Ses 15: Portfolio Theory III & The CAPM and APT 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 well let me pick up where we left off last time and give you just a very quick overview of where we're at now because we're on the brink of a very important set of results that I think will change your perspective permanently on risk and expected return last time remember we looked at this trade-off between expected return and volatility and we made the argument that when you combine a bunch of different securities that are not all perfectly correlated what you get is this bullet shaped curve in terms of the possible trade-offs between that expected return and riskiness of various different portfolios so every single dot on this bullet shaped curve corresponds to a specific portfolio or weighting or vector of portfolio weights Omega okay so now what I want to ask you to do for the next lecture or two is to exhibit a little bit of a split personality kind of a perspective I'm going to ask you to look at the geometry of risk and expected return but at the same time in the back of your brain I want you to keep in mind the analytics of that set of geometries in other words I want you to keep in mind how we got this bullet shaped curve the way we got it was from taking different weighted averages of the securities that we have access to as investments right so every one of these points on the bullet corresponds to a specific weighting as you change those weightings you change the risk and return characteristics of your portfolio so the example that I gave after showing you this curve where I argued that the upper branch of this bullet is where any rational person would want to be and by rational i've defined that as somebody who prefers more expected return to less and somebody who prefers less risk to more right other things equal so if you've got those kind of preferences then you want to be in the Northeast you want to be as north sorry north west as possible and you would never want to be down in this lower branch when you could be in the upper branch because you'd have a higher expected return for the same level of risk right so after we developed this basic idea I gave you this numerical example where you've got three stocks in your universe General Motors IBM and Motorola and these are the parameters that we've estimated using historical data now there's going to be a question and we've already raised that question of how stable are these parameters are they really parameters or do they change over time and I told you in reality of course they change over time but for now let's play the game and assume that they are constant over time and see what we can do with those parameters so with the means the standard deviations and most importantly the covariance matrix so this is the matrix of variances and covariances with these data as inputs we can now construct that bullet shaped curve the way we do it is of course to recognize that the expected return of the portfolio is just a weighted average of the expected returns of the component securities where the weights are our choice variables that's what we are getting to pick is how we allocate the 100% of our wealth to these three different securities okay and the variance of course is going to be given by a somewhat more complicated expression where you have the individual security variances entering here from the diagonals but you also have the off diagonal terms entering in that same equation for that variance of the portfolio and when we put these two equations together the mean and the variance and we take the square root of the variance to get the standard deviation and we plot it on a graph we get this okay this is the curve the bullet shaped curve that we generate just from three securities and from their covariances and where we left off last time is that I pointed out a couple of things that was interesting about this curve one is that unlike the two asset example where when you start with two assets the curve the bullet goes through the the two assets in this case with three or more assets it's going to turn out that the bullet is actually going to include these assets as special cases but they won't be on the curve in other words what this curve suggests is that any rational person is going to want to be on this upper branch what that means is that it never makes sense to put all your money in one single security you see that in other words if we agree that any rational investor is going to want to be on that efficient frontier that upper branch right why would you ever want to be off of that branch you'd like to be northwest of that but you can't you'd never want to be below that branch or to the right of that branch because you can do better by being on that branch so what this suggests is that we never are going to want to hold 100% of IBM or 100% of General Motors or 100% of Motorola if we did we be on those dots and those dots would lie on that efficient frontier but in fact they don't so right away we have now departed from warren buffett's world of i want to pick a few stocks and watch them very very carefully yeah right um that's not necessarily true there are points on this line where and they may be pathological so in other words they may be very rare but there may be points on the line where you are holding two stocks but not the third so you got to be careful about that but but those are exceptions as a generic statement you're absolutely right the typical portfolio is going to have some of all three of them and if you had four stocks the typical portfolio would have sum of all four yeah yes well let me put it another way that maybe a little bit more intuitive what this diagram suggests you guys are already groping towards is the insight that the more the merrier as you add more stocks you cannot make this investor worse off okay so in other words I've now shown you an example with three stocks we used to do two right is it possible that by giving you an extra stock to invest in I've made you worse off yeah why exactly I can never make you worse off in a world where you're free to choose that is because you always have the option of getting rid of the stock that you don't like you can always put zero on it so to your point on as I add more stocks first of all my risk reward trade-off curve will get better what does it mean to get better what does it mean for the risk reward trade-off to be better yes that's right a higher return for the same level risk or a lower risk for the same level of return in other words your upper branch actually moves to the northwest that's what it means to get better as I add more stocks this will move to the northwest and therefore you have available all of the opportunities to the south and to the east but you would never take those because you're rational in the sense that you always prefer less risk to more and more return to less okay yeah yeah and if we look at all the possible combinations that not at the same time we can look at them all at the same time but then all the subsets that you can think of yes and you must come up with some most efficient frontier exactly in that market right hold on to that thought for for ten minutes we're gonna come back to that all right let's let's do three first and then we could do all of them yeah correlation of these stocks do are going to just psychological factors of the market or is it due to intrinsic correlation and then the following is when Buffett says investing in one stock and just watching carefully is necessary shooting at the market will determine at some point that the stock is undervalued so so those are two good questions let me take each of them separately let's first talk about the correlation why is there correlation we haven't really talked much about it but it turns out that there are many different arguments for why there is correlation probably the most compelling is that a rising tide lifts all boats and vice versa in other words when business conditions are good then that helps all companies just like when business conditions are bad it hurts all companies so there's some macroeconomic type of commonality among businesses that create correlation that's one reason but the second reason is something you pointed out which is quite apt particularly over the last few weeks which is a psychological factor when the entire economy is under stress and people are scared to death about what's going to happen to the market what what they will do is withdraw money and mass from equities and put them into safer assets like cash or or Treasury bills or money market funds or whatever they can do to get to safety so I would say that the answer is both there are good economic reasons where correlations should exist among different companies but they're also psychological or behavioral reasons that exacerbate those kinds of commonalities now your second question about Buffett versus this approach there's one fundamental difference between what Buffett would say about a company that he decides to buy versus how we're approaching it the fundamental difference is that Buffett would say that he's been able to identify a severe mispricing in other words he would argue that markets are not in equilibrium he would argue that Goldman Sachs is dramatically undervalued where it is today and seven years from now he may be right and that's the kind of time frame he has in mind if not longer okay so far I've made no such argument at all about deriving these analyses right I've not made any argument about why the prices are good or bad in fact I'm arguing in a way that these prices I'm taking as given and the question is what can I do to construct a good portfolio irrespective of whether the markets are crazy or markets are rational in a few minutes I'm going to argue that when markets are rational and in equilibrium then there is something that we can say about the relationship between risk and reward that's extraordinarily sharp and meaningful from the perspective of financial decision making and then at the end of the course I'm going to try to explain to you what the limitations of that set of assumptions are that is well you know from a risk reward perspective let's take a look okay IBM has a higher expected rate of return and it's got a higher level of risk so you really can't say that you would never prefer a GM over IBM because GM has lower risk and lower expected return if on the other hand GM were over here then then you would be right because any point to the direct north-west of a particular point on this curve is strictly preferred and GM and IBM don't have that relationship right in other words the way you can identify securities that are dominated in both dimensions is so this is your risk dimension this is your expected return to mention pick a point in this space and ask the question what are the other portfolios that are strictly preferred to that point well the answer is pretty simple any portfolio that has higher expected rate of return for the same level of risk so the vertical line any portfolio that has less risk for the same level of expected return so the Eastern the the western direction and anything in this segment in that net orthe –nt our quadrant is strictly preferred so in the case of IBM you draw the vertical and the horizontal and ask the question does GM lie in that area no if you do GM and you draw the vertical and then the horizontal and ask does IBM lie in that restrict Lee preferred quadrant the answer is no so the answer to your question about IBM versus GM no there isn't any strict relationship that would say one would always dominate the other but if GM were here then IBM is clearly contained in that preferred quadrant so then the answer to your question would be yes okay yeah just so the answer is it depends on other things going on everyone would not do that everyone would do something else I'm about to tell you so I'm about to give you the tools to make that exact conclusion and the reason is that when you when I show you what people will do that's going too far dominate what what you think people want to do just with pears so instead of doing it with pears let's do it with all the securities as Zeke wanted to do ok we're going to do that in just a minute but I want to make sure everybody understands this basic framework first because we're going to now start making this a little bit more complex ok where we left off at the very last moment of Wednesday's lecture was we I showed you this diagram with the tangency portfolio but we hadn't really gotten to talking about it remember the case where we had only one risky asset and one riskless asset Treasury bills and in that case when you are combining a portfolio with one risky asset and one risk list you got a straight line it turns out that that is much more general you get a straight line anytime you can combine a riskless asset with any number of risky assets so let me give an example suppose we picked an arbitrary portfolio which is this red dot P and I wanted you to tell me what is the risk reward possibilities that you could achieve by mixing P with Treasury bills well you get that straight line right we derived that last time so any point along the straight line is what you could achieve right anybody tell me where the portfolio would be that invests a hundred percent of your assets and t-bills where is that on this graph right this knot right here how about a hundred percent in portfolio P right the red dot over there how about the 25% in t-bills 75% in P where would that way would that lie it would be a long go on Beware here 25% t-bills 75% right exactly it would be 3/4 of the way up towards this dot right because it's 75 of the risky 25 of the risk lists so you're going to get more you're going to get closer to the risky asset okay great so we've now demonstrated that what I can achieve as an investor just mixing portfolio P with the risk-free rate is anywhere along that line now this analysis applies to any portfolio P so for example suppose I wanted to ask you what risk reward trade-offs could i generate by mixing risk free the risk-free rate with General Motors what would that look like yeah exactly that's right if I wanted to mix Tebow's to general and I get that straight line right through that dot if I wanted to mix t-bills with IBM I'd go through that dot with IBM if I wanted to makes t-bills with Motorola I'd go through Motorola and if I wanted to mix t-bills with any portfolio on that frontier on that upper branch it would just be a line between t-bills and that point on the upper branch right so question if I were to give you the choice of mixing t-bills with only one portfolio just one which would it be which would you prefer blue the one where the line is tangent so you talking about right around here right somewhere here that's where the line is just tangent to that curve now why is that how'd you come up with that yeah exactly if you picked any other portfolio besides the tangency portfolio let's let's pick one and see if you pick though I don't know let's say this one right here if you drew a line between this point and that portfolio it's going to turn out that there are other points over here that are strictly in the northwest of that line that you could do better there exists only one portfolio that you can mix with t-bills such that you can never ever do better in terms of generating risk/reward trade-offs for everybody I that likes expected return and doesn't like risk and it turns out that that portfolio happens to be the tangency portfolio that's the portfolio that all of you in this room would love to have I don't know anything about you I don't know your backgrounds I don't know your versions but I don't have to know as long as I know that you like expected return and you don't like risk those are the only assumptions that I need then I know all of you in this room are going to want that port for you may not be at that portfolio for example some of you who don't like risk you're going to be down here those of you who are budding hedge fund managers you're going to be up here but the point is you're going to be on this line you're not going to be on this line down here why because why be on that line when you could get higher return for a given level of risk or lower risk for a given level of return you're giving up something for no good reason okay so this is a remarkable insight of modern portfolio theory this basically tells us that regardless of our differences in preferences as long as we satisfy the hypothesis that we like expect the return and we don't like risk that means that everybody in this room will agree that the only line that they would ever want to be on is that tangency line ok now questions Ingrid yes yes in fact this tangency portfolio is one very particular and special portfolio so in other words it's it's a particular weighting of IBM General Motors and Motorola that gives you this particular portfolio it's it's something you can solve analytically yes it has a solution and if we were using matrix algebra I can actually solve it for you but it's it's a little bit complicated so I'm not requiring that people know how to do that only that you know that it exists yeah exactly yes yes it if you assume that there are borrowing and lending differences then obviously these these analyses don't apply so in particular you know here you know if you're if you're here you're actually lending right if you're here you're fully invested in the stock market if you're here you're borrowing if you're borrowing and lending rates are different then it turns out that the curve that you want to be on actually has a kink in it and that means that there is a potential for being on this curve and then there's another tangency line that goes out at a different slope that's possible but that's more complicated than what we want to talk about at this point so here I'm assuming borrowing lending rates are the same okay Zeek and then Romney so if I had a choice of if I had control over the world view of the market yes then if the yield goes down like of you know what they of the bills yes I would want to have a more volatile market so that I can intercept the curve at a higher return point well okay hold on you're changing the the assumptions here why are you controlling the volatility of the market the volatility of the market is a data point that you're basically using as an input okay trying to figure out like what I am what I am because you're missing connecting I see this as a connection between the yield curve and the market because it's not only it's not only retrospective in the sense that if the yield goes down there is cash flowing from the market let's not worry about the dynamics this is not meant to be a dynamic story all right I didn't say anything about this happening over time and there are lots of different changes going on this is a static snapshot today versus next period these returns and covariances and all that apply to the returns from this period to the next whether it's monthly or annual that's a static snapshot as of today so we're not talking about any term structure effects yet okay yeah yeah yes yes absolutely if you start adding more stocks to this cocktail what's going to happen is the bullet is going to shift to the left and it's going to shift up and so the tangency point will change but the curve that that's straight line the tangent line what you're going to see is that tangent line is going to go like that the slope that's right you're going to get more expected return per unit risk and that is something we're going to take as a measure of how good this particular trade-off is we're going to look at that slope of this line and the slope of this line will give us a measure of the expected rate of return per unit risk it's exactly what it's going to do for us right beyond the folio of that line yep then you're borrowing that's correct if you look if you were to extend the line left words down to the left would you be then shorting the market to invest of tables yes and if that happens you know what you would do it would not go this way because of course standard deviation can't be negative it would go like this it would go this way right because standard deviation is always non-negative it's the square root of the variance which is always positive so if you decided to short the tangency portfolio and put it in tea pills well you'd be a knucklehead but-but-but you would you would be on this line right here you would have higher and higher risk because you're taking a short position on equities and you have a lower and lower return because you're shorting the high-yield asset and buying the low yield asset right ok any other questions about the geometry of this point it's very important right this this is a major insight yeah yes and is that the reason why the shift is more towards left because as we add more and more portfolios that end dominates the Rhoden that's right that's right as we add more securities you get more and more impact of diversification so that increases your expected rate of return per unit risk because you can't make somebody worse off by giving them choices right they can always put a zero for the new stocks that you give them if they don't like it so the only thing you can do is to make you better off meaning the only thing you can do is to give you a higher level of expected return per unit risk for a lower level of risk per unit of expected return so by adding more securities you're basically increasing the slope of this line so let's talk about the slope of the line the slope of that line is equal to the expected return of that tangency portfolio – the t-bill rate divided by the volatility of the tangency portfolio if you just calculate you know rise over run that's what you get as the slope there's a name for this the name for this is called the Sharpe ratio you may have heard of this particularly those of you who have interest in hedge fund investments hedge fund managers will often quote their Sharpe ratio very proudly the Sharpe ratio is simply a measure of that risk reward trade-off the higher the Sharpe ratio the better you're doing if you're a mean variance optimizer meaning you prefer more risk a more expected return and less risk okay so the idea behind the tangency portfolio is that it is the one that will give you the highest Sharpe ratio right let's look at it again if you pick a portfolio like here take a look took a look at the slope the slope is going to be lower take a point over here in the inefficient branch of the bullet then the slope is going to be even lower than the upper branch the biggest slope occurs when you invest between t-bills and that tangency portfolio that's what you're optimizing okay yeah yes yes yes between these additional stocks how would the correlation increase by adding another stock yeah how could that be well you've got 20 stocks okay and they've got a correlation among those 20 stocks now I want you to think about adding a 20 first stock when you add that 20 first stock you don't affect the existing correlations right I mean it's it is whatever it is those are parameters at least for now we're going to call them parameters when I add my twenty first stock I'm giving the investor an extra degree of freedom now instead of investing among 20 securities I'm gonna let you invest about 21 you don't have to invest in the 21st or another way of thinking about it is that when you only had 20 stocks you really had 21 portfolio weights but the 21st weight I've arbitrarily constrained to be 0 now I'm going to loosen the constraint and I'm going to say ok now you can invest in the 21 the 21st stock you won't affect the existing correlations but the new stock that you add in can benefit in providing additional diversification benefits yeah oh well actually you don't need a negative correlation to make this go to the left you just need to have something less than 1 right remember from the last lecture this is the case where you had perfect correlation anything less than perfect correlation brings you to the left ok so as long as when my 21st stock is not perfectly correlated with the existing stocks in that portfolio of 20 I'm going to move things to the left in general that is true you would but but what this suggests is that negative correlation is a very rare thing it's very difficult it's extremely difficult now that's the that's from the analytical perspective we can conclude it's very difficult let me ask you from an economic perspective why is it difficult to find a stock that's negatively correlated with all other stocks anybody give me a business rationale for that yeah right let's actually spend a little bit more time thinking about this I want you guys to tell me right now give me a stock that you would put your money in right now today SP has gone down by 45 percent since the high several several months ago right stock market's doing terribly and it doesn't look like it's getting any better so you tell me what stock would you put your money in right now today yeah Terry Campbell soup why is that okay but on the other hand if people are poorer all around might not they start consuming even less of canned soup and try to make their own soup from you know little packages of ketchup and hot water I saw that and I Love Lucy episode years ago it's pretty cool so are you sure are you sure that Campbell's soup is going to go up over the next few months in response to the current crisis it'll stay stable ah but that's not that's not negative correlation that's zero correlation I want something that's going to go the opposite direction of where the economy is heading tell me where that is yeah okay fine so you're gonna short the market that's it that's a cheap cheap answer sorry you don't get any credit in that I want I want to I want to answer the question that was raised by David which is you'll show me a stock that can get me even more to that left I want a negative negatively correlated stock yeah not Walmart well that's not the same thing as saying that it is going to go up over the next several months in response to this economic crisis you don't think that there's going to be a decline in consumer spending that'll affect retail as well well that's what I'm asking you where you gonna go so you telling me now that you believe that wart mark Walmart is the answer you think it'll be negatively correlated historically just to let you know retail has not been negatively Carla with the business cycle okay so yeah Z Freddie Mac if you like that investment I have something else for you yeah I don't I don't I don't know if you want to argue that Freddie Mac is negatively correlated with market downturns I mean the reason that Freddie my god is the trouble that it did was because of the economic downturn alright one more of bika philip morris that's an interesting one you know obviously people are very nervous that when you're nervous I'm going to be you're gonna be smoking on the other hand on the other hand again one could argue that it's not negatively correlated it might be either slightly positively correlated uh but even there you know people have argued that cigarettes are you know a consumption good that can get hit with a downturn in in markets the bottom line is that it's really hard to come up with negative correlated stocks because let me let me tell you if you found one that was negatively correlated if you found one that was really really negatively correlated what would all of you do exactly you'd buy it you know what that would have let the effect of that would be to depress to increase the price and depress the expected return right the expected return remember what the expected return is it's the expected future price divided by the current price if now all of you go out and buy Walmart or whatever stock you think is negatively correlated that would have the impact of increase in the current price and therefore decreasing the expected return now if you have a stock that's got a negative covariance and a negative return that doesn't help because in fact that was a suggestion that was put forward here let's just take the S&P and short it and then you get a negatively correlated stock the problem is that it's also got a negative expected return and then you're not helping things the key is to find negative correlation with a positive return if you can find that then you really found something worthwhile but my guess is it won't last for exactly this reason well the question okay so now let's go back and ask the question what does this mean if it if we agree that all of us want to be on that tangency portfolio what is that what does that tell us well that allows us to then make an argument that managers that are trying to provide value-added services for us they need to be doing something above and beyond what we can do ourselves okay now here's where Warren Buffett meets modern finance there in a way if I want to see whether or not Warren Buffett or any other managers adding value one simple criterion that I can put forward is this this is what I can do on my own I can get that line pretty much by just using my you know basic finance skills that I've learned here at MIT if you're going to manage my money and charge me two-and-twenty show me what you can do above and beyond this I want you to tell me where you can get me on this graph can you get me up here can you get me over here can you get me anywhere either to the left or above that curve we can use that as a measure of performance and there's a name for that it's called alpha typically when people talk about alpha they're talking about deviations from a line like this we're going to get to that more formally you don't have to write it down or make note of it just yet it's on the next slide but we're going to show you how to measure that explicitly so now not only is this a good idea for you as a baseline to manage your own portfolio but you can then use it as a metric to gauge whether other people are adding value to you so Warren Buffett would say no problem I think I've got alpha so I'm not going to bother with this I think I can get you up here that is if you want to invest with me and in fact if you looked at Warren Buffett's performance over the last 25 years that he's been doing it a 30 years his Sharpe ratio is a lot better than the tangency portfolios so he actually has added value if you use this as a criteria but the problem is you have to identify the Warren Buffett's before they become Warren Buffett's because after they become Warren Buffett's it's not clear that there adding the same amount of value right it's already the cats out of the bag yeah Vica yes so yes so the dynamics of this are very complex and this is right now a static Theory static meaning today versus next period we're not looking at the dynamics over time in order to do that there's lots of different effects that are much much more complicated for that you really got to take 15 for 33 and even for 33 won't cover those kinds of questions in complete detail because they rely on some very complex kinds of analysis but I'm going to get to that at the end so if I don't please bring it up again I want to I want to make a comment about that and how you can take this relatively simple static theory and make it dynamic in an informal way even though the analytics become very hard when you try to do it formally okay so the key points of this lecture are oh sorry question no no not at all not at all indifferent any point on this line is a different risk reward combination so in other words it depends on your preferences yes yes now we haven't talked about utility functions yet but we're going to in a little while let me preview that since you asked okay you all remember what indifference curves are from basic economics right an indifference curve when I first came across that you know I I was you know rather offended because you know I don't view myself as an indifferent individual I have lots of passions and so just you know why should we you know be indifferent about two choices in fact that's an economic term it simply means that you're just as well off between two combinations and therefore these two combinations you're indifferent to ok you're indifferent between so if I have to ask you to draw on this graph an indifference curve of risk/reward trade-offs for you the typical individual what would it look like can anybody give me a sense of what different kinds of risk/reward trade-offs you would be indifferent among and to make the question a little bit simpler let's start off with a particular point okay so let's suppose that this point right here is the point that I want you to draw the indifferent curve from which is a standard deviation on a monthly basis of about six percent and an expected return of about say what is it 1.4 percent or something so you've got a monthly return of 1.4 percent and a risk of about 6 percent give me another point that you would be indifferent between versus that one anybody any volunteers yeah okay all right okay so you have a particular number in mind in other words let's let's look let me ask you this if I if I cranked up your volatility from 6% to 8% how much extra return what I have to give you in order for you to be just as well off as you were at six and one and point one point four okay higher than one point eight okay okay this is one point four one and if I said now I want to be at eight percent how much risk do I have to give you how much expected return drive to give you to make you just as well off as this point right but how much higher that's the question it's a personal question Rami you would have to have an increase in 33% of your expected return even though I'm only giving you a 25% increase in the risk that's a third you're right so I'm increasing the risk by a third you want me to increase the expected return by a third so your trade-off is linear is that right you're looking at it linearly anybody else you know you may want to translate this into annual numbers right because I'm sensing that you may not have a good email out of a good feel for what your own preferences are and by the way this is a challenge you know not everybody understands what their own personal preferences are for these numbers this is not a natural act you know of human nature that we automatically have preferences on these numbers but the bottom line is that if I make you take more risk I'm going to have to compensate you and give you more expected return it's got every reason why you want to take that risk okay for some people it's linear for other people it's much more than later they don't want to take any more risk in fact right now most investors don't even want to answer that question because they don't want to take more risk you say well what if you did well I don't want to well but just what if you I don't want to what if I just don't want to take the risk so they can't even answer that question but if they could my guess is that it would be way up here so you'd have to give them a lot of expected return to make people take more risk today alternatively if you want to give people less risk my guess is that you can actually subtract a lot of return in order to take away a little bit of risk how do I know that take a look at the yield on the three-month Treasury right so an indifference curve it's going to look like this it's going to look like it'll be increasing but it'll actually be bowed this way and the theory behind why it's going to be convex holds water as opposed to spills water the reason it's going to be convex is because the decreasing or diminishing marginal utility between risk and expected rate of return like anything else you know economists have this notion of diminishing marginal utility between any two commodities right if you've got you know ice cream sundaes and basketballs you know there's only so many basketballs that you could enjoy before the next incremental basketball provides relatively little pleasure for you the same thing with ice cream sundaes you can only consume so many ice cream sundaes before the next incremental sundae provides somewhat less benefit to you right that kind of diminishing marginal utility gives you this kind of a bode curve so where you are on this straight line depends upon how bold your curve is somebody that's really risk-averse has a curve that looks let me draw this because it's a little bit easier to see rather than trying to follow my laser pointer somebody that is so here's the the trade-off this is the line somebody that's extremely risk-averse is going to have curves like this those are indifference curves and as you go to the northwest you're happier and happier somebody who so the optimal point is where this indifference curve hits this particular line on the other hand if you're very risk seeking if you don't need a lot of compensation of expected return per unit risk then your difference curves not going to look like that it's going to look like this in which case your tangency point will be farther to the to the north east you'll be taking more risk and getting more expected way to return okay but the bottom line for this graph and this lecture is that everybody no matter what your risk preferences are everybody's going to want to be on that line that tangency line and it turns out that that insight is going to translate into a remarkable remarkable conclusion about risk reward trade-offs okay so the key points for this lecture are diversification reduces risk in diversified portfolios covariances are the most important characteristics of that portfolio it's not the variances but the covariances investors should try to hold portfolios on the efficient frontier that upper branch and with the riskless asset everybody is going to want to be on the tangency line okay those are the major conclusions from this analysis and you can work all of this out analytically using the mathematics of optimization theory but in fact all of this could be done graphically as we have geometrically okay question okay so we're gonna get to that we're going to talk about benchmarks because you're right that most investments today are all benchmarked against something right and you're probably wondering how that got to be that whole direction of analysis and performance attribution that came out of this in other words it was because of this particular academic framework that was developed by Harry Markowitz and Bill sharp and others that indexation and benchmarking came to be so I'm going to get to that let me put that off for another lecture or so after we derive the implications of everybody wanting to hold the tangency portfolio it's going to turn out that that tension see portfolio happens to be the benchmark so we'll get to that yeah yes so so you would think that it would but in fact I'm going to show you that there's exactly one case where it doesn't and that's the case of the equilibrium that I'm about to describe so let me turn to that right now there any other questions right yeah well there is I mean there is although I would have to say that if you have a preference about the down side so not making anything as you point out then that changes this analysis alright so this analysis really requires that you use standard deviation as the sum total of your perception of the risk of a portfolio if you have other kinds of sensitivities then you need to bring them into the analysis and that will change these outcomes all right higher they resonate very higher yeah may end up being higher yeah it was still a larger deviation yep that's true but again you've made an assumption there that I'm not making which is you're assuming companies are outperforming or underperforming I'm assuming that the data are given and I'm not making a bet on whether any companies are likely to succeed or fail I'm merely looking at companies as investment opportunities that provide certain expected returns volatilities and covariances you want to go down the path of warren buffett and I'm resisting that because I don't have the skills of a Warren Buffett so I don't know what's a good value what's not a good value and the case in point is a discussion we just had today you tell me what is a good value today you really believe that Campbell's soup or Walmart should be the companies you invest in today I don't know I mean another argument is entertainment you know why don't you invest in movie theaters lots of people now are going to see the James Bond movie and you know they want to escape from reality wouldn't that be a growth industry given market conditions well that's true but you know how many people have 12 dollars to spend on a movie plus you got to get the popcorn and the bonbons and all those and you know by the time you're done it's like a $60 evening I mean I don't know so the point is it unless you are willing to make predictions this is the only alternative that provides a disciplined approach to investing in so-called good portfolios so it's a different it's a different approach okay so now let me turn to the next lectures lectures 15 through 17 where we're now going to talk about equilibrium we've already identified that all of us in this room assuming we have mean variance preferences that's a that's an important assumption I grant you but it's not an a reasonable one it's just it is an important assumption we've all agreed that we're going to take on portfolios that lie on that line and therefore the portfolio that is the tangency portfolio I'm going to give it a special name I'm going to call it M portfolio M okay what we now know is that given a choice between holding n securities and t-bills versus holding t-bills and a single portfolio all of you would be indifferent between those two choices if that single portfolio were M the tangency portfolio do we agree on that okay so therefore I could in principle construct a mutual fund called M this mutual fund holds stocks in exact proportion to the weights given by that tangency portfolio in other words it is the tangency portfolio so what that suggests is that all of you in this room would be absolutely indifferent between investing among the n stocks and t-bills on the one hand versus investing in two securities on the other one security is t-bills and the other security is shares of mutual fund M do we agree on that any controversy there I know I've made a number of assumptions to get us here but given mean variants preferences which is not an unreasonable assumption and given that we've assumed these parameters are stable over time that's where we are I mean forget about these there are fees no matter what you do so for now I'm going to forget about fees I'll put fees back in later and if I do that then it's going to look even more compelling for you to want to invest in M mutual fund M versus n stocks I don't know how many have traded individual stocks but if you ever tried to manage a portfolio of you know a thousand stocks it's actually fairly time-consuming right okay and by the way there are more than a thousand securities I mean the S&P 500 you can think of as being M but that's an approximation right there's probably seven or eight thousand securities that trade today probably only two or three thousand that you really take seriously and probably only fifteen hundred that you really need from a diversification perspective fifteen hundred stocks would you want to trade in that or would you want to trade in one mutual fund yeah yeah because Warren Buffett beat it in the past do you think he's going to beat it in the future I don't know that's right good question good question I mean you know if you're thinking about Warren Buffett as a 10-year investment I think I might short that I mean you know he seems healthy but you know those cherry those cherry cokes have to have an impact I'm sorry you know you eat enough steaks at that Omaha restaurant I know what it is and those cherry cokes I don't know so so okay so fine let's let's not do Warren Buffett let's do somebody else fine you tell me who that is tell me who the next Warren Buffett is can anybody tell me I'll be happy to do that I'd be happy to invest in them who is it thank you thank you but those who can't do teach those who can't teach teach gym and at least I don't teach gym so the point is that we don't know who the next Warren Buffett's going to be and I don't want to have to figure that out I mean that's a pretty tall order to tell an investor that they've got to figure out who the next investment genius is if they knew they wouldn't have to ask them to invest they did invest themselves right so what I'm showing you is a simple way of investing that may not be as good as Warren Buffett but it's certainly better than trying to pick the next Warren Buffett if you don't know what you're doing Jen thank you that's another way of looking at it if you ask the question is it easier to try it if is the historical covariances and variances and expected returns more predictive of the future than your ability to find the next Warren Buffett then yes that's another good argument that in other words this framework relies on less ability to forecast it doesn't it doesn't completely rule it out because as I said these parameters they change over time and you have to think about that impact so it's not totally trivial but from the theoretical perspective it seems like it's a very internally consistent approach now let me go on for a little while longer because if it were just this then this would be an interesting rule of thumb but this is not a theory of financial markets just yet right I haven't really done anything truly astounding because you're still left with the question of what's the appropriate risk reward trade-off you know what what should I use for my discount rate you know a lot of financial decision making is not just picking stocks and making good investments but it's whether or not should I invest in nanotechnology as a corporate officer of a particular tech company or should I invest in green technologies what discount rates should I use how should I engage in capital budgeting or project financing all of these questions seem like they have nothing to do with investments so I don't want to make this course into an investments course there's a lot about corporate financial management that relies on being able to understand these markets okay so let me let me show you where we go next because we're very close now to the big payoff we've already identified the tangency portfolio as being special I'm going to call that portfolio M and I'm going to argue that everybody in their right minds are going to be indifferent between picking among these two investment opportunities t-bills and M versus the n plus one investment opportunities of all stocks plus t-bills okay it turns out that portfolio M therefore has to be a very specific portfolio and it turns out that that portfolio is the portfolio of all assets in the entire economy in proportion to their market capitalizations now what I just said is an incredibly deep result so I don't expect you to just get it let me say it again first of all I want you to understand it and then I'm going to try to give you the intuition for it okay if it's true that everybody not only in this room but in the world if everybody in the world is indifferent between investing in those n plus-1 securities and in – then we can argue that those two securities play a very special role in particular think about what that mutual fund M has to be everybody in the world wants to hold em so you can actually let's make the leap of faith that everybody does hold em so in other words now we're in a world where everybody is already mean-variance optimizers and they already hold two assets in their portfolio the Treasury bill asset and the mutual fund M so you hold them you hold them you hold them you hold them you hold them everybody holds em we hold different amounts of it so as a hedge fund manager you're holding a large amount of em in fact you're holding twice as much as your wealth allows and you're borrowing t bills to do so somebody who's very conservative is holding a very tiny little bit of em mostly that person is invested in t bills but the point is that every single person's portfolio you look at when you look at their portfolio its M if that's true if what I just said is true what portfolio does M have to be there's only one that it can possibly and that is the portfolio of all equities in the marketplace held in proportion to their market value do you see the beauty of that now let me try to explain it okay I hope you understand it let me explain it why does that have to be this has to do with supply equaling demand I'm going to make an argument about equilibrium I haven't done so up until now up until now I haven't said anything about supply equaling demand but I'm about to do so if everybody is holding this portfolio M that's the demand side right everybody is demanding M on the supply side I'm assuming that all stocks that are being supplied are held if all stocks that are being supplied are held by somebody but if everybody in the world is holding the same portfolio m when you aggregate all of the demands so I'm going to add up your demand and your demand and your demand and your going to go through the class and go through the world we're going to add up everybody's demand in every single case your weights are identical you're holding the same portfolio m so when i a granade the entire world and i get the portfolio m what does it have to equal it can only equal the sum total of all assets in the world right supply equals demand and therefore when i a granade all of your holdings of m into one big fat m that big fat m can only be equal to one thing which is all the equities in the world and the waiting's are just simply their market caps right there's only so much of General Motors take the entire sum total of that that's your the global investment in General Motors and then you do that for every single stock and you divide that by the total market capital of the stock of all stocks you get the market portfolio M so this shockingly simple but extraordinarily powerful result is due to Bill sharp Harry Markowitz came up with portfolio optimization he applied mean-variance analysis to portfolio optimization and argued that everybody has to be on the line Bill sharp looked at this and said aha if everybody's on that line that means that everybody's going to be either holding em or t-bills or both and therefore the only thing that M could possibly be is the market portfolio and now we have a proxy for the market portfolio russell 2000 or the SP 500 both of those are very well diversified stock that have lot they don't have everything in it but they have a lot of things in it that proxy for everything the russell 2000 has 2000 stocks weighted by market cap that's as close as you're going to get to everything that you care about okay so now you'll see we're benchmarking is coming from but I'm going to get back to that in more detail so this this equilibrium result that says supply equals demand identifies this portfolio M and what it says is that if everybody does this if everybody takes finance here and learns how to do this it's not going to kill the idea it's going to lead to a very well-defined portfolio M now let me take it one step farther they're going to ask you to ask questions if I know what that portfolio m is then I've got an equation for this line okay I can write down a relationship between the expected return and risk of a portfolio on this line and this is it the expected rate of return of an efficient portfolio by efficient I mean portfolio that's on that line anything that's not on that line if it's below that line its inefficient right you're not taking you're not getting as much expected return per unit risk and you're not reducing your risk as much as you can per you of expected return the expected return of an efficient portfolio is equal to the risk-free rate plus the ratio of the standard deviation of that portfolio divided by the standard deviation of the tangency portfolio or the market multiplied by the excess return of the market portfolio this result is a risk reward trade-off between risk and expected return you see what it says is really something quite astounding it's telling you that here's the risk-free rate that's the base return for your portfolio and what this is telling you is that what you should expect for your portfolio is that base return plus something extra and the extra is the market's excess return multiplied by a factor and the factor is simply how risky your portfolio is relative to the market let's do a simple example suppose that your portfolio is the exact same risk as the market well if that's the case then what is your expected rate of return it's the market so it's the risk-free rate plus the market excess return which when you add it together is just the market suppose you're holding a port felt it's more risky than the market is your rate of return greater or less than the rate of return on the market greater suppose that your portfolio has no risk suppose that Sigma P is zero then what's your rate of return exactly makes sense right this is very intuitive what this tells us now is that we can figure out what the fair rate of return is for an efficient portfolio for any portfolio on this line I can tell you what my fair rate of return is and it's an objective measure it's not just theory now now I can go into the market place I can measure the expected return on the market you know what that is historically not including the last few months it's about 7% historically over the last hundred years 7% the expected return the market okay sorry the expected risk premium the access rate of return about 7% what about the volatility of the market it's been about 15% historically so according to this relationship I've already figured out what this number is it's like 7% I've already figured out what this number is it's like 15% so now you should be able to get a benchmark for what to expect when you've got a particular level of risk in an efficient portfolio you've got all the ingredients what about risk-free rate well it depends on what risk-free rate but let's talk about over a one year period right now we're looking at somewhere between I don't know 1 to 3 percent depending on what day of the week you're looking at all right so one year t-bill rate is about 1 percent or so yeah yes no the unsystematic risk is risk that it's not measured by Sigma P so we're going to come back to that let me let me hold off on that for now because I want to come back to after I finished developing this there's going to be a connection which is systematic and unsystematic risk that's going to come right out of this relationship okay Yeah right SP 500 as the SN here market portfolio yep and the capitalization is the weight so you've got nonzero weights for all different stocks there does that imply that there's no stocks in the S&P 500 that are southeast of any others no no there could be why would you have them because we said those are strictly non preferred well that's that's if you're looking at a pairwise comparison if now you're trying to create an entire collection of these portfolios of securities that's a different story that's why I answered in response to Justin's question Justin said why not just trade off those two why not it's because you can do far better by using all of them in this way you see by looking at pairwise you can no doubt do better but if I use all of them I get this entire line and you can't get that entire line just from looking at two of these stocks you can use all of them you kick GM over to the right a little bit and made it strictly non preferred to IBM yeah you still might have a positive portfolio wait well you you might but more likely it'll be a negative portfolio wait it'll be negative and you'll be shorting it somewhere along the line here however the tangency portfolio by assumption if it's the market portfolio cannot have negative weights okay and so there what will happen is that all of the stocks will change in their relationship based upon various different kinds of equilibrium so that you won't get into a lot of those situations where you're going to be shorting these negative stocks yeah yeah that's right that's right not every stock has some benefit in adding to this particular risk reward trade-off and the sum total benefit is summarized by this line okay that's the ultimate objective sorry if you if you if you hold a specific stock oh if you hold if you put zero weight yes what what Sharpe would argue based upon this theory is that you want to hold as many stocks as you can to get the most diversification now that's the theory in practice it may well be that the benefits do not outweigh the cost because when you hold multiple stocks you have to manage them and so it may cost more so a mutual fund that has 3000 stocks may have a higher expense ratio than a mutual fund with 500 it may not nowadays actually the technology is so good that probably it doesn't but 15 years ago that was not true but apart from the transactions cost the theory suggests more is better because it will always give you more opportunities and it can never hurt you because you could always put a zero weight on them if you don't like them okay now it turns out that this is a trade-off between the expect return of an efficient portfolio and the risk of that portfolio in other words this applies only to portfolios on that tangency line what if you want to know what the expected rate of return is for Walmart we just said that no individual stock is going to be likely to be on that efficient frontier and therefore no individual stock is likely to be on this line so this is great if what you're talking about is investing in efficient portfolios but how does that help the corporate financial officer that's trying to figure out how to do capital budgeting for a particular pharmaceutical project it turns out it doesn't it doesn't help this doesn't answer that question it turns out you need to have an additional piece of theory that allows you to derive the same results not just for the efficient portfolios here but for any portfolio and this is another innovation of bill Sharpe this is actually why bill Sharpe won the Nobel Prize it was not for this little picture here but it was for this equation right here what bill Sharpe discovered is after computing the equilibrium relationships among various different securities he demonstrated that there has to be a linear relationship between any stocks expected return and the market risk premium just like here where you've got the risk-free rate plus some extra premium so this is the premium the second term but what bill Sharpe showed that was it if this portfolio is not an efficient portfolio if it's not on that line the linear relationship still holds but it turns out that this particular multiplier is no longer the right one to use it turns out that the right parameter to plug in there is something called beta and you've heard all about beta I'm sure now I'm telling you exactly what beta is beta is the multiplier that is defined by the covariance between the return on the market and the return on the individual asset divided by the variance of that market return okay if the portfolio happens to be on that efficient frontier then this beta reduces to this previous measure so this is a special case of the more general relationship where beta is used as the multiplier so let me repeat what beta is beta is the ratio of the covariance between the return on the particular asset or portfolio that may or may not be efficient it's any asset with the return on the market portfolio so this numerator is a measure of the Co variability between the particular asset that you're trying to measure the expected return of and that tangency portfolio divided by the variance of that tangency portfolio okay beta it turns out is the right measure of risk in the sense that it is the beta that determines what the multiplier is going to be on the market risk premium which is to be added to your assets expected rate of return above and beyond the risk-free rate that's how the cost of capital is determined for your asset okay so I think you all saw how I derive this but I didn't derive this I'm just telling you this is really where Sharp's ideas became extraordinarily compelling and in order to understand how to derive that I'm going to refer you to 433 because in that investments course we really delve into the underpinnings of that kind of calculation it's a little bit more involved it involves a matrix algebra but it's not a terribly difficult or challenging and certainly be happy to give you references if any of you are that I believe it's in really Myers and Allen but the bottom line is that this gives you an extraordinarily important conclusion now to the several weeks that we've been working towards this goal which is now finally after you know eight or nine weeks I can tell you how to come up with the appropriate discount rate for various NPV calculations the answer is the expected rate of return the appropriate fair rate of return or the market equilibrium rate of return is simply given by the beta of that security multiplied by the expected excess return on the market portfolio so now we in this has a lot of assumptions granted we're going to talk about those assumptions over the next couple of lectures but what we've done today is move the theory forward by quite a bit because we've identified a particular method for coming up with the appropriate cost of capital as a function of the risk where the risk is measured not by volatility anymore but by the covariance between an asset and the market portfolio and next time I'm going to try to give you some intuition for why this should be why this makes sense and why in a mean variance efficient set of portfolios why it reduces to something that we know and love okay any questions okay I'll stop here and I'll see you on Wednesday

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