** Finance** is all about **time** and **risk**. It’s basically __a study of how people make decisions regarding the allocation of resources over time and the handling of risks of them__. Playing with it requires some very fundamental techniques and strategies, which are all indispensable if not enough for success in financial markets. And the idea of **present value** is one of the most important that will help you value financial assets over time thus making choices between current resources and future gains.

First off, money today is always more valuable than the same amount of money future. This is because you can always deposit that money in bank and roll it into a bigger amount by earning interest. In this sense, with $100 deposited in bank today on a interest rate of 4% or 0.04, in 3 years you will have

$100 * (1+0.04)^{3} = $112.5

Therefore, the **present value** of ** $112.5** 3 years from now is

**. You get the idea.**

__$100__**Present value** is __the amount of money today that would be needed to produce, using prevailing interest rates, a given future amount of money__.

Conversely, **future value** is __the amount of money in future that a certain amount of money today will yield, given prevailing interest rates__.

In the example of $100, the **future value** of ** $100** after 3 years is

**.**

__$112.5__More generally, with interest rate denoted by **r**, after **N** years, **M _{PV}** will yield

**M**, that is

_{FV}M_{FV} = (1 + r)^{N} * M_{PV}

Now suppose you are going to be paid $1200 10 years from now, what is the present value of this future payment? Considering the above formula and guessing with all the information available that the average annual interest rate will probably remain at 5%, we have

M_{PV} = M_{FV} / (1 + r)^{N} = $1200 / (1 + 0.05)^{10} = $736.6

So, the present value of the $1200 payment after 10 years is $736.6. In other words, it’s only worth of $736.6 in today’s money.

To help you smoothly use the concept in real situations, let’s walk through 2 examples.

#### Example 1

The concept of **present value** is very important and applied frequently in financial decision-making. For instance, Microsoft is thinking of investing $100 millions today to the development of Windows Live Search to rival Google, and estimates that the yield of the project in 5 years is $140 millions. Should the software giant go ahead with the investment?

The company’s decision, is very much dependent on the **estimation of future interest rate**. If the interest rate is 4%, the **present value** of $140 millions 5 years from now would be

M_{PV} = M_{FV} / (1 + r)^{N} = 140 / (1 + 0.04)^{5} = 115.1, in million

Apparently, the project is profitable and they should choose to invest the $100 millions now because the future gains(converted to present value of 115.1 millions, which is bigger than 100 millions) is bigger than the current investment cost. However, if they have guessed an annual average interest rate of 7%, it will be hardly profitable because,

M_{PV} = M_{FV} / (1 + r)^{N} = 140 / (1 + 0.07)^{5} = 99.8, in million

Thus, with interest rate of 7% and higher, the present value of the return will not be able to cover the cost today. It’s wise of them to keep away from the project.

#### Example 2

In this example, things are a little more complicated in that you will have to determine the present value of **a stream of payments** rather than one, which is more common in reality.

Imagine that you have won a million-dollar lottery and is asked to choose between **being paid $25,000 a year for 40 years** or **an immediate payment of $500,000**. So which is the right choice at an annual interest rate of 4% for the next 40 years?

With a stream of simple calculations of present value, you will make the right choice in no time. Suppose the payment starts the following year for 40 years, we have

1st year: M_{PV1} = M_{FV1} / (1 + r)^{N} = 25,000 / (1 + 0.04)^{1} = 24,038

2nd year: M_{PV2} = M_{FV2} / (1 + r)^{N} = 25,000 / (1 + 0.04)^{2} = 23,114

3rd year: M_{PV3} = M_{FV3} / (1 + r)^{N} = 25,000 / (1 + 0.04)^{3} = 22,225

…

40th year: M_{PV40} = M_{FV40}/(1 + r)^{N} = 25,000/(1 + 0.04)^{40} =5,207

Total: 494,819

Adding 24,038 + 23,114 + 22,225 + … + 5,207, we have the final present value of the 40 years’ stream of payments, $494,819, which is a little smaller than $500,000.

Therefore, based on a series of present value calculations, an immediate payment of $500,000 is more favorable over getting paid $25,000 a year for 40 years.