Question

Your friend Ellen is celebrating her 25^th birthday (i.e., she is 25 today) and wants to start saving for her anticipated retirement at age 55. She wants to be able to withdraw $10,000 from her savings account on each birthday for 10 years following her retirement (the first withdrawal will be on her 56^th birthday). Ellen intends to invest her money in the local saving bank, which offers 8% (EAR) interest per year.

Suppose Ellen wants to make 24 deposits with the same amount C to cover her retirement needs. She plans to start making these deposits on her 26^th birthday and continues to make deposits on every birthday until she is 50 (the last deposit will be on her 50^th birthday). The only exception is she will skip the deposits on her 40^th birthday as she will spend the money to travel in Europe. What is the minimal amount of C will Ellen have to deposit?

Answer 1

PV of retirement corpus
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream	To be computed
PMT = the dollar amount of each annuity payment	10000
r = the effective interest rate (also known as the discount rate)	8%
n = the number of periods in which payments will be made	10
Retirement corpus value=	PMT x (((1-(1 + r) ^- n)) / r)
Retirement corpus value=	10000* (((1-(1 + 8%) ^- 10)) / 8%)
Retirement corpus value=	$                                       67,100.81
This corpus will be accumulated by depositing similar amount every year in below explained way.
	1st stream	2nd stream
FV of annuity	Deposit 26 to 39	Deposit 41 to 50
	Corpus remain invested till 55	Corpus remain invested till 55
The formula for the future value of an ordinary annuity, as opposed to an annuity due, is as follows:
P = PMT x ((((1 + r) ^ n) - 1) / r)
Where:
P = the future value of an annuity stream	To be computed	To be computed
PMT = the dollar amount of each annuity payment	To be computed	To be computed
r = the effective interest rate (also known as the discount rate)	8%	8%
n = the number of periods in which payments will be made	14	10
Value at the end of annuity period	C*((((1+8%)^14)-1)/8%)	C*((((1+8%)^10)-1)/8%)
Time in years from end of annuity to age 55=	16	5
Value at age 55=	(C*((((1+8%)^14)-1)/8%))*(1+8%)^16	(C*((((1+8%)^10)-1)/8%))*(1+8%)^5
Value at age 55=	(C*((((1+8%)^14)-1)/8%))*(1+8%)^16	(C*((((1+8%)^10)-1)/8%))*(1+8%)^5
Value at age 55=	C*82.9589280717414	C*21.2855129674784
Total value should be equal to retirement corpus
C*82.9589280717414 + C*21.2855129674784=	$                                       67,100.81
C*104.24444103922=	$                                       67,100.81
C=	67100.81/104.24444103922
C=	$                                            643.69