The perpetuity's duration is approximately 104,166,666.67 years.
The zero-coupon bond's weight and investment are approximately 0.65% and $40,307.17, respectively.
The perpetuity's weight and investment are approximately 99.35% and $6,209,692.83, respectively.
This combination of bonds immunizes the liability against interest rate changes, ensuring you have the funds to pay it off when it matures.
Immunizing a Liability with Bonds
Here's how to immunize a $500,000 liability due in 6 years using a 3-year zero-coupon bond and a perpetuity:
1. Perpetuity Duration:
The perpetuity's price is its present value divided by its yield: $500,000 / 8% = $6,250,000.
Since the perpetuity pays a constant coupon, its duration equals its price divided by the coupon: $6,250,000 / 6% = $104,166,666.67.
2. Weighting and Investment in Zero-Coupon Bond:
To immunize the liability, the combined duration of the bonds should equal the liability's duration (6 years).
Let X be the dollar investment in the zero-coupon bond.
The zero-coupon bond's duration is 3 years, so its contribution to total duration is 3X.
The perpetuity's contribution is its duration multiplied by its investment weight: 104,166,666.67 * (1 - X/6,250,000).
Setting the combined duration equal to 6 years: 3X + 104,166,666.67 * (1 - X/6,250,000) = 6,000,000.
Solving for X, the investment in the zero-coupon bond is approximately $40,307.17.
Its weight as a percentage of the total investment is (40,307.17 / (40,307.17 + 6,250,000)) * 100% ≈ 0.65%.
3. Weighting and Investment in Perpetuity:
The investment in the perpetuity is $6,250,000 - $40,307.17 = $6,209,692.83.
Its weight as a percentage of the total investment is (6,209,692.83 / (40,307.17 + 6,250,000)) * 100% ≈ 99.35%.