Time value of money is one of the most basic fundamentals in all of finance. The underlying principle is that a dollar in your hand today is worth more than a dollar you will receive in the future because a dollar in hand today can be invested to turn into more money in the future. Additionally, there is always a risk that a dollar that you are supposed to receive in the future won't actually be paid to you. 

An example of time value of money
Suppose that a friend offers to pay you $1,000 today or $1,050 one year from today. This promise comes from someone that you trust very much, and thus you do not believe there is any risk that you will not be paid.

Which offer should you accept?

It all depends on the return you can earn on your money. If you can earn 6% on your money, for instance, then you should accept the $1,000 today. If invested for one year, it would grow to $1,060, beating the option of receiving $1,050 one year from now.

However, if you can only earn a 4% return on your money, you should accept the offer of $1,050 paid one year from now. If you accept the $1,000 and invest it at a 4% return, it will only grow to $1,040 in one year, compared to receiving $1,050 after one year from your friend.

Using time value of money
The underlying principles of time value of money are used in finance to value investments like stocks and bonds.

The basic formula for the time value of money is as follows:

PV = FV ÷ (1+I)^N, where:

PV is the present value
FV is the future value
I is the required return
N is the number of time periods before receiving the money

But let's not get too far into the weeds just yet. With a simple example, we can better understand how the time value of money can be used to value investments.

Valuing a stock with the time value of money
Suppose you have the opportunity to invest in a stock. You believe that the business will improve over the next four years, and pay you four $2 annual dividends, after which you believe you'll be able to sell the stock for $40 per share. You want to achieve an annual return of at least 15% on your investment.

How much should you pay for this stock, assuming you need a 15% return?

Let's first lay out all the cash flows:

Time 0: Unknown investment
Year 1: $2 dividend
Year 2: $2 dividend
Year 3: $2 dividend
Year 4: $42 ($2 dividend plus sale of stock for $40 per share)

Now that we have this information, we can start valuing each cash flow.

We plug in the first cash flow ($2 dividend), discount rate (15% annual required return), and time (1 year) into the formula above to get the following:

PV = $2 ÷ (1+0.15)^1

We can further simplify and solve this equation, step by step:

PV = $2 ÷ (1.15)^1
PV = $2 ÷ (1.15)
PV = $1.74

After doing the math, we find that the first dividend is worth $1.74 in present dollars.

Now we have to calculate the present value of the second dividend. We plug in the second cash flow ($2 dividend), discount rate (15% annual required return), and time (2 years) into the formula above to get the following:

PV = $2 ÷ (1+0.15)^2

We can further simplify and solve this equation, step by step:

PV = $2 ÷ (1.15)^2
PV = $2 ÷ (1.3225)
PV = $1.51

We calculate that the present value of the second $2 dividend to be $1.51.

The second dividend is worth less ($1.51) than the first dividend ($1.74) in present dollars because it will come later in time. Cash flows in later periods have to be discounted by a greater amount to ensure we receive the required 15% annual return on our investment. 

To complete this problem, we need to continue discounting all the remaining cash flows, then sum up the present values of all the cash flows. We'll skip the present value steps for the two remaining cash flows for brevity, but the answers are shown in the table below. See what you come up with and compare it to the table (if you're off by a penny or two, it's probably just due to rounding).

Time Period

Future Value

Present Value

Time 1

$2.00

$1.74

Time 2

$2.00

$1.51

Time 3

$2.00

$1.32

Time 4

$42.00

$24.01

Total

$48.00

$28.58

The final step is to sum up the present values of each cash flow to arrive at a value of $28.58 per share for this stock.

If our assumptions are correct about the dividend the stock will pay, and its value at the end of four years, we'll generate a 15% annual return on our investment in this stock provided that we do not pay a penny more than $28.58 per share.

Of course, the outputs generated by time value of money formulas are only as good as our inputs. If the company ceases to pay a dividend, or fails to trade for $40 per share in year 4, then $28.58 will prove to be a much too expensive price for the company's stock. That's not a weakness in the mathematics; the math is objective. The subjective inputs are the most important part of the puzzle -- and the hardest part to get right. 

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