Question

Following her 18th birthday, Madison began investing $50 at the end of each week in an account earning 4% per year. She plans to continue making weekly investments until she turns 68. If she hadn't started investing until she turned 60, how much would she have to invest each week in order to have the same retirement nest egg at age 68? Round to the nearest cent. [Hint: Find the size of the retirement nest egg under the first long horizon scenario, then use that number to solve for CF under the short investment horizon scenario.]

An asset is projected to generate 20 annual cash flows of $1,000 starting 8 years from today. If the discount rate is 6%, how much is this asset worth today? Round to the nearest cent. [Hint: This is a deferred annuity. Remember the rule about where on the timeline PV annuity goes when you have a deferred annuity.

Answer 1

FV 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
n - No. of periods

Assuming 52 weeks per anum

Particulars	Amount
Cash Flow	$                 50.00
Int Rate	0.077%
Periods	2600

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 50 * [ [ ( 1 + 0.00077 ) ^ 2600 ] - 1 ] / 0.00077
= $ 50 * [ [ ( 1.000769 ) ^ 2600 ] - 1 ] / 0.000769
= $ 50 * [ [7.3834] - 1 ] / 0.000769
= $ 50 * [6.3834] /0.000769
= $ 414919.52
Weekly Pyt if She started investing from 60th bday:

Particulars	Amount
FV of Annuity	$   4,14,919.52
Int Rate	0.0769%
Periods	416

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$414919.52 = Cash Flow * [ [ ( 1 + 0.0008 ) ^ 416 ] - 1 ] / 0.0008
$414919.52 = Cash Flow * [ [ ( 1.0008 ) ^ 416 ] - 1 ] / 0.0008
$414919.52 = Cash Flow * [ [ ( 1.377 ] - 1 ] / 0.0008
$414919.52 = Cash Flow * [ 0.377 ] / 0.0008
Cash Flow = $ 414919.52 * 0.0008 / 0.377
Cash Flow = $ 846.7
Weekly deposit required is $ 846.70

Part B:

PV of Annuity:

Annuity is series of cash flows that are deposited 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

Value at the end of 7th Year :

Particulars	Amount
Cash Flow	$            1,000.00
Int Rate	6.0000%
Periods	20

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 1000 * [ 1 - [(1+0.06)^-20]] /0.06
= $ 1000 * [ 1 - [(1.06)^-20]] /0.06
= $ 1000 * [ 1 - [0.3118]] /0.06
= $ 1000 * [0.6882]] /0.06
= $ 11469.92
Value Today :

Present Value:
PV = FV / (1+r)^n
Where r is Int rate per period
n - No. of periods

Particulars	Amount
Future Value	$            11,469.92
Int Rate	6.0000%
Periods	7

Present Value = Future Value / ( 1 + r )^n
= $ 11469.92 / ( 1 + 0.06 ) ^ 7
= $ 11469.92 / ( 1.06 ) ^ 7
= $ 11469.92 / 1.5036
= $ 7628.15

Value of Annuity Cfs today is $ 7628.15