Question

You will need $100,000 per year for 25 years starting in year 45.    You are able to save $3000 per year from year 3 to year 15, inclusive.  How much must you save from years 16 through 37, inclusive, if interest rates will be 4% from years 0 through the end of year 25 and 7% starting at the beginning of year 26 and on?    Solve for the unknown payments:

Please show work: I do not understand my professors work

Answer 1

First, let's compute the PV of corpus required to fund $ 100,000 withdrawl for 25 years
PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream	To be calculated
PMT = the dollar amount of each annuity payment	$                                       100,000
r = the effective interest rate (also known as the discount rate)	7%
n = the number of periods in which payments will be made	25
Corpus required=	PMT x (((1-(1 + r) ^- n)) / r)
Corpus required=	100000* (((1-(1 + 7%) ^- 25)) / 7%)
Corpus required=	$                               1,165,358.32
This will be funded with annuities so there are 3 annuities
	I	II	III
From Year	3-15	16-25	26-37
Annuity amount	$                                           3,000	P	P
Interest rate	4%	4%	7%
Total payments	13	10	12
So first we will calculate the future value of these annuities which will be the value at the end of annuity year. Then that future value remains invested till year 45
Years from annuity final year till year 45	30	20	8
FV of annuity
P = PMT x ((((1 + r) ^ n) - 1) / r)
Where:
P = the future value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
n = the number of periods in which payments will be made
	I	II	III
Future value of annuity at the end of annuity year=	3000* ((((1 + 4%) ^ 13) - 1) / 4%)	P* ((((1 + 4%) ^ 10) - 1) / 4%)	P* ((((1 + 7%) ^ 12) - 1) / 7%)
Future value of annuity at the end of annuity year=	$                                    49,880.51	P* 12.0061071229586	P* 17.8884512708689
Future value from year 15 to year 45	49880.51*((1+4%)^10)*(1+7%)^20	P* 12.0061071229586*(1+7%)^20	P* 17.8884512708689*(1+7%)^8
	From year 15-Year 25 @ 4%
	From year 25-Year 45 @ 7%	From year 25-Year 45 @ 7%	From year 37-Year 45 @ 7%
Future value from year 15 to year 45	$                                  285,719.47	P* 46.4598461886575	P* 30.7356897522037
Total of all these payment should be equal to corpus required
285719.47+P*46.4598+P*30.735=	$                               1,165,358.32
285719.47+P*46.4598+P*30.735=	$                               1,165,358.32
285719.47+P*77.1955359408612=	$                               1,165,358.32
P*77.1955359408612=	1165358.32-285719.47
P*77.1955359408612=	879638.85
P=	$                                    11,394.94
So the saving from year 16 to Year 37 should be $ 11,394.94