Question

Question 6: This is a classic retirement problem. A friend is celebrating her birthday and wants to start saving for her anticipated retirement. She has the following years to retirement and retirement spending goals:

Years until retirement	35
Amount to withdraw each year	$85,000
Years to withdraw in retirement	25
Interest rate	7.5%

Because your friend is planning ahead, the first withdrawal will not take place until one year after she retires. She wants to make equal annual deposits into her account for her retirement fund.

a) If she starts making these deposits in one year and makes her last deposit on the day she retires, what amount must she deposit annually to be able to make the desired withdrawals at retirement?

b) Suppose your friend just inherited a large sum of money. Rather than making equal annual payments, she decided to make one lump-sum deposit today to cover her retirement needs. What amount does she have to deposit today?

c) Suppose your friend's employer will contribute to the account each year as part of the company's profit-sharing plan. In addition, your friend expects a distribution from a family trust several years from now. What amount must she deposit annually now to be able to make the desired withdrawals at retirement?

Employer’s annual contribution	$1,300
Years until trust fund distribution	15
Amount of trust fund distribution	$20,000

Answer 1

Base case:

1. Find the PV of the annuity of withdrawal amount at the end of 35 years
   • We are given the following information:

   • PMT	$            85,000.00
     r	7.50%
     n	25
     frequency	1

   • We need to solve the following equation to arrive at the required PV
   • 
     1-(1+r)-n PV = PMT X-
     
   • 
     PV = 85000 x 1-(1 +0.075) -25 0.075
     
   • 
     PV = 947490.40
     
   • So by the end of 35 years She should have $947490.40 in her retirement fund
2. Next we find annual PMT made for next 35 years to accumulate the required amount $947490.40 in the fund
   • We are given the following information

   • r	7.50%
     n	35
     T	1
     FV	$        9,47,790.40

   • We need to solve the following equation to arrive at the required PMT
   • 
     (1+r)" - 1 FV = PMT
     
   • 
     947790.4 = PMT X (1 + 0.075)35 - 1 0.075
     
   • 
     PMT =6,144.44
     
   • So annual deposit = $6144.44

Part a) Now considering that she withdraws at the same pace however starts depositing one year later, the accumulated amount required is the same, however, the period for which deposits are made is reduced by 1 si annually she needs to deposit higher amount:

We are given the following information

r	7.50%
n	34
T	1
FV	$9,47,790.40

We need to solve the following equation to arrive at the required FV

(1+r)" - 1 FV = PMT

947790.4 = PMT X (1 + 0.075) 34 - 1 0.075

PMT =6,648.38

So annual PMT =  $6,648.38

Part b) Here we basically find the PV of the required amount after 35 years so that is calculated below:

We are given the following information

r	7.50%
n	35
frequency	1
FV	$        9,47,790.40

We need to solve the following equation to arrive at the required PV

FV = PV X (1+r)"

947790.4 = PV X (1+0.075)35

PV = 75407.76

So the lumpsum to be deposited today will be 75407.76

C)

1. Find the FV of the employer's contribution
   • We are given the following information

   • PMT	$              1,300.00
     r	7.50%
     n	35
     T	1

   • We need to solve the following equation to arrive at the required FV
   • 
     (1+r)" - 1 FV = PMT
     
   • 
     FV = 1300 (1 + 0.075) 35 - 1 0,075
     
   • 
     FV = 200527.10
     
2. Find the FV of inheritance
   • We are given the following information

   • PV	20000
     r	7.50%
     n	35-15 = 20
     frequency	1

   • We need to solve the following equation to arrive at the required FV
   • 
     FV = PV X (1+r)"
     
   • 
     FV = 20000 x (1 + 0.075 20
     
   • 
     FV = 84957.02
     
3. Reduce both from the required fund amount
   • Total amount required =  $9,47,490.40
   • Employer's contribution =  $2,00,527.09
   • Inheritance=  $84,957.02
   • Required amount =  $6,62,006.29
4. Find the annual PMT for the balance
   • We are given the following information

   • r	7.50%
     n	35
     T	1
     FV	$        6,62,006.29

   • We need to solve the following equation to arrive at the required PMT
   • 
     (1+r)" - 1 FV = PMT
     
   • 
     662006.09 - Pu (1 + 0.075)35 - 1 2.075
     
   • PMT = $4,291.73
   • So now she has to deposit only  $4,291.73 per year to achieve the requirement