We can use present value calculations to determine the number of periods (or payments) in an ordinary annuity if we know the other components: present value, interest rate, and the amount of each recurring payment. Exercises 7 and 8 below demonstrate how to solve for the number of periods (or payments).



Exercise #7. Janine has an investment account with a present value of $5,053. The account earns interest at 6% per year compounded annually. She intends to withdraw $500 at the end of each year. Assume that the present time (period 0) is January 1, 2011 and the first withdrawal will take place on December 31, 2011. How many withdrawals can Janine make before her account balance shrinks to $0?

PVOA=	$5,053
	$500	$500	$500	$500		$500		$500
						.....
	← 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/??	12/31/??
0	1	2	3	4			?	n=??
n = ?? years;  i = 6% per year

Calculation of Exercise #7 using the PVOA Table
Solving for n (the number of periods/payments in an ordinary annuity) is done with the following equation:

	PVOA   =	PMT times [ PVOA factor for n = ?? years; i = 6% per year ]
	$5,053   =	$500 times [ PVOA factor for n = ?? years; i = 6% per year ]
	$5,053 / $500   =	[ PVOA factor for n = ?? years; i = 6% per year ]
	10.106   =	[ PVOA factor for n = ?? years; i = 6% per year ]
	10.106   =	PVOA factor for n = 16 years; i = 6% per year

Let's review this calculation. We insert into the equation the components that we know: the present value, the interest rate, and the recurring payment amount. In line four, we calculate our factor to be 10.106. We now know both the PVOA factor (10.106) and the interest rate (6%). We go to the PVOA Table and look down the 6% interest column until we come to the factor 10.106. Tracking across the row, we see that at this point, n = 16. Since the periods in question are annual periods, the answer is 16 years.

Here is the proof of this calculation:

Investment Account Activity

W/D No.	Date	Interest Added*	Withdrawal	Account Balance
	Jan. 1, 2011			$ 5,053.00
1	Dec. 31, 2011	$ 303.18	$ 500.00	$ 4,856.18
2	Dec. 31, 2012	291.37	500.00	4,647.55
3	Dec. 31, 2013	278.85	500.00	4,426.40
4	Dec. 31, 2014	265.58	500.00	4,191.99
5	Dec. 31, 2015	251.52	500.00	3,943.51
6	Dec. 31, 2016	236.61	500.00	3,680.12
7	Dec. 31, 2017	220.81	500.00	3,400.92
8	Dec. 31, 2018	204.06	500.00	3,104.98
9	Dec. 31, 2019	186.30	500.00	2,791.28
10	Dec. 31, 2020	167.48	500.00	2,458.76
11	Dec. 31, 2021	147.53	500.00	2,106.28
12	Dec. 31, 2022	126.38	500.00	1,732.66
13	Dec. 31, 2023	103.96	500.00	1,336.62
14	Dec. 31, 2024	80.20	500.00	916.81
15	Dec. 31, 2025	55.01	500.00	471.82
16	Dec. 31, 2026	28.31	500.00	0.13

   * 6% times the previous account balance

Exercise #8. Jeremy borrows $4,461 at an interest rate of 8% per year, compounded semiannually on each July 1 and January 1. He plans to make a $600 loan payment at the end of each semiannual period. Assuming that the present time (period 0) is January 1, 2011 and the first loan payment will take place on July 1, 2011, when will the loan balance be $0?

PVOA=	$4,461
	$600	$600	$600	$600		$600		$600
						.....
	← 6 months →	← 6 months →	← 6 months →	← 6 months →				← 6 months →
1/1/11	7/1/11	1/1/12	7/1/12	1/1/13			??	??
0	1	2	3	4			?	n=??
n = ?? semiannual periods;  i = 4% per semiannual period

Calculation of Exercise #8 using the PVOA Table
The number of semiannual periods/payments in the ordinary annuity can be computed with the PVOA equation:

	PVOA   =	PMT times [ PVOA factor for n = ?? semiannual periods; i = 4% per semiannual period ]
	$4,461   =	$600 times [ PVOA factor for n = ?? semiannual periods; i = 4% per semiannual period ]
	$4,461 / $600   =	[ PVOA factor for n = ?? semiannual periods; i = 4% per semiannual period ]
	7.435   =	[ PVOA factor for n = ?? semiannual periods; i = 4% per semiannual period ]
	7.435   =	PVOA factor for n = 9 semiannual periods; i = 6% per semiannual period

Let's review this calculation. We insert into the equation the components that we know: the present value, the interest rate, and the recurring payment amount. In line four, we calculate our factor to be 7.435. We now know both the PVOA factor (7.435) and the interest rate (4%). We go to the PVOA Table and look down the 4% interest column until we come to the factor 7.435. Tracking across the row, we see that at this point, n = 9. Since the periods in question are semiannual periods, the answer is 9 semiannual periods.

The question we were asked is "when will the loan balance be $0?" The answer is July 1, 2015. Here is the proof of this calculation:

   Loan Amortization Schedule
   (For $4,461 at 8% per year with 9 semiannual payments.)

Pmt No.	Date	Total Payment	Interest Payment*	Principal Payment**	Principal Balance
	Jan. 1, 2011				$ 4,461.00
1	July 1, 2011	$ 600.00	$ 178.44	$ 421.56	4,039.44
2	Jan. 1, 2012	600.00	161.58	438.42	3,601.02
3	July 1, 2012	600.00	144.04	455.96	3,145.06
4	Jan. 1, 2013	600.00	125.80	474.20	2,670.86
5	July 1, 2013	600.00	106.83	493.17	2,177.70
6	Jan. 1, 2014	600.00	87.11	512.89	1,664.80
7	July 1, 2014	600.00	66.59	533.41	1,131.39
8	Jan. 1, 2015	600.00	45.26	554.74	576.65
9	July 1, 2015	600.00	23.07	576.93	(0.28)

   * Interest payment equals 4% of the previous principal balance.
   ** Principal payment equals $600 minus interest payment.

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