Question

Your client is 33 years old. She wants to begin saving for retirement, with the first payment to come one year from now. She can save $10,000 per year, and you advise her to invest it in the stock market, which you expect to provide an average return of 12% in the future.

1. If she follows your advice, how much money will she have at 65? Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

2. How much will she have at 70? Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

3. She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70. If her investments continue to earn the same rate, how much will she be able to withdraw at the end of each year after retirement at each retirement age? Do not round intermediate calculations. Round your answers to the nearest cent.

   Annual withdrawals if she retires at 65: $

   Annual withdrawals if she retires at 70: $

Answer 1

Future Value of Annuity :
Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period = 12 % or 0.12
n - No. of periods

a. if she retires at 65

from 33 to 65 years = 32 years

Particulars	Amount
Cash Flow	$          10,000.00
Int Rate	12.0000%
Periods	32

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 10000 * [ [ ( 1 + 0.12 ) ^ 32 ] - 1 ] / 0.12
= $ 10000 * [ [ ( 1.12 ) ^ 32 ] - 1 ] / 0.12
= $ 10000 * [ [37.5817] - 1 ] / 0.12
= $ 10000 * [36.5817] /0.12
= $ 3048477.19
if she retires at 65 years she will have $ 3048477.19

b. if she retires at 70

from 33 to 70 years = 37 years

Particulars	Amount
Cash Flow	$          10,000.00
Int Rate	12.0000%
Periods	37

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 10000 * [ [ ( 1 + 0.12 ) ^ 37 ] - 1 ] / 0.12
= $ 10000 * [ [ ( 1.12 ) ^ 37 ] - 1 ] / 0.12
= $ 10000 * [ [66.2318] - 1 ] / 0.12
= $ 10000 * [65.2318] /0.12
= $ 5435986.9
  if she retires at 65 years she will have $ 5435986.9

c. Annual withdrawls

PV of Annuity:

Annuity is series of cash flows that are deposited / withdrawn at regular intervals for specific period of time.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

1) if she retires at 65 years

Periods of withdrawl = 20 years

Particulars	Amount
PV Annuity	$    3,048,477.19
Int Rate	12.0000%
Periods	20

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 3048477.19 / [ 1 - [(1+0.12)^-5]] /0.12
= $ 3048477.19 / [ 1 - [(1.12)^-5]] /0.12
= $ 3048477.19 / [ 1 - 0.1037 ] /0.12
= $ 3048477.19 / [0.8963 / 0.12 ]
= $ 3048477.19 / 7.4694
= $ 408126.41
if she retires at 65 years , she will be withdrawn $ 408126.41 each year for 20 years

2) if she retires at 70 years

Periods of withdrawl = 20 years

Particulars	Amount
PV Annuity	$    5,435,986.90
Int Rate	12.0000%
Periods	15

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 5435986.9 / [ 1 - [(1+0.12)^-5]] /0.12
= $ 5435986.9 / [ 1 - [(1.12)^-5]] /0.12
= $ 5435986.9 / [ 1 - 0.1827 ] /0.12
= $ 5435986.9 / [0.8173 / 0.12 ]
= $ 5435986.9 / 6.8109
= $ 798134.64
if she retires at 70 years , she will be withdrawn $ 798134.64 each year for 15 years