Question

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 71.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 71.0 when he fully retires, he will begin to make annual withdrawals of $135,618.00 from his retirement account until he turns 88.00. After this final withdrawal, he wants $1.52 million remaining in his account. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 10.00% interest rate.

Answer 1

Number of withdrawals from age 71 to age 88 (nper)= 18, FV=$1.52mn, PMT=135618, rate=10%, PV=?

The account balance at the age of 70 should be 1,385,644.47

Value of account on his 66th birthday= 1385644.47/1.1^(70-66)=$946413.8

Now, to make this amount on his 66th birthday, FV= 946413.8, nper=66-26=40, rate=10%, PV=0, PMT=?

Hence, his contribution must be $1944 (Approx)

Answer 2

SOLUTION :

Interest rate, r = 10% = 0.1

=> 1+r = 1.1

FV of withdrawals from age 72 to age 88 ( for 17 years)

= 135618.00(1.1^17 - 1) / 0.1

= 5498591.51 ($)

Balance in the account at age 88

= 1520000 ($)

Total FV including balance had withdrawal not done

= 5498591.51 + 1520000 

= 7018591.51 ($)

PV of this at age 71

= 7018591.51 / 1.1^17

= 1388590.91 ($).

Amount in the account at age 64 (65th birthday) 

= PV of the above at age 64 (65th birthday)

=  1388590.91 / 1.1^7

= 712566.70 ($)

Let contributions be $A per year from age 25 (26th birthday) to age 64 (65th birthday)  I.e. for 40 years.

So,

FV at age 64 of deposits for 39 years  = amount in the account at age 64.

=> A*(1.1^40 - 1) / 0.1 = 712566.70

=> A = 1609.98 = 1610 ($) approx.  = amount of deposit from 26th birthday to 65th birthday.  (ANSWER)