# Time value of money

**Time value of money** is a concept that states the earlier you receive money the more it is worth. Sometimes called "present discounted value".

## Contents

## Expanded Definition

The time value of money is a concept most people inherently understand. It is similar to "a bird in the hand is worth two in the bush," but our version reads "$1 today is worth more than $1 tomorrow."

For example, if you ask someone would they rather receive their pay today or next month, assuming the amount would be the same in each case, they almost universally would say "today." Why? First, if you control the money *today*, you can put it to work earning more money for you -- earning interest, invested in stocks, real estate or other assets that can appreciate in value. Thus for the investor, time is money because it represents opportunity to earn more. Second, inflation, decreases the purchasing value of that money as time goes on. It takes more money in the future to buy the same amount of stuff as you can buy today (10-cent loaf of bread, anyone?).

### An example of how the concept works

Let's say you have a choice of getting $1,000 today or $1,000 one year from today. Suppose you picked the first option and you bought a CD that offered 4% interest. The next year you would have $1,040.00 instead of the $1,000.00 you would have had if you had opted to wait a year. Clearly the first option is superior.

This is even more important when you consider inflation. Let's take the first example again but this time factor in 2% inflation. The $1,040.00 you had from your CD would be worth about $1,020 in today's dollars while if you had waited a year, the $1,000 you'd receive would only be worth $980.39 in today's dollars ($1,000 divided 1.02 {for 2% inflation}) !!! Thus time really is money as the ageless cliche says.

Clearly investors need to understand that there is an opportunity cost and a devaluing cost due to inflation in waiting to act when making investing decisions. While one should never act in haste, an investor would be wise not to procrastinate after receiving pay or other income. This especially applies to IRA contributions. You should always make contributions as early as you are able for each calendar year.

All of these are inter-related concepts and derive from the concept of the time value of money. These answer the question, "What *is* the time value of money?"

**CAGR**

Suppose you have $100. Woo hoo! Time to go buy a copy of *Poor Charlie's Almanack* with enough left over for some coffee to sip while reading it. But, if you don't want to buy anything right now, you can invest that money and have even more next year.

So, you decide to invest that $100 in a 1-year CD that pays 8% (hey, this is my fantasy, I'll set the rules). When that CD matures, you'll have $108. $100 * 1.08 = $108 Enough to buy two more cups of coffee at Starbucks than you could before. Not bad. But say you let that CD roll over and pull the money out two years from now. How much would you have then? Easy. $116.64. $100 * 1.08 * 1.08 = $116.64.

Three years from now, $125.97. Four years from now, $136.05, and so on.

To calculate this, you could do $100 * 1.08 * 1.08 * 1.08 * ... and go as many times as you have years. But to save your poor fingers (not to mention the calculator's computing power -- you do know that a calculator only has a certain number of calculations it can perform correctly over its lifetime, right?), you can use a collapsed version. $100 * 1.08^n, where " ^ " means "raised to the power of" and "n" = number of years. In more general terms, this can be expressed:

<math>FV = PV * (1 + G)^n</math> Eqn 1 where FV = future value, PV = present value, G = growth rate (interest rate), n = number of years

This is one view of the CAGR (Compound Annual Growth Rate) formula.

**Discount rate**

But suppose you wanted to have $100 two years from now. How much would you have to invest at 8% per year in order to have that? Well, I can tell you that the answer is not $100. At 8% per year, we've already seen that you'd get $116.64, not the $100 we want to have. What's the least amount of money you could invest today to get $100 two years from now?

Well, lets rearrange Eqn 1 above and solve for PV:

<math>PV = \frac{FV}{(1 + G)^n}</math> Eqn 2

Now plug in $100 for FV, 0.08 for G, and 2 for n. You should get $85.73. You can do this as follows, using the scientific calculator on your computer (open the calculator and look for all the extra buttons -- if you don't see them, change the "View"):

1.08

x^2

1/x

*

100

=

Or, you could do it as: 100 / 1.08 / 1.08 =

So, you need to invest $85.73 at 8% per year to end up with $100 two years from now. **Note** that this also happens to show that $100 two years from now is worth less than $100 today (getting back to the time value of money).

In this case, that 8% is called the discount rate. You are discounting $100 by that amount over a two year time span.

**Required rate of return**

The discount rate is also the required rate of return. Let me show you.

If you have $100 today, and you want to have $200 four years from now, you need to earn how much interest per year? In other words, what is the return per year that will double your money in four years?

Once more, we go back to Eqn 1, but this time, we solve for G:

<math>G = ( \frac{FV}{PV} ) ^{\frac{1}{n}} - 1</math> Eqn 3

For this, you do need your scientific calculator. FV = 200, PV = 100, n = 4. Here are the steps:

200

/

100

=

x^y

0.25 (this is 1/4)

=

- 1

=

You should end up with 0.189207... Rounding this off and converting to percent, and you must invest that $100 at 18.92% per year in order to double it in four years.

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