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Future Value – Multiple Periods

Future Value - Multiple Periods
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So we introduced you to the future value concept with a very simple problem. So let me keep the problem simple in 1 dimension, and that is we'll give you 1,000 bucks today, like Plan A. But we'll make it a little more complex, and you'll see how quickly finance becomes just fascinating. So let's say that we want to know what is the value of $1,000, 2 years from now. Remember, we first started with one period, so let's draw a timeline. So what I'm going to do is, I'm going to go very slowly, because as I said, I don't want you to feel like you're being shortchanged on the basics, which are extremely important. So where am I giving you 1,000 bucks?
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And what do you do with it? I'm giving you 1,000 bucks. [LAUGH] You put it in the bank, and what is the interest rate? A very healthy 10%. You ain't gonna get anything like that these days, but let's assume for simplicity. Now, I'm asking you, how much will this be?
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At which future value, time 2. So two years have passed, and this will make the problem very, very interesting, and introduce you to the concept of compounding. So $1,000, 2 periods from now. Now, think about it. What will happen to your bank account? Remember, imagine it's sitting in the bank after one year. How much will you have? Over and above 1,000. Do you agree you'll get 100 bucks? Why? Because think of the 1,000 working as for you.
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And the rate of return is 10%. So this is r x P. So you get how many 1 r x P? And P is conveniently how much? 1,000 bucks. But then what will happen at period 2? You'll also get another 100 bucks, which is another r x P. So remember r is always per period. I think I tried to make this very clear. P is that a point in time but r is for period. And that's the way it's called it in the financial press and everywhere. So you now have at least 1,200 future value.
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Is at least 1,200. Here, 1,000 plus 100 plus 100.
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But if this were the case, it would be called simple interest. And that's almost unheard of. So the question you all have to ask yourself, what did this poor guy do? Why is he not getting interest? The only problem is, he or she cannot get interest for two periods. But now your bank account has gone from 1,000 to 1,100 so this 100 guy also needs to get interest. But over which period? One period. And how much is that money? Because 100 at 10% you get 10 bucks. So the amount becomes 1,210.
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So basically, if you look at this it looks like 10 bucks, who cares? But you'll see as N, which is the number of periods, becomes more and more and more the amount you're getting every year, just because of the interest that year is fixed at 100, 100, 100. But what's happening? Compounding is going on and whole process generates a lot more money than you had visualized. And it becomes dynamic. So let me just formalize this very quickly and write up the formula. So the future value formula will become,
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future value over 2 periods will become P(1+r ) [BLANK AUDIO] after how many periods? After 1 period, times 1+r again, and this becomes P(1+r) all square. Let me spend a minute on this. So you started off at time zero, and it'll give you another way to think about it. With 1,000 bucks.
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After one year how much is it?
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P (1+r). And that is 1,100 bucks. But then, if your sitting in the bank, what will happen? You now have more money than you invested, because money earns money. So you have 1,100. But it needs to be carried forward for 1 period until period 2. So what will you do to it? Multiply it again by 1+r. So that's why the formula becomes P(1 + r) whole squared. So suppose I write it out in terms of numbers, $1,000 (1.1) whole squared. And it's the squaring, not the adding, that's key here. So how much will it become if this becomes n, n periods? Meaning not two, but n, what will change? It's quite simple.
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But dramatic image impact, so let me first introduce the concept and then do some examples, let’s do it in reverse because they need go back and forth and think about it. This number changes from 2 to n. But remember its raise to the power, so when we come back what I'm going to do is, I'm going to do examples, which will kinda blow you away a little bit. And when Einstein saw this, Einstein kind of said, wow. Compounding is unbelievably powerful. And I joke about it in class. It's because Einstein stopped at square. He said E=mc2.
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So the silly joke, which I, again, repeat often and laugh every time, myself, nobody else laughs with me, is that he didn't go beyond two. So that's why it's so fascinating. Take a little break, think about it. Think about what I've just said, and we'll come back.
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