An annuity is a stream of equal payments made at regular intervals.

We have learned how to calculate the future value of the ordinary annuity and annuity due.

In this post, we will learn how to calculate the present value of ordinary annuity and annuity at due.

**The Present Value Of An Ordinary Annuity**

The present value of an ordinary annuity is the amount that would have to be deposited in one lump sum today to produce the same balance as the future value of a stream of equal payments (given the same compound interest rate and given that payment was made at the **end of the period**)

In other words, the present value of an ordinary annuity is the current cash value of the streams of payments of the ordinary annuity.

The present value of the annuity is calculated using the following formula:

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]$$

Where,

C is periodic cash flow

i is the interest rate

m is the number of times computing occurs in a year

n is the number of years

__Example 1__

Find the lump sum deposited today that will yield the same total amount as Payment of N890 at the end of each year for 16 years at 6% compounded annually.

**Solution:**

The present value of an annuity tells you the amount you need to invest now to earn the same balance as that of a future value of a stream of payment.

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]$$

Here, c=890, i=0.06 and n=16, m=1 because compounding occurs annually (once a year)

$PV=890\left[\frac{1-\left(1+\frac{0.06}{1}\right)^{-16(1)}}{\frac{0.06}{1}}\right]$

$PV=890\left[\frac{1-\left(1+0.06\right)^{-16}}{0.06}\right]$

$PV=890\left[\frac{1-0.393646)}{0.06}\right]$

$PV=890\left[\frac{0.605354}{0.06}\right]=N8994.251$

__Example 2__

Find the present value of a 5-year ordinary annuity with the periodic payment of N2000 and an interest rate of 8% compounded quarterly.

**Solution:**

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]$$

$PV=2000\left[\frac{1-\left(1+\frac{0.08}{4}\right)^{-5(4)}}{\frac{0.08}{4}}\right]$

** Note:** m is 4 because there are 4 quarters in a year

$PV=2000\left[\frac{1-\left(1+0.02\right)^{-20}}{0.02}\right]$

$PV=2000\left[\frac{1-0.67297}{0.02}\right]$

$PV=2000\left[\frac{0.32703}{0.02}\right]=N32703$

__Example 3__

What lump sum deposited today will yield the same amount balance as a periodic payment of N800 at the end of the year for 5 years at 9% compounded semi-annually.

**Solution:**

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]$$

$PV=800\left[\frac{1-\left(1+\frac{0.09}{2}\right)^{-5(2)}}{\frac{0.09}{2}}\right]$

** Note:** m is 2 because a year is equivalent to two quarters

$PV=800\left[\frac{1-\left(1+0.045\right)^{-10}}{0.045}\right]$

$PV=800\left[\frac{1-0.63928}{0.045}\right]$

$PV=800\left[\frac{0.356072}{0.045}\right]=N6330.17$

**Present Value Of An Annuity Due**

The present value of an annuity due is the amount that would have to be deposited in one lump sum today to produce the same balance as the future value of a stream of equal payments (given the same compound interest rate and given that payment was made at the **beginning of the period**).

In other words, the present value of an ordinary annuity is the current cash value of the streams of payments of the annuity due.

The present value of the annuity is calculated using the following formula:

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]\left(1+\frac{i}{m}\right)$$

Where,

C is periodic cash flow

i is the interest rate

m is the number of times computing occurs in a year

n is the number of years

**Note:** The formula for the present value of ordinary annuity and annuity due is nearly the same. The only difference is that $\left(1+\frac{i}{m}\right)$ is added to the present value of an annuity due.

To better appreciate the formula for the present value of annuity due, let's take one more example

__Example 4__

Mr Samuel deposited N1000 at the beginning of each year for 5 years at a rate of 6% compounded annually. Calculator the present value of the annuity due.

**Solution:**

Remember that the present value of an annuity due is calculated using:

$$PV=C\left[\frac{1-\left(1+\frac{i}{m}\right)^{-n m}}{\frac{i}{m}}\right]\left(1+\frac{i}{m}\right)$$

Here, C=1000, n=5, i=0.06, m=1 because interest compounds annually (yearly)

$PV=1000\left[\frac{1-\left(1+\frac{0.06}{1}\right)^{-5(1)}}{\frac{0.06}{1}}\right]\left(1+\frac{0.06}{1}\right)$

$PV=1000\left[\frac{1-\left(1.06 \right)^{-5}}{0.06}\right]\left(1.06\right)$

$PV=1000\left[\frac{1-0.747258}{0.06}\right]\left(1.06\right)$

$PV=1000\left[\frac{0.252742}{0.06}\right]\left(1.06\right)=N4465.11$

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