Company ABC is required to pay their customers $20,000 after 3 years. Based on an annual effective interest rate of 4%, Andy, the company’s actuary, uses full immunization strategy to construct a portfolio of assets using a 2-year zero-coupon bond and a 4-year zero-coupon bond. Calculate the par amount for the 2-year zero-coupon bond assuming full immunization is met.

Par amount = $9,615.39

Explanation:

The condition that must hold in order to meet full immunization are as follows:

Condition 1: PV(assets) = PV(liabilities)

Condition 2: MD(assets) = MD(liabilities) or P'assets = P'liabilities

Condition 3: There is one asset cash inflow before the liability cash outflow, and there is also one asset cash inflow after the liability cash outflow.

Where PV denotes Present Value and MD denotes Macaulay Duration.

PV(liabilities) = Amount required to pay / (1 + i)^n (1)

Where;

Amount required to pay = $20,000

i = interest rate = 4%

n = number of years after = 3 years

Substituting the values into equation (1), we have:

PV(liabilities) = $20,000 / (1 + 4%)^3 = 17,779.93

Let;

A = Weight of two-year-zero-coupon bond in the portfolio

n = Macaulay Duration of n-year-zero-coupon bond

Therefore, we can construct a portfolio of assets using a 2-year zero-coupon bond and a 4-year zero-coupon bond as follows:

A(2) + (1 – A)(4) = 3

2A + 4 – 4A = 3

2A – 4A = 3 – 4

-2A = - 1

A = -1/-2

A = 0.5

We can now calculate the par amount as follows:

Par amount = PV(liabilities) * A * (1 + i)^t (2)

Where t = 2 as the duration of the bond

Substituting the values into equation (2), we have:

Par amount = 17,779.93 * 0.5 * (1 + 4%)^2

Par amount = 17,779.93 * 0.5 * 1.04^2

Par amount = 17,779.93 * 0.5 * 1.0816

Par amount = $9,615.39

Therefore, the par amount for the 2-year zero-coupon bond assuming full immunization is met is $9,615.39.

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