[Maths Class Notes] on Annuities Pdf for Exam

Many people have had the experience of making a series of fixed payments over a course of time – such as rent, premium or vehicle payments – or obtaining a series of payments for a course of time, such as the certificate of deposit (CD) or interest from a bond or lending money. These ongoing or recurring payments are technically called “annuities”. Note that there is also a financial product referred to as an annuity, but both are not just similar though the two are related.

Types of Annuities

Annuities, in this sense of the word, are divided into 2 basic types: ordinary annuities and annuities due.

1. Ordinary Annuities: An ordinary annuity makes (or needs) payments at the termination of each period. For example, bonds usually pay interest at the termination of every 6 months.

2. Annuities Due: With an annuity due, payments, on the contrary come at the start of each time period. Rent, which landlords typically need at the initiation of each month, is one of the common annuity examples.

How to Calculate Annuities

There are various ways to measure the annuity rate changes or the cost of making such payments or what they’re ultimately worth. However, it is first better to know about calculating the present value of the annuity or the future value of the annuity.

Formula to Calculate Present Value Annuities

The formula for the present value of an ordinary annuity:

PV ordinary annuity = P * 1 – (1 + r)^-n/ r

Where,

PV = present value of an ordinary annuity

P = value of each payment

R = interest rate/ period

N = total number of periods

The formula for calculating the present value of an annuity due is:

PV Annuity Due​ = C × [i₁ − (1 + i)⁻ⁿ​] × (1 + i)​

Formula to Calculate Future Value Annuities

Instead of calculating each payment separately and then adding them all up, you can instead apply the following formula, which will tell you the amount of money you’d have in the end:

FV Ordinary Annuity = C × [i(1 + i)^{n −1}]

Where:

C = cash flow/period

i = rate of interest

n = total number of payments

The formula for the future value of an annuity due is:

FV Annuity Due​​ = C × [i(1 + i)ⁿ⁻¹​] × (1 + i)​

Solved Examples

Example:

Calculate the future value of the ordinary annuity and the present value of an annuity due where cash flow per period amounts to rs. 1000 and interest rate is charged at 0.05%.

Solution:

Using the formula to calculate future value of ordinary annuity = C × [(1 + i)^{n – 1/i}

= Rs. 1,000 × [0.05 (1 + 0.05)⁵⁻¹]

=Rs.1, 000 × 5.53

=Rs. 5,525.63

Note that the one-cent difference in these outcomes, Rs. 5,525.64 vs. Rs. 5,525.63, is because of rounding in the first calculation.

Now to calculate the present value of an annuity due:

Use the formula

PV Annuity Due​ = C × [i₁ − (1 + i)⁻ⁿ​] × (1 + i)​

Plugging in the values:

= Rs. 1,000 × [0.05(1− (1 + 0.05)⁻⁵] × (1 + 0.05)

= Rs. 1,000 × 4.33 × 1.05

= Rs. 4,545.95​

Did You Know?

• Annuities are applicable when you are saving money.

• Generally in an annuity problem, your account begins empty but has money in the future.

• Annuities suppose that you put money in the account on a routine basis (every month, quarter year, etc.) and let it remain to earn interest.

• If you’re putting money into the account on a regular basis, then you’re looking at a basic annuity problem.

• Recurring payments, such as the rent or interest are sometimes referred to as “annuities”.

• The present value of the annuity is the amount of money that would be needed now to generate those future payments.

• The future value of the annuity is the total value of payments at a particular point in time.

• In ordinary annuities, payments are released at the end of each time period.

• With annuities due, they’re made at the commencement of the period.

Conclusion:

The annuity method formula makes it possible – and comparatively easy, – to identify the present or future value of both the ordinary annuity and the annuity due. The future value of the annuity calculator also has the competency to calculate these annuity rate changes for you with the correct inputs.