[MUSIC] A picture is worth a thousand words. [MUSIC] So let's continue trying to understand fund performance through pictures. [MUSIC] Here is a time series of returns, for a fund. [MUSIC] Every month, there's a different outcome. And you see the average return, and the expected return of this fund assuming that we can expect the future to be like the past is the average return. Expected return is the average, it's the best guess, it's common sense. And if you plot these returns, It probably looks something like a normal distribution. So if every fund is viewed in this fashion, as a distribution of returns, how do you choose good funds? So here we have two graphs. An orange one and a blue one. Visually looking at them, which fund would you employ? Most people would choose the orange one. Because it has a lower variability of return. What that mean is, the expected return, you're going to get an outcome that is much closer to the expected return, in the orange fund, and then in that blue fund. This makes natural sense and certainly jibes with sort of finance theory, which sort of says that volatility is a proxy for risk. And in this case, what they're trying to say is it's proxy for how far away from the expected return you're going to be. Now, real life is not as cleanly distributed as in this graph. There are rare events, there are fat tails, and so if you look at real returns, it is not a nice, normal distribution, it's actually quite different. And in this case, what you can see looking at the orange chart versus the blue chart, the orange chart has fat tails. In this case, we call it kurtosis, but, just think of it from a common sense point of view, is that there are fat tails. And so if you have these two charts, which would you prefer, the orange one or the blue one? And again most rational people would say they want the blue one because the odds of a fat tail or a rare event are far from the average, is much lower in the blue one. So kurtosis is not an attractive phenomenon when you look at distributions. You want to have funds with low kurtosis or low fat tails. And now here is another set of graphs, two funds, the blue one and the orange one. And what do you notice about these two? The orange distribution has positive skewness. Positive skewness suggests that extreme positive events are more likely than extreme negative events. From a common sense point of view, this is also an attractive feature. You would like to have positive skewed funds because you have a low probability, but if you have a nice event, it's a positive surprise. And by the same token, most investors should detest negatively skewed funds, because when you get a surprise it can be a very, very bad negative surprise. Lastly, I want to put on a graph which tells you more of what actually happens in the real world. In the real world, none of these distributions actually take place, instead you have a lot of fat tails and the fat tails are not in the middle of the distribution, but actually, they're literally in the tails of the distribution. And financial markets tend to have a lot of fat tails. Keep that in mind as you evaluate funds and as you evaluate strategies and as you develop your own strategies. But with this introduction to sort of distributions, let's go back to what would you like to see in a fund manager? What is a desirable property from a distribution point of view, of a good fund. It has high expected value, the ideal fund, whatever, high expected value. It would have low volatility, it would be positively skewed and would have minimal kurtosis. If you can find such an investment, that would be absolutely fabulous. One way to think about hedge funds, and this a philosophical type of view, is that you're paying a manager to take the natural characteristics of an asset class and improve upon them. So for example, the dotted line here tells you the natural distribution of the S&P 500. And a skilled hedge fund manager operating on that asset class is able to change the distribution so that it has a higher return, it has a lower risk, hopefully, it has positive skewness and low kurtosis. If he can do that, he is a skilled hedge fund manager. That is what all of us are looking for, skilled managers who can take the natural distribution of an asset class and improve upon it. You have now seen a variety of techniques and tools that use historical data to make influences about the quality of a manager. Now some caveats and some practical experience on judging the distribution of a manager. A manager's performance is difficult to evaluate in isolation. All we can see really is the distribution of returns. And we can tell quite a bit from it through these exercises as you've just seen. But the first thing we must do is we must compare it to the distribution of the asset class that the manager operates on. Otherwise, you cannot tell what nice features of the distribution are attributable to the manager's skill as opposed to embedded in the asset class itself. Using some of the concepts we've talked about before, we're trying to isolate beta from alpha. Beta is the distribution of the asset class. Alpha is the properties that the manager adds to the asset class. Oftentimes, particularly in sophisticated strategies like hedge funds, there are instruments like options, there's derivatives, option as a derivative, there's leverage. A lot of these things can change distributions but keep you from figuring out whether it's skill or luck that produce these nice distributions. Options are naturally asymmetric and if there are options in a portfolio, it can change the character of the portfolio so that you may think that there's skill when actually it may be luck. [MUSIC]