Money managers are professional investors who are paid to generate a return on other people's money. The money management industry has its own specialized metrics for investment returns in general and for manager's performance in particular. We will now discuss returns and volatility of returns. Asset returns are calculated in three main ways. For a one time investment, that pays off in the future, the absolute and annual rates of return can be calculated, either as a continually compounded, or as a discreet rate of return. Both continuously compounded, and discreet are acceptable metrics, but because the two metrics will not give the same result. In order to get a meaningful apples to apples comparison, always use the same method when comparing two or more returns. The first method, which calculates the continuously compounded return involves taking the log, to the base e. Usually written as a natural log ln of the ratio of the final price divided by the first price. And this will give us a value for the absolute return. So, if we started with $100 and we ended up with $130, natural log of 1.3, this gives us .2624 or 26.24% for the absolute return over the two year period. Then, if we want to calculate the one year or annual return, we simply divided this value by the number of years, 2 years at a value of 13.12% per year. The second, or discreet method. Involves taking the ratio of the final price divided by the first price. Minus 1. So the way you would this, is to simply say, $130 divided by $100- 1 would be equal to .3 which is equal to 30%. So note that the exact same return of $130 gives us 26.24% using the continuously compounded method and 30% using the discrete method. Now, if we wish to calculate the discrete annualized return, we will need to take the geometric mean of our ratio of final price over first price, and then subtract 1 from that. So, in other words, we're going to take our ratio, $130, divided by $100 and we're going to take it to the power that reflects how many time intervals we want to have. So, if we want to have two time intervals for 2 years, we are going to take this to the one-half -1 is equal to 1.1402- 1 is equal to 14.02%. So, again, notice that we get a different value for the geometric mean or discrete annualized return of 14.02% versus the annualized continuously compounded return of 13.12%. When an investment is not made all at once, but cash is invested at several different times, the metric used to evaluate the overall return is to identify a single fixed discrete, annual rate of return to apply to each of the payments that, if summed, would result in the final pay out. This value is called the internal rate of return, or IRR. For example, assume an investment of $1 million at year 0, and an additional $1 million at year 1, to develop a property that is sold for $5 million at year 4. We set up the problem as an algebra problem where we solve for x. (1+x)^4 + (1+x)^3 = 5. Most pocket calculators have an IRR function, which will find that x = 29.62%. In other words, (1 + .2962)4 + (1.2962)3 = 5. Next, we're gonna look at calculating the geometric mean of a series of annual returns. So, let's suppose that I have four annual returns, +25%, -18%, +10%, and -4%. What I want to know is if I have $1 at the beginning of the four year period, how many dollars will I have at the end of the four year period? The way that I can calculate this is by looking at each year at a time, and then taking the product. So, I can say for year one, $1 would be $1.25. For year two, $1 would be 0.92 cents, excuse me 0.82 cents. For year three, $1 would be $1.10. And year four, $1 would be 0.96 cents. If I multiply all these together, I'll find that I have 1.0824 or $1.08 and a little bit of extra, for an absolute return over the four year period of a little over 8%. And if I want to convert this to an annualized return, I'm going to take the geometric mean over four time intervals of the value 1.0824. And the way that I do that is I take 1.0824. One-fourth power, 4 because I have 4 years -1, and this gives me a geometric mean return of 2%. So, as we've just seen, the geometric mean return of our four different annual returns is just about 2%. So we could have had an investment that returned exactly 2% here for 4 years. And it would've achieved the same outcome as our much more scattered returns, ranging from plus 25% all the way down to minus 18%. So, something that we very much want to know is how scattered are our returns. It's a basic assumption of finance that all else being equal, if two investment opportunities have the same long-term return, we would prefer the one that doesn't have the huge range or scatter of returns and is much more consistent. So, the metric that we use to calculate how clustered together or spread out our returns are is called the standard deviation. The way that we calculate the standard deviation is we first take the arithmetic mean of our four values. Then we're gonna subtract that mean from each individual value. That gives us the distance from the mean of each value. So, for our values here, first step is to calculate the mean. This gives us 3.25% Now we're gonna calculate for each of these values, we're gonna subtract 3.25% from it, so this is going to give us. 21.75%- 21.25%. 6.75%- 7.25%. Next we take each of these distances or differences from the mean and we square them. And this is gonna give us .0473, .04515, .00455, .005. And then we are going to add these together and take their average. So this plus this, plus this, plus this, divided by 4 and that is going to give us .0255. And then we take the square root of that result, and that is going to give us 16%. What we're actually saying here is if we have a very tightly clustered group of outcomes. So for example, all the outcomes are the same, then the standard deviation, which we traditionally represent by the Greek letter Sigma would be equal to 0. If we have a much more spread out set of events, here we have a standard deviation that's equal to 16%. Standard deviation of returns is known in finance as volatility of returns. And it is a standard measure of risk. Because, as I said before, all else being equal, the greater the volatility of returns, the riskier an investment is. An investment with a fixed pay-out, like a bond, that also has effectively zero chance of loss is known as a risk-free investment. It has zero volatility of returns. The annual interest rate paid by financially strong, national governments. When they borrow in their own currency, is known as the risk-free rate of return. Typically, the U.S Government three-month treasury bill rate is used. This rate is currently 0.08%, far lower than its long-term historical average of 4.55%. The risk-free rate is the rate for lending without risk. But large companies and financial institutions can also borrow at rates quite close to this rate, so it's often used for simplification of our problems.
