0:00

One of the places that these models for growth come in really useful,

Â is in the ideas of present and future values.

Â So the present or future value of key ideas in business, and

Â I'm going to illustrate them through an example here.

Â So, lets imagine that there's no inflation in the economy and

Â there's a prevailing interest rate of 4%.

Â By which I mean, that if you have some money, you can invest it, and

Â be sure of receiving a 4% return on it, annually.

Â Here are two investment options.

Â Number one, $1000 today, or number two, $1500 in ten years.

Â Now, given that that $1000 is going to grow by 4%.

Â Each year, and I'm thinking here of compound, so we're going to grow according

Â to a multiplicative or proportional growth type model, which would you prefer?

Â Thousands a day, or $1500 in 10 years?

Â And the key feature of this question,

Â is that you are comparing values at two different time points.

Â 1000 today or 1500 in ten years.

Â And it's believed that there's a time value of money.

Â And, so in order to decide between which of these two investments,

Â I would prefer I can do one of two things.

Â I could take the 1,000 and see how it grows by 10 years.

Â If it's compounded at 4%.

Â Or, alternatively, I could take the 1500 and

Â back track it to today, and, basically, ask the question.

Â How much would I have to have invested today to get 1,500

Â dollars in ten years time?

Â So that idea of taking a value in the future, 1500, and bringing

Â it back today is the idea of calculating a present value of a future quantity.

Â So, could do these comparison, one of the approaches is to

Â find the present value of the $1,500 and so that's what I'm going to do.

Â 2:06

Let's have a look now, at the present value calculation.

Â So our formula for growth, our model for growth is that at time Pt in the future.

Â We're going to have the principle P0, times theta to the power t.

Â Now, that tells us how the future depends on the present value.

Â What we would like to do now, is make P0 the subject of the formula.

Â If we do that, we can restate this equation as P0 equals Pt,

Â times theta to the power minus t.

Â That's what happens if you go through and make P0 the subject of the formula.

Â And now this formula tells you how you can take a value in the future, Pt.

Â And discount it back to today's value.

Â How much is that worth now?

Â By multiplying through by theta to the power minus t.

Â Remember, theta is the constant proportional growth factor.

Â So, using this formula, we can see that $1500 in ten years time

Â in a 4% interest rate environment is going to be worth,

Â in today's money 1500 times 1 plus 0.04, that's 1.04,

Â that's the multiplier, if you've got 4% interest.

Â So that's our theta, and

Â now to the power of -10, because we're discounting it back, 10 time periods.

Â So that's how much it's worth in today's money.

Â If we work that out, again you can do that on your calculator or using a spreadsheet,

Â you're going to see that this equals $1,013, just a little bit over.

Â Now, $1,013 is worth more than $1,000 which was your alternative,

Â to get $1,000 today.

Â So a typical person, or a rational person would prefer the second

Â investment of $1500 received in ten years time,

Â because it's present value is greater than the $1000, the other option on offer.

Â And so, the great thing out of this simple, this straightforward

Â quantitative model for growth, the proportionate model for growth.

Â And it gives us a really simple discounting formula, and

Â discounting is one of the activities that businesses go through,

Â as they think about quantitative modeling.

Â because we'll often think about a value in the future and

Â make comparisons between objects at different points in time.

Â And we need to create a time baseline to do those comparisons, and

Â that's what the discounting is going to allow you to do.

Â To take a future value and bring it back to a current value, so

Â we can create a common baseline, typically to compare investments and do valuations.

Â So let me tell you of a couple places,

Â where you can see this idea of present value being used.

Â 4:53

It's certainly used as a discounting technique,

Â to discount investments as we've done in our example.

Â And example of where you'd want to understand the value of an investment,

Â would be what's called annuity.

Â An annuity is a schedule of fixed payments,

Â over a specified and finite time period.

Â So basically, someone says to you, I'm going to give you $100 every month for

Â the next 10 years, okay?

Â 5:21

But you're getting $100 this month, $100 next month, and $100 in 10 years time.

Â Now, what's the value of that complete income string?

Â Well, the money that you're going to receive in the future should be to

Â understand its current value, discounted back to the current time period.

Â And so, to value an annuity, you need to do a present value calculation.

Â You basically create the present value of each of the installments and

Â sum up those present values, and that gives you the present value of an annuity.

Â So that's one place, where the of present value is used.

Â Another place where you can it use importantly,

Â is in the process of customer value calculations.

Â So businesses are often trying to value their customer in some fashion.

Â 6:14

Often, you keep a customer for a while.

Â You might want to consider the lifetime of the customer.

Â Let's say, we're comparing two customers.

Â We want to compare them in some fashion, but a lot of the income

Â that's been generated from these customers will be in the future.

Â And so, if we want to do like to like.

Â Apples to apples comparisons of those two customs, we're going to need to discount

Â back those future revenues to today's time period.

Â And, so there's a lot of discounting that goes on in lifetime customer value.

Â Calculations, and this growth model

Â that I presented is one of the ways of getting at the idea of discounting.

Â So, lots of uses.

Â 6:55

When you compound investments, there's in fact a choice of the compounding period.

Â Now, typically we'll talk about compounding on a yearly basis.

Â At the end of each year, your amount of money now get's hit by a multiplier.

Â So it's a 4% interest, then you're going to multiply by 1.04.

Â There are alternatives though.

Â Rather than compounding on a yearly basis,

Â you could compound potentially on a monthly basis.

Â So at the end of each month, you're money grows by a little bit.

Â You could even possibly do it on a weekly basis.

Â You could do it on a daily basis.

Â Minute by minute basis.

Â A second by second basis.

Â 7:45

Now, the nice thing about thinking of the continuous time

Â version of the quantitative model is that there's a very straightforward,

Â somewhat elegant formula that tells you exactly how much your money is going to,

Â over a time period, t.

Â So if your money is growing at a nominal annual interest rate of R%,

Â I'm using the letter capital R there, then it turns

Â out that the amount of money you've got at time t, Pt, is just equal to P0,

Â your principle, times e, that's the exponential function coming in there.

Â To the power RT.

Â Now, note that's a little r there, because I've taken the interest rate, capital R,

Â and turned it into an out of a hundred.

Â I've divided through by a hundred.

Â And so, for example, if your interest rate, the nominal interest rate is 4%,

Â that little r would be .04,

Â so there's a very nice formula for continuous compounding.

Â So that's an alternative way of modeling a growth or decline process rather than

Â doing it in discrete time, we could do it in continuous time and we end up with

Â a very neat formula that, interestingly, involves the exponential function.

Â That was one of the reasons why I said, in the introductory

Â module, it was one of the functions you needed to know.

Â It comes up naturally here.

Â I'm going to do a quick example with continuous compounding, show you,

Â how you would do a calculation.

Â The important thing to note though with continuous compounding is that

Â the value T, now can aptly take on any value.

Â Remember when we were talking about discreet?

Â It could only take on specific values.

Â The end of each year or the end of each month.

Â Now that we're in continuous time, T can take on any value, inside an interval.

Â So let's have a look what happened, if we were to continuously compound $1,000 at

Â a nominal annual interest rate of 4% after one year.

Â Make the calculation easy.

Â T you put a one in.

Â 9:41

Then, what you're going to end up with is a thousand times, e to the power of 0.04.

Â Again, you do a calculation like e to the power of 0.04 on your calculator or

Â using a spreadsheet, it turns out that if you do that calculation,

Â you'll end up with a $1040 and 80 cents after one year.

Â And notice that, that's a little bit different from the $1,040,

Â if you just compound it at a single point in time at the end of the year.

Â 4% of 1,000 gives you 40.

Â But if we continuously compound, then we end up with $1,040.80.

Â So it's a little bit different, the end result of continuously

Â compounding rather than discretely compounded.

Â And I talk about a nominal annual interest rate of 4%,

Â because of course at the end of the year, if it was continuously compounded,

Â you earned a little bit more than 4%.

Â So 4% is just called nominal.

Â You earn 4.08% to be more precise.

Â So.

Â That's the effected interest rate.

Â So there's a little bit about continuous compounding.

Â Now, I'm going to apply this exponential growth model,

Â now back to the epidemic we were talking about.

Â Sure, I introduced the continuous compounding in an investment context, but

Â these exponential models that they give rise to are much more general,

Â than just talking about money.

Â And at least in the early stages of an epidemic,

Â it's not unreasonable to think of an exponential model as a starting model.

Â So Let's consider modeling the epidemic with an exponential function.

Â So when we have these exponential models,

Â here I'm writing Pt=P0, that's a starting amount or starting number of infections,

Â starting number of cases times e to the power rt.

Â We call that exponential growth or decay.

Â And if the letter r, the number in practice is greater than zero.

Â Then, it's a growth process.

Â And if it's less than zero, if r is negative, then it's a decay process.

Â So these models can capture growth or decay.

Â Increasing or decreasing.

Â Functions.

Â 11:56

So let's just state a potential model for the epidemic and

Â a continuous time model for the initial stages of the epidemic.

Â Of course I want to say initial stages, because it would be a disaster if

Â the epidemic continued to grow in an exponential fashion,

Â because everyone on the planet would get sick pretty quickly.

Â But at least the for getting phases, it's not unreasonable.

Â So here's a model.

Â The number of infections at week T is 15 times e to the power of .15T.

Â So that's my growth model.

Â Now, I've got a question for you.

Â Half way through week 7, how many cases do you expect?

Â And because this is a continuous time model,

Â I can put in any value of T that I want, that I think is reasonable.

Â I don't have to put in the whole number values,

Â as when we are talking about discrete time.

Â And, so half way through week 7 is actually 7, and a half 7.5.

Â So let's, first of all, have a look at the function.

Â So notice this is definitely not a linear function,

Â there's what we call curvature there.

Â This is what the exponential function looks like, and

Â if you remember the of the exponential function, for every one,

Â unit change in X you get the, constant proportional increase in, Y.

Â 13:17

That's what's going on here.

Â And, in fact, that .15 in the exponent, is telling you, T was measured in

Â weeks that you're getting an approximate 15% increase from week to week.

Â So, a 15% increase, approximately.

Â So, that's a interpretation, of the exponential function.

Â Remember, I said interpretation is key, that exponent the .15 has

Â an interpretation as the percent change here, in cases from week to week.

Â 13:48

Let's do the calculation now.

Â So we'll calculate the expected number of cases at week 7.5.

Â Remember half way through week 7 is equal to 7.5.

Â I simply take my quantitative model 15 times e.

Â To the power 0.15 now times t, but t is 7.5.

Â Comes to be 46.2.

Â Reasonable rounding takes that to 46.

Â So, at the beginning of the epidemic, I'm expecting 15.

Â I have 15 cases, by 7 and a half weeks.

Â Halfway through week 7, I'm able to expect about 46 cases.

Â And of course, one could calculate this for any value of T that you wanted to, and

Â practically speaking this sort of forecast would enable someone to do some

Â resource planning, if you were in charge of trying to cope with that epidemic,

Â how many physicians do I need, how many medical centers do I need to put in place?

Â You need a projection.

Â You need a model to be able to do that.

Â So there's a continuous time growth model.

Â Going back to the the interpretation of the 0.15, here it is.

Â There's a approximant 15% weekly growth rate.

Â And I say weekly, because time T was measured in weeks.

Â And a reminder of the difference between continuous time and discrete time modules,

Â the graph on the left are reproduced from the fishing example.

Â Where we were talking about how many fish would be caught on any particular year.

Â And on the right-hand side we've got our continuous time model.

Â Notice, how that's that smooth function.

Â It fills in all the gaps for the discrete model.

Â You've got specific instances that you're evaluating the function.

Â So that's the difference between discrete and

Â continues, again just remember the two source of waters you can choose to have

Â a digital watch, that's when you want to have a discrete version of time.

Â Or you could choose to have of an analog watch with hands on it, and

Â then you're going to be looking at a continuous version of time.

Â Its your choice, it's not typically that one is right or

Â one is wrong, but they are both used in practice.

Â