Using the PVOA equation, we can calculate the interest rate (i) needed to discount a series of equal payments back to the present value. In order to solve for (i), we need to know the present value amount, the amount of the equal payments, and the length of time (n).



Exercise #9. Sylvia has an investment account that shows a balance of $2,523.50 on January 1, 2011. She wants to make five withdrawals of $700 each on December 31 of years 2011 through 2015. Sylvia wants the account to have a balance of $0 on December 31, 2015. In order to proceed with her plans, what annual interest rate does Sylvia need on her account, assuming that annual interest earnings are added to the principal on December 31 of each year?

PVOA=	$2,523.50
	$700	$700	$700	$700	$700
	← 1 year →	← 1 year →	← 1 year →	← 1 year →	← 1 year →
1/1/11	12/31/11	12/31/12	12/31/13	12/31/14	12/31/15
0	1	2	3	4	5
n = 5 years;  i = ??% per year

Calculation of Exercise #9 using the PVOA Table
The interest rate for the ordinary annuity described above can be computed with the following equation:

	PVOA   =	PMT times [ PVOA factor for n = 5 years; i = ?? per year ]
	$2,523.50   =	$700 times [ PVOA factor for n = 5 years; i = ?? per year ]
	$2,523.50 / $700   =	[ PVOA factor for n = 5 years; i = ?? per year ]
	3.605   =	[ PVOA factor for n = 5 years; i = ?? per year ]
	3.605   =	PVOA factor for n = 5 years; i = 12% per year

Let's review this calculation. We insert into the equation the components that we know: the present value, payment amount, and the number of periods. In line four, we calculate our factor to be 3.605. We now know both the PVOA factor (3.605) and the number of years (n = 5). We go to the PVOA Table and look across the n = 5 row until we come to the factor 3.605. Tracking up the column, we see that the factor is in the column with the heading of 12%. Since the periods in question are annual periods, the answer of i = 12% means the investment has to earn 12% per year.

Here is the proof of this answer:

Investment Account Activity

W/D No.	Date	Interest Added*	Withdrawal	Account Balance
	Jan. 1, 2011			$ 2,523.50
1	Dec. 31, 2011	$ 302.82	$ 700.00	$ 2,126.32
2	Dec. 31, 2012	255.16	700.00	1,681.48
3	Dec. 31, 2013	201.78	700.00	1,183.26
4	Dec. 31, 2014	141.99	700.00	625.25
5	Dec. 31, 2015	75.03	700.00	0.28

*12% times the previous account balance

Exercise #10. Matt is moving to Texas and needs to borrow $5,616 on January 1, 2011. His budget will allow him to make quarterly payments of $800 on the first day of January, April, July, and October. Matt's loan includes 8 quarterly payments; the first payment is due on April 1, 2011. What is the rate (compounded quarterly) that Matt will be paying (and the lender will be receiving) under this arrangement?

Before calculating the interest rate, we organize the information on a timeline:

PVOA=	$5,616
	$800	$800	$800	$800		$800		$800
						.....
	← 3 months →	← 3 months →	← 3 months →	← 3 months →				← 3 months →
1/1/11	4/1/11	7/1/11	10/1/11	1/1/12		10/1/12		1/1/13
0	1	2	3	4			7	8
n = 8 three-month periods; i = ?? per quarter

Calculation of Exercise #10 using the PVOA Table
The number of periods/payments in the ordinary annuity described above can be computed with the following PVOA equation:

	PVOA   =	PMT times [ PVOA factor for n = 8 quarters; i = ?? per quarter ]
	$5,616   =	$800 times [ PVOA factor for n = 8 quarters; i = ?? per quarter ]
	$5,616 / $800   =	[ PVOA factor for n = 8 quarters; i = ?? per quarter ]
	7.02   =	[ PVOA factor for n = 8 quarters; i = ?? per quarter ]
	7.02   =	PVOA factor for n = 8 quarters; i = 3% per quarter

Let's review this calculation. We insert into the equation the components that we know: the present value, the recurring payment amount, and the number of periods. In line four, we calculate our factor to be 7.02. We now know both the PVOA factor (7.02) and the number of periods (n = 8). We go to the PVOA Table and look across the n = 8 row until we come to the factor 7.02. Tracking up the column, we see that we are in the 3% column. Since the periods in question are quarterly periods, the answer of i = 3% means the loan has an annual rate of 12% per year (3% times 4 quarters per year).

Here is the proof of this calculation:

   Loan Amortization Schedule
   (For $5,616.00 at 12% per year with 8 quarterly payments.)

Pmt No.	Date	Total Payment	Interest Payment*	Principal Payment**	Principal Balance
	Jan. 1, 2011				$ 5,616.00
1	Apr. 1, 2011	$ 800.00	$ 168.48	$ 631.52	4,984.48
2	July 1, 2011	800.00	149.53	650.47	4,334.01
3	Oct. 1, 2011	800.00	130.02	669.98	3,664.03
4	Jan. 1, 2012	800.00	109.92	690.08	2,973.96
5	Apr. 1, 2012	800.00	89.22	710.78	2,263.17
6	July 1, 2012	800.00	67.90	732.10	1,531.07
7	Oct. 1, 2012	800.00	45.93	754.07	777.00
8	Jan. 1, 2013	800.00	23.31	776.69	0.31

   * Interest payment equals 3% of the previous principal balance.
   ** Principal payment equals $800 minus interest payment.

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