[MUSIC] Welcome back. Today we are going to talk about expected return estimates, and we're going argue that sample based information is, unfortunately, close to useless when it comes to expected return estimation. Let me try and explain this unfortunate finding by looking at a very simple example. So, in this graph we see two portfolio values. In both cases, portfolio starts at $100, and after five years the final value is $150. Now, there's two contrasted situations. The portfolio in red is a portfolio that goes pretty smoothly, goes up from 100 to 150, in a fairly smooth manner. The other portfolio, which is depicted in black, is a portfolio value which goes around quiet low. Now, dramatic decreases and dramatic increases in portfolio value, and in the end happens to end up at the same value, which is $150, Now, here is the problem. The problem is, if we are given those two time series, and if we are told that we should use a sample based estimator for what's the expected return on those two time series. We are going to conclude that in both cases the best expected return estimate we can come up with is plus 50%. Well, that's 50% over five years, that will mean plus 10% per year if you will. Now, we find the exact same outcome for those two portfolios even though the behavior was very different, one had a low volatility and the other one had high volatility. Now, let me try and explain that this parameter estimate, 10% annual return rate per year, can be severely misleading, especially in the second case, in the case of the portfolio with a high volatility. Well, let's take a look at that portfolio, and let's assume that now we are repeating the experiment, and just splitting the sample in half. And we're going to try and estimate expected return on those two halves samples. Well, if you do so, by the half the end of the first half sample, what happens is the stock return or stock price will have gone down a lot, right? So if we stop at the point where there's like this valley. So you're power to estimate that this point in time for expected return is actually not plus 10%, it's severely negative. Which doesn't make really good sense, right? Expected return is probably positive at this point in time, otherwise, nobody will be holding stocks if expected returns will be negative. So your expected return is just a reflection of the past performance, doesn't tell you much information looking forward. And then if you look at the second half of the sample starting from this low point, then you go back up all the way to 150, that's a huge kind of improvement. So the bottom line is your sample based expected return will be very sample dependent. Small changes in the sample will lead to large changes in the sample based estimate for expected returns. And that's not going to be so much the case for the red portfolio because that one is pretty smooth. Conclusion, sample based expected return estimators are extremely, extremely noisy, especially for high volatility portfolios. So, the confidence intervals are very large, we have very little confidence that the estimator that we come up with is of any meaningfulness. There was a very interesting paper by Bob Martin 1980 that actually shows this problem in a very explicit and mathematical way. Okay, so what do we do then if we are stuck in this situation where statistics is closed to useless? Well, we're going to do something, we're going to forget about statistic based on looking at sample. We're going to try and move forward and use another way of thinking about statistics. This other way of thinking about statistics is called Bayesian statistics. Now, here's the difference between the standard, also known as frequentist statistics analysis, and the Bayesian statistical analysis. In frequentist or standard statistical analysis, we assume that we know nothing about the outcome of the experiment before looking at the sample. In other words, if I have a coin, and someone's tells me what's the probability of a head or a tail, I'm going to respond, well, I don't know. It is only if you allow me to flip the coin a few times that I can tell you about the probability. And if you flip the coin let's say ten times, the guy says, yeah, why don't you flip the coin ten times? And if I find six heads, for example and four tails, I'm going to come to the conclusion as a frequentist statistician, that my best estimate is 60% heads, 40% tail. That's the frequentist statistician, everything they know about the data, they learn from the sample. Well, in contrast the Bayesian statistician will recognize that before coming up to the statistical experiment, they have some prior knowledge. Well if you give me a coin, I'm going to assume that it's most likely than not a fair coin, right? And so I'm going to assume that as a default option, if you will, before flipping the coin I'm going to assume it's 50/50. And now if you'll let me flip the coin 10 times, and if I find 6 heads and 4 tails, I'm not going to come to the conclusion that this is a 60/40 outcome. I'm going to come to the conclusion that still a 50/50?% outcome, that this is a fair coin. Because the outcome of the statistical experiment is not sufficiently dramatic, to kind of lead me to upgrade or update my prior, I'm going to stick to my prior. Now, if you allow me to do to flip the coin not 10 times but 1000 times. And if I find 600 heads and 400 tails, well then the situation is different. I'm now going to start to say well, maybe it's very unlikely, starting from a fair coin, to find 60/40 on that large a sample. So, what I'm going to be doing then is I'm going to be okay in terms of upgrading my prior, and mixing my prior with sample based information. So what we're seeing here has very important implications for expected return estimates. If sample based information is close to useless, we need to rely on priors, otherwise we're stuck, we have nothing to rely upon. And given that sample based formation will not be very meaningful, the upgraders of our prior will be very minimum. So we'd better come up with good priors, when it comes to expecting return estimation. Well, let me give you an example, not of a very good prior, but an example of very robust and agnostic prior. This agnostic prior is known as shrinkage mean methodology. It's a shrinkage methodology is known as shrinkage towards s the grand mean. So let me assume I'm looking at N stocks, and I need to estimate expected return for each one of these N stocks. Let me call that mu i hat, the expected return on each one of them. And what I'm going to do is I'm first going to be estimating the average performance of every stock in the sample, and I get an estimate that I call mu i bar. Which is noisy and I know it's noisy. And then I'm going to look at the grand mean estimate, mu bar, which is going to be the average over all parameter estimates for the mean values of the return on each stock. So mu bar is the average of all the mu i bar. And then my mu hat estimate for stock I, will be a combination, a mixture, of the mu i bar, which is the sample based estimate for its stock, which I know is noisy, and also the grand mean. And by shrinking the parameter estimates towards the grand mean, especially if you increase the shrinkage intensity, then you're going to get more robust estimation. Of course, if you let the importance of the grand mean convert to one, the shrinkage intensity factor converts to one, then in this case, you're going to force all parameter estimates to be identical. Well, that's the agnostic prior that we talked about, which would lead you to hold the minimum variance portfolio as a proxy for the maximum Sharpe ratio portfolio. Wrapping up, sample based information is close to useless when it comes to expected return estimate. In this context, we've gotta work hard and think hard to come up with meaningful priors because that's about everything we can rely upon, and that's the only things we can rely upon. We've been looking at a very simple prior which is kind of shrinking towards the grand mean. And next time we're going to be using more economically motivated priors, trying to relate our expected return estimate toward some kind of meaningful risk estimate [MUSIC]