# Time Value Of Money

Updated: Mar 24, 2021

Time value of money is one of the most important concepts in finance. If you don't understand time value your financial education will come to a dead stop much sooner than you think.

## Lesson One: Getting Started

Time value of money allows us to move money through time. Let me ask you a series of questions to demonstrate the idea

1) Would you rather \$1,000 now or \$1,000 in 5 years?

"I will take the money now"

2) Would you rather \$1000 now or \$1,000,000 a year from now?

"I'll take the million!"

3) Would you rather \$10,000 now or \$12,000 a year from now?

"I'm not sure..."

What do these questions teach us?

1) People prefer money sooner rather than later.

2) People are willing to wait if they can get sufficiently more money in the future.

But what about the third case? How can we decide between \$10,000 now and \$12,000 later? For starters let's simplify the question and assume we are independently wealthy, so we have no need to take the money and buy something. We are just looking to maximize our wealth.

Let's focus on getting the most bang for our buck. If we want to be as wealthy as possible a year from now we have to ask ourselves "could I turn the \$10,000 into more than \$12,000 if I took the money now and invested it?".

That depends on what you do with it of course. You could invest in the stock market and maybe make or lose a lot in a year or you could invest in a t-bill and make .5%.

In this case, the \$12,000 a year from now is guaranteed. It is as safe as a t-bill. So in this case, if we took the \$10,000 and invested it by taking the same amount of risk we would only have \$10,050. Clearly, we should take \$12,000 dollars in a year. In effect, we are being offered a 20% return and .5% return for the same risk. Of course, we choose the 20% return.

We calculate the value after one year by using the following equation.

Fututre value = principal *(1+ interest rate)

Remember to convert the interest rate to a decimal by dividing by 100.

Another question, how much would I have to offer today to make you indifferent between the two options?

In order to find out how much money they would have to offer today so that we really didn't care if we got the money now or later let's reuse the equation above. By re-arranging our future value equation we get the following.

Present Value = Future value / (1+interest rate)

* we switched the word principal for "present value"

Solve:

Present value = \$10,200/(1+.005)

Present Value = \$10,149.25

Let's test our solution by using our future value equation.

Fututre value = principal *(1+ interest rate)

Future value = 10,149.25 *(1+.005)

Future value =\$12,000

By using the future value and present value equations we can find the value of sums of money at different periods of time.

New tools: future value and present value.

## Lesson Two: Building Up

In the real world, we aren't always dealing with one-year periods. How can we find the time value of money over periods of more or less than a year?

Let's find the future value of 100 dollars grown at 4% for two years. We can do it in three different ways, each simpler than the last.

Now let's find the present value of \$108.16 in two years assuming r = 4%.

New Tool: future value and present value calculations over multiple time periods.

## Lesson Three: Special Situations

Single payments are fairly easy to deal with, what about multiple payments?

1) You can find the present value of a stream of never-ending payments. We can refer to this never-ending stream as a no-growth perpetuity.

PV of no-growth perpetuity = payment / r

Example: What is the present value of a never-ending annual payment of 5\$ if r= 5%

2) You can find the value of a never-ending payment stream growing at a constant rate.

PV of constantly growing perpetuity = payemnt / (r-g)

where g= growth rate

Example: What is the present value of a never-ending annual payment of 5\$ if r= 5% and g= 4%?

3) An annuity is a stream of cash flows for n years. You can find the present value of an annuity using the following formula.

## Lesson Four: Using Excel

Excel has multiple functions which make TVM calculations simple.

1)PV

The PV function requires the following:

rate: discount rate

nper: number of periods

pmt: payment

fv: future value, for example, the future value of a bond is \$1000 dollars because you get the \$100 par value in addition to the last coupon payment.

For example, you purchase an annual coupon bond that pays 60 dollars annually and matures in 20 years. The market rate is 10% on comparable bonds. Value the bonds using the PV function.

2) You can solve for any one of the variables seen above using the variable as the function key.

3) Uneven Cash Flows

Suppose I offer you the opportunity to buy 3 cash flows for 100 dollars. The cash flows are;

15 dollar in year one

20 dollars in a year two

105 dollars in year three

If you purchased the cash flows what would your rate of return be?

In order to calculate our rate of return, which is known as internal rate of return or "IRR" we will use the IRR function in excel.

This function will discount out three cash flows such that the NPV is zero. Net present value (NPV) is the present value of cash flows less the cost of the cash flows. IRR calculates the discount rate which makes the PV of the cash flows = the cot of said cash flows.

This means that if we discount each flow by 14% and all of them up the PV will be \$100. If the present value of the cash flows is \$100 and the cost of the cash flows is \$100 then NPV is zero.