A pension plan is obligated to make disbursements of $3.0 million, $3.8 million,

Question

A pension plan is obligated to make disbursements of $3.0 million, $3.8 million, and $3.0 million at the end of each of the next three years, respectively. The annual interest rate is 9%. If the plan wants to fully fund and immunize its position, how much of its portfolio should it allocate to one-year zero-coupon bonds and perpetuities, respectively, if these are the only two assets funding the plan? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Portfolio Investment in one-year zero-coupon bonds % Investment in perpetuity %

Explanation / Answer

Duration of the obligation = Sum of PVF x C x N / Sum of PVF x C = 1.95

Here, PVF - Present Value Factor = 1 / (1 + 9%)^N, C - Cash Flows, N - Period

Now, assume y% allocation to zero coupon bond and 1-y% to perpetuity

Duration of 1-year zero coupon = 1 and Duration of perpetuity = 1 + 9% / 9% = 12.11

=> 1.95 = y x 1 + (1 - y) x 12.11

=> y = 0.9145, i.e. 91.45% allocation to zero coupon bond and 8.55% to perpetuity.

N C PVF PVFxCxN PVFxC 1 3 0.917431 2.752 2.752 2 3.8 0.84168 6.397 3.198 3 3 0.772183 6.950 2.317 Sum 16.099 8.267 Duration 1.9473

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