One of the places that these models for growth comes in really useful is in the ideas of present and future values. So present and future value are key ideas in business and I'm going to illustrate them through an example here. So let's imagine that there's no inflation in the economy and there's a prevailing interest rate of 4%, by which I mean that if you have some money you can invest it and be sure of receiving a 4% return on it annually. Here are two investment options. Number one, $1000 today, or number two, $1500 in ten years. Now, given that that $1000 is going to grow by 4% each year, and I'm thinking here of compound, so we're going to grow according to multiplicative or proportional growth type model, which would you prefer, 1,000 today or 1,500 in ten years? And the key feature of this question is that you are comparing values at two different time points, 1,000 today or 1,500 in ten years. And it's believed that there's a time value of money. And so, in order to decide between which of these two investments I would prefer, I can do one of two things. I could take the thousand and see how it grows by ten years if its compounded at 4% or alternatively, I could take the 1,500 and back track it to today and basically ask the question, how much do I have to invest in today to get 15,000 in 10 years time? So, that idea of taking a value in the future, 1,500, and bringing it back today, is the idea of calculating a present value of a future quantity. So to do these comparison, one of the approaches is to find the present value of $1,500, and so that's what I'm going to do. Let's have a look now at the present value calculation. So, our formula for growth, our model for growth, is that at time Pt in the future, we're going to have the principle P0 times theta to the power t. Now, that tells us how the future depends on the present value. What we would like to do now Is make p zero the subject of the formula. If we do that, we can restate this equation as p zero equals pt times theta to the power minus t. That's what happens if you go through and make p zero the subject of the formula. And now this formula tells you how you can take a value in the future, Pt, and discount it back to today's value, how much is that worth now by multiplying 3 by theta to the power -t. Remember, theta is the constant proportional growth factor. So using this formula, we can see that $1500 in ten year's time, in a 4% interest rate environment, is going to be worth in today's money 1500 times 1 plus .04, that's 1.04, that's the multiplied, we've got 4% interest. So, that's our theta and now to the power minus 10, because we're discounting it back 10 time periods. So, that's how much it's worth in today's money. If we work that out, again, you can do that on your calculator or using a spreadsheet, you're going to see that this equals $1013, just a little bit over. Now, $1,013 is worth more than a $1,000, which was your alternative to get a $1000 today. So, a typical person or a rational person would prefer the second investment, a $1500 received In ten years time, because its present value is greater than the $1,000, the other option on offer. And so, the great thing out of this simple, this straight forward quantitative model for growth, the proportional model for growth, is it gives a really simple discounting formula. And discounting is one of the activities that businesses go through as they think about quantitative modelling because we'll often think about a value in the future and make comparisons between objects at different points in time, and we need to create a time baseline to do those comparisons. And that's what the discounting is going to allow you to do, to take a future value, and bring it back to a current value, so we can create a common baseline, typically, to compare investments and do valuations. So let me tell you of a couple places where you can see this idea or present value being used. So it's certainly used as a discounting technique to discount investments, as we've done in our example. An example of where you'd want to understand the value of an investment will be what's called an annuity. So an annuity is a schedule of fixed payments over a specified and finite time period. So basically someone says to you, I'm going to give $100 every month for the next 10 years, okay? But you're getting $100 this month, $100 next month, and $100 in 10 years time. Now what's the value of that complete income stream? Well, the money that you're going to receive in the future should be, to understand its current value, discounted back to the current time period. And so to value an annuity, you need to do a present value calculation. You basically create the present value of each of the installments and sum up those present values, and that gives you the present value of an annuity. So, that's one place where this idea of present values is used. Another place where you can see it used, importantly, is in the process of customer value calculation. So, businesses are often trying to value their customer in some fashion. Often you'll keep a customer for a while, you might want to consider the lifetime of a customer. Let's say we're comparing two customers. We want to compare them in some fashion but a lot of the income that's been generated from those customers will be in the future. And so, if we want to do like to like, apples to apples comparisons of those two customers, we're going to need to discount back those future revenues to today's time period. And so, there's a lot of discounting that goes on in lifetime customer value calculations and this growth model I've presented is one of the ways of getting at the idea of discounting, so lots of uses. When you compound investments, there's, in fact, the choice of the compounding period. Now typically, we'll talk about compounding on a yearly basis. At the end of each year, your amount of money, now, gets hit by a multiplier. So it's a 4% interest, then you're going to multiply by 1.04. There are alternatives, though. Rather than compounding on a yearly basis, you could compound, potentially, on a monthly basis. So at the end of each month, your money grows by a little bit. You could even possibly do it on a weekly basis. You could do it on a daily basis, minute by minute basis, a second by second basis. If you let the compounding interval get smaller and smaller and smaller, in the limit, what you end up with is a process that we call continuous compounding. And that provides an alternative modelling framework to the discrete geometric series approach that we just saw. Now, the nice thing about thinking of the continuous time version of the quantitative model is that there's a very straight forward, somewhat elegant formula that tells you exactly how much your money is growing to over a time period t. So, if your money is growing at a nominal annual interest rate of R%, I'm using the letter capital R there, then it turns out that the amount of money you've got at time t, Pt is just equal to P0 your principal times e, that's the exponential function coming in there, to the power RT. Now, note that's a little r there because I've taken the interest rate, capital R, and turned it into an out of a hundred. I've divide it through by a hundred. And so, for example, if you're interest rate, the nominal interest rate, was 4%, that little r would be 0.04. So there is a very nice formula for continuous compounding. So that's an alternative way of modelling a growth or decline process rather than doing it in discreet time, we could do it in continuous time. And we end up with a very neat formula that interestingly involved the exponential function. That was one of the reasons why I said in the introductory module that it was one of the functions you needed to know. It comes up naturally here. I'm going to do a quick example with continuous compounding, show you how you would do a calculation. The important thing to note, though, with continuous compounding, is that the value t now can actually take on any value. Remember when we were talking about discrete, it could only take on specific values, the end of each year or the end of each month. Now that we're in continuous time, t can take on any value inside an interval. So let's have a look what happened if we would continuously compound a $1000 at a nominal annual interest rate of 4%. After one year, make the calculation easy, we'll put to t over one year. Then, what you're going to end up with is a 1000 times e to the power 0.04. Again, you'd do a calculation like e to the power 0.04 on your calculator, or using a spreadsheet. It turns out that if you do that calculation, you'll end up with $1,040.80 after one year, and notice that that's a little bit different from the $1,040 if you just compounded at a single point in time, at the end of the year, 4% of 1,000 gives you 40. But if we continuously compound, then we end up with $1,040 and 80 cents. So it's a little bit different, the end result of continuously compounding rather than discreetly compounding. And I talk about a nominal annual interest rate of 4% because, of course, at the end of the year, if it was continuously compounded, you earned a little bit more than 4%. So 4% is just called nominal. You earn 4.08% to be more precise. So, that's the effective interests rate. So there's a little bit about continuous compounding. Now, I'm going to apply this exponential growth model, now, back to the epidemic we were talking about. Sure, I introduced the continuous compounding in a investment context but this exponential models that they give rise to are much more general than just talking about money. And, at least in the early stages of an epidemic, it's not unreasonable to think of a exponential model as a starting model. So, let's consider modelling the epidemic with an exponential function. So, when we have this exponential models, here I'm writing Pt = P0, that's a starting amount or starting number of infections, starting number of cases times e to the power rt, we call that exponential growth or decay. And if the letter r, the number in practice, is greater than zero, then it's a growth process, and if it's less than zero, if r is negative, then it's a decay process. So, these models can capture growth or decay, increasing or decreasing functions. So let's just state a potential model for the epidemic, and a continuous time model for the initial stages of the epidemic. Of course, I want to say initial stages because it would be a disaster if the epidemic continue to grow in an exponential fashion, because everyone in the planet would get sick pretty quickly. But at least at the beginning phases, it's not unreasonable. So here's a model, the number of infections at week t is 15 times e to the power 0.15t. So that's my growth model. Now I've got a question for you. Halfway through the week 7, how many cases do you expect? And because this is a continuous time model, I can put in any value of t that I want, that I think is reasonable. I don't have to put in the whole number values as when we're talking about discreet time. And so halfway through week seven is actually seven and a half to equal to 7.5. So let's, first of all, have a look at the function. Notice, this is definitely not a linear function, there's what we call curvature there. This is what the exponential function looks like. And if you remember the characterization of the exponential function, for every one unit change in x you get the constant proportional increase in y. That's what's going on here. And in fact, that 0.15 in the exponent is telling you t was measured in weeks, that you're getting an approximate 15% increase from week to week, so, a 15% increase, approximately. So, that's interpretation of the exponential function. Remember I said interpretation is key, that exponent of .15 has an interpretation as the percent change here in cases from week to week. Let's do the calculation now. So we'll calculate the expected number of cases at week 7.5, remember halfway through week seven is equal to 7.5. I simply take my quantitative model, 15 times e to the power of 0.15 now times t, but t is 7.5. It comes out to be 46.2, reasonable rounding takes that to 46. So at the beginning of the epidemic, I'm expecting 15, I have 15 cases by seven and a half weeks. Half way through week seven, I'm able to expect about 46 cases. And of course, one could calculate this for any value of t that you wanted to. And, practically speaking, this sort of all forecast would enable someone to do some resource planning if you were in charge of trying to cope with that epidemic, how many physicians do I need, how many medical centers do I need to put in place? You need a projection, you need a model to be able to do that. So, there's our continuous time growth model. Going back to the interpretation of the .15, here it is. There's a approximate 15% weekly growth rate. And I say weekly because time t was measured in weeks. And a reminder of the difference between continuous time and discrete time models, the graph on the left reproduced from the fishing example, where we were talking about how many fish would be caught on any particular year, and on the right hand side, we've got our continuous time model. Notice how that smooth function, it fills in all the gaps. For the discrete model you've got specific instances that you're evaluating the function. So, that's the difference between discrete and continuous again. Just remember, the two sorts of watches, you can choose to have a digital watch, that's when you want to have a discrete version of time, or you could choose to have analog watch with hands on it and then you'll be looking at a continuous version of time. It's your choice. It's not typically that one is right, or one is wrong. But they are both used in practice.