Welcome back to Corporate Finance. Last time, we introduced the time value of money. We started with some intuition. We introduced the tools, namely the timeline and the discount factor, and then we showed how to move money back in time via discounting. Today, I wanna go the other direction. I wanna move money forward in time via a process called compounding. Let's get started. Hey, everyone. Welcome back to Corporate Finance. Today, we're gonna be talking 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 time value of money, notably, the time line, 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 wanna 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 CF0, 1, 2, and 3 and moving them to period 4 via compounding. So focusing on cash flow 2, I move that forward to period 4 by taking cashflow 2, multiplying it by my discount factor raised to 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. Now, I can add all of these cash flows now because they're all in the same time units. I can add all of these cash flows. They're all in date 4 time units. Now, these cash flows, once they've been moved forward are referred to as future values, right? So, again, this is just notation, like with present values. This is the future value as of period 4 of cash flow 3, the future value as of period 4 of cash flow 2, and likewise for cash flows 1 and 0. Let's do an example. How much money will I have after three years if I invest $1,000 in a savings account paying 3.5% interest per annum? Well, step one, put down a timeline. Okay, put the cash flows on a timeline. So, I'm gonna invest $1000 today, period 0. And the question's asking how much money will I have in three years? Well, all we're gonna do is move the cash flow forward in time by compounding. I'm gonna 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 3 of the cash flow in period 0 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 1, put the cash flows on a timeline. That's exactly right. So, we're saving $100 a year beginning today for the next three years. We're gonna earn 5%, and I wanna know how much I have after four years. Well, to do that we're gonna have to move each cash flow forward in time to period 4. So look at the cash flow in period 3, 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 gonna multiply that by our discount factor raised to the power positive 2. And likewise, for the cash flows in period 0 today, and one year from today. If we do the arithmetic, we get these future values of the cash flows, right, 105, 110, 115, 121. We can now add all of these cash flows, cuz they're all on the same time for period 4 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 2, 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 gonna deposit $100 today. That's gonna earn 5% interest and give us an additional $5 at the end of year one. So our pre-deposit balance, that's 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, 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 2, and I can move cash flows 3 and 4 back in time. And by applying a discount factor rates to a negative value, I can move the cash flows today and the cashflow one year from today 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 gives us future values. So what I wanna 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 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.