Question

Three employees of the Horizon Distributing Company will receive annual pension payments from the company when they retire. The employees will receive their annual payments for as long as they live. Life expectancy for each employee is 13 years beyond retirement. Their names, the amount of their annual pension payments, and the date they will receive their first payment are shown below:Employee Annual Payment Date of First Payment Tinkers $ 32,000 12/31/21 Evers 37,000 12/31/22 Chance 42,000 12/31/23 Required: 1. Compute the present value of the pension obligation to these three employees as of December 31, 2021. Assume a 12% interest rate. 2. The company wants to have enough cash invested at December 31, 2024, to provide for all three employees. To accumulate enough cash, they will make three equal annual contributions to a fund that will earn 12% interest compounded annually. The first contribution will be made on December 31, 2021. Compute the amount of this required annual contribution.

Answer 1

Answer:

Present value of the obligations $ 697,064

Cash contributions (annuity-due):

Annual payment $ 259,126.684

Explanation:

We solve for the PV of each annuity:

$C \times \frac{1-(1+r)^{-time} }{rate} = PV$

Tinkers

$32000 \times \frac{1-(1+0.12)^{-13} }{0.12} = PV$

PV $205,553.5493

Evers:

$37000 \times \frac{1-(1+0.12)^{-13} }{0.12} = PV$

PV $237,671.2914

we now discount for the deferred year:

$\frac{Maturity}{(1 + rate)^{time} } = PV$  

Maturity  $237,671.00

time  1.00

rate  0.12000

$\frac{237671}{(1 + 0.12)^{1} } = PV$  

PV   212,206.2500

Chance

$42000 \times \frac{1-(1+0.12)^{-13} }{0.12} = PV$

PV $269,789.0335

now, we deffer for the two year period

$\frac{Maturity}{(1 + rate)^{time} } = PV$  

Maturity  $269,783.00

time  2.00

rate  0.12000

$\frac{269783}{(1 + 0.12)^{2} } = PV$  

PV   215,069.3559

We add then:

215,069   +  269,789   +   212,206.2500 =  697,064

The firm will make the first payment on Dec 31,2024 thus this will be an annuity due.

$PV \div \frac{1-(1+r)^{-time} }{rate}(1 + r) = C$

PV 697,064.00

time 3

rate 0.12

$697064 \div \frac{1-(1+0.12)^{-3} }{0.12}(1 + 0.12) = C$

C  $ 259,126.684