Question

please answer all parts of the question! i also need to find the equivalent effective annual interest rate for this $10, 000 investment over this 10 year period.

You purchase a 10 year annuity with payments at the end of each year for $10,000 (where for this annuity effective annual interest is 4%). Immediately after you receive payments, you deposit the payment into an account earning 5% effective annual interest. How much is in this account at the end of 10 years? Use this to find the equivalent effective annual interest rate for this $10, 000 investment over this 10 year period.

Answer 1

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end of the period. PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

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

Particulars	Amount
Cash Flow	$          10,000.00
Int Rate	4.0000%
Periods	10

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 10000 * [ 1 - [(1+0.04)^-10]] /0.04
= $ 10000 * [ 1 - [(1.04)^-10]] /0.04
= $ 10000 * [ 1 - [0.6756]] /0.04
= $ 10000 * [0.3244]] /0.04

= $ 81108.96

Calculation of amount after 10 Years:

  

Year	Bal Years	CF	FVF @5 %	FV of CFs
1	9	$ 10,000.00	1.5513	$   15,513.28
2	8	$ 10,000.00	1.4775	$   14,774.55
3	7	$ 10,000.00	1.4071	$   14,071.00
4	6	$ 10,000.00	1.3401	$   13,400.96
5	5	$ 10,000.00	1.2763	$   12,762.82
6	4	$ 10,000.00	1.2155	$   12,155.06
7	3	$ 10,000.00	1.1576	$   11,576.25
8	2	$ 10,000.00	1.1025	$   11,025.00
9	1	$ 10,000.00	1.0500	$   10,500.00
10	0	$ 10,000.00	1.0000	$   10,000.00
Future Value of CFs				$ 125,778.93

Amount after 10 Years is $125778.93

Effective Rate:

Thus $81108.96 has become $125778.93 over a period of 10 Years
Future Value = Cash Flow * ( 1 + r )^n
$ 125778.93 = $ 81108.96 ( 1 + r) ^ 10
( 1 + r) ^ 10 = $125778.93 / $ 81108.96
( 1 + r) ^ 10 = 1.5507
( 1 + r) = 1.5507 ^ ( 1 / 10 )
( 1 + r) = 1.0448
r = 1.0448 -1
r = 0.0448
i.e Effective Rate over 10 Years is 4.48 %