Welcome back. This is mega problem or mega application number 4. The reason I want to do this one is to show you how you can do things smarter. I hope you enjoy this one. It looks simple but can get very complicated if you start worrying about how to do it. You can do it in a very simple way or a very complicated way. By the way, all the problems I'm using now should help you with solving the assignments, because the assignments can get tricky, and if you work through these, it'll really help you do the assignments. Let's get started. Please stare at the screen. You have invested $75,000 in a trust fund at 7.5 percent for your child's education. Your child will draw $12,000 per year from this fund for four years, starting at the end of year seven. What will be the amount that will be left over in this fund at end of year 10 after the child has withdrawn the fourth time? This sounds similar to the first problem. I'm taking away a little bit of the complexity to show you a way of going quickly in solving problems. The complexity I've removed is that tuition is not changing with inflation. But what I've thrown in is, I'm asking you a question where your trust fund will have some money left after 10 years rather than taking a loan and paying it all off. Let's see whether you have money left or not. Let's draw the timeline as always. How many points in time? You start off with zero and then 10 more? What do we know? Let's place that on the board. We know you have $75,000 at this point, plus. However, we know starting in year seven, in some senses, we did a more complex problem first because of the complexity of inflation, you start withdrawing 12K. The question is this, how much is left at the end of 10 years? Now you can do a very simple way of thinking about it. The simple way is ignore all time value of money and make me feel very sad about it. But remember the interest time of money is 7.5 percent, so ignoring it is much more drastic than if it was close to zero percent, which is basically no time value of money. The simplistic way of saying this would be, "I have 75, and then let's subtract 12,000 four times," which is about 48. Twelve times four is 48. You're left with what? Seventy-five minus 48,000 is $27,000. That's the very simplistic, naive way. But let's do it smarter way. Many people start doing it one step at a time. They'll take this to year seven, then take this to year eight and start subtracting. We did something like that to show you the confirmation of the result. Let's do it in a way that's pretty simple. Break up the problem into two pieces, and the two pieces don't mix the value of the trust fund with the payments being made annually starting in year seven. Even if they were being made starting in year one, it would be complex. Do what? Separate the negatives from the positives. Let's break it up into two pieces. The positive piece, which I'll call positive, is 75K carried forward how many years? Ten years. Can you do this? Answer is yes. It's a future value problem. What is the interest rate? 0.075. What is n? Ten. What is PMT? Zero, what is PV? Seventy-five thousand. How much does this work out to be? If my numbers are right press Enter, you'll get a 154,577.37. This is the amount of money you'll have. Now, clearly more than double. The reason is quite simple. Actually, there are two reasons. One is the interest rate is high, the second is the 10 is not a small number. Ten years is a fairly long time for the money to work. Then do the negative. What is the negative? Draw the timeline. Negative, first one is 12K. Second one is time period 12K, 12K, negative 12K. What are you seeing over here is, if you mix the two life becomes very, very complex. You can do that for fun like I did it earlier to see how you can confirm your results and we'll do a little bit of that here. But this is pretty straightforward. Do the future value, of what? It's not a one-time, it's a annuity, so what is the interest rate? 0.075. Remember, I'm going to press in a positive number, but I know it's negative that I'm paying, right? So 0.075, what is n? N is four. Remember I'm doing future value because I want to be in time, 10. N is four, PMT is how much? Now here you can press the negative 12,000, but I'm not going to, as I said, worry about it. Why did I close the parenthesis? Because there's nothing else. There's nothing else. You can make it more interesting but I won't right now. It turns out to be $53,675.6. You now know the two pieces. One piece is $154,577.37, the other piece is negative and now I'll make it negative because I know I'm paying. What is the net fund balance? In which year? Year 10. It turns out to be take 154,577.37 subtract $53,675.6 and I think the answer will work out to be $100,902.36, I'm a boozer, I keep laughing at myself. How much will you have? You will have $100,000, almost $101,000 even after you've paid off 12k, four times and this shows you the power of compounding because you had a reasonable amount of money. Just stick with me for a minute and we'll be done for today. Do this for me. Now let's confirm that this is indeed true. Do this for me and I'll be thrilled. Let's start and do it the more complicated way. Let's start with $75,000 and carry it forward. To which year? Year 7. Can you do that? Let's call that x_7. Then from that, what will you subtract? 12k. You're left with net x_7. Don't you like that? Net x_7. Then what will you do in year 8? You'll carry this forward at 7.5 percent to get y_7, which is the amount in your bank or your trust fund after you've paid off the first 12k and you are left with whatever remains, then subtract what? Another 12k. Keep going and in year 10, what should we have here? Approximately a 101k. I'm saying approximately because it's slightly less than that. Please do this and the force will be with you. We have done a lot of problems, and I have done them purposely, even with my bad handwriting and not going to Excel on purpose so that you understand the logic of timelines. But remember, go to Excel, especially if you ought to. You know that your cash flows are changing over time. You know you have to use functions different than PMT, PV, and so on for which there are simple formulas. When will that happen? When the cash flows are changing in nature. For example, the growth intuition if this were happening, now, try this out. Assuming that the 12k is increasing over time. It's 12k today and it will increase at say, 3 percent a year. You'll go back to the first application and you can redo it all. May the force be with you and next time, next module, what we'll do is a real-world application of this, and it would be called bonds. But don't worry about the name. Bonds basically is a very sophisticated name for loans and there are all kinds of loans. Loans taken by governments, loans taken by corporations. We have largely focused on loans taken by individuals here. But next time we'll get excited and do more interesting loans taken by different entities. See you.