Finance - Annuities - Future value

Definition and concept.

An annuity is an investment for which:

deposits are made at regular intervals - usually (but not always) at the end a defined period of time;
- equal amounts are deposited to the account each interval.
  - interest is compounded at regular intervals - usually with the same regularity for which deposits are made.

The value of an annuity is returned to the owner of the account at the end of the number of periods agreed.
Returns of the annuity can be made as a lump sum, a series of lump sums or on a regular basis.

There are two broad types of annuity depending on when contributions to an annuity are due:

when payments are made at the end of a given period, the annuity is called an ordinary annuity;
when payments are due at the beginning of a given period, the annuity is called annuity due.

In almost every case of annuities to be discussed here, payment is made at the end of the nominated period.

Ensure any question DOES NOT confuse you with ordinary compound interest:

annuities require there must be a regular payment mentioned in the question;
compound interest questions do not mention multiple payments.

Determining the future value of an ordinary annuity.

We can construct the pattern for an investment $1 at the end of each period (e.g. month, year, etc) as follows:

End of period 1	Deposit $1.
End of period 2	Deposit receives interest - so $1 × 1.01 = $1.01
	Then make another deposit of $1.
	So the total at the end of period 2 = $2.01
End of period 3:	The total of $2.01 receives interest - so $2.01 × 1.1 = $2.0301
	Then make another deposit of $1.
	Total at the end of period 3 = $3.0301.

The totals we are calculating for the balance at the end of each period for our $1 contributions are referred to as "interest factors".

Calculating the future value of an Annuity using a table of interest factors.

The three interest factors calculated above are representative of the many interest rates available, the many periods for contributing to an annuity and the many different amounts we can deposit. They form the basis for calculating all possible combinations and the variety of values for the interest factors are recorded in special tables to facilitate the easy calculation of the total future value of an annuity.

An example of such a table is as follows:

Table of interest factors to calculate
the future value of an ordinary annuity of $1

n	1%	2%	3%	4%	5%	6%	8%	10%	12%
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	2.0100	2.0200	2.0300	2.0400	2.0500	2.0600	2.0800	2.1000	2.1200
3	3.0301	3.0604	3.0909	3.1216	3.1525	3.1836	3.2464	3.3100	3.3744
4	4.0604	4.1216	4.1836	4.2465	4.3101	4.3746	4.5061	4.6410	4.7793
5	5.1010	5.2040	5.3091	5.4163	5.5256	5.6371	5.8666	6.1051	6.3529
6	6.1520	6.3081	6.4684	6.6330	6.8019	6.9753	7.3359	7.7156	8.1152
7	7.2135	7.4343	7.6625	7.8983	8.1420	8.3938	8.9228	9.4872	10.0890
8	8.2857	8.5830	8.8923	9.2142	9.5491	9.8975	10.6366	11.4359	12.2997
9	9.3685	9.7546	10.1591	10.5828	11.0266	11.4913	12.4876	13.5795	14.7757
10	10.4622	10.9497	11.4639	12.0061	12.5779	13.1808	14.4866	15.9374	17.5487
11	11.5668	12.1687	12.8078	13.4864	14.2068	14.9716	16.6455	18.5312	20.6546
12	12.6825	13.4121	14.1920	15.0258	15.9171	16.8699	18.9771	21.3843	24.1331
13	13.8093	14.6803	15.6178	16.6268	17.7130	18.8821	21.4953	24.5227	28.0291
14	14.9474	15.9739	17.0863	18.2919	19.5986	21.0151	24.2149	27.9750	32.3926
15	16.0969	17.2934	18.5989	20.0236	21.5786	23.2760	27.1521	31.7725	37.2797
16	17.2579	18.6393	20.1569	21.8245	23.6575	25.6725	30.3243	35.9497	42.7533
17	18.4304	20.0121	21.7616	23.6975	25.8404	28.2129	33.7502	40.5447	48.8837
18	19.6148	21.4123	23.4144	25.6454	28.1324	30.9057	37.4502	45.5992	55.7497
19	20.8109	22.8406	25.1169	27.6712	30.5390	33.7600	41.4463	51.1591	63.4397
20	22.0190	24.2974	26.8704	29.7781	33.0660	36.7856	45.7620	57.2750	72.0524
21	23.2392	25.7833	28.6765	31.9692	35.7193	39.9927	50.4229	64.0025	81.6987
22	24.4716	27.2990	30.5368	34.2480	38.5052	43.3923	55.4568	71.4028	92.5026
23	25.7163	28.8450	32.4529	36.6179	41.4305	46.9958	60.8933	79.5430	104.6029
24	26.9735	30.4219	34.4265	39.0826	44.5020	50.8156	66.7648	88.4973	118.1552

Notice that the first three values in the 1% column (for n = 1, 2 and 3) are the three values calculated above.

The value of each interest factors relates to a specific combination of periods and interest rates.

How to use the interest factor table to calculate the value of a particular annuity.

			Example: Calculate the future value of an annuity of 6 years paying 8% p.a. with payments of $100 made at the end of each quarter.
1.	Determine the number of periods.	This number is really the number of payments. Calculate according to the question. If there are monthly payments for 4 years then n = 4 × 12 = 48	n = 6 years × 4 quarters = 24 periods.
2.	Determine the interest rate per period.	The question usually provides a 'per annum' interest rate. So divide that number by the number of payments in one year.	Interest rate per quarter: 8% ÷ 4 = 2%
3.	Look up the required value (or interest factor) in the table using the values from Step 1 and Step 2.		Table value from Row 24 periods, Column 2% = 30.4219
4.	Multiply the table value (i.e. the interest factor) by the amount of each payment or deposit.		Value = 30.4219 × $100 = $3,042.19.