Question

In: Finance

A pension plan is obligated to make disbursements of $2.4 million, $3.4 million, and $2.4 million...

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.)

Solutions

Expert Solution

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.


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