Welcome back Corporate Finance. Last time we introduced the time value of money. We started with some intuition, we introduced the tools, namely the timelines and the discount factor, and then we showed how to move money back in time via discounting. Today, I want to go the other direction. I want to move money forward in time via a process called compounding. Let's get started. Hey everyone, welcome back to Corporate Finance. [SOUND] Whoa. Today we're going to talk about compounding but let me start off with a brief recap of what we did in the last lecture. Last time we introduced the time value of money, the concept. We started off with some intuition and showed that money has a time unit that prevents money arriving at different points in time from being aggregated, or added together. Then we introduced some tools associated with the time value of money, notably the timeline, which is just a visual representation of when money is moving in or out. And the discount factor, which was our exchange rate for the time value of money. It allowed us to convert the time units on money, moving it forward or backward. And then we applied those tools to move cash flows back in time via discounting, and the big lesson was don't add cash flows with different time units ever. Today I want to go the other direction and talk about compounding, or moving cash flows forward in time. So let's get started. So, compounding just refers to moving cash flows forward in time. And here's our familiar timeline, and what I'm doing is I'm taking each cash flow, CF 0, 1, 2, and 3, moving them forward to period 4, via compounding. So, focusing on cash flow 2, I move that to period 4 by taking cash flow 2, multiplying it by my discount factor raised by the power of 2. Because I'm moving it two periods forward, and it's positive because I'm moving it forward. More generally, all of the exponents are positive. Again, because we're moving all of the cash flows forward in time. I can add all of these cash flows now, because they're all in the same time units namely. I can add all of these cash flows. They're all in date four, time units. Now these cash flows, once they have been moved forward, are referred to as future values. This is just a notation like with present values, this is the future value as of period four of cash flow three. The future value as of period four of cash flow two. And likewise for cash flows one and zero. Let's do an example. How much money will I have after three years if I invest $1000 in a savings account paying 3.5% interest per annum? Well, step one. Put down a cash line. [LAUGH] Put down a timeline. Okay, put the cash flows on a timeline. So I'm going to invest $1000 today, period 0, and the question's asking how much money am I going to have, in three years? Well, all we're going to do is move the cash flow forward in time by compounding. I'm going to multiply by 1 + R, where R is 3.5% in this case, and I'm moving the cash flow three years forward in time. So that's a positive 3 exponent on my discount factor. If we do the arithmetic we get that the $1,000 is worth $1,108.72, or just under $0.72. And I should also mention that this is just the future value of the $1,000. In particular, it's the future value as of period three of the cash flow in period zero, which was $1,000. Let's do a second example. How much money will we have four years from today if we save $1,000 a year, beginning today, for the next three years assuming we earn 5% per annum? Step one, put the cash flows on a time line, that's exactly right. So we're saving $100 a year beginning today for the next three years. We're going to earn 5%, and I want to know how much I have after four years. Well to do that, we're going to have to move each cash flow forward in time to period four. So look at the cash flow of the cash flow in period three. I need to move that forward one period. So I multiply by 1 + R raised to the positive power 1. We're going forward one period. Cash flow two has to go forward two periods. So we're going to multiply that by our discount factor raised to the power positive 2, and likewise for the cash flows in period zero today and one year from today. If we do the arithmetic, we get these future values of cash flows 105, 110, 115, 121. We can now add all of these cash flows, because they're all in the same time for period four time units, and if we do that, I get 452.564. So what does this mean? How do we interpret that? Well, we will have $452.56 at the end of four years if we save $100 starting today for the next three years, and our money earns 5% per annum. Interpretation two, the future value four years from today of saving $100 starting today for the next three years at 5% per annum is $452.56. So what's going on here, what's going on behind the scenes, well, we're going to deposit $100 today. That's going to earn 5% interest and give us an additional $5 at the end of year one. So our predeposit balance is pre before our next deposit is just $105, which by the way, is also equal to the future value one period, hence of the $100, all right? The future value of this$100 one period hence, is just the 100 x 1 + R to the 1. I deposit another $100, and I've got$205 after the first year. We continue this process for four years, and lo and behold, we're left with $452.56 at the end of the fourth year. More generally, there's nothing special about moving the cash flows to the end of the timeline or the beginning. You can move them anywhere, just as long as you're consistent. We can pick any point in time, such as period two. And I can move cash flows three and four back in time, right? going to by applying discount factor raised to a negative value, I can move the cash flows today. And the cash flow one year from today, one year forward in time, by applying a positive exponent to the discount factor. And now these cash flows, here, are all in the same time 2 period units. All right, so let's summarize this up. We use compounding to move cash flows forward. We apply a discount factor with a positive exponent to move them forward in time, and that gets us future values. So what I want to see you do now is work on the problem set. And then coming up next in our next lecture, I want to talk about some useful shortcuts for our present value and future value of common streams of cash flows. So thanks again for listening, and I look forward to seeing you in the next lecture.