One of the places that these models for growth come in really useful, is in the ideas of present and future values. So the present or future value of key ideas in business, and I'm going to illustrate them through an example here. So, lets 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 a multiplicative or proportional growth type model, which would you prefer? Thousands a day, or $1500 in 10 years? And the key feature of this question, is that you are comparing values at two different time points. 1000 today or 1500 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 1,000 and see how it grows by 10 years. If it's compounded at 4%. Or, alternatively, I could take the 1500 and back track it to today, and, basically, ask the question. How much would I have to have invested today to get 1,500 dollars in ten years time? So that idea of taking a value in the future, 1500, and bringing it back today is the idea of calculating a present value of a future quantity. So, could do these comparison, one of the approaches is to find the present value of the $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 P0 the subject of the formula. If we do that, we can restate this equation as P0 equals Pt, times theta to the power minus t. That's what happens if you go through and make P0 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 through by theta to the power minus t. Remember, theta is the constant proportional growth factor. So, using this formula, we can see that $1500 in ten years time in a 4% interest rate environment is going to be worth, in today's money 1500 times 1 plus 0.04, that's 1.04, that's the multiplier, if you've got 4% interest. So that's our theta, and now to the power of -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 $1,013, just a little bit over. Now, $1,013 is worth more than $1,000 which was your alternative, to get $1,000 today. So a typical person, or a rational person would prefer the second investment of $1500 received in ten years time, because it's present value is greater than the $1000, the other option on offer. And so, the great thing out of this simple, this straightforward quantitative model for growth, the proportionate model for growth. And it gives us a really simple discounting formula, and discounting is one of the activities that businesses go through, as they think about quantitative modeling. 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 of present value being used. It's certainly used as a discounting technique, to discount investments as we've done in our example. And example of where you'd want to understand the value of an investment, would be what's called annuity. 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 you $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 string? 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 the of present value is used. Another place where you can it use importantly, is in the process of customer value calculations. So businesses are often trying to value their customer in some fashion. Often, you keep a customer for a while. You might want to consider the lifetime of the 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 these customers will be in the future. And so, if we want to do like to like. Apples to apples comparisons of those two customs, 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 that I 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 a 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 get's 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, you're 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 straightforward, somewhat elegant formula that tells you exactly how much your money is going 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 principle, 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 divided through by a hundred. And so, for example, if your interest rate, the nominal interest rate is 4%, that little r would be .04, so there's a very nice formula for continuous compounding. So that's an alternative way of modeling a growth or decline process rather than doing it in discrete time, we could do it in continuous time and we end up with a very neat formula that, interestingly, involves the exponential function. That was one of the reasons why I said, in the introductory module, 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 aptly take on any value. Remember when we were talking about discreet? 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 were to continuously compound $1,000 at a nominal annual interest rate of 4% after one year. Make the calculation easy. T you put a one in. Then, what you're going to end up with is a thousand times, e to the power of 0.04. Again, you do a calculation like e to the power of 0.04 on your calculator or using a spreadsheet, it turns out that if you do that calculation, you'll end up with a $1040 and 80 cents after one year. And notice that, that's a little bit different from the $1,040, if you just compound it 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 $1,040.80. So it's a little bit different, the end result of continuously compounding rather than discretely compounded. 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 effected interest 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 an investment context, but these 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 an exponential model as a starting model. So Let's consider modeling the epidemic with an exponential function. So when we have these 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 continued to grow in an exponential fashion, because everyone on the planet would get sick pretty quickly. But at least the for getting phases, it's not unreasonable. So here's a model. The number of infections at week T is 15 times e to the power of .15T. So that's my growth model. Now, I've got a question for you. Half way through 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 are talking about discrete time. And, so half way through week 7 is actually 7, and a half 7.5. So let's, first of all, have a look at the function. So 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 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 .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 a interpretation, of the exponential function. Remember, I said interpretation is key, that exponent the .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 half way through week 7 is equal to 7.5. I simply take my quantitative model 15 times e. To the power 0.15 now times t, but t is 7.5. Comes 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 7 and a half weeks. Halfway through week 7, 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 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 a continuous time growth model. Going back to the the interpretation of the 0.15, here it is. There's a approximant 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 modules, the graph on the left are 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's 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 continues, again just remember the two source of waters 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 of an analog watch with hands on it, and then you're going to be looking at a continuous version of time. Its your choice, it's not typically that one is right or one is wrong, but they are both used in practice.