Ses 13: Risk and Return II & Portfolio Theory I

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 what I wanted to today is to continue where we left off last time in talking about the empirical properties of stocks and bonds I want you to develop an intuition for how to think about markets we've already done that over the course of the last last few lectures by looking at market prices and understanding how to price them but I'd like you to get some kind of a historical perspective now on specific asset classes because we're going to be relying on market prices to make inferences about other kinds of securities and other decisions you're going to make as I told you at the very beginning of the course we're going to rely on markets for information because it's the wisdom of crowds that really gets us the information we need in order to make good financial decisions so I want to begin that process of now giving you the intuition about the wisdom of crowds by looking at the historical performance of stocks and bonds and then we're going to talk about how to quantify risk more analytically and put it all together in the very the very basics of modern portfolio theory so I want to start by asking the question first of all what characterizes US equity returns you know how do we get our arms around the behavior of that asset class and the way I'm going to do that is to give you some performance statistics about the volatility about the average return about how predictable they are and also patterns of returns across different kinds of stocks so we're going to look at some empirical anomalies before actually turning to the analytical work of trying to figure out how to make sense of this from a more formal mathematical framework before I do that let me ask you to think about the following question which is if you were designed a market for stocks what properties would you want that market to have and I'm going to argue that there are a few properties that all of us I think can recognize as being good properties for stock prices so the first is that stock market prices are random and unpredictable now that might seem like a little counterintuitive and certainly I think you would acknowledge that over the last several weeks markets have been supremely unpredictable and that doesn't feel so good it doesn't seem like that's a good thing but in a minute I'm going to try to make that a little bit more clear by looking at the alternative of predictable okay or unpredictable which is predictable so let me come back to that point the second property that I think you'll agree is a reasonable one for us to expect is that prices should react quickly to new information it should adjust to new information really without any kind of delay and finally we'd like to see that investors shouldn't be able to earn abnormal returns after you adjust for risk so in other words once risk adjustment is taken into account there shouldn't be any additional return left over that's what we think of as a well-functioning market another way of putting it is the market is highly competitive it's hard to make money in those markets okay now they may not be markets that you would enjoy trading in but that's not the question the question is what would be a good market and efficient market so let me talk about predictability for a minute because I said that it seems a little counterintuitive that a good market is one that's not predictable so let's pretend that this is the stock market okay this is the sp500 that looks nice right nice regular curve anybody come up with a prediction for this how would you go about predicting the behavior of this kind of a stock market what's that cyclical what kind of curve would you fit to this yeah sine wave in fact that's how I generated this I used a sine wave and then I add a little noise okay now why might this not be a good model for a market if this were the stock market what would you do yeah exactly you know after after you know a few of these cycles you sort of get the idea right and if you're down here you're going to think well gee I think it's likely to go back up so I'm going to buy a ton over here and when you get right up there you'll say gee you know I think it's time for me to sell a ton and you don't have to go through too many bees before you get richer than your wildest dreams yeah exactly that that's exactly right so as soon as you start doing this as soon as you try to do this what happens to the pattern the pattern disappears exactly you see this is one of the reasons why finance is a lot more challenging than physics in physics if you try to drop a ball in a gravitational field it won't change its mind and say gee now I'm going to change the gravitational constant on you just because you're testing me but in financial markets the moment you try to take advantage of this pattern the pattern changes in fact the more you try to take advantage of it the more quickly the pattern changes in fact if you do this a lot if there are a lot of people trying to predict patterns then you know what you get you get no pattern you get randomness that's the idea behind an efficient market being random if it were not random then that means that there aren't enough people who are bothering to try to forecast the price and incorporate information into the price now I said two things that at first seem different but in fact they're opposite sides of the same coin when you are forecasting market prices you know what you're doing you're actually helping markets become more efficient by incorporating information into that price how do you do that well if for example you think that having a presidential election will cause volatility to decline then if you know that there's a presidential election coming up you will start trading in a way that will ultimately be betting on volatility declining as you start that trading you force that volatility index to go down so the fact that you've got information and you think you can forecast prices when you use it when you use the information what does it mean to use the information when you buy or sell securities on the basis of that information then the price of the security ultimately reflects the information right so an efficient market is one where you don't have this you don't have a very strong predictability if it is strongly predictable then most likely either the market is rigged or there aren't enough people that are trading in order to make prices fully reflective of all available information now this is the way markets really look these are random walks with with drift drift meaning there's a positive trend or in some cases a negative trend but otherwise it's random around that trend so you can't really easily forecast it and you can see that prices go up they go down there are long periods where they go up but they're also for other stocks long periods where they go down and you don't know what's going to happen next this is a sign of a very efficient market a while ago there was a academic study that was done to try to test for efficiency and one of the tests was that if the underlying price series was not very volatile that was considered an efficient market but it turned out that was roundly criticized because of the point that just because a market is not volatile it doesn't mean that it's working well and an example was at the time this is like 20 or 30 years ago the Chinese stock market the Shanghai Stock Exchange it was a relatively young market and at that time there were only two stocks that traded on it was the national railroad company in the Bank of China and at that time which is again about 15 or 20 years ago it was considered unpatriotic to sell the security if you had bought it so you could buy it but you weren't allowed to sell it and so the price went way up and up and up and and that's not an example of an efficient market it was not at all volatile but as a result there was no real information reflected in that price yeah well it's not volatility per se but rather the combination of the predictability per unit volatility that's really what you want to focus on we're going to come back to that when we talk about portfolio theory and look at this trade-off between risk and expected return but no I wouldn't say that the indian market is inefficient it's undergoing some pretty significant changes as is the US and as is the world but that's because the global economy is contracting as we know because of this financial crisis so I wouldn't characterize it as inefficient at this point but you know that remains to be seen yeah there isn't any hard and fast rule no but if you take a look at the trade-off of risk to reward in other words that ratio of expected return to volatility you can come up with rules of thumb that will give you a sense of whether or not a market is efficient or inefficient so we're going to come back to that in fact wouldn't get it in just a minute let me let me actually turn to some data now and then we can see exactly what those trade-offs look like okay now what's going to be interesting about this part of the talk is that when I tell you about these numbers these numbers are based upon data from I think it was at 1946 to 2001 in fact a lot of the data that has been collected over the last year is very very different from this so you know will will be interesting to sort of compare the two all right so there are four empirical facts that I want you to take with you about the US stock market the first is that interest rates in general have been slightly positive on average but not by much in other words the real interest rate that the nominal interest rate minus the inflation rate is been pretty low over the course of history so the first fact is that real rates have been slightly positive so you can see for example the average rate of earn for the one-year t-bill is about 38 basis points on a monthly basis this is monthly I have an annualized that what inflation over that same period is about 32 basis points so when you subtract the two you're going to get 6 basis points on a monthly basis as the real rate of interest okay on the other hand if you take a look at the stock market which is represented by VW stock index VW doesn't mean Volkswagen VW stands for the value weighted index it's an index of all the stocks on the NYS AMX and Nasdaq weighted according to their outstanding market capitalization and you can see that the valuated stock market over this period is about 1% per month the equal weighted stock market II W is a little bit higher 1.18 and motorola over this period had a expected rate of return of about 1.6 6% per month so the return has been higher for these indexes and if you want to get a sense of why that might be take a look at the next column which is the standard deviation this remember is a measure of the riskiness of the security right it's a measure of the fluctuations and if you take a look at the equal weighted evaluated the volatility is both they're both a lot higher than for t-bills so this is one of the reasons we we got the idea in finance that there's a risk reward trade-off the more risky the higher the expected rate of return and if you look at motorola the riskiness of motorola is much larger than that of any of the stock indexes instead of a 5 percent monthly standard deviation you're looking at double or 10% the monthly standard deviation but look at the rate of return the rate of return is commensurately higher yeah I mean oh because it fluctuates from month-to-month so the idea is if you're buying a t-bill and you're holding it it still has price fluctuation right yeah okay the other thing that I want you to see is something along the lines of the minimum and the maximum this is another way of representing the riskiness of the security so t-bills are bounded between 0.03 and 1.3 for in terms of their return very narrow band Treasury notes which are 10-year instruments obviously are going to be swinging around much more much more than a one-year t-bill right so the longer the the longer the maturity the longer the duration the riskier is the instrument right but if you take a look at the evaluated return and the equal weighted return and then Motorola you can see progressively more and more risk involved in these kinds of securities okay now if you look at their compound growth rates you get what you pay for right in the sense that you're looking at t-bills down here so $1 invested in one of these guys will give you maybe $10 at the end of 2001 but you're looking at a much much larger return for either evaluated or equal weighted indexes right more risk more expected return that's the message that you get from looking at the basic data here now just to give you a sense of where interest rates have been I think we discuss this when we did fixed income interest rates have really been all over the map there was a point in our history not that long ago where the short-term interest rate the one-year t-bill was you know something like 16 to 17 percent per year that's one year t-bill right it's a astonishing but that was a period where there was a large amount of inflation in the United States on the other hand if you take a look at the more recent period interest rates have been extremely low and that's part of the reason we're in a credit crisis is because credit is very cheap so you can get yourself into a lot of trouble when it's relatively easy for you to borrow and it doesn't cost you that much in terms of the payments now let me show you what the total returns look like for these different asset classes by total returns I mean if you bought one of these instruments and you held it a month at a time and you computed the return for holding that instrument so for a bond it includes whatever coupons get paid plus the price fluctuations it's the total for stocks it'll include the dividends that got paid as well as the price fluctuations and I'm going to do this on the exact same scale from minus 25 percent to 25 percent okay so these are monthly returns now that I'm plotting from 1946 to 2001 and so you can see that that their total return for the US 10-year bond actually has different periods where in some cases it's not very risky but in other cases it bounces around a great deal when there's a fair amount of interest rate uncertainty you get a lot of volatility but when markets are not moving around that much on the interest rate side you get periods that are relatively calm okay yes that interest rates our yeah that's right that's right correct yeah these are yield to maturity okay these are these are total returns though these are what you get as an investor this is what you get if you hold a particular security to maturity on basically on a given on a given day you will get these spot rates okay so a question yeah well and also inflation axial in that or they felt that you know that plus whatever interest rate expectations was going to be was going to be what you're going to get over a 10-year period yep that's right apparently I mean and there wasn't time in fact when the stock market did yield that for quite a bit of time right absolutely on the other hand let me turn it around and ask you now that interest rates are at I don't know the the ten year I think is that the thirty year is at 417 this morning do you think for the next 40 years or 30 years that Treasury bills are only going to return 4% a year is that realistic I mean it's not so easy to say is it when you're in the midst of it it's not so easy to say based upon the historical evidence it seems crazy to think that we could possibly be in such a low interest rate environment over the next 30 years especially given that you know we're printing money like it's going out of style now and we're going to be doing that over the next couple of years we got to because somebody's got to pay for all of these uh you know rescue packages and and so we basically have to you know engage in some kind of inflationary monetary and fiscal policy but it's still saying 4.17 as this morning so the market does the best it can given the data but it's hard to forecast and we just said at the beginning of this class that it better be hard to forecast because if it's not hard to forecast then something's wrong right then it means that it's not reflecting all available information yeah the fitted slide of the stocks going went out I'm wondering how much of that is due to the states yeah that's definitely a factor so I'm not trying to explain the numbers and I'm not trying to justify them you're absolutely right this is a very special country in a very special time so you can either thank your good fortune that you're here or you can argue that well it's not going to persist and you know time to move to you know wherever but I don't know I don't know which that is right yeah but but it is very unusual if you look at other countries there are other countries that are having difficulties during this time period but there are other countries that are growing you know even faster right if you look at China over the last ten years the growth rate of the Chinese economy is double to triple what the US economy is that it was a smaller economy but still over an extended period of time it's got tremendous growth rate okay so that really is the challenge is to try to understand what's going on in the context of you know where where we're living and how we're living so let me go through and show you some more numbers and then we can talk about some of the interpretations so this is the total return for the u.s.

Ten year you get a sense of the scale right – 25 – 25 now this is the return of the US stock market during that same period using that same scale right more risky so you could see the difference in fluctuations and if this weren't exciting enough for you this is Motorola and there are many stocks like Motorola so when you invest in an individual stock you're getting not just the fluctuations of the economy but you're getting the fluctuations that affect that specific company so you should expect if you're taking on more risk that you're going to be getting a reward for these kinds of incredible bouncing around and in fact you do the average return of Motorola is quite a bit higher than that of the of the market now let me show you a little bit about predictability we talked about a market that's random as one that is is a good or efficient market this plots the return today versus tomorrow or yesterday versus today if you want to think about that pairs of returns for the aggregate stock market and this is done on a daily basis now if you look at it on a monthly basis for the S&P 500 from 1926 to 1997 you get something that also looks kind of random right so not much bricht ability here not much breathability there but suppose we were to graph the return of General Motors against the S&P 500 well now all of a sudden it looks like there's a little bit of a pattern right not totally random sort of looks like there is a kind of a line that goes through that that scatter of points right there's a relationship on a given day between General Motors and the broad market index but over the course of two days or two months there's very little predictability okay now let me talk about volatility oh yeah question that's right GM is one of the stocks in the S&P 500 in other words it's not any one stock that gives this random scatter of points in other words this random scatter points is really all 500 stocks put into a portfolio but remember we're asking a different question this is a question about the relationship between the SMB environment last month versus this month there's no real relationship this on the other hand is a question between a question about the relationship between S&P this month and GM this month the same month okay now you're right that SP has GM as one of the components but it's only one of them if it weren't in there you would actually still see this kind of a relationship yep yeah I there's no particular significance it just means that you know that one year rate happens to be identical to the ten-year rate so people are just assuming that that rate is going to continue over a period of time so there's no particular economic significance to when they cross these are all annualized remember right so you're asking the question over a 10-year period what is the average interest rate you're going to get paid by the US government versus over a one-year period what is the interest rate you're going to get paid by the US government that's all okay now let me talk about volatility these are monthly estimates of US stock market daily volatility from 1926 to 1997 so every month I've got 20 days or 21 days I'm going to take the month the daily returns and calculate the standard deviation for those daily returns and I plot it and it looks like this now these are monthly so if you want to annualize it you have to multiply it by the square root of 12 to get the annualized volatility but the point of this is that there are periods of time where the market is dream Li volatile and there are periods of time where it's relatively quiet and if I would have extended this to the most recent period you would see that within the last several weeks we had a spike that was about as big as this spike over here anybody know what this spike is you can sort of guess yeah October 1987 right that was the big stock market crash that occurred during that that month volatility shot way up during that month lots of risk and right now we're in a situation where there is lots of risk as well okay now I'm going to just conclude my overview with a few interesting anomalies that I want to tease you with to get you to think a little bit more carefully about stock markets what I'm going to show you are just a bunch of factoids factoids meaning that they are empirical facts in the data but if you change some of the assumptions or the sample that you use these facts could change all right so these are not Universal constants that for some theoretical reason has to be true this is just properties of the data what I've done here is to take all the stocks in the NYC MX and Nasdaq from 1964 to 2004 and on a monthly basis I'm going to rank them by market capitalization break them up into ten equal numbers of securities and then average the returns in those deciles and then compute it over time and then average those decile returns okay that's a little little complicated but pretty straightforward let me tell you for example what this means here this particular bin over here is the bin of all stocks that have the largest market capitalizations I know among all of the stocks in my sample I've divided up into ten equal groups this the largest tenth and I'm going to compute the returns monthly for that bin and then average that across my entire sample and I get an annual return of close to ten percent this is annualized now this is not monthly annualized okay if I now compute that same rate of return for the smallest stocks the stocks that are the smallest part of my sample that's down here I get a return of something like fifteen percent per year fifteen percent versus ten percent that's a huge difference that's like a six six percentage point difference right no it's more like nine percent versus fifteen plus something five percentage points five hundred basis points a year that's the gap between small stocks and large stocks okay this is known as the size effect or the size premium small stocks seem to do better than large stocks now there are lots of stories you can tell about why I'm not going to tell those stories but I just want you to see the data now you want to know what's weird about this what's weird about this picture is that I'm going to change the sample ever-so-slightly what I'm going to do is I'm going to compute this graph only for the month of January and then for all the other months outside of January okay and I'll show you what happens the yellow bars those are the January returns and the blue bars are all of the non January returns so I don't really think there's that much of a size effect once you delete the January's you get a little bit of a difference between the biggest and the smallest but the difference that we're talking about now is very small but look at the January effect that's big and it turns out that this seems to be a phenomenon that has become a little bit less pronounced recently but for many years it was quite strong and reliable and people actually traded on this particular pattern buying stocks in January holding them in December and holding them till January or doing a spread where you basically try to go long small stocks in December and short stocks large stocks in December and hold that spread and let it widen so this is a really puzzling anomaly again you can come up with explanations for this I'm not going to tell you what they are you can think about them and possibly even trade on them over the next month or so so all right now the second anomaly that I want you to be aware of and this depending on who you speak to may not be considered an anomaly for example Warren Buffett would call this genius and in fact this is the value premium what this suggests is that there are certain stocks that for whatever reason are just simply undervalued systematically or alternatively other stocks that are systematically overvalued the value premium is where you take the same universe of stocks that I showed you before but instead of sorting according to market capitalization you sort it according to another characteristic and the characteristic is the price equity ratio or the river that is what you usually hear about book to market right book value divided by market value now you all know what that difference is by now right book value is the value of the company as it was initially formed and as it accrues either cash or profitability but based upon its accounting Book value whereas the market value is what the market thinks the company is worth and in many cases with technology stocks and other growth stocks the price gets way way way ahead of the book value of the company's assets and so those are situations where you've got a very large price to book ratio if you have a very large price to book ratio that means that you're going to be on the right-hand of this spectrum and that says according to this chart that the expected rate of return is generally pretty low on the other hand low book to market or high book the price oh sorry the other way around high book to market low price to book is on the left hand side these are what Warren Buffett and Graham and Dodd would call value stocks right these are stocks where you've got lots of good book value but somehow the market doesn't really appreciate it and so the price is low relative to the book value in other words the price to book ratio is low and look at the returns there that the the value premium is the difference between the high price to book and the low price to book and you're looking at a premium of about six to seven hundred basis points on an annualized basis that's a big difference because in both of these bins it turns out the risks are actually roughly comparable so it's not as if you know the stocks on the Left are way more risky than the stocks on the right that actually is true of the market capitalization effect small stocks actually have higher volatility than large stocks but that's not true a value in growth so that's another puzzle or depending on who you talk to this is a way to invest okay momentum this is something that academics discovered maybe 10 or 15 years ago which is also really anomalous by momentum we mean simply last month's return or last year's return if that's positive does it tend to persist over the next 12 months so the stocks with low momentum on the left-hand side seem to do worse than the stocks that have high momentum so the momentum effect seems to be really strong and again look at that difference that difference is something on the order of a 15% spread if not more it's a very very big spread so this might lead you to try to construct the trading strategy based upon this and there are a bunch of other anomalies that have been reported in the academic literature in fact for a while certain academic journals were accused of never meeting an anomaly that they didn't love because they just kept publishing one after another and in a way you know it you have to be a bit skeptical about this right because there are so many different ways of looking at stocks so many characteristics and you know that in a sample of a hundred a hundred random variables 5% of them are going to be statistically significant right so even if none of them are you know in in terms of being significantly different from zero so you got to take these anomalies with a grain of salt but what I've presented to you are the ones that seem to be the most persistent the ones that people spin stories about the ones that people construct mutual funds around and so you'll have to think a little bit about whether or not you believe any of these anomalies but I wanted to make you aware that they're there and you know if you take 15 for 33 investments you'd actually end up spending a fair bit of time digging through each one of these to see whether or not there's something in there that you could use for investment purposes the the last thing I'm going to mention with this introductory lecture is mutual funds these anomalies uh were obviously very very exciting from the perspective of active portfolio management because once you identify one of these anomalies you could argue I'm going to take advantage of it of course then the argument that was raised earlier comes into play if you're gonna take advantage of it isn't that going to disappear and the answer is in general yes it will but it may take a while and along the way you'll do quite well so the question was asked well if that's the case if they're all these anomalies and if you could take advantage of them well then mutual fund managers ought to be able to outperform simple buy and hold strategies right because they can take advantage of these anomalies if you do a histogram of mutual fund returns that are in excess of their risks so you make some kind of a simple risk adjustment and you look at the mutual funds additional value added above and beyond those risk adjustments those excess returns are given by this histogram from data from 1972 to 1991 the histogram of excess returns has basically this kind of a distribution you've got some positives you've got some negatives you've got more negatives than positives and on average it's actually less than zero mutual funds net of fees are actually losing money for you on average that was the conclusion by these academics as of a few years ago yeah yeah yeah example you you might be able to get well so first of all that's not necessarily the objective of every mutual fund right there are mutual funds that are not trying to give you lower volatility but rather they're trying to give you access to broader investment vehicles and and instruments so the original index fund that was set up by Wells Fargo in the 1970s the purpose was to allow an investor to get access to a hundred securities without having to actually go out and buy one hundred securities right by diversifying right so but it doesn't lower the volatility except through diversification all right so you're right diversification will lower it versus buying Motorola but the question is how does this do versus buying 100 stocks or rather buying a mutual fund like Vanguard where the Vanguard 500 index trust where you're not trying to outperform the market you're trying to match the market okay and the argument is that the mutual funds that have been trying to beat the market on average they don't they don't they don't beat the market some of them do some of them don't but as a group they don't add any extra value that's the argument that was made now that's not to say that there aren't good mutual funds and there aren't bad mutual funds there may be so you know somewhere in here is Peter Lynch's Magellan fund terrific fun very talented portfolio manager but on the other hand if you can't tell in advance who is going to be the next Peter Lynch and who's going to be the next I don't know who I won't cast any aspersions but if you can't tell in advance who's good and who's bad then you're essentially throwing a dart at this histogram right you may be lucky and you'll get on the right side or you may be unlucky you get the left side but on average you should do better by putting your money in a passive index fund now that's the argument of Vanguard and all of the passive investment vehicles I'm not going to take a stand on that because we're going to come back and talk a bit about that at the end of the course and then as part of investments you're going to really look into that I just want you to get a feeling for the data that's out there and the data that's out there says it's very hard to tell whether or not mutual funds as an aggregate are adding any value by the way you realize that there are actually more mutual funds out there then there are stocks right you know that yeah there are about ten thousand mutual funds they're about eight thousand stocks out there including you know the penny stocks and pink sheet stocks there are probably only four or five thousand stocks that you would actually ever invest in as a as a retail investor yourself and so the number of mutual funds far exceeds the number of stocks the way that mutual fund managers justify that is by saying look you know there are 31 flavors at baskin-robbins and so we want to provide investors with lots of different possibilities right not everybody wants the S&P 500 some people want the SP 100 some people want the S&P 250 some people want the S&P 385 and so I'm going to construct the fund for every clientele that's out there that's that's a legitimate argument as long as investors understand that when you buy into a mutual fund you're buying you know something that may cost you more than if you try to do this on your own okay so the key points that I want you to take away is that the average return on US stocks from 1926 to 2004 was 11.2% now that's considered the good old days so no more the average risk premium was about 8% again good old days that's probably not going to happen for a while stocks are quite risky a standard deviation of returns for the market is about 16 percent annually ha that that isn't risky anymore right that again good old days the the market today using the implied volatility of the VIX index right the implied volatility of SP at the money options was about 49 percent so the annual forward-looking SP 500 stock market volatility right now is about 49 percent which by the way is down from 80 percent a couple weeks ago so as I predicted volatility was going to decline once the election was clear that eliminated a piece of uncertainty but there's still a remaining piece which is what's going to happen to our economy that's why the volatility is at 49 percent versus a historical average of 16 to 20 percent stocks on an individual basis are clearly much more risky than as a group so you're absolutely right ramie that when you put it into a portfolio you reduce the risk and so the S&P 500 is a lot less risky than Motorola also stocks tend to move together over time over time from one day to the next there's very little relationship but on a given day stocks tend to move together in groups General Motors seemingly to be correlated with the S&P as well as other stocks and obviously market volatility changes over time and financial ratios that can be used to create these different kinds of bins for sorting stocks and constructing these anomalies they actually do seem to have some kind of predictive value why we don't know in many cases but the anomalies are there and so that's something to be aware of okay other other questions all right now that you have a feel for the data I want to take a step back and ask the question how do we make use of this information in a way that can help the typical investor okay and also help the individual trying to decide on a corporate financial decision how do we use this these kinds of empirical insights in our theory so to do that I'm going to turn now to lectures 13 and 14 and focus on how to measure risk and return in a more systematic way and then incorporate that into portfolios so we're going to talk a bit about motivating the idea behind portfolio analysis and then use some of the theoretical concepts that we introduced last time like mean variance and covariances and use it to piece together a good portfolio so the motivation for what we're trying to do now is to figure out how to combine securities into a group that will have attractive properties if you're an investor or a corporate financial manager this is a decision that you've got to do in almost every day for example all of you have already made that decision today whether you know it or not because if you didn't do anything between yesterday and today to rebalance your portfolio you've made an affirmative decision to let the bet ride right let it ride you're going to be taking another roll of the dice at 4 o'clock today and let's hope it turns out well for you but you've made a decision every day you don't do anything with your portfolio you are making a decision so what we want to do is to see if we can think about making that decision a little bit more systematically to do that I want to define what I mean by a portfolio so the definition in very simple terms is just a specific weighting or combination of securities such that the weights add up to 1 right it's just a way to divide up the pie if you've got a certain amount of wealth you're going to allocate it among different securities into a portfolio a portfolio is defined as a set of weights of those securities where the weights add up to 1 now this is sort of the framework that we use in terms of the notation so Omega the Greek letter Omega when we write it as a vector it denotes the set of weights for your portfolio so if you get n securities then this vector Omega which is equal to Omega 1 comma Omega 2 Omega n that is a portfolio and if you want to be clear about how to define it it's simply the number of shares of security I multiplied by that price divided by the sum of all of the values of your securities in your entire collection right any questions about this very basic stuff but it's important to get it right up front okay now by the way the number of shares and I I'm going to let that be a real number meaning it could be 0 it could be 5 it could be 500 it could be – 200 – 200 means that you have short sold 200 shares of that security so these weights they all sum to 1 but they don't have to be all non-negative some of these weights could be 0 some of these weights could be negative okay if some of these weights are negative then what do you know about some of the other weights well yes they have to be positive because ever after one but you know something more about the weights greater than 1 right because in order for it to add up to 1 if some of these things are less than 0 then some of these other things are greater than 1 what does it mean for a security to be greater than 1 a weight to be greater than 1 that makes sense what's interpretation yeah Lucas it means that you're leveraged that's right leverage meaning that you're buying more than you have money where you're getting that money from your your your basically getting a loan right but who's loaning you the money Annie that's right so nobody loaned you money but somebody loaned you something they loaned you a stock so this simple little framework already has given us one interesting insight which is that when we short sell a stock and buy another one we're actually getting a loan from somebody who's lending the stock to us that we've sold we've taken that cash we're putting it into another stock so we're getting a loan of one security and turning around and using those funds to buy another security okay that's a new transaction as far as we're concerned but it's one that's going to be very important in how we think about portfolio construction all right if you didn't understand that go back and take a look at these notes and try to work out a numerical example for yourself and if you still don't understand it ask again next time or ask the TA during recitation okay it's a very important point all right now there's a case that I'm not going to talk about in this class where the weights can actually sum to zero I don't want to talk about that because that's a much more complex situation where basically you have a portfolio with no money down this is sort of like the arbitrage portfolios that I described to you in earlier settings we're not going to analyze that in the context of stocks but there are a very large number of hedge fund strategies that are based upon just these portfolios and so this will be a very important concept that you'll cover in 433 but we're not going to talk now I just want to make you aware of that that you can have the weights summing not to one but but actually to zero okay now the assumption that I'm making implicitly is that the portfolio weights are summarizing everything there is to know about your investment okay so once you know the portfolio weights and you know the stocks then you know what your portfolio is about okay so as an example you have an investment account with a hundred thousand dollars and you've got three stocks in there two hundred shares of a thousand shares of B seven hundred fifty shares of C so that your portfolio is summarized by the weights 1060 and thirty percent okay so from now on we're not going to worry about prices and shares anymore we're going to focus just on portfolio weights and the returns of your securities multiplied by those weights all right it's a simplification Yammy in yeah sure so you're already asking a question that's a quite a bit more advanced than what we're going to cover what's a 130 30 portfolio can you explain that or have you just heard that in the news right that's right exactly so let me describe a product that's out there that's been developed just over the last few years it's called a one thirty thirty portfolio and what one thirty thirty stands for is 130 percent long and 30 percent short okay now you can have a 120 2400 or 188 e-portfolio but the idea is that you have weights that add up to a hundred percent but the long positions are no greater than 130 percent and the short positions are no less than minus thirty percent now the reason that this is an interesting product is that you have to you a little bit of history this is getting a little out of the our area but I'll give you a preview of of investments 433 typically when institutional investors like pension funds or mutual funds when they invest they're not allowed to short sell this is nothing to do with the SEC it's nothing to do with law it has to do with the particular entities that are investing because short selling was viewed way back as being a very risky endeavor because you could lose everything and more there's unlimited amounts that you could lose right because you're a short selling a stock and go way up and you could lose tremendous amounts so mutual funds were originally not allowed to short sell at all so for a mutual fund the portfolio weights were restricted to be non-negative and certain pension funds were not allowed to short sell so if you were a pension plan for a state university and you gave your money to investment manager XYZ that investment manager would be required not to short sell for your portfolio okay it was discovered over the last several years that this kind of constraint artificially dampened the return of a portfolio not surprisingly because when the market goes down if you're long only you're going down with it all right but if you have a short position then at least the shorts would be able to buffer some of the losses on the long side so institutions have started getting more and more sophisticated thanks to hedge funds pushing them into this area because of course hedge funds can do anything right they can short they can long they can go sideways whatever so hedge funds led the pack by saying you know we're going to actually short sell some of your long only portfolio to help you get a little bit of extra return on the downside and so pretty soon institutional investors said well you know what I actually like the idea of you short selling a little bit but I don't want you to do it too much because I don't really know what the risks are I'm new to this I don't want to get the risks out of control so I'm going to limit how much you can short sell the limit of how much you can short sell imposes a limit on how much more long you can go than 100 percent just like we said if something's negative then some of the weights have got to be greater than one right well they got to add up to greater than one so when you have that situation where you have a limit on the total negative positions you can have that limit puts a symmetric upper bound beyond the number one of what you can go long and so for reasons that are probably a little bit too far afield to get into here 130 30 seems to be a bit of a sweet spot for managers out there so they say we will limit our short positions to no more than 30 percent of the capital so you give us a hundred million dollars to manage we will take no more than 30 million dollars of shorts and therefore we will take no more than 130 million dollars of lungs but when you add them up you still get back to a hundred percent so that's one 30 30 it's very popular and it's something that is likely to grow particularly given this current market environment because one 30 30 have has done better than the SP not surprisingly because of that 30 percent short position okay so here's the example of your own portfolio and how you get those weights that's pretty straightforward here's another example where you've now got some short positions here the short positions are not from short-selling but the short positions are in riskless bonds in other words now instead of shorting a stock you're shorting a bond or selling a bond or borrowing money from a broker and getting leverage so this is leverage where the security that you're levering up is using bonds and your levering up the equity positions so your portfolio weights look like this your equity positions when you add up the equity positions those portfolio weights you get 200% but your riskless bonds you shorted 50,000 dollars you're borrowing $50,000 from the broker so you've got – 100% when you add those two you get back 100% or in this case $50,000 so you start out with $50,000 cash and then you buy a hundred thousand dollars worth of stocks by borrowing an extra 50 from your broker okay yeah but I still have different this and manager yes so when you're actually starting portfolios you have this is pretty one because I I would just be me well the reason that I have them here is I want to show you exactly how you would compute the portfolio weights of your entire portfolio so basically what this is is your equity portfolio but in addition your fixed income positions added in right I mean your your portfolio could be anything it could be stocks bonds options currencies real estate so I'm just including all of this in the portfolio itself yes yes absolutely and you and you're doing that stocks a B and C have different risks as does the bond and so you're mixing and matching and putting them together into what hopefully will be an attractive portfolio okay now we mentioned before that when you get a mortgage that's leveraged to so this is an example of a situation where you buy a home for $500,000 but you only have a $100,000 downpayment right your equity in the home is only a hundred thousand dollars the bank has loaned you four hundred thousand dollars your leverage ratio is 5 to one so if you were to look at your portfolio weights it would be five hundred percent house – four hundred percent bank right or bonds or mortgage right that's very high leverage and in that case when you're leveraged five to one if the house price goes down by something like I don't know 2% you've lost 10% of your the value of your home okay so of the value of your equity rather if the house price declines by 15% you know that's really bad news okay so leverage is a two-edged sword when things are working well it gives you a boost when things are not working well it can hurt you on the downside as well okay here's another example where you've got a zero net investment strategy you can work that out for yourself this is a little bit trickier because the portfolio weights now add up to zero you've got to think a little bit about what it means to have portfolio weights at all so I'll leave that for you to look at that's something that as I said we won't cover in this course in great detail okay so now motivation what we're trying to do now that you know the basic language of portfolio weights and how to manipulate them to some degree I want to ask the question why why bother with the portfolio we've already gotten a couple of comments about why you want a portfolio you want to have stocks with different kinds of risks so you have diversification but there's another approach and the other approach is a championed by Warren Buffett right Warren Buffett has criticized this idea of diversification not putting all your eggs in one basket by saying you should put all your eggs in one basket and then simply just watch that basket really carefully isn't that better well that sounds good but you know what if it's the case that you don't really know how to pick the right basket and and therefore whatever basket you're watching may not be particularly attractive because you pick the wrong basket so that's really the idea behind portfolio theory it's that we're not all of us are Warren Buffett's not all of us want to become Warren Buffett's we want to have a relatively systematic approach to making a good investment decision we don't want to try to beat the market we want to figure out whether we can come up with a responsible and attractive way of investing that has some kind of economic logic to it so the point is that we don't know which stock is best and so we don't want to pick just one stock like Motorola because there are periods where Motorola looks fantastic and periods where Motorola looks horrible so we want to be able to pick a portfolio that's got good characteristics so diversification is one way to do that it's to basically spread your risk across a number of securities and portfolios can do that but at the same time they can also create focused bets so it's not just the case that you have to buy every possible stock there is out there in order to diversify for example you may have information or you may have conviction that information technology is going to do really well over the next couple of years because somebody's going to figure out how to process all of these bad loans and problem banks and IT is going to ultimately be the solution well if that's the case you can make a better than IT without having to make a bet on any one firm or one stock the way you do that is to form a portfolio of stocks that are all in the IT sector and so you get diversification but at the same time you're able to make a bet in an area where you think you have particular expertise okay and finally portfolios can customize and manage your own personal risk reward trade-offs so for some of you you want a lot of risk you want it concentrated in a small number of industries and you want to do it with relatively small price stocks you can do that somebody else might have a very different set of preferences portfolios allow you to tailor the risk reward trade-offs to your particular preferences okay so now we have a motivation for portfolios then the next question is that sounds great now tell me how do i construct one of these good portfolios and in order to answer that question I've got to tell you what good means or you've got to tell me what good means so typically what we say about a good portfolio is it's a portfolio that has high mean and low risk right that's what good means there are two characteristics that we tend to focus on for purely statistical reasons it's because those are easy to compute and they are the first two statistics that one would look at when you're looking at an investment the mean and the standard deviation so you might think that naturally it would make sense to pick a portfolio that's got high mean and low standard deviation okay that's an assumption in other words we're assuming that we're going to measure risk by standard deviation and we're assuming that we're measuring return by the actual expected rate of return for certain investors those are not appropriate for example there are some investors that have that are really keen on social socially aware investing so they don't want to invest in companies that pollute the environment they don't want to invest in companies that engage in you know non-union workers or they don't want to invest in companies that happen to be exploiting labor in unregulated markets those are examples of non-pecuniary characteristics that figure into this choice of stocks we're going to abstract from that so for our purposes the characteristics that we're going to look at for a good portfolio is does it have a high return does it have low risk and the way we're going to measure risk is in terms of the volatility or standard deviation now here again there's lots of ways of measuring risk we can measure by the upper quartile the 5% loss or spread but in fact what we're going to use is this standard deviation measure for symmetric distributions like the normal it turns out that that's not a bad measure but some people have argued that by looking at spread you're confusing the upside with the downside right nobody has any problem with upside risk or upside you know kind of distribution I haven't run across anybody that's a G you know this year I'm really making too much money and that's just not a good thing if you mean anybody like that you know introduce me I'll I'll help them out with their problem but the point is that for a symmetric distribution it doesn't matter and you know in more advanced approaches to investments people have used one-sided measures but we're not going to do that in this course so we're going to focus on variance or standard deviation as the measure of risk okay and the assumptions that I'm going to make for the remainder of the course is that all investors like higher mean and all investors dislike higher variance okay now that's a really reasonable assumption but you could challenge it if you wanted to argue that investors care about other things so just be aware that I'm making an approximation and the approximation is exactly that that mean and variants are the only things that our prototypical investor is going to care about okay so now we actually are pretty close to being able to come up with an answer to the question what's a good portfolio and how do we pick stocks one of the things we're going to answer over the course of the next few slides is how much does a stock contribute to the risk and the expected return of a portfolio so if you're thinking about investing in a new stock it's like you know inviting somebody you know into your you know into your club you want to ask well what are you going to contribute to my club what are you going to contribute to the portfolio what will you add to what I already have are you going to help me with my expected return are you going to help me lower my risk and if the answer is no to both of them then I don't want you you're not going to do anything for me why should you be in my portfolio so that's the kind of argument we're going to make to be able to construct a good portfolio so let's get a little bit more specific about that here's a graph and you're going to see this a lot this graph is going to be one that we use for all of portfolio analysis it's where we plot on two-dimensional space the average return of the stock as well as its risk where risk is now being measured by standard deviation okay so I've got five assets plotted here Merck is one and General Motors is another one Motorola is a third McDonald's a fourth and I've got t-bills down there on the lower left this gives you a sense of the different trade-offs there are right clear General Motors is lower risk that Motorola but it's also lower return and McDonald's is definitely going to be higher risk than Merck but notice that McDonald's is also lower return than Merck so at least in this setting nobody in their right mind by that I mean no rational investor would ever want to hold McDonald's over Merck right by our assumption we ever we are assuming we're assuming that investors like expected return and they don't like risk now question Yeah right exactly that's right so I was waiting for somebody to say that Warren Buffett would say that's a stupidest thing you ever heard because all you're doing is plotting history on this chart and this tells you nothing about what might happen over the next 12-24 months you know it could be that you know health care is going to just become a real problem pharmaceutical companies are going to get battered because of the Democratic administration that's going to force them to reduce the prices and so over the next six to twelve months the only thing that people will be able to do is to you know go to their neighborhood McDonald's and just enjoy a nice hamburger and you know complain about what's been going on with the pharmaceutical industry in that case McDonald's is a great bet and Merck is a terrible investment we're going to extract from all of that we are not in the business of forecasting stock returns why because I just showed you in the previous set of slides that it's hard to forecast in fact you told me that in an efficient market it's actually hard to tell what's going to happen with these stocks and if you could tell then people are going to start using that information and then the information is worthless because it'll have already been taken into account so you see you know this is a very important philosophical difference between warren buffett and academics warren buffett believes that there are static miss pricings out there that can be found and taken advantage of academic finance as of the 1960s and 70s when this theory was developed started from the point that that you just came to very quickly which is that you think there are no patterns in the data if there were someone would have already done it which by the way Warren Buffett would answer by saying you know what that sounds like the joke about the economists walking down the road sees $100 bill and walks right by it and when somebody says why didn't you pick up the hundred-dollar bill they said well if it were real someone would have already picked it up right I mean that's the argument that we made together we made that argument that if there was a pattern somebody would take advantage of it and then the pattern can't be there so you know and then again Warren Buffett would say that's a stupidest thing I've ever heard because in fact I've done it I saw the patterns I took advantage of it and I have a bit more money than you do so there who do you believe well you know it's kind of hard to argue with a 40-something billionaire right I think that's as well forty billion but that's not the perspective of this analysis like expected return was perfect information still be rational Bob McDonald versus mercy long making sure McDonald's yeah that's right exactly so now I'm going to talk about that for a little while but you're right so if you could short then what you'd want to do is exactly what you said you want to basically long the low-risk high-yield asset short the high-risk low-yield asset make that spread and make it as riskless as you can by including other securities yes and then it should that's right so the argument that economists would make is that this picture is the equilibrium of where these returns should be given what the market determines their fair rates of return are relative to their risks so that's a very again a very big philosophical difference economists would say all of these securities are exactly where they should be and they may change over time but at every point in time they are where they should be supply equals demand markets clear everything is equilibrium and our decision is simply to figure out what to make of the portfolio of these securities what is the best portfolio of these securities so I'm just warning you this is a philosophical departure from what you're used to thinking and reading in the newspapers because the newspapers would say well let's take a look at the earnings of McDonald's like let's take a look at Merck let's talk to the macro economists and see what's going to happen over the next twelve months let's talk to the earnings analysts and see whether they forecast higher earnings lower earnings the whole point of the academic infrastructure that we set up is it you can't predict these things and if you believe that then basically warren buffett is just one really lucky guy okay so I'm going to have to justify this academic position to you and I won't do that till the end of the semester because first of all have a lot of material to cover and I want to cover all of the material in the basic form and then at the end I'm going to give you a sense of where things really stand it's a fiction it's a fiction that you can't forecast stock prices but it's a fiction that actually is pretty close to reality for 99 percent of the public now you guys are not 99 percent of the public but for the people that will someday be your clients or your investors they will it will be true that the typical individual has no hope of being able to out forecast Warren Buffett and if you can't out forecast somebody then you may as well assume that they're random and they're perfectly priced and then you still have the problem okay if you assume that then what do you do that's all we're gonna try to figure out I'm going to tell you what you do with portfolio theory okay now since we're almost out of time I want to just tell you where we're going what we're going to do is look at this graph and ask the question what do people want they want higher return they want to go and they want lower risk they want to go west so the Northwest is where we're going to be heading in this graph and the question is how can we get there how can we get as Northwest as possible using these securities and the answer will shock you I think because you're going to see that by doing a very simple little bit of high school algebra we can actually create a portfolio that beats all of these things that is if you didn't know anything about portfolio theory you would be severely worse off because you'd be stuck having to be on one of these five points and if you knew a little bit of high school algebra and some finance you can actually do a lot lot better so we'll see that on Monday

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