# Ses 14: Portfolio Theory II

the following content is provided under a Creative Commons license your support will help MIT OpenCourseWare continue to offer high quality educational resources for free to make a donation or to view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu well if you remember last time where we left off we were talking about risk and return and we said that we're going to make the following simplifying assumption which is that we're going to assume that investors like expected return and they do not like risk as measured by volatility all right and so the way that we depicted graphically is to use a graph where the x-axis is standard deviation of your entire portfolio and the y-axis is the expected return of that portfolio and the question is where on this graph can we get to given the securities that we have access to that will maximize our level of happiness where happiness again is assumed to mean higher expected rate of return and lower risk as measured by variance or standard deviation so for example you if you take a look at this simple graph and you ask the question where on the graph do you want to be you would like to be go always going in the northwest direction right because north means higher expected return and west means a lower risk so obviously if we could we'd love to be on this axis all the way way up right no risk lots of return that's an example of an arbitrage and we know that that can't happen very easily because otherwise everybody would be there and pretty soon it would wipe out that opportunity so the question from the portfolio construction perspective is now a little bit sharper than it was last week when we started down this path now we want to construct a portfolio we want to take a collection of securities and wait them in order to be as happy as possible meaning we want to be as both west as possible so let's see how we go about doing that one thing we could do is just pick an individual stock so if you have these four stocks to pick from then to go as northwest as possible you're sort of looking at Merck as you know the you know the extreme but it's not at all clear whether or not that's something that you really want because for example General Motors while it has a lower expected return than Merck it does have a bit of a lower risk and for some people that might actually be preferred so at this point we don't have a lot of hard recommendations to provide you with without any further analysis so we're going to do some further analysis today the analysis is to ask the question all right what are the properties of mean and variance for a given portfolio not just for an individual security so it turns out that there's a relatively straightforward way of answering this and let me just go through the calculations and then we can see what that implies for where we want to be in that mean standard deviation graph so the mean of a particular stock I'm going to write as expectation of RI or mu I for short the variance of a particular stock I'm going to write as Sigma squared I or the standard deviation is then just Sigma I okay and it turns out you can show this rigorously I won't do that but you can take a look at that if you are unconvinced it turns out that if you construct a weighted average of stocks so that the return of the portfolio is given by that top line R sub P then when you take the expected value of that top line what you get is the middle line in other words the expected return of a portfolio is just equal to the exact same weighted averages of the expected rates of return of the individual components so you understand the difference between the second and the first line in that red box it's very important distinction the the top line is basically an accounting identity it says that when you want to compute the actual realized return in your portfolio you just take a weighted average of what you did on each stock what your return is on each stock that's an accounting identity the second line is not an accounting identity it comes from an accounting identity but what it says is that on average the rate of return of your portfolio is equal to a weighted average of the average rates of return on each of the components of your portfolio okay that's a very important principle any questions about that before we fully move on okay so I'm going to write that as a shorthand for the portfolio mu sub P so moosa' pees just equal to this whole expression right here all right so what we've now deduced is that the mean of my portfolio is simply the weighted average of the means of each of the securities in the portfolio what I'm going to turn to next is a much more complicated calculation which is what is the variance of my portfolio it turns out the variance of the portfolio is not a simple weighted average of the variances of my individual securities this is where it gets complicated and also where it gets really interesting and valuable from the investor's perspective okay let's let's do the calculation you're going to have to dredge dredge up your old BMD knowledge here of how to compute variances of sums of random variables the variance of my portfolio return R sub P is simply equal to the expected value of the excess return of that portfolio in excess of this mean right squared and then take the expectation of that and remember that the return of the portfolio is just a weighted average of the returns of the individual securities and the mean of the portfolio is just the same weighted average of the means of the securities so when you plug the relationships in what you get is that the variance is simply equal to the expected value of the square of this long weighted average so you've got a weighted average a bunch of terms right and then you square that and then you take the expected value well if you've got n terms in that weighted average and you square that n terms how many terms comes out of that square anybody n terms and you square those n terms so those n terms multiplied by itself how many terms do you get when you do that yeah well why not just n squared you're thinking about the unique elements maybe the off diagonal I'm talking about all of them so with n terms when you square it n terms multiplied by n terms you get n times n terms n squared terms and so these n squared terms all look like this they all look like Omega I times Omega J multiplied by the excess return of I times the excess return of J some times I equals J and when that happens you get Omega I squared times the variance of security I but when I does not equal to J then you're going to get Omega I times Omega J times the covariance between the return on I and J okay and another way of writing that covariance is equal to the correlation between I and J multiplied by the standard deviation of I and the standard deviation of J the point is that when we look at the variance of a portfolio it's not just a simple weighted average of the variances of the component stocks it's actually a weighted average of all the cross products where the weights also the cross products of the weights so it's it's these it's all of these and how many of these are there well I goes from 1 to n J goes from 1 to n n times n you get N squared of these so this actually has a nice representation it actually comes out of a table alright so you can think of all of the portfolio weights multiplied by the excess returns on one dimension of the table the columns and the exact same entries in the rows you've got n columns and rows n times N or N squared elements that make up the variance of your portfolio so this is where it gets really complicated but this is why we have computers and spreadsheets and things like that so in order to figure out the variance of your portfolio you've got to basically add all of the elements in this table you've got n squared things to add up to to get the variance of your portfolio and the insight of modern finance theory the reason that Harry Markowitz won the nobel prize in economics for this idea is that when you add up all of these different elements you can get results that are very different from just looking at one or two stocks because in particular there are some cases where these cross products are either small or maybe even negative and in that case it actually helps you to reduce the overall riskiness of your portfolio this is the intuition this is the mathematics underlying don't put all your eggs in one basket when you don't put all your eggs in one basket what you're getting what you're benefiting from are these cross products that can either be small or negative and will ultimately reduce the fluctuations of your portfolio a note another way of thinking about it is that some stocks go up some stocks go down and if they don't go up and down in the exact same way if they are unrelated or less related than when you put it all together in a portfolio it dampens the riskiness of your overall holdings okay so there are N squared elements that you have to add up in order to get the portfolio variance it's literally the sum of all of the entries in this table and the point of the the next few bullet points is that there are a lot more covariances than there are variances the variances are along the diagonal of that table right that's the variance of the first stock the variance of the second stock dot the variance of the nth stock those are how volatile the individual stocks are but there are only n of those there are N squared minus n of the other stuff that's where you were thinking about n minus 1 right it's N squared minus n or n times n minus 1 other non diagonal entries so what this suggests is that the covariances are a lot more important in determining the riskiness of your portfolio than the riskiness of the individual stocks you know intel looks like a scary stock because it's really volatile but when you've got n Intel's in your portfolio even though each of the individual stocks is scary what you have to keep your eye on is how they are correlated because the correlations in a portfolio of n stocks is actually more important than the individual variances the individual variance is matter but they don't matter nearly as much as the covariances okay so now I'm going to do an example it's hard to get intuition for you know n by n matrices unless you're you know Rain Man or some you know incredible genius the way that I think about this is let's look at two assets okay if I can understand two assets I can sort of generalize and think about n assets so I gave you the general result just to tell you that this is how you do it but now to understand it really let's focus just on two assets okay suppose we only have two stocks a and B and I'm going to calculate the expected return and standard deviation of a portfolio with just those two stocks so the portfolio weights are going to be Omega a for stock a and Omega B for stock B and Omega a plus Omega B adds up to one so just keep that in mind I don't put it there just because I want to save a little space on the slide but these are the weights for the two stocks and those are the only two things you hold all right the expected value is as we said it's a weighted average of your holdings of a and B right multiplied by the mean of a and the mean of B now the variance the variance of these two securities when they are put into your portfolio is going to be simply Omega a squared times the variance of a plus Omega B squared times the variance of B and now you only have one cross product to worry about because you've only got two stocks right and that is 2 times Omega a times Omega B times the covariance between a and B now you can do this in a two-by-two table just like the N by n table and with a two-by-two table you know that you've got two diagonal elements and two off diagonal elements and that's where you get that number two for the two times Omega a Omega B times the covariance okay and I rewrite it in terms of correlations because it's easier to think in terms of correlations why because correlations we know have to be between minus 1 and 1 and so if I tell you a stock has a correlation with another of 30% you can actually get your arms around that you can get intuition for that all right so it's a little easier to interpret but either way is fine so we now have an expression for the mean and the variance of a portfolio of two assets any any questions about this everybody understand this how I got it no tricks up my sleeve it's really meant to be relatively straightforward but the implications will be dramatic and I'm going to show you in a minute ok so let me go through at least one numerical example and then I'm going to show you why this is actually so important from 1946 to 2001 motorola had an average monthly return of 1 and 3/4 percent and a standard deviation of nine point seven three percent General Motors had an average return of 1.0 eight percent and a standard deviation of six point two three percent that has obviously since gone up a great deal the standard deviation that is and their correlation is point three seven or thirty-seven percent how would a portfolio of the two stocks perform well it depends one of the weights right if you change the weight you're going to change the performance so let's just take a look if you put all your weight on General Motors and nothing on motorola not surprisingly you're going to get General Motors return at the other end of the extreme if you put all your weight on motorola and nothing on General Motors you're going to get Motorola's characteristics all right nothing mysterious about that however if you wait the two so if you put 50/50 on General Motors and motorola you're going to get a return that is 50/50 of each of those returns but you're going to get a risk that is not 50/50 right because the risks the risks don't aggregate in a linear way the risks actually aggregate nonlinearly because of those covariances or correlations okay so what you see here is that by taking a weighted average of the two what you can do is you can boost your return to one point for two but the extra risk you're taking is from six point two three to six point six eight percent per month that's not a big increase in risk but that is a pretty significant increase in return now again I'm not telling you that's what you all should do because it depends on your risk preferences and for the next lecture or two I'm going to be seeing that over and over again I'm going to be saying it depends upon your risk preferences but at the end of the next two lectures I'm going to be able to tell you something that doesn't depend on your wrist preference I want to be able to tell you something that all of you should be willing to do if you're rational and if you're not I will trade with you personally to help you learn rationality okay but for now I don't have a story to tell you about which of these rows you ought to take there is no ought to here it's just a matter of what you prefer risk versus return right now interestingly I put here one more case which lies outside of the band of zero one this is a case where you put 125 percent of your wealth into Motorola and you've put negative 25 percent of your wealth into General Motors remember we talked about this at the very beginning you're using short selling to short sell General Motors which you might think is not a bad idea nowadays given how how trouble they are you're going too short General Motors take the proceeds and put that plus the hundred percent that you started with you're going to put that in Motorola so you really want to make a big bet on communications and micro processors and you're going to leverage that bet by putting a negative bet on the auto industry that's what that last row implies and what you get is a much much higher rate of return than anything else in the in the other examples right much higher rate of return but look at the risk now the risk is close to double what you would have gotten had you put all your money in General Motors are you are you willing to do that well you can easily imagine there are people in this class that would be delighted to do that and there are others of you that would be scared to death that that's way too much volatility these are monthly numbers by the way so if you want to calculate the annual equivalent how would you do that how would you get an annual standard deviation from a monthly standard deviation any any idea well let's do that let's do the return first how do you annualize the monthly return to get an annual return let's forget about compounding what would you do what multiply by 12 that's right if you forget about compounding you multiply by 12 what if you didn't forget about compounding then what do you do add 1 raise it to the 12th power then subtract 1 that's right okay what about risk I see this is tricky what's the variance of a 12-month return in relationship to the variance of a one-month return we haven't talked about that what is it yeah no no it doesn't compound in that way variance is actually a little simpler under certain assumptions Yeah right there's no correlation between one month to the next that's right agreed so let's assume that away then it's constant but the monthly variance is constant but what's the annual variance what's the riskiness of a 12-month return if you know what the riskiness of a one-month return is yeah well it can depending on your assumptions but I want a simple set of assumptions Andy very variance yes multiply by 12 now why is that it's because and you know all this you know this from your d-md least you I think you do you may not think you do but you do the variance of a plus B is equal to what who can tell me what's the variance of a plus B two random variables variants of april's variant 2b plus 2 times the covariance of a and B if the covariance is 0 then it is the variance of a plus the variance of B absolutely that's right so if we assume there are no cycles there are no predictability there's no regularities there's no correlation every month is independent of every other month then the variance of a 12 month return is literally just 12 times the variance of a one month return assuming the variance to stay constant throughout the month the monthly variance doesn't go up or down okay now that's the variance what about the standard deviation the standard deviations are square root of the variance so that means that the standard deviation on an annualized basis is equal to square root of 12 times the standard deviation of a monthly okay so finally we're almost there what's the square root of 12 something between 3 and 4 right so I don't know call it three and a half whatever we have to take this number and multiply it by about three and a half and that will get us an approximate annual standard deviation so a twelve percent volatility multiplied by three and a half you're talking about some crazy volatile stock or portfolio but actually nowadays you know we're used to that kind of volatility that's no big deal but relative to the individual stock it's quite a bit more volatile okay so this is an example of how you compute the riskiness and the return of a portfolio of two stocks and when you do this you get a whole bunch of possibilities and the idea behind modern portfolio theory is you pick the one that you like the best and that's optimal for you okay now let me graph this ah in that beam standard deviation graph and you'll see something really interesting so here's General Motors and there's Motorola and what I graph with the red dots is just the different combinations of 20 575 5050 75 25 and so on and so the red dots that are strictly between 0 & 1 where the weights are between 0 & 1 those are the dots contained in that arc between GM and Motorola so the first point I want to make is that when you graph the risk/reward trade-off as I told you it's not linear it's nonlinear in fact it's it's curved it looks like a bullet okay that tells you right away that there's something more subtle about portfolio theory than just taking weighted averages the means are weighted averages but the variances are not because of the covariance between them and then when you take the square root to get a standard deviation it looks a little bit more complicated yet again it looks like this now that red dot that goes beyond Motorola that red dot is the case where you're shorting General Motors and you're using the proceeds to take an extra large position in Motorola okay so you're going beyond the Motorola risk reward point so I need you to spend a little bit of time now twisting your brain in a way that makes this more understandable what I mean by that is that you have to think about the weights of the various different portfolios but the weights don't show up on this graph but I want you to keep those weights in the back of your head so while you're looking at this graph you have to remember that as you move along that bullet the omegas the weights are changing so at the General Motors dot your weight for General Motors is a hundred percent and your weight for Motorola is zero at the Motorola dot your weight for General Motors is zero and your weight for Motorola is a hundred percent and as you vary the weights you trace this arc this curve okay and as you go beyond either General Motors or Motorola in other words as you short sell one to invest more than 100 percent in the other you then go outside of the two points in either direction right any questions about this graph yeah okay so how do you calculate the correlation that's a good question historically the way you would do it is exactly the way that these correlations look like they're defined so in other words you would calculate the covariance by essentially taking a historical average of these cross products so going back to the first slide in this particular lecture I give you a formula that shows you how to do that let me go back and take a look at it so that you refresh your memory let's see oh is that it was not in this lecture was in the previous lecture in the previous lecture I give you a formula that shows you how to calculate the weighted average of this or the historical average it's basically just taking historical data and then estimating this quantity right here there's this particular cross product the Omega weights come out because that's what you get to decide but the inside stuff how to calculate that is simply to take historically your actual return minus the mean of i rj t minus the mean of j and then take 1 over t and that's your estimator for the sigma hat i j so you get historical data and you would estimate the mean estimate the mean take the cross products over time this t goes from 1 to t and that gets you an estimator is that the best estimator not necessarily because things change over time so you have to worry about that and that's what you learn in 15 4 33 how to actually implement a lot of these ideas using data yep okay okay so the question is is there any recommended length of time for which you would go about estimating the data estimating these quantities using the data the answer is no there's no recommended time simply because you have to trade off two things the longer time you have the more accurate your estimator is going to be in the limit as T goes to infinity you'll get a perfect estimator okay however that's under the assumption that nothing ever changes and I'm pretty sure that beyond some T over let's say 150 or 200 years you know we didn't exist you know in terms of stock markets and and data I don't think that you know for example General Motors goes back 200 years so beyond some T this is going to look very boring but the other thing you have to trade off is the fact that that market conditions change so General Motors today is not what General Motors was even 10 years ago so if you use lots of data you're going to build into your results what are called non-stationarity z' and the bottom line is you have to balance off the non-stationarity z' against the error that you introduce by not using enough data and that really is where 15 4:33 comes in so there are methods that we have developed for balancing those two but the bottom line is that there isn't one single answer so it depends on how unstable the markets are and how volatile the underlying estimates are given the data yep well sure you could try to include a full cycle but then you're left with the question the full cycle of what for example there are these things whole Kondratiev cycles that claim to be something like 50 year periods so if you want to include one of those you got to include you know 50 years of data but do you really think that General Motors is the same company over those 50 years you're building in a lot of bias in terms of the non-stationarity of one one aspect while incorporating the stationarity of another aspect so how do you balance off those two right this is where it gets more complex complex and you have to take a stand on the kind of non-stationarity that are in the data other questions yeah other studies that show how accurate well first of all we're not trying to predict the stock market remember we're not trying to predict where markets are going to go what we're suggesting is that there are underlying parameters of the data that are stable over time so in other words the stocks can go up or down right but on average they have some level of return that people expect given their risk that's what we're assuming is stable similarly covariances stocks can go up or down in relationship to other stocks going down or up we don't know what's going to happen day to day but over a period of time it seems like we there's a pattern where most stocks on the new york and american stock exchanges and as that they seem to go up together and they seem to go down together on average so that's what really what we're trying to distill from the data it's very different than trying to forecast what's going to happen with the stock market next week and this I told you is the fundamental difference between academic finance and Warren Buffett Warren Buffett is all about prediction he's not about creating a good portfolio that will be worth something reasonable over a period of time what he wants to do is he wants to beat the market he wants to find undervalued stocks invest in them and then sell them when they actually reach equilibrium or become overvalued right that's a very different approach and perspective than what we're doing here so that's a good question and it highlights that that distinction yeah yes there is and there have been many studies that have been that have been done on it and but that is beyond the scope of this course so I'm going to refer you again to 15 433 but I will tell you at the end of this once I get through this make sure everybody's with me I will tell you how this works in practice okay so I'll talk a bit about applications but first I want to get through the theory to make sure we all understand it all right so we now have this bullet and the bullet obviously depends on all the parameters but in particular it depends upon the correlation the reason that there is a bullet shape is because there's a correlation between these two stocks and by the way this correlation and this bullet shape is really important okay I'll tell you why it's really important take a look at a vertical slice a vertical line going through the GM dot okay that vertical line is the riskiness of GM right it's the standard deviation of GM one of the things that you'll notice about this graph is that if you only had two stocks available to GM and Motorola the least amount of risk that you could possibly create for yourself is if you put a hundred percent in General Motors in other words if all you cared about was going west the the most extreme west you can go other than putting your money in t-bills over here the most extreme west you can go is General Motors you can't get any less risky than that all right yeah question what I'm saying is if you whoops let's see yeah if you are trying to get less risky as possible and you only had General Motors or Motorola then the most west you can go is General Motors right that's all however however this is the important point if now I let you take weighted averages of the two if I give you the right to form a portfolio then then you can get a dot which is this dot right here that dot everybody in this room should prefer that dot to General Motors why because it's less risk than General Motors but it's also higher return there's no downside at least from the perspective of mean and standard deviation there may be other reasons you don't like that dot but if all you care about is mean in standard deviation that dot is strictly preferred so I've just made all of you better off with this piece of knowledge just by telling you how to weight these two stocks and the fact that they've got some kind of correlation I've given you a mechanism of reducing your risk and and increasing your expected rate of return both okay so if you were really risk-averse if you said to yourself you know what I don't want all this fancy stock picking just give me the least risky stock of the two then you'd be at General Motors and then if I came to you and said you know what I can do even better than that I can get you even less risk than General Motors and at the same time I'm going to give you a higher return so right there the value of portfolio theory is pretty clear it gives you options you did not have yeah yeah that's right that's right when you when you have correlations that are unstable over time and you didn't account for them you can get into trouble okay again let's let's let's come back to that after I go through all of this all right so just if this is a clarifying question I'll answer but extensions let's wait until we actually go through this okay so let me talk about the general motors and motor oil example but now where I am going to tell you that correlations are changing over time so let's do the example that Lewis wants us to do let's actually assume the correlation is I don't know 0 or 1 or minus 1 let's go through all three cases and see what happens you'll see some remarkable things coming out of this if the correlation is 0 then it's pretty easy for you to just plug in 0 for that correlation in that second equation and all you get are the first two terms of that variance expression right so when you assume that the COPE when you assume that the correlation is 0 when this thing is 0 what happens is that this entire term goes away and all you have are the first two terms that's this column right over here ok so I've just produced the previous table but with the assumption of 0 correlation and I get that now I'm not going to graph it for you yet I want to just show you what happens when you choose different assumptions so that's one assumption on the other hand if I choose a different assumption if I choose the assumption that the correlation is let's say 1 so in other words they are perfectly correlated then I'm going to get I'm going to get this whoops turn on this I'm going to get this becoming 1 and then I'll get another simplification of this expression and similarly if I assume that the correlation is minus 1 then I'm going to get yet simplification of this expression some algebra that I'll show you in a minute but but this is what I'm doing in these three columns I'm assuming different values of correlation between General Motors and Motorola and then computing the the standard deviation of the portfolio using this formula okay so you can all do this at home do this in a spreadsheet it's going to be very easy to to check my work but I want to show you the graph because that's really I think the insightful intuition here look at the graphs these are the graphs of the three different cases of zero correlation perfect correlation and perfect negative correlation I'm going to go through each one of these with you so the case of perfect positive correlation it turns out that the risk/reward trade-off is actually just a straight line it gets really really simple just a straight line so in particular there is no non-linearity because if they're perfectly correlated you know what you basically have the virtually the same stock the only difference is that they're different scales of each other right but if they're perfectly correlated that means that there's a linear relationship between those two stocks and so when you do this mean variance analysis in mean variance space you basically get no no nonlinear you don't get that little bump the little bullet that we saw before right so there's no way you can get less risk than General Motors unless you end up shorting motorola and putting it in General Motors and then your expected return goes down as you're expected as well as your risk all right now let's take the case where you've got no correlation it turns out that if you have no correlation you get the bullet but the bullet is even more wedge-shaped what that means is that you can reduce even more risk without getting rid of return remember the red dot that we saw in the previous slide this red dot well in the case where you've got negative zero correlation this is a case where the correlation is 0.3 37% right that's what the data tells us but if you assume that the correlation was zero you would get even more of a savings of volatility for a given level of expected return okay so it would look like this you see how this bullet sticks out a lot more than the previous red dot that was somewhere over here okay now let's continue suppose the correlation is minus 50% then you get an even wider bullet a more that sticks out even more that saves you more standard deviation you're getting to the west even more and finally and here's where you get a really remarkable result if the correlation is minus 100 percent you know what you get you get a piecewise linear trade-off you get this and that it doesn't turn into a wedge it turns into a triangle that actually hits the x-axis and the reason this is such a startling result is it tells us that there exists a way to construct the portfolio that gives us a rate of return of like 1.3 9% with no risk zero risk 1.3 9% let's just call it 1.3 percent to be conservative multiply that by 12 and you're going to get something like what 16 percent a year you tell me if you know of any investment opportunities that gives you a return of 16 percent a year with no risk and I'll examine that for you carefully okay it doesn't exist of course the reason it doesn't exist is because you can't find two assets that have perfect negative correlation if you could there are wondrous things you could achieve with that combination and portfolio Theory basically tells us how right this is a rest book for how to exploit correlation all right now again I can't tell you where you should be on this curve other than if it's really minus one then I would be here okay lots of return no risk there that's an arbitrage we know that that can't possibly happen there's no free lunch there's no arbitrage on average over circumstances however if the bullet is minus 0.5 then you've got lots of opportunity to create really attractive portfolios that doesn't require Warren Buffett's skills there's no forecasting here right we're not trying to pick stocks we're not trying to see how the markets going to do next month who knows all we're assuming is that means and variances are stable over time and the correlation is stable over time those are those are non-trivial assumptions I grant you but if you believe in those assumptions more than you believe in your ability to forecast what's going to happen in General Motors six months from now then this might be a good way to construct the portfolio but I haven't told you where on this curve you ought to be I've just told you how to construct that curve all right it's your job to look at the curve and say I like this point or I like that point based upon your own personal preferences for risk and reward so we're not there yet where I can tell you how to behave I will get there in about a lecture and a half but we're going to build up the infrastructure to be able to get us there okay so what this tells us is that we need to know what the correlation is in order to figure out where we're going to be on these different curves which curve is going to apply when you have lots of correlation a correlation of 1 there really isn't much of a risk savings per unit return we can't get a lower risk for a given level of return but we can if there is less correlation than perfect and these are the different curves that illustrate that ok so there are some other examples that I'd like you to work through on your own this is another portfolio calculation just go through the same calculations that we did here and you know you'll graph the different risk reward trade-off between these two general dynamics and Motorola and you can get exactly the same analysis with those two stocks General Dynamics and Motorola okay now what about if you've got a risk-free rate so suppose that the two assets that I want you to look at is not General Motors and Motorola but rather the stock market and Treasury bills then what is your risk reward trade-off look like well it turns out that in the case where Treasury bills are in question the volatility of Treasury bills is virtually zero it's not exactly zero because there may be some kind of randomness in the underlying rates of return because of inflationary expectations but as an approximation if it's a risk-free rate and you know that you're going to get that risk-free rate then the volatility of that return is in fact zero over that period of time so in that case the expected rate of return of a portfolio between a stock market and t-bills that's the weighted average but the variance is going to be very simple because it's going to be the variance of 1 which is zero plus the variance of the other plus two times the weighted average times the covariance but there is no covariance right there's no correlation because one of the things is non-random and so when you work out the weights for the two and you graph them you get this this is a beautiful thing nice and simple no weird curves or any kind of bullet shape you got t-bills here you got the stock market here and the weights as you vary them will bring you anywhere along this line or possibly up over here if you're in the middle of the line so literally if you have the same distance between here and here and here and here that actually gives you a 50/50 waiting on those portfolio weights so geometrically this actually corresponds to a 50/50 weighting of t-bills and the SP okay now question what happens when you are over here let's suppose you're at this point at this point what would your portfolio weights look like how would you characterize that yeah right surety bills to surety bills to buy into the stock market that's right short t-bills what does it mean to short tea bills so what are you doing yeah you're borrowing you're borrowing money you're leveraging when you're shorting t-bills you're basically borrowing and getting cash up front you're going to pay back later with interest so shorting t-bills is just borrowing if you're borrowing money and you're putting it into the stock market in addition to 100% of your own wealth you've borrowed additional money to put it in the stock market then you're going to be way up here higher return much higher up here then down here but you're going to get higher risk as well so leverage this idea of borrowing and putting your money in the stock market that increases your expected rate of return but it also increases your risk okay leverage increases your risk and now getting back to the question that Lewis asked us is this where we got into trouble with the current crisis yes in a nutshell it is but it's more complicated because the underlying securities are more complex but the basic idea is if you if you leverage up if you leverage up way up here you're up maybe out there and and all of a sudden there's a bump in the road and what you are leveraging this thing that you're investing in is not nearly as smooth and as riskless as you thought it was it could wipe you out and one of those elements that could cause such a wipeout is if you somehow forgot about the fact that correlations can change so you thought that you were I don't know somewhere here and all of a sudden correlations go to one and now you're actually over here you see how risk can change really quickly correlations don't have to be stable over time and that's the lesson that most people in industry who don't have a finance background who've never taken this course they won't know these are physicists or mathematicians or computer scientists they estimate the correlation it's a parameter it's like the gravitational constant or Avogadro's number let's plug it in hey you know nine point oh eight times 10 to the 23rd that's what it should be and you know nobody ever told them that it could change and when it changes bad things can happen really quickly so we're going to come back to that but let's get the standard theory down first and then we'll talk a bit about applications and ah yeah Yeah right here of course because you might want to get more returned and you're willing to take on risk let's take a look at this this bullet point here says that you're going to be at approximately something if it's if the correlation let's be realistic about it okay the correlation is not going to be minus 0.5 it's going to be more like point three seven so this bullet here is going to be like one point I don't know what let's call it one point three percent just to make it easy and you got a standard deviation of about six percent okay so on an annualized basis that's giving you a return one point three times twelve is something like what 16 percent 16 percent return but the risk on an annualized basis multiplied by three and a half is about let's say 20 percent so twenty twenty two percent annual standard deviation for a 16 percent rate of return now some of you might like that but I have a few friends here for whom that return is just boring that's just not going to get their attention what they want is they want to be up here you know like at two percent rate of return two percent rate of return per month is about 24 percent a year so if you're a hedge fund manager 24 percent is when you start to begin to feel alive you know that's when things really start to happen for you and at 24 percent annual return you can't have volatility of like fifteen or twenty percent unless you're doing something you know really different than the standard market portfolio so a hedge fund manager is going to say Anand give me a break you know this is going to put me to sleep I want to be up here but you know you've got a family three kids you have to worry about you know making mortgage payments you've got that kind of lifestyles not for you so in both cases though in both cases we can agree that where we want to be is on this curve right in other words you would never want to be here at this point why because if you were here you can either for the same level of risk increase your return or for the same level of return decrease your risk so what we can agree on even though we don't agree on where we want to be on that curve we can all agree we want to be on that curve as opposed to inside the curve right exactly great point why would we want to be on the bottom half nobody would want to do that because if you want on the bottom half you can just easily move up to the top and you have a higher expected rate of return for the same level of risk so you're exactly right that bottom half of the curve you can just throw it away alright if the only people that are going to be down there are knuckleheads all right so we don't want to be down there so in fact the only part of the curve that really matters is this point going all the way up and that's we're going to give a name to that in a minute that's going to be called the efficient frontier because you would never want to do anything else actually that's not quite true what everybody would like to do is they'd like to be up here unfortunately we can't get there we can't get there with just General Motors and motorola in a few minutes I'm going to show you how you might be able to get there with other stocks if you introduce other assets then you might be able to get there but you can't get there right now okay so here we are with stocks and t-bills and we know that you'll get this linear combination here what I'd like to do next is to make this yet more complicated all right and before I do let me just summarize what we've done so far what we've done is to show that with two assets we can do all of this analytically and illustrate graphically all of the intuition that holds for the more general case with two assets we expect the rate of return is just a simple weight of average but the variance is not just a simple weighted average it's more complicated and it depends in particular on the correlation for different correlations we get different shapes of trade-offs in mean variance space and you have to understand what those trade-offs are if it turns out that one of the two assets is t-bills for example then the trade-off is really straight line okay but if both assets are risky then you get the bullet shape until you either get minus 1 or 1 in terms of core relations and then you get straight lines of different stripes in those two different cases okay now we're ready to talk about the general case the general case works exactly the same as the two asset case you've got your means you've got your variances and you've got your covariances and you add up all of these different covariances to get the total variance of the portfolio and if you consider a couple of simple cases like for example an equal weighted portfolio so you've got n assets and you put each of your it for each of your assets you put one over n of your wealth in them then you can show that the variance of your entire portfolio is equal to the average variance plus n times n minus one times the average covariance when n gets large then it turns out that the average variance doesn't matter what is driving the risk of your portfolio has nothing to do with the variance of the individual components what it has to do with is what the average covariance is that's what's driving the risk of your portfolio because you've got a lot more covariance as than you do variances okay so this is kind of a neat insight because it says that it's really important how things are related and as those relationships change the risk of your portfolio is going to change so getting back to Lewis's point about what's going on in current markets and what caught a lot of portfolio managers by surprise with the subprime markets if the risk of your portfolio is approximately given by the average covariance and you've been assuming all along that you've got this big pool of mortgages and the mortgages are all uncorrelated you've essentially assumed that you've got virtually no risk because the average covariance by assumption and by historical analysis is close to zero but when the real estate market goes down nationally then everybody starts to default and for closures become very highly correlated and so you can see how overnight literally overnight your risks can shoot up and you're not prepared for that unless you know that this is what's going on in your portfolio yeah okay we never thought of this scenario that everything will does go down but I don't see great question why should things changing in terms of correlation well you know what I'll give you an example I'll give you a personal example correlation is a function of human behavior right I mean prices are being formed by you and I investors and so correlation simply means that all of us end up doing the same thing around the same time right I'm going to be heading to the airport later on this evening and I don't expect it there's going to be much of a big deal getting on my flight I'll probably get to the airport about half an hour 45 minutes ahead of time it's just a shuttle so I'm heading to Washington so it's not a big deal what's going to happen in two weeks from today anything anything going on two weeks from today that you can think of Thanksgiving so if I went to the airport two weeks from today and tried to get on that shuttle half an hour before you think I can get on that flight why not because everybody else is going to do that well why should everything be correlated on that day isn't that Wednesday like every other Wednesday well it's not it's because somehow we've all decided that we're going to take off at the same time on the same day in that in that year right well well first of all I don't think Thanksgiving is that lower probability okay but but to your point is that the correlation changing well it is correlation is a is a statistical measure of two objects and what we're trying to capture is when they move up or down at the same time right so what I'm trying to get at is that there are certain periods of time where human behavior all of a sudden becomes very highly correlated and there are many reasons for that in particular Thanksgiving is a reason we all decide that Thursday is this national holiday and therefore on that Wednesday we actually have to travel in order to get to where we need to go by Thursday right but any arbitrary Wednesday is not going to be necessarily highly correlated so when I go to the airport on a typical Wednesday I don't expect it to be so bad but on Wednesday before Thanksgiving it's going to be a madhouse right that's an example of a change in correlation because of a particular coordination that we've all agreed upon now that's a relatively artificial example but I did it to make a point now let me give you a real answer to your question about why things may be correlated when we are all scared about the value of our investments when our fear circuitry gets triggered what's the natural instinct for all of us because it's hardwired into our brains it's to get the safety it's to get out of those bad assets and get into the good assets right you saw that three-month t-bill yield at 10 basis points or 5 basis points that's a sign that we're all scared to death and we want to get to safety that's an example of a correlation why because everybody can be selling at the same time in a crowded theater if you smell smoke and somebody shouts fire what are you going to do it's not rocket science to predict that the four exits that are out there will be a bit crowded after that right we court so correlation is not a physical quantity that's the problem with physics and biology physics has parameters that don't change over time I wish I wish we could have that in finance we don't we have parameters that are not parameters or random variables they depend a lot of things correlation is one of them and until we start recognizing that correlations are really part and parcel of human interactions we're going to continue to make mistakes like we've made over the last ten years and that's a simple an example of that other question yeah yes yeah yeah perhaps it is the one person yeah that's true that's possible so you have to decide whether or not what you're looking at is an aberration or whether it's something that's systemic so for example if I didn't know anything about Thanksgiving and I happen to travel on Wednesdays on a regular basis to Washington then you know next in two weeks I'll get it's really crowded and then it won't be crowded won't be crowded it won't be crowded and pretty soon after a while I'll say well that was just a 1% event of course next year it'll happen again and then I'll say gee you know well that's another 1% kind of but pretty soon I'm going to realize that gee in it seems like there's a pattern here and that 1% it's not just so simple as a 1% error statistics remember is a mathematical quantification of our stupidity right I mean what we don't know we don't know why things happen the way they happen so we put a distribution on it we say that it's normal and we say that there's you know 5% this way for a person that way but that just isn't representation of our ignorance the more we know the less we have to rely on statistics so I don't need statistics to tell me that two weeks from now it's going to be really crowded at Logan but if I didn't know about Thanksgiving if I came from Mars and I was doing a study of airport congestion it would take me a while to get enough data to figure out that once a year on the Wednesday before the Thursday in November it gets crowded so the challenge for all of you is how much intelligence can you bring to the analysis what I'm showing you is very simple mathematics mathematics is not enough if you just had the mathematics you would be losing money you know continuously because there's just much more to financial markets than just the math okay the math is trivial this is high school algebra but the real key is to put together the framework of economic and financial analysis with the mathematics so let me continue doing that this is another example of computing the average variance and then looking at the volatility of your particular holdings given different correlations so I want you to do lots of exercises where you look at different correlations I don't ever want you to take correlation as a parameter that is fixed over time when you apply this stuff when you're doing problem sets and final exams that's fine you know the correlation is whatever it is but recognize that in practice these things change a lot over time okay now the idea behind correlation is that they help you reduce the ups and the downs of the variance of your portfolio but it turns out that there's a limit to how much of the risk you can reduce and this graph basically shows that as you add more and more securities even though the correlations are bouncing around as you add more and more securities there ends up being some kind of a steady state limit to what the variance of your overall portfolio is after 20 30 40 50 100 stocks you'll notice that the variance of your portfolio doesn't go down any more and it turns out that that limit whatever that is is what we consider to be the systematic risk that is implicit in the economy in other words that's the risk that no matter how well diversified you are you've got to bear that risk everybody all of us we have we can't get any less risky than that unless we start putting our money in the mattress there aren t bills okay that limit is known as the systemic or market risk of a portfolio so see here's the graph of your portfolio variance as you increase more and more securities but at some point it asymptotes to this level which is what I'm going to be focusing on as undiversified ball it's the risk that's left over after all of the various correlations have done what they can to dampen the ups and the downs of your collection of securities and obviously there's going to be some calculations you'll need to do to figure out what that number is but I want to give you the intuition first and then we'll do the calculations and then we're going to study the properties of this hard limit okay now I'm about to proceed to the next stage of our analysis where we start asking the question how do we pick the very best possible portfolio but before I do I want to make sure that everybody is comfortable with the analytics we've developed if you have any questions now would be a good time to ask because from this point on I'm going to assume that you understand how portfolio theory works the mechanics of portfolio weights and how to compute means variances covariances and what they imply for the portfolio question yeah you said that we start from the assumption that the markets are pretty much stable and why would nobody ever work hard curve for example you know if there was a big scandal tomorrow like a corruption thinking about I'm rolling for Parliament Anthony okay why would I not want to short sell you know the higher return stock and buy you know the lower returns talk and you might if there are other elements that are in this analysis so as I said this analysis assumes there are only two dimensions mean and standard deviation right you've now introduced the third which is corporate responsibility or corporate governance or fraud or something like that then you would need three dimensions and then there's something there be something sticking outside the screen and then you have to choose among the three possibilities so it's possible that you want to be down here because this company has a better reputation then then this one up here but then what you're telling me is that reputation matters to you beyond net standard deviation and expect the return what's that stable means that these parameters are likely to stay the same over time there's a crisis every right well the argument that a financial economist would make and I'm not saying that I believe in this I'm going to tell you what I believe at the end of this class but I'm telling you now what the party line is and what's in your textbook what's in your textbook and what the party line is is that when you use historical amounts of data those kind of crisis periods are in there already so if I estimate the correlations including October 87 including March 2000 the bursting of the internet bubble August 1998 LTCM if I include all of those in my data then it's captured in there right it's not captured today in other words this point here for general dynamics doesn't reflect that there's a crisis today what it reflects is on average over a long period of time some of which includes crises that's what the expected return looks like and so I've already incorporated that into the parameters and if it's if it so happens that today is a bad day for general dynamics that shouldn't influence whether you buy it or not because what you should be thinking about is over the next 15 or 20 years how will my portfolio do so that's another difference in perspective from us versus Warren Buffett although actually not as much as you think Warren Buffett typically takes a long-term perspective right his investment in goldman sachs we all thought it was a great deal he's lost money in it since he put that money in a few weeks ago but is he worried about it I don't think so because he's invested in it for the next 15 to 20 years and I believe as he does that he got a great deal over that time period so if you care about what's going to happen tomorrow then all of the things that I'm telling you you should not take that as seriously because these kind of parameters aren't made for day-to-day forecasting right then you need to get hedge fund models and for that you need to take not only 433 but a number of other courses and courses that aren't even offered yet because frankly if I told you how to do it I'd have to kill you right he's highly proprietary but this is the approach where we're not trying to forecast markets and so I'm acknowledging that you're right that there are instabilities but as long as I'm using a long enough amount of data that it's in there it's captured in there in that set of parameters okay if it's not if you know of something that's not in here obviously you can't expect this to give you the right answer and you have to adjust your use of this technology accordingly other other questions yeah yeah even the news interest rates go up then you can expect that you know people in variable rates right will begin to quarterly came these are not just a correlation and then decide to start shorting them yeah based on that's right so yeah you I guess you know you can't have too many z axes based on those well you have to know how to build those other variables into this kind of a framework that's another way of putting it so in other words this framework is still useful in the case where you've got other variables but you just need to know how those other variables will impact the parameters for this analysis if you can figure that out then then you're hedge fund manager right that's what hedge funds try to do they try to figure out how all of these relations impact on how to construct a good portfolio all right that's right for example that would be one way of looking at it but there are other ways it doesn't have to be quantitative there are a lot of talented hedge fund managers that don't know how to use a calculator make a lot of money okay so don't don't think that it's all about quantitative analysis that's what we think that's that's how I think that's how MIT may think but that's not the only way to make money out there you know Warren Buffett I don't think knows how to calculate these covariances but he's done okay so so but but I think what we can do with this framework is to analyze how he does what he does and understand it in the context of this kind of a framework all right so let's now go to the next topic which is all right how do we choose a good portfolio given what we now know about this framework well we said before Brian pointed out that we never want to be on the lower part of that frontier so that's one thing we know right we don't want to be on the bottom part so we want to be on the efficient frontier now this entire bullet can be viewed as what's called the minimum variance boundary meaning for any given investment over here we have the absolute minimum variance that has the same level of expected return so when I go looks when I go horizontally I'm looking at the the smallest amount of risk I can take for that same level of expected rate of return that's what this bullet can be viewed as giving you and it turns out that because we like expected return and we don't like risk we want to be on the upper part of that bullet so this upper part is known as the efficient frontier so that's the one thing we can tell about portfolio theory it's that we want to be on the upper branch but we're on the upper branch depends upon our risk preferences okay so here's a concrete example suppose you can invest in any combination of General Motors IBM and Motorola what portfolio would you choose so these are the data that you would start with okay the means and the standard deviations and then of course the covariances you've got to have the covariances to be able to calculate that bullet so you can invest in any combination what would you choose let's calculate the expected rate of return and the variance of that portfolio and you want to ask the question what looks good to you okay well it turns out that when you calculate this kind of a bullet you find out something that's yet again amazing what you find is that the bullet actually is better than any of the three stocks that you start it with so look at where the three stocks are General Motors IBM and Motorola and the bullet look at where the bullet is the bullet is strictly to the northwest of these stocks right in other words you can do better than any one of these either by going here or by going here you for the same level of risk you can get a higher return for the same level of expected return you can get lower risk okay so the first point is that portfolio theory for multiple stocks is even more compelling because now with those two at least your the two stocks were on that minimum variance boundary now with three stocks and more it's possible that none of the stocks are going to be on the minimum variance boundary yeah you know this is backwards looking to you people Katrine companies and so on all make this graph and they all decide to strategize based on this graph yes the price of these stocks next day are going to be correlated in a different way than you expect because it's going to affect a very good point but it turns out your conclusion is false but that's a good question asked and I'm going to answer that not today but on Monday when we go over the question what happens when everybody does it I'm going to deal with that question head-on okay that's going to be a very important point and this is going to be one of the very few instances where when everybody does what I tell you they're going to do that's actually going to give you an equilibrium and everything will work out in just the right way that's the magic of the cab but you're right that Warren Buffett could not answer that criticism if you talk to Warren Buffett and said Warren if everybody did what you did then it would work and he would say you know what that's right that's why nobody can ever do what I do I'm just smarter than everybody else and so I'm sorry that's the way it goes here I'm not appealing to everybody being smarter than everybody else because frankly we're not what I'm appealing to what I will appeal to is if everybody does the right thing by the right thing I'm about to tell you what that right thing is everybody is on that upper branch and maximizes their risk reward trade off a very special thing happens all right I'm going to keep that as a surprise for Monday okay question a question yeah yeah well first of all it's not clear that warren buffett's number one all right he's kind of long track worker and he's got the biggest pie but in terms of actual track record he doesn't have the best track record there are people that you've never heard of that have a better track record than Warren Buffett for example there's a fella by the name of James Simon's who I may have mentioned earlier on as a hedge fund manager who also happens to be a first-rate mathematician who started up a hedge fund called Renaissance Technologies that is probably the single best track record of any manager in the history of investments and he does it completely quantitatively completely automated he hires something like 100 PhDs that work nothing nothing else but how to forecast you know the next minute as well as the next hour the next year it's an extraordinary track record so there are a number of folks like that let me not dwell on that but I'll come back to that you know in a few lectures when we talk about performance attribution okay so I'm going to leave you with one final thought since we're out of time there is going to be a very special role played by a portfolio called the tangency portfolio and I want you to think about how your risk reward trade-off would look when you mix your t-bill risk-free asset with arbitrary portfolios on this bullet think about what that trade-off would look like and ask yourself the question does there exist a special portfolio on that efficient frontier that everybody in this classroom is going to want to have alright I want you to identify them and I ask you that question on Monday and I expect an answer all right see you on Monday test attribution text