"Now, I want to move to another example, which is too risky assets." "Now there's, when we move past 1602 and now we have two stocks." "And for the moment, I'm going to forget about leverage, and let's just say you can put x1 in the first risky asset that's stock number one and I can put 1 minus x1 in the second risky asset." "That's stock number two." "Okay, so what, what do I get here?" "The expected, the, the portfolio expected return is just the linear combination of the two expected returns." "So r1 is the expected return on the first stock, and r2 is the expected return on the second stock." "And it, you just, or actually, I'm assuming, I'm assuming you have $1 to invest in this example." "And so I was assuming you had 100 gliders over there." "Now, I've just made it $1." "Unrealistically small amount." "But I just wanted a nice, a nice number, okay?" "So you, let's say $1 is 100 gliders and then I haven't changed anything." "Okay, so so I have one, see, I start out with $1, so if I put x1 dollars in the first one," "I have 1 minus x1 left for the other one, so it's very simple." "And this is the, the formula for the variance of the portfolio, which we saw essentially we saw that in the second lecture." "So what I can do is go through the same sort of exercise I did there with two risky assets." "Alright, and so I, what I want to do is draw a, a curve, something like this, but" "I'll, I'll solve for x1 in terms of r just like I did for the riskless asset and I'll plug it into the equation for the variance." "I'll have to take the square root of that and I can plot that, okay." "And, you might think it would look something like that." "Well, its not going to look exactly like that because it's risky." "Something's risky." "Well I did that, in the calcu, and, and see that in your problem set you're going to have to think about issues like this." "But what I did is I took data on the average return for the US stock market as measured by the SP 500." "And the variance." "And then the alternative investment I took was ten year treasuries for the United States government." "Long term because they're ten years." "But we're only investing for one year." "So they're risky because the market price goes up and down." "They're not riskless." "I call those bonds." "There's other kinds of bonds." "And I computed the relationship between the standard deviation of the portfolio and the standard deviation and the expected return, just as I showed you" "again using data from 1983 to 2006." "And it kind of looks like this curve, doesn't it, except this is a degenerate problem, but it's, since it, it looks like this, right?" "I said parabola, I said it wrong." "Hyperbola." "You know how hyperbola, you remember this from math?" "Hyperbolas, well, they look like that, and they approach asymptotes." "Which are straight lines." "So here is here is the hyperbola for stocks and bonds." "Though, just as I had a point here which represented a 100% VOC, I can have over here a point which represents 100% US stocks, okay?" "And I can take another point which is 100% bonds, that's here." "This point is 25% stocks, 75% bonds, this point is 50% stocks, 50% bonds, okay?" "And I can see that this is the choice set that, that I as an investor have between stocks and bonds." "so, is that clear?" "Or, is there any, so again there's no, you see that all these are different portfolios." "If, if you're going to do just stocks and bonds and nothing else what you choose to do depends on your taste." "And your risk tolerance." "I could go 100% stocks, but I'm going to have a lot of risk, I'm going to have, I'm going to have a nice expected return, it looks like it's about 13," "14%, but I'm going to have a high variance, looks like it's about 18%." "This is the SP 500 stock market, and so, it has a lot of variance." "I could be safe, and I could go all in bonds." "I could be here." "Then I'd have, you know, a lower, much lower return, but I'd have a lower variance." "So what should I do?" "Well, what do you learn from this?" "First of all, you learn, there isn't any single, optimal portfolio, but there is something." "Let, let's talk about being 100% bond investor." "What do you think of that?" "Is that a good idea?" "You'd get this point right here." "Well, right, you definitely should not be 100% bond investor." "That's one thing we just learned." "Why is that?" "Because, if I go up here, I have no more." "See that's the same standard deviation, the same risk but I have a higher return." "Right, higher expected return." "So what have we just learned?" "We learned that if we just stay in this space of Stocks and Bonds, maybe you could be 100% stock investor, but never, in a million years, should you ever even think of being a 100% bond investor, okay?" "Because, look, it's just simple math." "I can figure it out." "I can figure out that I get a higher expected return, and no more risk." "So this is lesson number one, that Markowitz showed us." "Amazing, it's so simple and obvious, right?" "It's not so simple, because at the time Markowitz wrote, Yale university was probably a 100% bond invested, believe it or not." "They couldn't figure it out in those days." "The So we've made progress." "That's why I think Markowitz is among the most deserving of the Nobel Prize winners in economics." "This is really basic." "It actually intrigues me." "I don't know how much you like math." "But going back to my childhood, I was interested in geometry and these these, simple mathematical curiosities like hyperbolas are just fascinating to me." "It goes back to, Apollonius of Perga, writing in around 200 B.C., wrote a book on conic sections." "And he invented the word hyperbola, parabola, ellipse." "So I was thinking of looking back at his book." "I think it still survives and seeing what he says about finance." "But I, I can be sure he had no idea that his theory would apply to finance." "I wish I could go back in a time machine and talk to him." "He would be so happy to know that his theory of conic section, you know, ended up applied to astronomy by Kepler and Newton." "And now it hits into finance." "Isn't in amazing how there's a unity of thought?" "And, and this simple diagram has just taught us something about investing that's not obvious, not obvious until you think about, I've just told you, never invest only in bonds." "But it doesn't tell you how much stocks and how much bonds to, once you, you know, once you're above the, once you're above this point, it seems to be a matter of taste." "There isn't any, any single decision that you can make." "So, now" "I want to move to a more complicated world, where we have three assets, okay?" "We're, we're doing, we're starting from." "We had one risky asset." "Now, then we had two." "Now let's go even further." "Let's say three risky assets." "Well, the the expected return is the same." "It's the weighted average." "Now, now we have three weights, x1, x2, and x3." "And they have to sum to $1." "I could, I could have written x3 as 1 minus x1 minus x2, but I wrote it differently here." "It looked messy to write it the other way." "And this is the formula for the portfolio variance." "It's the x1 squared times the variance of the return on the first risky asset plus x2 squared times the variance of the return on the second risky asset." "Plus x3 square times the variance of the return on the third risky asset." "And then you have three more terms representing covariances." "You have to take account of the covariances of the asset because if they move together they, if they all go in the same direction at the same time, that's going to make your portfolio riskier." "And, and so that's the portfolio variance, and the portfolio expected return is just, why didn't I write it there, it's x1 r1 plus x2 r2 plus x3 r3, where the sum of the x's is one, $1." "So it's something that you can do to calculate." "What is the optimal portfolio." "So, I've decided to add a third asset to my diagram." "The, the pink line up here is the same I would call that an efficient portfolio frontier." "I've, have that in the title of the slide, for stocks and bonds." "That's the pink line here." "But I've added the efficient portfolio frontier for three stocks." "Three assets, stocks, bonds, and oil." "Oil is an important investment, because it's it, our economy runs on it." "And the total value of oil in the ground is, is comparable to the value of the stock markets of the world." "So, it's big and important, so let's put that in." "And what, what I have actually here is the minimum, the minimum variance mixture for any given return." "Expected return for the three assets." "And you can see that it's possible when you add a third asset, oil, to bring the efficient portfolio frontier to the left." "Okay?" "Because we've got another asset." "And it's also paying a good return." "And it's not correlated, oil doesn't correlate very much with the stock market." "So we're spreading the risk out over more assets." "We're putting, we're, we're putting more eggs in our basket, okay?" "And and so then we are, we have a better, a better, a better choice set now." "We can pick any point on that blue line." "And so we shouldn't just have stocks and bonds." "We've learned we should have stocks, bonds, and oil." "We're, we're leaning, we're leading toward a fundamental insight." "Which is due to Markowitz." "Which is, the more the merrier." "The more different kinds of assets you can put in, the lower you can get the standard deviation of your return." "For any given, for any given expected return." "And so the better off you are, so this is diversification." "So while diversification was applauded in the 19th century, no one had ever done the math like this before." "And now we can see that that you, when you do the math, you want to have all three in your portfolio." "And yet people don't know that." "They don't, there's an, there's a emotional resistance to this implication." "So I have shown here three assets." "The, the pink line is irrelevant once we realize we have three assets." "We have stocks, bonds, and oil." "So you should choose on this curve." "And of course you should never take down here." "Even though that's possible." "In other words, you can say, what portfolio would guaran, would give me 9% return with the least risk?" "Well, turns out it's a 100% bonds." "But I just told you, never do 100% bonds." "because you can go up to this point." "Alright." "So you never go down here." "So the efficient portfolio frontier is really the part of the hyperbola that's above the minimum variance and you don't want to do minimum variance either." "Right, this is the lowest possible risk portfolio." "You can't get down to zero risk if all of your assets are risky." "So you're stuck here." "But that's not necessarily the best thing, because people allow some risk." "This is having the minimum risk." "So, I can get my return up much higher without taking much risk." "So I'd probably do that." "Okay." "Now, I can do this with more than three assets, I can do it with 1,000 assets." "Now that we have computers, back in 1952, I erased it, but" "I ad 1952 here, Marcowitz had to do it all by hand." "But now that we have computers, it's so easy, you know there all kinds of programs, in fact on your problem set, we have Wolfram Alpha, which will do all these calculations for you, for its own data." "So these are easy to do now." "But what I want to do now, is add the riskless asset." "So what we've done with the blue line takes three risky assets." "It looks at only at assets with a standard deviation greater than zero." "Now I want to do the optimal portfolio, when there are four assets." "I've got stocks, bonds, oil." "All risky." "And now I have the thing that isn't risky would be your one-year governments." "Right, it's not risky because the maturity matches my investment horizon." "I know exactly what I'm going to get." "It's 5%, let's say." "So what can I do investing in these four assets." "Well here it goes back to what I did over here with this simple diagram." "I can pick any portfolio on the Efficient Portfolio" "Frontier and consider that as if that were VOC, right?" "And then I can, I can compute just how leverage allows me to combine that with the riskless asset, and that portfolio." "So I can pick a point, like I can pick this point here and then I could achieve by combining that portfolio, which is 15% oil, 53% stocks, and 32% bonds." "I could combine that with any amount of risky debt and" "I would get a straight line going between, actually this diagram doesn't show zero on it." "I maybe should have done it differently." "But between 5%, so actually that point right here," "I can do it on this diagram, that point here." "Is like 12% expected return and 8% variance so it would be some form, well it would be here, except this would be 12% and this would be," "8%." "I can pick any point and this would be 5% any point along the straight line connecting those points is possible." "So what do I want to do if I have, I want to get the highest expected return for any standard deviation?" "I want to take a line that goes through 5% on the, on the y-axis, and is as high as possible." "So I'm taking a point." "Right over here, at 5% and trying to get as high as I can, it turns out then, that I want to pick the point which has a tangency with the efficient portfolio frontier." "And so, that means the highest straight line that touches the efficient portfolio frontier." "And so now I can achieve any point on that line." "And that's again Markowitz's insight." "So if I were to pick that point, I would be taking what does it look like?" "I don't have it indicated." "Probably something like 11% oil, 30% stocks, 50% something like that, it doesn't add up and that would be holding no debt, right?" "But I could, I could get even higher return, if my client wants that, by leveraging." "I would borrow and buy even more of this risky portfolio." "So this portfolio here is called the Tangency Portfolio." "Now, what, what Markowitz theory shows is that." "Once you add the risky asset, the relevant efficient portfolio frontier is now really this tangency line, and so I want to do a mixtures of the riskless asset and the tangency portfolio" "that, that." "The cards with my risk preferences, but" "I don't, I don't want to ever just move to one of these other portfolios, because these other portfolios, like 15% oil," "53% stock, 32% bond is dominating, has a higher expected return for the same risk." "By a portfolio of the tangency portfolio leveraged up a little bit by borrowing." "And similarly, well yeah, and so this is, and then it comes out and this is the" "I don't know if it was marked clear in Markowitz's paper but it became clear soon after." "There really is, in a sense, an optimal portfolio." "It's the tangency portfolio." "Because everyone wants to invest at, on this line." "And, and that point is a, any point on this line is a mixture of the riskless asset and the tangency portfolio." "And so, everyone wants to invest in the same portfolio." "So there is an optimal portfolio, in a sense." "It and, which it's in a sense, that everybody wants to do the same risky investments." "People will differ in their risk preferences." "And so some of them will want to do a riskier, a more leveraged version, and some of them will do a less leveraged version, of the, of the risky portfolio, of the tangency portfolio." "But everyone wants to do the tangency portfolio so that is the key idea of Markowitz's portfolio management, and it's been expressed by some as the mutual fund theorem." "First of all, I have to just define for you, what is a mutual fund." "You might not know that." "A mutual fund is a certain kind of investment company aimed at a retail audience." "They could have just called this the investment company theorem but history, I can't tell you the history of thought on this." "It's it's" "a mutual fund is a certain kind of investment company." "That is mutual, that means that the owners of the shares in the fund are, there's no other owners, it's just one class of investors all of, they're all equal." "So it's mutual." "But that's irrelevant because the idea is that all we need is one mutual fund, there's thousands of mutual funds." "To serve investors, because I had, that everyone is investing in the Tangency Portfolio." "So they should call their fund, the Tangency Portfolio Fund and, and our fund is the optimal mix of, of stock, bonds, oil and whatever else And then you don't necessarily want to own only that mutual fund, but you want to own," "mixtures of that mutual fund and the riskless asset." "So everyone holds, you only need one investment company." "See, this is the other, I, I told you this story." "I said imagine that you were mathematically inclined and you have all the statistics." "And you're going to figure it out, what's the best thing to do?" "We've just figured it out." "And I haven't gone through all the math details." "There is a best thing to do." "You should offer, as your investment product, the tangency portfolio." "And that's it." "Once you've figured it out, there's nothing more to do." "There's no need to hire any more finance people." "You've figured it out." "According to Marcowitz theory, and all the investors in the world will just invest in this one." "And that's case closed." "We don't need thousands of mutual funds, under the assumptions of Marcowitz, which is that we're, we're agreed on the variances and co-variances and expected returns." "There's a single optimal, risky portfolio." "And then the instructions to investors are very simple." "All you need is two assets in your portfolio." "The mutual fund, that owns the tangency portfolio, and whatever amount of debt you want." "So if you're footloose and fancy free, you can even leverage it." "You can borrow and, two to one, three to one, it's up to your tastes." "But you don't need to look at anything other than the mutual fund." "So that, that's an important insight." "And what it means, then, is, that leads to something else." "So Markowitz didn't get this idea." "It came out later." "But somebody was thinking, well, if everyone should be investing in the same portfolio, it doesn't add up unless that is equal, that portfolio, is equal to the total assets out there in the world, right?" "If, if the if, if there's twice as much oil as there is stock then there has to be twice as much oil as stock in the, in the tangency portfolio, otherwise it doesn't add up, right?" "because everyone has to own everything." "It's supply and demand have to equal." "So, it means the mutual fund theory implies that the market portfolio equals the tangency portfolio." "Okay, and now I've, I've pretty much finished the theory." "All right, I should say it implies, if investors follow this model that we're having, that they all want to the Markowitz model." "If all investors think like Markowitz says, they all want to do the same thing, they all want to invest in the same best portfolio." "So that has to be proportional to the market portfolio, so the tangency portfolio equals the market portfolio." "So I was saying earlier, why is it that everyone doesn't invest in VOC stock?" "How does it add up, right?" "If, if VOC stock is just better than something else." "Then, that, that suggests everyone wants to put all their money in VOC stock." "But we're realizing they don't." "Because they're concerned about, they, they see this tradeoff between risk and return." "And they want to hold some proportion of VOC stock, and the riskless asset." "It has to add up so that the market is cleared and all the VOC stock is owned." "And more generally, if there are many assets, all the assets have to end up owned by someone." "And so, the, the, the cardinal implication of this theory is that the, the market portfolio, which is everything that's out there in the world to invest in, has to be proportional to the tangency portfolio." "And so, one of the implications is if that's true and I'm done with, I have a couple more slides here." "And we go to here." "The capital asset, now it's called the capital asset pricing model in finance." "So that's capital asset, which was, pricing model, which was not invented by Markowitz." "But was invented by Sharpe and Lintner and somewhat shortly after Markowitz." "The capital asset pricing model and I'm not going to derive this equation, but it says the expected return on any asset, the ith asset, equals the risk free rate plus the beta of that asset times the difference between" "the expected return on the market, and the expected return on the riskless asset." "I was just going to try to explain this intuitively." "And then I'll, I'll be done." "Alright, one more slide about this Sharp ratio." "But the intuitive idea." "Let me just say, everything should have a very simple explanation." "The intuitive idea is this." "The, starting from Markowitz, we got an understanding of what risk is." "And people didn't clearly appreciate that." "People used to think that risk was uncertainty, right, in, in finance." "If, if a stock has a lot of uncertainty, that uncertainty means that it is a dangerous stock, and people will demand a high expected return." "Otherwise they won't hold the stock." "But the CAPM says, no." "People don't care about the uncertainty of a stock." "Because if it's one stock out of many, they'll put it in their portfolio." "And if it's independent of everything else, it all gets averaged out." "And so, who cares?" "So people don't care about variance." "But what is it that people care about?" "People care about covariance." "Risk, this is basic insight that followed from Markowitz." "People care about how much a stock moves with the market because that's what costs me something." "I don't care if, I could, I could own a million little stocks that all have independent risk, it all averages out, doesn't mean anything to me." "I'll put them in tiny quantities in my portfolio." "But if they correlate with the market, I can't get rid of the risk." "because it's the whole, the big picture risk." "That's what insurance companies, that's what everyone cares about." "It's this market risk." "The big risk." "You only care about how much a stock correlates with the big picture in it's risk." "So that's measured by beta." "The beta is the regression, the slope coefficient when you, you regress the return on the ith asset on the return on the market." "So high beta stocks are stocks that go with the market." "That we found out that Apple has a beta of 1.5 or roughly that, that means they respond in an exaggerated way." "It's not one, it's greater than one." "They more than move with the market." "And so, investors will demand a higher return on Apple stock, because its beta is greater than for other stocks." "That's the core idea that underlies it." "You see that intuitively?" "So you have to change your idea of what risk is." "Risk is covariance." "It's co movements in that I have just one more slide here." "Its named after William Sharpe who is the inventor of the, with Lintner, of the capital asset pricing model." "The Sharp ratio is for any portfolio." "The average return on the portfolio minus the risk-free rate divided by the standard deviation of the portfolio." "And if you take the CAPM model, the, the Sharpe ratio is constant along the tangency line." "This is a way of correcting, the ex, the average return from some investment, for leverage." "The idea is, some companies used to advertise, we've had a 15% average return." "And then investors would say, but wait a minute!" "You didn't tell me what your leverage is." "That, that's the first thing you should learn from this course." "Someone advertises that they had 15% return, you say hah," "I want to know what your leverage was." "I want to know, really, you know, how to, you were just leveraging it up and taking big risks, and on average you'll do well but it's risky." "So, this is the correction you make, so how do you correct for leverage?" "You might say." "Well I want to look at what fraction of the investor, company investment portfolio is in the risky asset, and what fraction is in the risk-less asset." "But it's not so easy to do that because the company can cover up its tracks, it can invest in a company that's leveraged, right, and so you have to go one step further and undo the leverage for that company." "It's hard to do that." "But the easy thing to do is just calculate the Sharpe ratio for the investment company." "So if some guy is investing and claiming to have done 15% of return per year on his portfolio," "well, I'm going to look at the standard deviation of the portfolio." "That's evidence of how leveraged this guy was." "And I'll compute the Sharpe ratio and unless it's bigger than the Sharpe ratio for the, you know, the typical stock, I, I don't, I'm not impressed." "anyway, so, I think I've come to an end in this lec, so what you should have gotten from this lecture is a concept of risk return tradeoff." "A concept of of optimal portfolio as being something subtle and related to Apollonius of Perga in difficult ways." "But there's also very simple things about how to evaluate portfolios and portfolio management that comes out of it." "[MUSIC]"