Definitions

An annuity is a sequence of periodic payments, usually equal, made at equal intervals of time. Premium on insurance, mortgage payments, interest payment on bonds and debentures, payments of rent, payments of hire purchase and dividends are just few examples of annuities.

The time between successive payments is called payment period. The time from the beginning of the first payment period to the end of the last payment period is called the term of an annuity. When the term of an annuity is fixed, i.e. the dates of the first and the last payment is fixed, the annuity is called an annuity certain. When the term of an annuity depends on some uncertain event, the annuity is called contingent annuity. Bond interest payments form an annuity certain: life-insurance premiums form a contingent annuity( they cease with the death of the insured). Unless otherwise specified the word annuity will refer to an annuity certain.

When the payments are made at end of each payment period, the annuity is called ordinary annuity. Examples include loan repayments and interest payment on bonds and debentures. When the payments are made at the beginning of each payment period is called annuity due. Insurance premium are a good example of this type of annuity.

When the payment period and interest rate coincide, the annuity is called a simple annuity, otherwise it is a general annuity.

The accumulated value of an annuity at the equivalent dated value of a set of payments due at the end of the term. Similarly, the discounted value of an annuity is defined as the equivalent dated value of a set of payments due at the beginning of the term. In this article I will use the following notations as follows:

R = the periodic payment of the annuity

n = the number of interest period during the term of the annuity

i = the interest rate per payment period

S = the accumulated value of an annuity

A= Present value of an annuity.

Accumulated value of an ordinary simple annuity

The accumulated value “S” of an ordinary simple annuity is defined as the amount due at the end of the term equivalent to the dated values of the payments comprising the annuity with the date of the last payments as the focal date. As ordinary simple annuity is shown on the time diagram below with the interest period ( which equals the payment period) as the unit of measure.

_______R_______R__________R___________________________________R_____________R

0             1              2                    3                                                                   n-1                         n

————————————————term—————————————————————

Let suppose R =1 then using the geometric progression of the sum of all terms with constant ratio of n periods where the constant factor is (1+i) and the initial term is 1. Using the formula for the sum of all terms of a geometric progression S i/n = 1(1+i)n -1/ (1+i)-1 = (1+i)n -1/i. There fore if the annuity is R then Accumulated value “S” = R* Si/n =R* (1+i)n-1/i. Where Si/n refers to the annuity factor relevant to the interest rate for the annuity period for an ordinary simple annuity where the interest period and the annuity period coincide.

Example

An employee invests $300 per year from his tax return. After 10 such payments, he increases his deposits to $400 per year. Assuming that he has been earning 12% per year, effective what accumulation there be after 15 payments.

Solution

This problem is best considered as combination of two annuities: one of $300 per year paid in year 1-10 and, and one of $400 per year paid in year 11-15.

Value of $300 annuity at time 10 = 300* (1.12)10-1/0.12 = 5264.62

This accumulated value must then be moved with compound interest to time 15. Hence value of $300 annuity at time 15 = 5264.62*(1.12)5= 9278.06

Value of $400 annuity at time 15 = 400*(1.12)5-1/0.12= 2541.14

There fore the accumulated value of the total annuity = 9278.06 + 2541.14 = 11819.20

Discounted value of an ordinary simple annuity

As depicted above, an ordinary simple annuity is an equivalent value of the annuities now. There fore, the present value “A” can be algebraically written as follows:

A= R/(1+i)+ R/(1+i)2……………………..R/(1+i)n

A= R*[ 1/(1+i) + 1/(1+i)2...............................1/(1+i)n]

Using the sum of a geometric progression one can calculate the sum of the discounted value of $1annuity at a particular rate of interest. In this instant, the first item of the series is 1/(1+i) and the common factor is also 1/(1+i). There fore, applying the geometric series sum formula the sum of

{1/(1+i)+ 1/(1+i)2…………………..1/(1+i)n] = 1/(1+i) [ 1- (1/1+i)n]/1-1/(1+i). Simplifying the righthad side of the formula gives a annuity factor for a ordinary simple annuity as follows:

Say the annuity factor is denoted by “a i/n”. There fore, “a i/n” = [ 1- (1+i)-n]/i. That is the the present value of an ordinary simple annuity A = R* “a i/n “= R* [ 1- (1+i)-n]/i

Example

An annuity pay $500 per year for 5 years. And then $300 per year for 4 years. Calculate the value of this annuity before the first payment using a annual interest rate of 11%.

Solution

First draw the time line due to the change part way through the term. The time line is as follows:

____500____500_____500______500______500_____300_____300_____300_____300

0        1            2              3                4                5               6              7             8               9

This problem can be thought of consisting of two annuities. The first is $500 per year in years 1-5 and the second is $300 per year in years 6-9.

If considers the the $500 per year annuity then

Value at time 0 = 500 a 11/5 = 500* [1- 91.11)-5/0.11 = 1847.95.

Now consider the the 4300 dollar per year annuity. The value of this at time 5 is as follows:

Value at time 5 = 300* [1-(1.11)-4/0.11 = 930.73

There fore value at time 0 = 930.73*(1.11)-5 = 552.35

Hence the total value of annuity = 1847.30 + 552.35 = 2400.30

Finding the periodic payment of an ordinary simple annuity

If the accumulated value or the present value of an annuity in known, as well one know the interest rate and the term of an annuity assuming the interest rate is constant and the Repayment is constant then one can rearrange the ordinary simple annuity formula to find the repayment.

If the accumulated value “S” is known then R = S/ s 1/n = S/ (1+i)n -1/i

If the present value is known then R = A/ a i/n = A/ 1- (1+i)-n/i

Example

Upon the birth of their first child, a couple decide to deposit $ X at the end of each month ( commencing in one months time) so that upon the child’s 18 th birth day, immediately after the final deposit, they may give her $ 100000. Calculate X, if the interest rate is (5 convertible monthly

As well, ,they alternatively deposit $ Y at a rate of 9 % convertible quarterly, Calculate Y. Explore why Y is greater than 3X and give possible reasons.

Solution

In this S = 100000, i= 0.09/12 =0.0075 n = 18*12 = 216

hence x = 100000/ s 0.0075/216 = $ 186.44.

In the second case, Y = 10000/ s 0.0225/72 = $ 567.73

Y is greater than 3X because in the first case the interest is 0.0075 and it is greater than the quarterly interest rate of 0.0225 as the equivalent rate of 0.0075 per month is approximately equal to 2.27%. In addition, Y is deposited end of the quarter such that it does not earn interest during that quarter. In contrast, the two payments of X deposited before the end of the quarter earn some interest in that quarter. Because of these reasons, Y is greater than 3X.

Finding the interest rate

A very practical applications of the above formulas is finding the interest rate. In many business transactions the effective interest rate is unknown or concealed by the complexities of the arrangements. In order to compare alternative investments or loans, it is necessary to determine the effective interest rate in respect of each proposition and make the decision based on these effective (or true) interest rates.

When R, n and either S or A are given the interest rate I may be determined approximately by linear interpolation. For most practical purposes linear interpolation gives sufficient accuracy. It is an approximate method.

Example

A used car sells for $600 cash or $100 deposit and $90 a month for 6 months. Find the effective interest rate if the purchaser buys the car on installment plan.

Solution

For the above transaction the following equation of value must hod true to have the cash option equivalent to the installment plan.

Cash price = deposit + discounted value of installment

There fore 600 = 100 + 90 i/6

That is,a i/6 = 500/90 = 5.5556

The relevant equations for linear interpolations are:

a 0.02/6 = 5.6014

a 1/6 = 5.5556

a 0.025/6 = 5.5081

That is, i– 0.02/ 0.025 – 0.02 = 5.556 – 5.6014/ 5.5081 – 5.6014

i-0.2/ 0.005 = -0.0458/ -0.0933 = 0.00245

i= 0.00245 per month

There fore effective rate per year = 0.02245*100*12 = 26.95% per year

Finding the term of an annuity

In some situations the accumulated value S, and present value A, the annuity payment R and interest rates are known. However one must determine the number of term of an ordinary simple annuity. In most circumstances for the formula S = R s i/n or A =R a i/n will not give a full number. In this instance a last payment for the next period is paid as to have an equivalent value different from R in most practical circumstances.

Example

A man dies and leaves his wife an estate of $50000. The money is invested at 12% per year. How many monthly payments of $750 would the widow receive and what would be the size of the concluding payment?

Solution

In this problem, A = 50000, R =750 per month i=12/12/100 = 0.01

There fore 750 * a0.01/n = 50000

a 0.01/n = 50000/750 = 66.666

1- (1.01)-n/0.01 = 66.666

1 – (1.01)-n = 0.6666

(1.01)-n = 0.3333

-n log 1.01 = log 0.3333

n = 110. 40963

In this n is not a full number. There fore one must be able have a payment for the 111 th payment. Say it is X.

There fore, 750* a 0.01/110* (1.01) + X = 50000* (1.01)111

150575.63 + X = 150863. 76

X = 308. 13

Changes in the interest rate and the calculation of accumulated value and present value of an annuity

In real life the interest rate can change over an annuities term. In this situation one must calculate for separate periods of the annuities and must have a focal date closer to the final term and compound the earlier periods to the final date or term. This systematic process is important sothat the focal dates are correctly chosen and the number of periods for separate annuities are correctly selected.

Example

An annuity is having interest rate change over its term. If that annuity has the rates for year 1,2,3 as 8% and for year 4 and 5 interest rate is 7% and 6, 7, 8, 9 and 10 having an interest rate of 6%. What is the present value of the 10 year annuity if the annuity is 50 monthly for 10 years?

Solution

First draw a time line for the above annuity. The time line is as follows for different periods of interest rate changes.

_50_50_50_50_50_50_50_50_50_50_50_50_________50_50

0 1    2   3     4    5    6   7     8   9   10 11     12                19   20

|———–0.04———|——0.035——|————0.03———-|

Intermediate value at the end of year 5 = 50 a 0.03/10 = 426.51

Intermediate value at year 3 = 50 a 0.035/4 + 426.51* (1+0.035)-4

= 183.65 + 371.68 = 555.33

Present value = 50 a 0.04/6 + 555.33* (1+0.04)-6

= 262.10+ 438.89 = 700.99