Question

A pension plan is obligated to make disbursements of $1.8 million, $2.8 million, and $1.8 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

Answer 1

We need to find the duration of the plan and choose the weights of zero coupon bond and perpetuity so that their combined weighted duration equals the duration of the plan.

Duration of zero coupon bond = no. of years to maturity = 1 year

Duration of perpetuity = (1+y)/ y = (1+0.09)/ 0.09 = 12.1111111111 years

Duration of plan

Year	PVIF@9%	Amount	Present Value	Weights(present value / Total)	Weight x year
1	0.91743119266	$1.8 m	$1.65137614678 m	0.30592311367	0.30592311367
2	0.84167999326	$2.8 m	$2.35670398112 m	0.43658752206	0.87317504412
3	0.77218348005	$1.8 m	$1.38993026409 m	0.2574893587	0.7724680761
		TOTAL	5.39801039199 m	Duration	1.95156623389 years

Now, let weight of perpetuity be 'w'. Then, weight of zero coupon bond is (1 - w).

Duration of plan = weight of zero coupon bond x duration of zero coupon bond + weight of perpetuity x duration of perpetuity

or, 1.95156623389 yrs = (1 - w) x 1 yr + w x 12.1111111111 yrs

or, w = 0.0856 or 8.56%

Weight of zero coupon bond = 1 - 0.0856 = 0.9144 or 91.44%

Weight of perpetuity = 8.56%