Question

You are trying to decide how much to save for retirement. Assume you plan to save

$6,500

per year with the first investment made one year from now. You think you can earn

8.5​%

per year on your investments and you plan to retire in  

32

​years, immediately after making your last

$6,500

investment.

a. How much will you have in your retirement account on the day you​ retire?

b.​ If, instead of investing

\$6,500

per​ year, you wanted to make one​ lump-sum investment today for your retirement that will result in the same retirement​ saving, how much would that lump sum need to​ be?

c. If you hope to live for

30

years in​ retirement, how much can you withdraw every year in retirement​ (starting one year after​ retirement) so that you will just exhaust your savings with the

30th

withdrawal​ (assume your savings will continue to earn

8.5​%

in​ retirement)?

d.​ If, instead, you decide to withdraw

$193,000

per year in retirement​ (again with the first withdrawal one year after​ retiring), how many years will it take until you exhaust your​ savings? (Use​ trial-and-error, a financial​ calculator: solve for​ "N", or​ Excel: function​ NPER)

e. Assuming the most you can afford to save is

$1,300

per​ year, but you want to retire with

$1,000,000

in your investment​ account, how high of a return do you need to earn on your​ investments? (Use​ trial-and-error, a financial​ calculator: solve for the interest​ rate, or​ Excel: function​ RATE)

Answer 1

a) Here Annuity = 6500$ , Rate of return = 8.5% , n = number of year = 32
here formula of Future value of annuity is to be used
FVIFA = Annuity[(1+r)^n - 1 /r]
= 6500[(1+8.5%)^32 - 1 /8.5%]
= 6500[(1+0.085)^32 - 1 /0.085]
= 6500[(1.085)^32 - 1 /0.085]
= 6500[13.6067 - 1 / 0.085]
= 6500[12.6067/0.085]
= 6500[148.3137]
= 964038.92

b) Here formula of PV can be used
PV = FV/(1+r)^n
r = Rate of return = 8.5% , n = number of year = 32 , FV = 964038.92 $
PV = 964038.92/(1+8.5%)^32
= 964038.92/(1+0.085)^32
= 964038.92/(1.085)^32
= 964038.92/13.6067
= 70850.50 $

c) Here formula of PV of annuity can be used
r = Rate of return = 8.5% , n = number of year = 30 , PV = 964038.92 $
PVIFA(r%,n) = [1-(1/(1+r)^n / r ]
PVIFA(8.5%,30) = [1-(1/(1+8.5%)^30 / 8.5%]
=[1-(1/(1+0.085)^30 / 0.085]
=[1-(1/(1.085)^30 / 0.085]
=[1-0.08652 / 0.085]
=0.9135/0.085
=10.7468

Thus Annuity can be found as
PV = Annuity x PVIFA(8.5%,30)
964038.92 = Annuity x 10.7468
Annuity = 89704.38 $

d)

Here formula of PV of annuity can be used
r = Rate of return = 8.5% , n = number of year = ?, PV = 964038.92 $ , Annuity = 193000

PV = Annuity x PVIFA(8.5%,n)
964038.92 = 193000 x PVIFA(8.5%,n)
PVIFA(8.5%,n)= 4.9950

Assume n = 7

PVIFA(8.5%,7) = 5.1185

Assume n = 6

PVIFA(8.5%,6) = 4.5536

Using interpolation we can find n

n	PVIFA
6	4.5536
7	5.1185
1	0.5649
?	0.4414

= 0.4414/0.5649

= 0.7814

Thus n = 6 + 0.7814

= 6.7814 years