# What is the formula for a present value of a sum of money?

Jul 4, 2015

It depends on whether we are dealing with a lump sum or an annuity. Let's talk.

#### Explanation:

A lump sum is a single amount of money, for example a single, one-time, deposit to a savings account. An annuity is money invested, or withdrawn, at regular intervals, for example $100 invested in your savings account every year for the next 10 years. We will use examples for both situations and see how to make the present value calculations. Simple Sum In 10 years, you would like to have money for a down payment on a house. After doing some "guesstimating" you think you will need a down payment of$20,000 in 10 years (n). What is the present value of that $20,000? Or, in other words, how much do you need to invest today to have$20,000 in 10 years? Let's assume you can earn 5% on your invested money (r).

Here is the formula for the present value of a simple sum:

$P V = \frac{f v}{1 + r} ^ n$

$P V = \frac{20 , 000}{1.05} ^ 10$

$P V = \frac{20 , 000}{1.6289}$

PV = 12,278

The answer tells us that $12,278 invested today at 5% will become$20,000 in 10 years.

Annuity
Let's change the question to make it the present value of an annuity .
Instead of purchasing a house with your money, you want to help pay your widowed mother's rent. You want to know how much you will have to invest today so you can withdraw $5,000 a year for the next 10 years to help your mother. All money invested can still receive a 5% return. This is the present value of an annuity because a yearly cash flow is involved. A different formula is required to solve the problem - the present value of an annuity. $P V = P m t \left[\frac{1 - \left(\frac{1}{1 + r} ^ n\right)}{r}\right]$$P V = 5 , 000 \left[\frac{1 - \left(\frac{1}{1.05} ^ 10\right)}{.05}\right]$$P V = 5 , 000 \left[\frac{1 - \left(.6139\right)}{.05}\right]$PV = 38,608 The answer tells us that if you invest$38,608 today, you will be able to take out \$5,000 a year for the next ten years to help you mother.