Hello again, thank you for joining us. So in previous lectures, we've been talking about how we would describe a return distribution, right? We talked about measuring the first moment and the second moments. Or average returns and volatility, as well as some other measures of risk, all right? So what I would like to do in this lecture is to look at some historical data to get a sense for what historical patterns on risk and return look like, all right? Okay, so one of the best documented facts in finance is that, on average, investors have historically earned high rates of returns for bearing greater risk, right? And this, of course, is consistent with our intuition, right? This is what we might expect. And this positive risk and return trade off is also something that will inform our models or portfolio choice and pricing of assets in later lectures. So the data that I'm showing to you right now covers the period from 1926 through up to 2009. So what you see is the average annual return and the standard deviation, volatility, for different asset classes together with the average inflation and its standard deviation. You also see the distribution of returns, right, for each asset group displays as a histogram, all right? So, each bar in the histogram indicates the number of years the return fell within that 5% bucket, right? So, what you see, of course, how dispersed, right, the realized returns have been for each asset class. Okay, so let's look at the data. What do you see? Well, we quickly see that, over this long window of time, all right, common stocks have, on average, provided a relatively higher rate of return, right? These returns are greater than the returns on long-term government bonds, corporate bonds, or government bonds, or Treasury bills, or the inflation rate over this period, the average inflation over this period, by a large margin, right? So this indicates that equities, right, on average have provided positive real returns, right? So that is after adjusting for inflation. Now the table also shows you, however, that the returns on common stocks are also, right, highly variable, right? Very volatile, as indicated by the standard deviation. So let me show you another histogram, right, so using slightly more recent data. All right, again, you see the different asset classes, Treasury bills, Treasury bonds, large stocks, and small stocks, right? Now the relative difference in the dispersions, right, in volatilities, is very apparent, right? The returns on equities, for example, have ranged from a gain, right? A positive gain of over 50%, to a loss of similar magnitude, minus 50%, right? By the way, they both happened in the 1930s. Now the extra returns, the premium provided by equities to investors, right, have clearly come at an expense, right? At an expense of considerably greater risk, right? Now you can also notice that the dispersion of returns is even greater for smaller stocks, right, than equities in general. All right, so the main points that I'm trying to make is that we see a clear positive relationship in historical return and risk. As shown in this graph as well, right? In this graph, what we see is the average annual return and the volatility plotted in the mean and volatility space for different asset classes. All right, so at this point, I would like to issue a warning about looking at historical data, right? Yes, it is true that over very long periods of time, the patterns that we see across different asset classes are very robust, right, in terms of the risk and return relationship, right? There is a positive risk and return relationship. There's no question about that. And so, yes, we expect a positive expected return, right, for bearing risk for the future. But the question is how do we come up with that expected return, right? So often, of course, we look at the past data to come out with expectations of what future return distributions might look like, right? Forecasts of future returns, right? We do that because we're human, and we like to extrapolate from past data. Now that is not necessarily a bad assumption, all right? If expected returns are constant over time, right? In other words, if expected returns on different asset classes and the risk and return relationship are constant over time. Sure, then long-run average realized returns should be a good estimate of expected future returns. But do you think you should think twice before using historical data as forecasts of future returns? Yes, yes! A resounding yes, all right? There are several problems with using historical data to come up with forecasts of future expected returns. Now the sample that you might be looking at, for example, be biased, right? I bet you would get very different estimates for average returns depending on the specific sample period you pick, the end points you pick. Or whether, for example, that period includes something like the financial crisis or not, right? So, okay, well, then you might say, okay, fine. So then, I will use the longest historical window possible, right? So then, my estimates will not suffer from bias. They will be less biased, they'll be more accurate. Well, that's true, but that only solves the problem partially, all right? Because you might wonder now how relevant those numbers are now, right? So if you had reliable data, for example, going back all the way to 1600s, right, there you have a long window of data. Would you want to include that data even if you had good quality data? You might think that maybe things are not the same anymore, right? Furthermore, expected returns may vary in cyclical fashion, right? In good times and bad times, right? And remember, higher valuations imply lower future expected returns. One final cautionary point about coming up with forecasts of expected returns on relatively new strategies or on some specific funds like hedge funds performance. Now, historical performance on these new strategies or specific funds are likely to be upwardly biased. Why? Well, for one, right, think about it. First of all, they are reported on a voluntary basis. Well, think about it, right? When would they be reported then? And two, you don't really see the ones that never make it, right? That do not survive, right? So then the historical performance data is going to be suffering from survivorship bias. Okay, so in this lecture we looked at some data to look at the historical patterns in risk and return, right? We've seen that, historically, riskier investments have, on average, had higher returns, right? There's a positive relationship between risk and return. I've also talked to you about being very careful with using historical data when you need to come up with forecast of expected returns.