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

##### 1 Answer

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 An annuity is money invested, or withdrawn, at regular intervals, for example $100 invested in your savings account every year for the next 10 years.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:

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.

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.