00:00[Music] So, in order to get into performance measures, we need to go over how we actually measure the return on assets, which is pretty straightforward

00:11So, we'll have a look at how we measure return over a single period of time, and then how we actually average the return over multiple periods

00:21So, for measuring the return over a single period, we have a couple of options, the first of which is our discrete return

00:29So, this is the most simple case

00:33Uh, basically, we say that the value of our asset at time T, so you could sort of consider this the price, is equal to the value at the time before it, so at T minus 1, multiplied by One plus the rate of return for that time period, and notice we have the subscript T, which is corresponding to the value at the end of the period, not before

01:01And so we just expand this out and you get V to the T minus one plus V, the T plus one RT, and if you just rearrange that, you get that the rate is equal to this fraction

01:17So it's the difference between the values at each time period divided by the initial value, so to get it in sort of a percentage terms

01:25So this is our most simple way of measuring return over one period, and what this is basically equivalent to is saying that we have our value at our previous time period, and then The value of our asset actually stays sort of constant throughout the time period, and then right at the end, we instantly receive interest proportional to our previous amount

01:51So this is what this is basically saying, is that right at the end, we receive this sort of R percent C interest

02:02Our second option is continuous return, so this is saying instead of assuming that we have kind of a constant value, and then we instantly receive some amount at the end, we say, which is a little bit more realistic, that over time, the value is sort of constantly increasing, instead of it just going up instantaneously, it's steadily

02:27Going up, so we're going to actually start out by taking a similar approach, or pretty much the same approach as we did with the discrete return

02:38So we'll say that we've got our previous, our previous return, we multiply by 1 plus, just put R there, but just imagine, imagine that we do not want to say that the interest kicks in over one period

02:54What if we were to say that the interest kicked in over two periods? What we would then do is we would take half of this rate, and we would say that we get that, we multiply by 1 + r/2, and then we would multiply it again by 1 + r/2, saying that this is your initial amount, this times that the

03:14First time gives you how much you've got after it kicks in the first time, and then you multiply by 1 + r/2 again to give you the amount you've got when interest kicks in the second time

03:27And if we were to say that interest kicks in three times over the single period, then we would divide the rate by three, and we would take this to the power of three, because you'd say multiply by 1 + R/3 first time, then multiply it again, and then multiply it again

03:42So we generalize this out, we say 1 + r / n to the power of n, and so this is saying that n * we receive interest, which is 1/n of the r that is applicable to the entire time period, you

04:02Know, think of it in terms of, if this were an annual rate, we say over one year, we take the annual rate divided by n, and we say, you know, for each one over n years, we get this much

04:17The last step we have to take in order to make this continuous, because so far it's still discrete, we've just broken it down into multiple time periods instead of one, to make it continuous, we have to take the limit of this as n goes to infinity, so we're breaking it down theoretically into an infinite number of parts, so interest is being added every single instant, and so what on Earth does this become? It turns out that this is equal to E to

05:00The power of RT, I should add that here

05:04Actually, this is a t subscript in there, so the limit of 1 + r/n to the power of n as n goes to infinity is equal to e to the RT, that's sort of by definition

05:20So once again, we just do a little bit of rearranging here and you've got that e to the RT is equal to this fraction, and then by definition, you can say that the rate is equal to the log of that fraction

05:32So that's just by the definition of a log, e to the RT is equal to that, then RT is equal to the log of that

05:40So now, this is the last thing we have to look at in this video, so what are our options, options for taking the average return over multiple periods? We've got two options, the first of which is the arithmetic average

06:07You'd have to be really familiar with this by now

06:08This is what you would probably just consider to be the average

06:13It's just taking the sum of all the rates and then dividing by the number of rates, so that's what you would be familiar with

06:20When you hear the term average, you would probably think of that, but then we have another type of average, which is the geometric average, and in this one, we are multiplying 1 plus the rate for all of the rates, and then we're taking it to the power of 1/t, so this is effectively saying assuming we've got sort of a Compounding of interest, kind of effect going on, where after the first year, you know, when you get interest, in the second year, you get interest not only on your initial amount, but also on the interest you received previously

06:51If we keep compounding like that over all of the years, then what rate do we need such that if the rate remained constant over all that time, and it once again did compound, then yeah, what rate would give us the same sort of total return? So while this one doesn't account for the kind of compounding effect, it just sort of assumes that all of the, yeah, it just takes the average of the rates as they

07:25Are this one accounts for compounding, and this is just an alternative way of expressing that, maybe a little easier to see

07:33These two are the exact same thing, and so in the following video, we'll have a look at actually how the choice of which averaging method we use is related to our choice of which single period return measure we use

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