0:00

One of the places that these models for

Â growth comes in really useful is in the ideas of present and future values.

Â So present and future value are key ideas in business and

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

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

Â there's a prevailing interest rate of 4%, by which I mean that if you have some

Â money you can invest it and 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% each year,

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

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

Â 1,000 today or 1,500 in ten years?

Â And the key feature of this question is that you are comparing

Â values at two different time points, 1,000 today or 1,500 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 I would prefer,

Â I can do one of two things.

Â I could take the thousand and see how it grows by ten years

Â if its compounded at 4% or alternatively, I could take the 1,500 and

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

Â how much do I have to invest in today to get 15,000 in 10 years time?

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

Â Pt in the future, 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 p zero the subject of the formula.

Â If we do that, we can restate this equation as p

Â zero equals pt times theta to the power minus t.

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

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

Â Pt, and discount it back to today's value,

Â how much is that worth now by multiplying 3 by theta to the power -t.

Â Remember, theta is the constant proportional growth factor.

Â So using this formula, we can see that $1500 in ten year's time,

Â in a 4% interest rate environment,

Â is going to be worth in today's money 1500 times 1 plus .04,

Â that's 1.04, that's the multiplied, we've got 4% interest.

Â So, that's our theta and now to the power minus 10,

Â 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

Â a spreadsheet, you're going to see that this equals $1013, just a little bit over.

Â Now, $1,013 is worth more than a $1,000,

Â which was your alternative to get a $1000 today.

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

Â investment, a $1500 received In ten years time,

Â because its present value is greater than the $1,000,

Â the other option on offer.

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

Â this straight forward quantitative model for growth,

Â the proportional model for growth, is it gives a really simple discounting formula.

Â And discounting is one of the activities that businesses go through as they think

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

Â and make comparisons between objects at different points in time, and

Â we need to create a time baseline to do those comparisons.

Â And that's what the discounting is going to allow you to do, to take a future

Â value, and bring it back to a current value, so we can create a common baseline,

Â typically, to compare investments and do valuations.

Â So let me tell you of a couple places where you can see this idea or

Â present value being used.

Â So it's certainly used as a discounting technique to discount investments,

Â as we've done in our example.

Â An example of where you'd want

Â to understand the value of an investment will be what's called an annuity.

Â So an annuity is a schedule of fixed payments over a specified and

Â finite time period.

Â So basically someone says to you, I'm going to give $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 stream?

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 this idea of present values is used.

Â Another place where you can see it used, importantly,

Â is in the process of customer value calculation.

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

Â 6:55

When you compound investments, there's, in fact,

Â the 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, gets 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, you could compound, potentially,

Â on a monthly basis.

Â So at the end of each month, your 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, minute by minute basis, 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 straight forward,

Â somewhat elegant formula that tells you exactly how much

Â your money is growing to 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 out that the amount of money

Â you've got at time t, Pt is just equal to P0 your principal times e,

Â that's the exponential function coming in there, 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 divide it through by a hundred.

Â And so, for example, if you're interest rate,

Â the nominal interest rate, was 4%, that little r would be 0.04.

Â So there is a very nice formula for continuous compounding.

Â So that's an alternative way of modelling a growth or decline

Â process rather than doing it in discreet time, we could do it in continuous time.

Â And we end up with a very neat formula

Â that interestingly involved the exponential function.

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

Â module that 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,

Â show you how you would do a calculation.

Â The important thing to note, though, with continuous compounding,

Â is that the value t now can actually take on any value.

Â Remember when we were talking about discrete,

Â it could only take on specific values, 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 would continuously compound

Â a $1000 at a nominal annual interest rate of 4%.

Â After one year, make the calculation easy, we'll put to t over one year.

Â Then, what you're going to end up with is a 1000 times e to the power 0.04.

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

Â or using a spreadsheet.

Â It turns out that if you do that calculation,

Â you'll end up with $1,040.80 after one year, and

Â notice that that's a little bit different from the $1,040 if you just compounded

Â at a single point in time, at the end of the year, 4% of 1,000 gives you 40.

Â But if we continuously compound, then we end up with $1,040 and 80 cents.

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

Â compounding rather than discreetly compounding.

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

Â 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, that's the effective interests rate.

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

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

Â back to the epidemic we were talking about.

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

Â this exponential models that they give rise to are much more general than

Â just talking about money.

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

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

Â So, let's consider modelling the epidemic with an exponential function.

Â So, when we have this exponential models, here I'm writing Pt = P0,

Â that's a starting amount or starting number of infections, starting number of

Â cases times e to the power rt, we call that exponential growth or decay.

Â And if the letter r, the number in practice,

Â is greater than zero, then it's a growth process, and

Â 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, increasing or decreasing functions.

Â 12:38

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 as when we're talking about

Â discreet time.

Â And so halfway through week seven is actually seven and a half to equal to 7.5.

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

Â 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.

Â And if you remember the characterization of the exponential function, for

Â every one unit change in x you get the constant proportional increase in y.

Â That's what's going on here.

Â And in fact, that 0.15 in the exponent is

Â telling you t was measured in weeks, that you're getting an approximate 15% increase

Â from week to week, so, a 15% increase, approximately.

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

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

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

Â 13:47

Let's do the calculation now.

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

Â remember halfway through week seven is equal to 7.5.

Â I simply take my quantitative model,

Â 15 times e to the power of 0.15 now times t, but t is 7.5.

Â It comes out to be 46.2, reasonable rounding takes that to 46.

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

Â I have 15 cases by seven and a half weeks.

Â Half way through week seven, 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.

Â And, practically speaking,

Â this sort of all 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, you need a model to be able to do that.

Â So, there's our continuous time growth model.

Â Going back to the interpretation of the .15, here it is.

Â There's a approximate 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 models,

Â the graph on the left reproduced from the fishing example,

Â where we were talking about how many fish would be caught on

Â any particular year, and on the right hand side, we've got our continuous time model.

Â Notice how that smooth function, it fills in all the gaps.

Â For the discrete model you've got specific instances that you're

Â evaluating the function.

Â So, that's the difference between discrete and continuous again.

Â Just remember, the two sorts of watches, you can choose to have a digital watch,

Â that's when you want to have a discrete version of time, or

Â you could choose to have analog watch with hands on it and

Â then you'll be looking at a continuous version of time.

Â It's your choice.

Â It's not typically that one is right, or one is wrong.

Â But they are both used in practice.

Â