=Intro bookend=

Hi. 

In this video we will discuss the basics of money, time, and interest. In today's world, money and interest rates are everywhere, so understanding the function of interest is of paramount importance. We will start with a simple example, and then move on to discuss the subject more mathematically.

=Learning objectives=

After this lesson, I hope you will be able to do the following:

* Explain the meaning of interest rate
* Calculate basic interest amounts
* Describe the nature of compound interest
* Calculate interest over multiple compounding periods

and

* Describe how interest works for you (against you) in investments (loans)

=Prerequisites=

To get the most out of this video, you should be comfortable with some middle school mathematics, namely percentages, exponents, and basic algebra. If you need a refresher, follow this link to the Khan Academy, which offers free video tutorials on many subjects.

=Time value of money=

So let's discuss the concept of the time value of money. It is the idea that getting to use someone else's money is valuable and requires compensation.

Let's meet Alice. She happens to have saved up some extra money. She can use this money to buy some fun stuff, or even just store it in a safe place for a rainy day. 
Then there's Bob. He doesn't have any spare money, needs it in order to maybe start a business, or pay for a car or a college education.
He looks over at Alice, and says hey, how about you give me some of your extra money, and I promise I'll give it back to you later.
But Alice is no chump, she knows that if she gives Bob money, even if she were 100% certain that he would give it back, she is forgoing the opportunity to use her own money for some of that fun stuff. Or having it stashed away just in case she needs it. 
On top of that, no matter how trustworthy Bob may be, it is possible that he will fail to give the money back - maybe his business goes bankrupt, or some other unfortunate circumstances materialize, such that Bob simply is not able to return the money.
As a result, Alice will demand compensation in exchange for forgoing the use of money, and taking on some risk that she will not get it back. Bob has to promise to give her more money in the future than she is giving him today.

And this is the nature of the time value of money - the fact that holding on to someone's money for a period of time has value.

=Rate of return=

The exact contractual arrangment between Alice and Bob can vary. The two broad classes are those of debt and equity.
With debt, Bob promises to return a certain fixed amount in the future, like say, 1 dollar and 20 cents for every dollar he took. No matter whether his business does really well or really poorly, he has that fixed obligation to Alice, and she is first in line to get paid whenever Bob has some money.
With equity, Alice takes a partial ownership stake in Bob's enterprise, and in return is promised a fraction of future profits. If Bob's company does really well - Alice does really well. If the company does poorly, she gets less money. The obligation is not fixed - Alice shares in the risk of the enterprise she invested in.

Regardless of the details of the arrangement, when Alice gives Bob some money, she expects a return on her money. This return is commonly expressed as a percentage of the amount invested, on an annualized basis. For instance, if Bob borrowed the money, he might promise something like 10% per year. Or if this was an equity arrangement, Bob might have said that he expects his business to earn a 25% rate of return. 

=Interest rate calculation=

Let's talk about the arithmetic of interest rates.
Say you invest 100 dollars, at a promised rate of 5% per year.
At the end of the year, you'd still have a claim on your original 100 dollars, plus, 5% of that 100 as payment for giving up your money for the year, in total adding up to 105 dollars.
Note, that we can factor out the initial amount of 100 dollars, and express this as 100 times 1 plus 5%, which of course is still 105.
This suggests a general algebraic expression to find how much money you will have after one year. You take your initial value, P (p stands for present), and multiply it by 1 plus the rate, to find the future value, F.

The reason it is helpful to factor out the initial value will become apparent once we extend this line of reasoning to multiple periods.

=Rates over time=

Let's see how it works over multiple periods. At time zero, which is the canonical way to denote the present, you have 100 dollars. At time 1, at the end of the first year, as we just discussed, you have 105.

Now, imagine that instead of taking your money back at the end of one year, you just let it ride for another year.
So at the start of year 2, you have 105 dollars invested, and you are expecting another 5% return on it, which comes out to 110 dollars and 25 cents. 
The extra 25 cents is the return you earned in year two, on the 5 dollars that you earned in the previous year. 
Notice that our final result is 100 multiplied by 1 + the rate, 1.05, and then multiplied by 1.05 again.
In the first year, we got 5% on 100, in the second year we get 5% of our new starting amount, which is the 105.

Extending this to year 3, we start with the 110.25 that we had at the end of year 2, earn 5 percent on it again, and get 115.76 or so. Notice how this is the same as starting with 100, and multiplying by 1.05 three times, or by 1.05 to the third power.

At this point, I hope you are starting to see the pattern.
We can write down a general-form algebraic expression for finding the value of our investment in the future, 
as our initial amount P, multiplied by 1 plus the rate, raised to the power of t, the number of years we let it sit, to get the future amount F.

=Quiz 1=

So here's a quick question for you - how much money do you end up with, if you start with 100 dollars, get a 5% rate of return per year, and wait for 5 years?
Take a moment and calculate it out - or at least write down the expression you need to calculate it.

Following the pattern we just developed, we'll need to start with 100, and multiply it by 1.05 - which is one plus the rate, 5 times, which will produce 127 dollars and 63 cents or so.

=Testing it out=

This recursive pattern lends itself very nicely to be implemented in a spreadsheet. Let's play around with this and see what we can discover.

First, let's make a timeline in the first column. Notice how the spreadsheet is smart enough to infer a simple arithmetic sequence based on the first two numbers. Though we could have made it ourselves using "previous cell plus 1" relative reference trick also.
Now in the second column, we'll track the value of our investment. So that we can easily change the rate and the initial investment amount, let's put these constants over here on the side. 

Our time 0 starting value is just the initial investment of 100.
Time 1 value is the previous amount, times 1 + the rate. 105, looks good.

Now if we fill this down, the reference to the previous cell will shift down with us, but so will the reference to the rate. We don't want that, so let's lock in the rate reference with a dollar sign in front of the row number.
Fill down once, 110.25, as we expect. Let's go all the way. Note the magic trick - double-clicking on the fill handle automatically fills down to the last cell with content in the column immediately to the left.
Spot check a random cell in the middle to make sure it looks as we expect, yep, looking good.

Let's calculate the dollar difference between successive values. Notice how every year our investment value grows by a bigger increment than the previous year. This is because our increment is a percentage of starting value every year, and every year's starting value is larger and larger. This is the nature of compound interest - accelerating growth over time, even though the percentage rate of return is fixed.

Let's put this on a graph - select the values we want to use, and insert a chart.

Notice that the growth is not linear, but curves up - that's precisely because the step from each period to the next is not equal, but rather grows larger every time.
What happens if we tweak the interest rate upwards? Let's try 20% for instance. All our future values go up.
What if our rate is 0? Then the value of our investment remains fixed at 100, we get no compensation whatsoever for giving up the use of our money.

I hope you are appreciating the power of rates of return. If you have invested assets, these returns are extra money that is being generated for you. Free money - although you are taking some degree of risk in exchange for it.
If you have debt, then the return works against you, because you have to give away your money to someone else to pay these returns.
It is much better and much more pleasant to have your money working for you, than it is to have you working for somebody else's money.

=Additional reading=

I hope you enjoyed this introduction to the core of the time value of money. Using a spreadsheet, we can model many real-life situations. You can ask questions like, what if I save 5000 dollars every year for the next 40 years, and invest at an average rate of return of 8%? Is this enough to retire on? What if I have 5000 dollars in debt at a 5% interest, and make a 500 dollar payment every year, how long will it take to pay it off? How much in interest will I have paid by the time I am done? We will discuss these and other questions in other videos.

As a fun extra reading suggestion, try "A History of Interest Rates". If you ever wanted to know about interest rates over the past few thousand years of economic history, this is your ticket.

Thank you for your attention.

=Attributions=