differences from the liability are symmetric (i.e., [math]t_L - t_A = t_B - t_L[/math]), the duration matching condition often simplifies to the present value of the earlier asset cash flow being equal to the present value of the later asset cash flow (e.g., [math]A(1+i)^{-t_A} = B(1+i)^{-t_B}[/math]), which further simplifies to a direct relationship like [math]A=B(1+i)^{-(t_B-t_A)}[/math].
• The immunization equations can be set up using either present values (at time 0) or accumulated values (at the immunization date, [math]t_L[/math]), as both approaches are mathematically equivalent and will yield the same solution.

This article was generated by AI and may contain errors. If permitted, please edit the article to improve it.

Solution: D

Set up the following two equations in the two unknowns:

[[math]] \begin{array}{cc}{{A(1.05)^{2}+B(1.05)^{-2}=6000}}\\ {{2A(1.05)^{1}-2B(1.05)^{-3}=0.}}\end{array} [[/math]]

Solving simultaneously gives:

[[math]] \begin{array}{l}{{A=2721.09}}\\ {{B=3307.50}}\\ {{|A-B|=586.41}}\end{array} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.