Question

1. John has started a new job and wants to open a retirement account. He is now 26 years old and plans to retire if he is over 65. For this, he is considering depositing $ 2815 retirement accounts annually. Annual interest yield is 2%.
a. How much money will he save when he retires? Calculate by drawing a cash flow diagram.
b. If John wanted to save $ 150,000, how much money would he have to deposit into his retirement account annually with equal payments from 26 to 65?
c. In how many years would John have saved $ 45000? (If the result is fractional, leave it that way, do not round it).

Answer 1

FV of Annuity :

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here deposits are made at the end of the period. FV of annuity is future value of cash flows deposited at regular intervals grown at specified int rate or Growth rate to future date.

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

Part A:

Particulars	Amount
Cash Flow	$            2,815.00
Int Rate	2.000%
Periods	39

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 2815 * [ [ ( 1 + 0.02 ) ^ 39 ] - 1 ] / 0.02
= $ 2815 * [ [ ( 1.02 ) ^ 39 ] - 1 ] / 0.02
= $ 2815 * [ [2.1647] - 1 ] / 0.02
= $ 2815 * [1.1647] /0.02
= $ 163937.83

Part B:

Particulars	Amount
FV of Annuity	$      150,000.00
Int Rate	2.0000%
Periods	39

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$150000 = Cash Flow * [ [ ( 1 + 0.02 ) ^ 39 ] - 1 ] / 0.02
$150000 = Cash Flow * [ [ ( 1.02 ) ^ 39 ] - 1 ] / 0.02
$150000 = Cash Flow * [ [ ( 2.1647 ] - 1 ] / 0.02
$150000 = Cash Flow * [ 1.1647 ] / 0.02
Cash Flow = $ 150000 * 0.02 / 1.1647
Cash Flow = $ 2575.67

Part C:

Particulars	Amount
FV of Annuity	$      45,000.00
Int Rate	2.0000%
Cash flow	2815

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$ 45000 = $ 2815 * [ [ ( 1 + 0.02 ) ^ n ] - 1 ] /0.02
$ 45000 = $ 2815 * [ [ ( 1.02 ) ^ n ] - 1 ] /0.02
[ [ ( 1.02 ) ^ n ] - 1 ] = 0.3197
( 1.02 ) ^ n = 1.3197
Take Log on both sides

Log ( ( 1.02 ) ^ n ) = Log ( 1.3197 )
Log ( a ^ b ) = b * Log ( a )

n * Log (1.02) = Log (1.3197 )
n * 0.0086 = 0.1205
n * 0.0086 = 0.1205
n = 14.0091

It takes 14 Years to be come 45000