Question

A pension plan is obligated to make disbursements of $2.4 million, $3.4 million, and $2.4 million at the end of each of the next three years, respectively. The annual interest rate is 10%. 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.)

Answer 1

Duration Calculation

Year	Pension Amount	Present Value of the Pension amount Present Value = Future Value * (1/(1+interest rate)^year)	Proportion of the Pension amount	Duration (Year * Proportion of the Pension amount)
1	$2.4 million	=$2.4*(1/(1+10%)^1) =$2.4*(1/(1.1)^1) =$2.4*0.9091 = $2.1818	=$2.1818/$6.7948 =0.3211	=1*0.3211 =0.3211
2	$3.4 million	=$3.4*(1/(1.1)^2) =$3.4*0.8264 = $2.8099	=$2.8099/$6.7948 =0.4135	=2*0.4135 =0.8270
3	$2.4 million	=$2.4*(1/(1.1)^3) =$2.4*0.7513 = $1.8031	=$1.8031/$6.7948 =0.2654	=3*0.2654 =0.7962
Total		$2.1818+$2.8099+$1.8031 = $6.7948	1.0000	1.9443

Consider, Zero-coupon bonds as "x" and perpetuities as "1-x".

Duration = x + ((1-x)*((1+interest rate)/interest rate))

1.9443 = x + ((1-x)*((1.10)/0.10))

1.9443 = x + ((1-x)*11)

1.9443 = x+11-11x

10x = 11-1.9443

x = 9.0557/10

x=0.9056 = 90.56%

So, Zero-coupon bonds should hold 90.56% of the portfolio and the remaining 9.44% should be perpetuities.

In terms of amount, $6.15 (90.56%*$6.7948) should be zero-coupon bonds and $0.64 should be perpetuities.