If we were to compute the optimal portfolio, given the risk and return of individual stocks, how do we know that you won't want to hold negative quantities of some assets? So that's what's called a short sale. You can, in most countries of the world, you can hold negative quantities of a stock just as well as you can hold positive quantities. If you get one of those online brokerage service sites, you have to maybe sign up for a special account to do short sales. But you can say I'd like to short this stock. So how do you own a negative quantity of a stock? What it means is you borrow the shares and you sell them. Now you owe the share to someone else. The stockbroker can help you do that, and it's very routine these days. For centuries it's been very routine. Most investors don't do it. So you could buy a negative quantity of a stock. Why would you do that? Well, you would do that disregarding the cap M that we're talking about. You might say you would do that if you think the price is going to go down. If the stock is overpriced, it doesn't look good. Instead of not buying it, I'll buy a negative quantity of it. Capital asset pricing model assumes that, yeah, you can buy positive or negative amounts. Let's find the optimal portfolio. But then it turns out, however, that if you really take the idea that everyone would do the same thing, no stock could ever have an optimal holding of a negative number because there would be a negative holding for everybody. And everyone would want to short it, and that can't add up. So we're going to allow short sales in our math, but we'll assume that they won't happen, not on average. >> Here's why short sales won't happen. In our model, the optimal portfolio decision of all investors in equilibrium will be symmetric or identical. Given that, we can conclude that no single investor will ever short a stock in equilibrium because then everybody in our model would be shorting that stock. Which brings up the question, who is providing the stock for you to short? >> So what I'm really coming up to is a kind of abstract model, and I'd like to talk about the real world. But the capital asset pricing model is a little abstract. So what it assumes is that everybody is rational. It assumes that nobody has any risks that are inherent to them. They all want to do the same thing except for maybe a variation of leverage. They all want to do the same thing. And so as a result, nobody will ever short a stock. Now this is an abstract model, and I have to apologize a little bit for it, because there are short sales. But it's a model, and it's actually a very important model. By the way, the United States government has historically allowed short sales, but they were briefly abolished. The US government got so scared that there would be a 1929-style stock market crash, that it made a temporary law against short sales. No one was allowed to short these 799 stocks. So we're going to develop a theory that puts no short sale restrictions on it. We're going to assume that everyone does careful calculations of the mean and standard deviation of their portfolio return. And let's see what that implies. What it does imply is that there will be a optimal portfolio and we should be on what's called an efficient portfolio frontier. I won't explain this. This is from David Swenson's book about how he moved Yale to the optimal portfolio. When he took over management of Yale's portfolio, it was originally not well managed, and it was not optimal. So here, this is his account of the movement of the L portfolio toward the efficient frontier of between 1990 and the year 2000. So we have scientific management of portfolios here at Yale. So we'll come back to understanding what this is. >> So just a general question about investing. How are sort of unsophisticated, middle-class people supposed to invest their money now with the interest rates so low? What is there to do? Like what can you do? >> Well, interest rates are starting to go up, so maybe wait five years. [LAUGH] Now here is a fundamental issue about rationality. And most people are not financial wizards, as your question implies. And so what should we do? Should we have the government invest for them? Somehow there's something wrong about that, too, because the governments don't have a particularly good record of deciding how to investments either. The United States has been an example of capitalist institutions going way back. We have had lively stock markets and other kinds of speculative markets. People have lost money and had been taken advantage of over the years. But on the other hand, it produces an atmosphere of attention to business that produces a general culture of sympathy to business. If you invest in businesses, then you'll be less likely to vote for a strong man leader who will corral businesses. So the US has set an example to the world about. Just letting some of these things happen. So you've got this guy, Thomas Edison, with these electric lights, he might be a nut, who knows. But some people invested in him. And there were other stories that didn't work out so well. But on balance, over 100 or 200 years the US system has looked pretty good and it's become spread. I mean, not just the US's but I'm saying having free markets and involving people at large in some investing decisions, it has worked out well, even though it doesn't work out well for everyone.