Independent solution

How to solve this Full Immunization question

Setup

Setup

Let A and B be the amounts assigned to the two zero-coupon positions. Express both positions as values at the liability date using the 5% accumulation and discount factors.

A(1.05)2+B(1.05)−2=6000A(1.05)^2+B(1.05)^{-2}=6000

Model

Model

Full immunization imposes two equations: total value equals 6,000 and the first derivative with respect to yield is zero. The symmetric time offsets give the displayed signed-duration equation.

2A(1.05)−2B(1.05)−3=02A(1.05)-2B(1.05)^{-3}=0

Compute

Compute

Solving simultaneously gives A = 2,721.09 and B = 3,307.50. Their absolute difference is 586.41, which rounds to 586.

A=2721.09, B=3307.50, ∣A−B∣=586.41A=2721.09,\ B=3307.50,\ |A-B|=586.41

Answer

Answer

The calculation gives 586 for full immunization, matching published choice D.

A−B=586(D)\boxed{A-B=586\quad\text{(D)}}