What I'm focusing on now is the proportionate growth or the blue step function. I want to introduce some notation for this sort of process, and remember, growth is such a fundamental process that you really do want to be comfortable in creating quantitative models for it. And this model that I'm about to show you is the most straightforward model for a processes growing in a proportionate fashion. So let's denote the initial amount now is P0. Zero to denote initial, that's what we call the principle. And the constant proportionate growth factor by the Greek letter theta. The example that I just showed you our growth factor was 10% which means multiplied by 1.1. You increase a number by 10% if you multiply by 1.1. So we're going theta, and this instance was 1.1. Now the growth progression, in general, is given to you in the table here. And so at time 0, the first cell, you've got P knot. But time one you're multiplying that by theta, 1.1 in our particular example. Time two you multiplied the previous time period, the value then which was the principle times theta. Which gives you P0 theta squared. In period 3, P0 theta cubed. And in general, when we've gone out T time periods, we've got P0 theta to the power T. So you can see these power functions coming in here. If the value theta is greater than one then your process is growing because if you multiply by a number greater than one you increase. On the other hand, if theta is less than one then you've got a declining or decaying process. because if you multiply any number by a number between zero and one and then you're going to make that number smaller. Half of a 100 is 50, the number is getting smaller. Then there's a special name that we give to this sort of progression or series, and it's called a geometric progression, sometimes a geometric series. And so that's our basic model for proportionate growth. It's certainly a discrete time model because we are looking at only period zero, one, two, three, etc. We're not looking what's happening between those time periods so it's inherently discreet. And that's why I showed the growth of the money that we had been talking about through a step function. The end of each time period, we get money from the bank or whoever we invested with and it jumps up and so that's the discreteness. So this is a progressional geometric series that we're looking at, and it is characterized by the change from one period to the next is a constant multiplicative factor, which we call in general theta. So let's do an example. With a geometric series as a quantitative model for growth. Consider an Indian Ocean nation that's big on fishing. They catch 200,000 tons of fish in this year. The catch is unfortunately projected to fall by a constant 5% factor each year for the next ten years. Question we might be interested in thinking about especially if we lived in this country would be how many fish are predicted to be caught 5 years from now? What does it look like? If it's an Indian Ocean country, it might well be focused around fishing and this is a major revenue source, so then we'd be really interested to know what's going to happen to our revenue. So what's going to happen? How many fish are going to be caught five years from now? And here's another, sort of, question. Including this year, what's the total expected catch over the next five years? And so, these are reasonable questions to ask, and with our geometric series quantitative model, we're going to be in a position to do so. We're going to be able to ask and answer. So, here's the constant multiplier. We were told that the catch was going to fall by 5%, constant 5%, each year. That means that our multiplier is 0.95. because to fall by 5% means to multiply .5 by 0.95. If you take 100, and you make it 5% smaller, you're essentially going to multiply that 100 by 0.95 to get 95 which is what it means to make it 5% smaller. Now in general, if our process is changing by R%, I've got a capital R% in each time period, then the appropriate multiplier is theta equals one plus R over 100. And that over 100 is going with that percentage, which means out of 100. So that's how you get the multiplier. So, I put in a five for the R, and if my process was increasing, it would be a positive five that I put in there, and my theta would be 1.05. And if it were decreasing by 5% each year, then I'd put in capital R as minus 5. And I will get theta equal to 0.95. So increasing positive R. Decreasing negative R. And so we go from the percent change to the constant multiplier. So what we're being told in the setup to this particular instance, is that we've got a multiplier of .95. That was was meant by each year, a catch is going to fall by 5%. Now we've got the problem set up, we could implement within a spreadsheet and simply work out what the fish catch is going to be. So think of the table here as a snap hot from a spread sheet. So we start off with P0. Before I'd use that to refer to how much money we started off with, but now it's how many fish we're catching right now which is 200,000. And we've got theta equal to 0.95. And so using the formula that is associated with this model. In five years the catch is going to be 200,000 times 0.95 to the power 5. Now you can do that calculation with a calculator. You could certainly do it in a spreadsheet. And if you'll do that, you'll find that your projected catch in five years time is 154,756. And so that's using the model. Over the next five years, well, to work that out I have to figure out how many fish we expect right now. Well, we know that, 200,000. How many we expect to catch next year? 190,000. Year two. 180,500. And so on over the first five years and I can simply add up those numbers to get a little over a million. Million tons. And so with the quantitative model, I'm in a position to answer these two questions. Now a little bit about rounding. When you do the calculations, if you're doing them in a spreadsheet or a calculator, you will start to get places of decimals occurring. And so in year four, the actual number is 162,901.25. So it's the .25. Of course, you can't really catch a core of a fish. So, practically, if we were presenting these numbers to somebody, we would certainly, round them to a whole number. And we might eve round to thousands. And why would we be very happy rounding? Well, first of all, as I say you can't have place decimals. And remember, all models are wrong, but some are useful. And so we're really not losing anything by some rounding of those numbers. But, formally, according to this model, I'm expecting to catch a little over 8 million tons over the next five years. So that's our geometric series model for growth in discreet time. Now here's a graph of the fish catch. Remember a picture is worth a thousand words. It's never a bad idea to present what is going on graphically and if you have a look at the height of each of these lines, the first one is at 200,000. The second line goes up to 190,000, which is 5% less than the first one. And then we keep going down by 5% from one period to the next time period. So, there's a graphical representation of our fish catch for each of those five years in addition to the current catch. And all that I was doing when I was saying what was the total fish catch is basically add up the heights of these six lines that I've got here. Now, one thing you should be aware of is that the step size on these lines is not the same. So we think of it as going downstairs now. It's not a linear model, it's not additive. In fact, as we go down from step to step the absolute difference is getting a little smaller each time from step to step. So this would be a sort of easy staircase to go down. The steps are getting smaller and smaller as we go down from one to the next. The same distance between the steps because our unit of analysis is time zero, one, two, three, but if you have a look at the difference in height, it's getting smaller each time because it's a proportionate growth as opposed to a linear or additive growth type model. In the additive then the distance from the top of one step to the bottom, would always have been the same. It turns out, that this particular quantitative model we have, the geometric series has some rather nice properties about it and I want to introduce here what we called the sum of the geometric series. So, the fish catch was projected to be a geometric series. And one of the questions was, how many fish have we caught over the first five years including the current year? So, that's a sum. Now, it turns out that there's a neat little formula that captures the sum of a geometric series. So, I'm just going to present it to you and make a couple comments about it. So, we need some notation, as ever, and we often write time in modeling. Parlance with the letter T, make sense? And we're going to write the sum up to time T, and including as S of T. So S of T denotes our sum. It turns out that if we've got a geometric series, then S of T is equal to T0, which is the initial amount or principle financial language times and then that's 1 minus theta to the power T plus 1 over 1 minus theta. And so now you can see why you need to know about these power functions if you're going to become useful in this quantitative modeling. You certainly need a certain set of mathematical skills, and I did present the functions that I think you you absolutely need to feel comfortable with. So, here's the panel function coming in. Now with a formula like this, I don't have to go through the process of working out each individual year's catch. I can just plug straight in to the formula. And if I do that for this particular geometric series where P0 is equal to 200,000 and theta was equal to .95, T I'm summing up to, in the fisheries example year five, so T is equal to 5, I have to put in 5 plus 1 and I get exactly the same number out as I got before, 1,059,632. So it's encouraging that it's the same. It has to be the same. But what you can see here is that one of the advantages of a quantitative model is it can potentially provide a much more efficient way of doing calculations than if you looked at things on a time by time period in the spreadsheet type world view. Where yes, you can do this in a spreadsheet and add it up. But imagine we have a situation where we wanted to go out a time period that was more rows than you could put in a spreadsheet. Your spreadsheet approach just wouldn't work anymore, but you've always got the formula to use. And so there are situations where taking the time to formalize the business process through a quantitative model will give you much more efficient ways of computation than through a spreadsheet. So I'm not knocking spreadsheets. I'm not saying there's no use for them. But I'm saying that there are certain sorts of problems that can be very efficiently solved through the formulae that we're able to generate through having taken the time to create a quantitative model. And here's an example of such a formula. So the sum of a geometric series.