Question

Problem 5-57 Spreadsheet Problem (LG5-2, LG5-9) Consider a person who begins contributing to a retirement plan at age 25 and contributes for 40 years until retirement at age 65. For the first ten years, she contributes $3.700 per year. She increases the contribution rate to $5,700 per year in years 11 through 20. This is followed by increases to $10,700 per year in years 21 through 30 and to $15,700 per year for the last ten years. This money earns a return of 10 percent First compute the value of the retirement plan when she turns age 65. (Round your answer to 2 decimal places.) Value Compute the annual payment she would receive over the next 40 years if the wealth was converted to an annuity payment at 9 percent. (Round your answer to 2 decimal places.) Annual payment
pls be simple and clear with the answer and explanation.

Answer 1

Solution 1
Here we have 4 annuities of 10 years each. We will calculate the amount of each annuity and will keep the same invested till age is 65
FV of annuity
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
PMT = the dollar amount of each annuity payment	Given
r = the effective interest rate (also known as the discount rate)	10%
n = the number of periods in which payments will be made	10	years
Annuity value factor	=((((1 + 10%) ^ 10) - 1) / 10%)
Annuity value factor	15.9374
Annuity	1	2	3	4
Annuity amount	$                                              3,700	$                             5,700	$                              10,700	$                           15,700
Interest rate	10%	10%	10%	10%
Amount remained invested for next years	30	20	10	0
Maturity value of each annuity=	Annuity amount * Annuity factor
Maturity value of each annuity=	3700*15.9374	5700*15.9374	10700*15.9374	15700*15.9374
Maturity value of each annuity=	$                                      58,968.38	$                     90,843.18	$                      170,530.18	$                   250,217.18
Maturity value of each annuity at age 65=	=58968.38*(1+10%)^30	=90843.18*(1+10%)^20	=170530.18*(1+10%)^10	=250217.18*(1+10%)^0
Maturity value of each annuity at age 65=	$                                 1,028,962.98	$                   611,147.49	$                      442,311.37	$                   250,217.18
Total fund value	=1028962.98+611147.49+442311.37+250217.18
Total fund value	$                                 2,332,639.02
Solution 2
Now this corpus is converted into an annuity so the PV of all annuity payments should be equal to 2,332,639.02
PV of annuity
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream	$                                 2,332,639.02
PMT = the dollar amount of each annuity payment	To be calculated
r = the effective interest rate (also known as the discount rate)	9%
n = the number of periods in which payments will be made	40
PV of annuity during retirement=	PMT x (((1-(1 + r) ^- n)) / r)
2332639.021=	PMT * (((1-(1 + 9%) ^- 40)) / 9%)
2332639.021=	PMT * 10.757360195239
Each annuity payment=	2332639.021/10.757360195239
Each annuity payment=	$                                    216,841.21