Welcome back. Today, we are going to talk about how to estimate expected returns not based on sample information, which as was argued last time is unreliable, but instead based on some meaningful economic prior. First of all, let us recognize that anyone attempting to maximize the Sharpe ratio will need some expected return estimates. So the Sharpe ratio for the portfolio given by mu p minus r, excess return on the portfolio divided by Sigma p portfolio volatility will require as inputs. The mu i, the expected return for individual components of the portfolio, and that's the problem. These are the parameters that are so difficult to obtain in a meaningful way based on sample information. So what we are going to try and do is to introduce economically motivated priors for these mu i parameters, so as to avoid relying on sample-based information. We've already looked at an example of these agnostic prior. One example of an agnostic prior, which is not so well motivated from an economic standpoint is to assume that all expected returns are equal. Well, if you're assuming that all expected returns are equal, then maximizing the Sharpe ratio is exactly equivalent to minimizing the denominator of the Sharpe ratio, which is volatility. In this case, your best proxy for the max Sharpe ratio portfolio is the minimum variance portfolio. Well, the problem though is that you may not feel very confident that in real-world, all assets have the same returns. Do we think that all assets have the same return? Well, that doesn't sound very consistent with economic intuition that suggests that riskier assets should actually have a higher return. So it sounds like we should be able to improve on these very basic and very agnostic set of prior assuming they're all equal expected returns by introducing differences in expected returns based on differences in these parameters. Let me give you a very simple examples. Let's assume that expected returns are proportional to volatility or excess expected returns are proportional to volatility. So the assumption I'm making is more volatile assets will get higher expected returns. Well, if this is the assumption we're making, we're saying that all assets have the same Sharpe ratio. In other words, excess return mu i minus r is proportional to sigma i, and the constant of proportionality is precisely what is known as the Sharpe ratio for these assets. Now, if we assume that all assets have the same Sharpe ratio, then we can very easily perform maximum Sharpe ratio optimization. If you're looking at the expression for the Sharpe ratio, the numerator can simply be expressed as a weighted average of the volatilities of the portfolio. So that's pretty convenient because now we know what the volatility parameters are. We do have very nice estimates for our best estimates for volatility parameters. So we can simply estimate the portfolio Sharpe ratio and maximize the portfolio Sharpe ratio by maximizing this quantity here. We don't even need to know by the way what's the common value for these assumed common Sharpe ratio, we just need to maximize the numerator given by the weighted average of the volatilities divided by portfolio volatility. This ratio is known as diversification ratio. So we would just maximize the diversification ratio in this case. Now, of course the outstanding question is, do we believe that all stocks and all components have the same Sharpe ratio? If we think that's what the world is like, then this approach is very meaningful. Well, here is the subtlety. We have reasons to believe that not all of risk is rewarded. Asset pricing theory actually suggests that only the systematic component is rewarded, or whatever is specific in the risk of any given security stocks or asset classes, that specific risk is not rewarded. It's not rewarded because it can be diversified away. In other words, asset pricing theory suggests that you should not be rewarded for a piece of risks that you could have, that you should have diversified away in the first place. Well, in this context, we may want to assume that all stocks have the same Sharpe ratio. We may want to assume that there's a relationship between excess expected return and not total risk, but the systematic part of volatility. So in other words, we may want decompose volatility in terms of specific risk and systematic risk, and relate and come up with a better estimate for expected returns by relating it to systematic risk as opposed to relating it to total risk. Well, that's exactly what you can do with the factor model. So let me wrap up. Sample-based information is close to useless when it comes to expected return parameter estimates. So we have to hope that we can rely on some meaningful economic intuition to get access to some reasonable prior for expected returns. It turns out that factor models can actually be very convenient at providing us with meaningful economically motivated prior for expected returns, and this is exactly what we are going to discuss next time.