Immunization and Duration
Here, I use two cases to show how duration can be used to immunize a portfolio from changes in interest rates. The first case is simple: consider an employer with a single pensioner who is to retire in 10 years and to whom is promised $1 million at that time—in this case, a single balloon payment. (See Luenberger, Example 3.10). The employer wishes to invest an amount of money now to pay this obligation 10 years hence. This is, in fact, a simple example of liability-driven investment that is popular in corporate defined benefit pension plan management. If there were a 10-year zero coupon bond, the problem would be solved immediately because the zero would have duration matching the duration of the obligation—just calculate the amount of the zero necessary to produce $1 million in 10 years. Suppose, however, there are no zeros available. Instead, suppose the choice is among three coupon bonds: a 30-year, a 20-year, and a 10-year bond. As we shall see, the employer need consider purchasing only two bonds.
Go to the companion website for more details (see Duration for specifics on each bond).
Intuitively, the portfolio of bonds should be selected such that the present value of the portfolio is exactly equal to the present value of the $1 million obligation and, secondly, that the duration of the portfolio is as close as possible to the duration of the obligation. ...