# Present Value of an Annuity

An annuity is a series of evenly spaced equal payments made for a certain amount of time. There are two basic types of annuity known as ordinary annuity and annuity due. Ordinary annuity is one in which periodic payments are made at the end of each period. Annuity due is the one in which periodic payments are made at the beginning of each period.

The present value an annuity is the sum of the periodic payments each discounted at the given rate of interest to reflect the time value of money. Alternatively defined, the present value of an annuity is the amount which if invested at the start of first period at the given rate of interest will equate the sum of the amount invested and the compound interest earned on the investment with the product of number of the periodic payments and the face value of each payment.

## Formula

Although the present value (PV) of an annuity can be calculated by discounting each periodic payment separately to the starting point and then adding up all the discounted figures, however, it is more convenient to use the 'one step' formulas given below.

PV of an Ordinary Annuity = R × | 1 − (1 + i)^{-n} |

i |

PV of an Annuity Due = R × | 1 − (1 + i)^{-n} | × (1 + i) |

i |

Where,

**i** is the interest rate per compounding period;

**n** are the number of compounding periods; and

**R** is the fixed periodic payment.

## Examples

**Example 1:** Calculate the present value on Jan 1, 2011 of an annuity of $500 paid at the end of each month of the calendar year 2011. The annual interest rate is 12%.

__Solution__

We have, Periodic Payment

R= $500 Number of Periodsn= 12 Interest Ratei= 12%/12 = 1% Present ValuePV= $500 × (1-(1+1%)^(-12))/1% = $500 × (1-1.01^-12)/1% ≈ $500 × (1-0.88745)/1% ≈ $500 × 0.11255/1% ≈ $500 × 11.255 ≈ $5,627.54

**Example 2:** A certain amount was invested on Jan 1, 2010 such that it generated a periodic payment of $1,000 at the beginning of each month of the calendar year 2010. The interest rate on the investment was 13.2%. Calculate the original investment and the interest earned.

__Solution__

Periodic Payment

R= $1,000 Number of Periodsn= 12 Interest Ratei= 13.2%/12 = 1.1% Original Investment = PV of annuity due on Jan 1, 2010 = $1,000 × (1-(1+1.1%)^(-12))/1.1% × (1+1.1%) = $1,000 × (1-1.011^-12)/0.011 × 1.011 ≈ $1,000 × (1-0.876973)/0.011 × 1.011 ≈ $1,000 × 0.123027/0.011 × 1.011 ≈ $1,000 × 11.184289 × 1.011 ≈ $11,307.32 Interest Earned ≈ $1,000 × 12 − $11,307.32 ≈ $692.68

Written by Irfanullah Jan