Question

Suppose in question 1, the tuition obligations have a Macaulay duration of 6.86 in years and a present value of 52,798. In order to immunize against the tuition payments by investing in some combination of two bonds with duration 2.70 and 8.22, what is the dollar amount that you should invest in the bond with duration 8.22? Assume annual compounding. Round your answer to 2 decimal places.

Answer 1

Macaulay Duration = 6.86 years | PV of Tuition obligations = 52,798

Available Options: Bond 1 with Duration = 2.70 | Bond 2 with Duration = 8.22

For immunization, we match the duration of investments with the obligations, which means that we invest in such proportions in the two bonds that its duration matches the duration of tuition obligation.

Let proportion of investment in Bond 1 be x and proportion of investment in Bond 2 be (1-x)

=> Duration of obligations = x * Duration of Bond 1 + (1-x) * Duration of Bond 2

=> 6.86 = x * 2.70 + (1-x)*8.22

=> 6.86 = 2.70x + 8.22 - 8.22x

=> 8.22x - 2.70x = 8.22 - 6.86

=> 5.52x = 1.36

=> x = 1.36 / 5.52 = 0.246377 or 24.64%

Proportion of Bond 1 = 24.64% and Proportion of Bond 2 = 1 - 0.246377 = 0.753623 or 75.36%

The amount that should be invested in Bond with Duration 8.22 = Proportion of Bond 2 * PV of Tuition Obligations

Amount to be invested in bond with Duration 8.22 = 0.753623 * 52,798 = 39,789.80 or $ 39,790