0:00

What I'm focusing on now is the proportionate growth or

Â the blue step function.

Â I want to introduce some notation for this sort of process, and remember,

Â growth is such a fundamental process that you really do want to be

Â comfortable in creating quantitative models for it.

Â And this model that I'm about to show you is the most straightforward

Â model for a processes growing in a proportionate fashion.

Â So let's denote the initial amount now is P0.

Â Zero to denote initial, that's what we call the principle.

Â And the constant proportionate growth factor by the Greek letter theta.

Â The example that I just showed you our growth factor was 10% which

Â means multiplied by 1.1.

Â You increase a number by 10% if you multiply by 1.1.

Â So we're going theta, and this instance was 1.1.

Â Now the growth progression, in general, is given to you in the table here.

Â And so at time 0, the first cell, you've got P knot.

Â But time one you're multiplying that by theta, 1.1 in our particular example.

Â Time two you multiplied the previous time

Â 1:22

In period 3, P0 theta cubed.

Â And in general, when we've gone out T time periods,

Â we've got P0 theta to the power T.

Â So you can see these power functions coming in here.

Â 1:35

If the value fader is greater than one then your process is growing because

Â if you multiply by a number greater than one you increase.

Â On the other hand, if theta is less than one then you've got a declining or

Â decaying process.

Â because if you multiply any number by a number between zero and

Â one and then you're going to make that number smaller.

Â Half of a 100 is 50, the number is getting smaller.

Â Then there's a special name that we give to this sort of progression or series, and

Â it's called a geometric progression, sometimes a geometric series.

Â And so that's our basic model for proportionate growth.

Â It's certainly a discreet time model because

Â we are looking at only period zero, one, two, three, etc.

Â We're not looking what's happening between those time periods so

Â it's inherently discreet.

Â And that's why I showed the growth

Â of the money that we had been talking about through a step function.

Â The end of each time period, we get money from the bank or

Â whoever we invested with and it jumps up and so that's the discreteness.

Â So this is a progressional geometric series that we're looking at, and

Â it is characterized by the change from one period to the next

Â is a constant multiplicative factor, which we call in general theta.

Â So let's do an example.

Â With a geometric series as a quantitative model for growth.

Â 3:24

Question we might be interested in thinking about especially if we lived in

Â this country would be how many fish are predicted to be caught 5 years from now?

Â What does it look like?

Â If it's an Indian Ocean country, it might well be focused around fishing and

Â this is a major revenue source, so

Â then we'd be really interested to know what's going to happen to our revenue.

Â So what's going to happen?

Â How many fish are going to be caught five years from now?

Â And here's another, sort of, question.

Â Including this year, what's the total expected catch over the next five years?

Â And so, these are reasonable questions to ask, and with our

Â geometric series quantitative model, we're going to be in a position to do so.

Â We're going to be able to ask and answer.

Â So, here's the constant multiplier.

Â 4:11

We were told that the catch was going to fall by 5%, constant 5%, each year.

Â That means that our multiplier is 0.95.

Â because to fall by 5% means to multiply .5 by 0.95.

Â If you take 100, and you make it 5% smaller, you're essentially going to

Â multiply that 100 by 0.95 to get 95 which is what it means to make it 5% smaller.

Â Now in general, if our process is changing by R%, I've got a capital R% in each time

Â period, then the appropriate multiplier is theta equals one plus R over 100.

Â And that over 100 is going with that percentage, which means out of 100.

Â So that's how you get the multiplier.

Â So, I put in a five for the R, and if my process was increasing,

Â it would be a positive five that I put in there, and my theta would be 1.05.

Â And if it were decreasing by 5% each year,

Â then I'd put in capital R as minus 5.

Â And I will get theta equal to 0.95.

Â So increasing positive R.

Â Decreasing negative R.

Â And so we go from the percent change to the constant multiplier.

Â So what we're being told in the setup to this particular instance,

Â is that we've got a multiplier of .95.

Â That was was meant by each year, a catch is going to fall by 5%.

Â Now we've got the problem set up, we could implement within a spreadsheet and

Â simply work out what the fish catch is going to be.

Â So think of the table here as a snap hot from a spread sheet.

Â So we start off with P0.

Â Before I'd use that to refer to how much money we started off with,

Â but now it's how many fish we're catching right now which is 200,000.

Â And we've got theta equal to 0.95.

Â And so using the formula that is associated with this model.

Â In five years the catch is going to be 200,000 times 0.95 to the power 5.

Â Now you can do that calculation with a calculator.

Â You could certainly do it in a spreadsheet.

Â And if you'll do that,

Â you'll find that your projected catch in five years time is 154,756.

Â And so that's using the model.

Â 6:24

Over the next five years,

Â well, to work that out I have to figure out how many fish we expect right now.

Â Well, we know that, 200,000.

Â How many we expect to catch next year?

Â 190,000.

Â Year two.

Â 180,500.

Â And so on over the first five years and

Â I can simply add up those numbers to get a little over a million.

Â Million tons.

Â And so with the quantitative model, I'm in a position to answer these two questions.

Â 6:52

Now a little bit about rounding.

Â When you do the calculations, if you're doing them in a spreadsheet or

Â a calculator, you will start to get places of decimals occurring.

Â And so in year four, the actual number is 162,901.25.

Â So it's the .25.

Â Of course, you can't really catch a core of a fish.

Â So, practically, if we were presenting these numbers to somebody,

Â we would certainly, round them to a whole number.

Â And we might eve round to thousands.

Â And why would we be very happy rounding?

Â Well, first of all, as I say you can't have place decimals.

Â And remember, all models are wrong, but some are useful.

Â And so we're really not losing anything by some rounding of those numbers.

Â But, formally, according to this model,

Â I'm expecting to catch a little over 8 million tons over the next five years.

Â So that's our geometric series model for growth in discreet time.

Â Now here's a graph of the fish catch.

Â Remember a picture is worth a thousand words.

Â It's never a bad idea to present what is going on graphically and if you have

Â a look at the height of each of these lines, the first one is at 200,000.

Â The second line goes up to 190,000, which is 5% less than the first one.

Â And then we keep going down by 5% from one period to the next time period.

Â So, there's a graphical representation of our fish catch for

Â each of those five years in addition to the current catch.

Â And all that I was doing when I was saying what was the total fish catch is

Â basically add up the heights of these six lines that I've got here.

Â 8:26

Now, one thing you should be aware of is that the step size

Â on these lines is not the same.

Â So we think of it as going downstairs now.

Â It's not a linear model, it's not additive.

Â In fact, as we go down from step to step the absolute

Â difference is getting a little smaller each time from step to step.

Â So this would be a sort of easy staircase to go down.

Â The steps are getting smaller and smaller as we go down from one to the next.

Â The same distance between the steps because our unit of analysis is time zero,

Â one, two, three, but if you have a look at the difference in height,

Â it's getting smaller each time because it's a proportionate growth

Â as opposed to a linear or additive growth type model.

Â In the additive then the distance from the top of one step to the bottom,

Â would always have been the same.

Â It turns out, that this particular quantitative model we have,

Â the geometric series has some rather nice properties about it and

Â I want to introduce here what we called the sum of the geometric series.

Â So, the fish catch was projected to be a geometric series.

Â And one of the questions was, how many fish have we

Â caught over the first five years including the current year?

Â So, that's a sum.

Â Now, it turns out that there's a neat little formula that captures the sum

Â of a geometric series.

Â So, I'm just going to present it to you and make a couple comments about it.

Â So, we need some notation, as ever, and we often write time in modeling.

Â Parlance with the letter T, make sense?

Â And we're going to write the sum up to time T, and including as S of T.

Â So S of T denotes our sum.

Â 10:11

It turns out that if we've got a geometric series, then S of T is equal to T0,

Â which is the initial amount or principle financial language times and

Â then that's 1 minus theta to the power T plus 1 over 1 minus theta.

Â And so now you can see why you need to know about these

Â power functions if you're going to become useful in this quantitative modeling.

Â You certainly need a certain set of mathematical skills, and I did

Â present the functions that I think you you absolutely need to feel comfortable with.

Â So, here's the panel function coming in.

Â Now with a formula like this,

Â I don't have to go through the process of working out each individual year's catch.

Â I can just plug straight in to the formula.

Â And if I do that for this particular geometric series where P0 is equal to

Â 200,000 and theta was equal to .95,

Â T I'm summing up to, in the fisheries example year five, so T is equal to 5,

Â I have to put in 5 plus 1 and I get exactly the same number out as I got

Â before, 1,059,632.

Â So it's encouraging that it's the same.

Â It has to be the same.

Â But what you can see here is that one of the advantages of

Â a quantitative model is it can potentially provide a much more efficient

Â way of doing calculations than if you looked at things

Â on a time by time period in the spreadsheet type world view.

Â Where yes, you can do this in a spreadsheet and add it up.

Â But imagine we have a situation where we wanted to go out a time period that was

Â more rows than you could put in a spreadsheet.

Â Your spreadsheet approach just wouldn't work anymore, but

Â you've always got the formula to use.

Â And so there are situations where taking the time to

Â formalize the business process through a quantitative model will give you much more

Â efficient ways of computation than through a spreadsheet.

Â So I'm not knocking spreadsheets.

Â I'm not saying there's no use for them.

Â But I'm saying that there are certain sorts of problems that can be very

Â efficiently solved through the formulae that we're able to generate

Â through having taken the time to create a quantitative model.

Â And here's an example of such a formula.

Â So the sum of a geometric series.

Â