Harry . Markwitze 1991.1.10 Nikkei Talk　　 Thank you. 　It is my honor to present a lecture on P.T. to the renowned Nikkei organization. The conditions for this lecture are a bit unusual. As you know 　I am a professor: usually I use a blackboard - in fact I find that students only pay attention to what's on the blackboard. However today there is no blackboard. I do have overhead projector transparencies 　but it was determined that they would not be visible from the rear of the room. So for the first time ever I will give a lecture on P.T. without a blackboard or OHP. On occasions I will ask you to help by imagining a big blackboard up here. I'll instruct you as to what to write or draw on our large imaginary blackboard. 　In the past 38 years P.T. has grown to be a very large subject to which many have made and continue to make important contributions. I will discuss certain parts of it in historical order. First 　some basic principles I worked out during the 1950s 　then certain further developments in the following decades. 　You will see that there is more than one way to use P.T. No way assures that you will make money when everyone else loses. There are certain questions you should ask. 　I'm not trying to discourage you from using P.T. in general or NIKKEI NEEDS in particular. P.T. is not a black box that gives you the right answer without effort on your part 　but constitutes a set of techniques which will assist your judgment and understanding. The reason why P.T. has continued to grow in the U.S. is that many of its users understand what it can and can't do. I would like to see similarly sophisticated investors here to assure long-run growth here. 　I developed P.T. about 40 years ago as my Ph.D. thesis at the University of Chicago. My findings were published in a 1952 article and a 1959 book 　and this was as far as it got in the 1950s. It seems to me that P.T. is just common sense put into a mathematical form. I proposed that investors should pick a portfolio which minimizes risk for a given level of return on the portfolio as a whole 　while also maximizing E.R. (earnings ratio) for a given risk. Now imagine a big blackboard. Draw an X-axis and a Y-axis Label one RISK and the other RETURN 　(I put RETURN on the X-axis. Nowadays everyone puts RISK on the X-axis 　so you'd best do it that way. Think about a big imaginary blackboard: you can do it your way I can do it mine. ) Now that you have drawn in the axes 　draw a curve that represents (a portfolio with the maximum potential earnings ratio for various risk levels) Now call it "efficient frontier" 　Propose that investors pick a security from (securities above the efficient frontier.) 　Risk and return were already used in financial practice 　but not in the financial theory of the day. (I won't go into it here: if you want to know ask at Q&A time.) 　People spoke as if returns on securities were chance events 　and adopted the "standard deviation" as the measure of the risk or uncertainty of a distribution. "Standard deviation 　 as many of you know, is a measure of how spread out is a probability distribution; for actual computation it is frequently more convenient to use 　variance" which is the square of standard deviation. 　We need to emphasize the portfolio as a whole. If you look at the formula for the standard deviation of a portfolio you will see that it depends not only on the standard deviation of the individual stocks 　but also on the correlation of the returns between stock. (When I speak of return 　I refer to capital gains as well as dividends.) 　Correlation is a measure of the extent to which returns tend to move up and down together. You know that one should not put all one's eggs in one basket. The correlation coefficient is a measure of the extent to which two securities are in the same "basket." 　Whether or not a particular security enters a particular portfolio on the efficient frontier depends on more than its own risk and return. It depends also on the risks and returns of other securities and on all the correlation coefficients. In particular 　it depends on the correlation between the security and others in the portfolio. 　Two problems with making P.T. practical: 　(1) estimating all the expected returns standard deviations and correlations; 　(2) figuring out which portfolios are efficient that is 　which give maximum return for a given risk. 　The second problem - how to compute efficient sets - was solved in the 1950s. The first problem was not really studied until later decades (and so will be discussed later in the talk.) 　In the mid-1950s I developed a computer program called the critical line algorithm (CLA) 　which traced out the efficient frontier for large numbers of securities. With modern technology a personal computer using the CLA can trace out an efficient frontier in moderate time for a universe of hundreds of securities. Exact numbers depend on which PC and how many hundred securities and certain other characteristics of the problem. Also 　now other algorithms are available. We need not go into the pros and cons: the important point is that the problem of tracing out an efficient frontier is solvable quite economically once the requisite estimates are made for anyone with access to modern computers even PCs. 　There is one important feature you should understand concerning these portfolio selection programs: they allow you to specify constraints on choice of portfolios 　e.g. upper bounds on individual holdings upper bounds on groupings 　upper bounds on turnover. Simulations and actual practice show that the settings of these constraints are quite important. For example 　using the same methods of estimation but setting the turnover constraint a bit tighter can make the difference between a strategy which beats the market nicely in the long run and one which doesn't. 　The problem of computing the efficient frontier from estimates 　perhaps subject to your choice of constraints on the portfolio 　was solved in the 50s. I did almost nothing to solve the problem of how to form estimates. Where did I think these estimates would come from? In my 1952 article (I envisioned 2 strategies: First Security Analysis 　which would be the responsibility of security analysts and then Portfolio Analysis 　which would calculate the efficient frontier based on the data provided by the analysts.) However 　security analysts of the day were not interested in estimating Eri much less V 　or cov. (I have been) told a story of a firm which operated a buy/hold/sell policy 　but couldn't possibly know it it's a buy even if they were willing to estimate Eri. It all depends on other opportunities (Eri + Vri of other securities and their correlation). 　Since security analysts were not interested 　the persons interested in portfolio analysis had to do it for themselves. The first work concerned estimation of covariances. This seemed to be the most urgent problem since there are so many n = 200 => 200 Eri 200 Gri 19 900 Cov. 　It seemed in the late 50s and early 60s that we could hope to persuade security analysts to provide 200 200 but not 19 900. (i.e.: for 200 securities there will be 200 Eris and 200 Gris but 19 　900 covariances. Security analysts would tell you the Eris and Gris but not the covariances.) In my 1959 book pp 99-100 　I suggest that what is needed is some kind of model of covariance. I proposed what is now called the "one factor model." This was extensively explored by William Sharpe in his Ph.D. dissertation 　and became the industry standard for about a decade from the middle 60s to 70s. What is the one factor model? Erase your imaginary blackboard. 　Draw X- and Y-axes; label the X-axis % change in market index and the Y-axis return on security. 　Draw a line marking a combination of % change in index and return on security for some period - usually for the past month. It is common to use 60 months. 　The slope of the line is the 'beta' of the security. 　Since return for a period of one month is mostly capital gain we can interpret this as : 　% change in price of stock / % change in index 　The one factor model assumed that this is the only source of correlation between pairs of securities. During the late 60s 　there was some evidence that there were other sources of correlation 　but it was considered that they were minor and could be ignored. In the mid 1970s strong evidence presented itself that there were other sources of systematic risk that could not be ignored. 　This was demonstrated by what happened to Merrill Lynch and Oliphant. They worked on the principle that "if you think the market will go up 　buy high beta." The market did go up as measured by the S+P 500 　but high beta stocks went down. Clearly there were other sources of systematic risk which couldn't be ignored. 　Barr Rosenberg's dissertation advocated a many factor model which became the industry standard for many years and perhaps still is. 　There are other methods of estimating covariances that have been proposed over the years. 　E.g.1: Steven Ross uses a different method of obtaining factors for a many factor model. 　E.g.2: Historical covariance matrix. There are theoretical objections about this method: there is not enough data to estimate that many individual covariances. (Less data is required for estimating coefficients of factor models.) This method is true in principle. However recently Hackanson and Trower and especially N. Haugen 　have shown that historical covariance works quite well if you put upper bounds on holdings: i.e. 　it forces the portfolio to be well diversified. Our own research at Daiwa confirms this and adds the note that it also helps to constrain turnover: i.e. it forces diversity and won't let the portfolio move too much from one day to next. The above results (Hackanson Trower 　Haugen and Daiwa) are based on what is called "back tracking" or "historical simulations." This is standard procedure in the P.T. industry before using procedures in practice. We would like to know 'How will (a portfolio perform)?' but can only ask 'How would (the portfolio have performed)?' 　Back testing: (this allows us to avoid a survival basis and look ahead.) 　There are important questions to ask about any specific back tests. 　(If I were in a classroom I would go to the blackboard and write down some questions: e.g.): 　(1) What transaction cost (T.C.) is used in the simulation? T.C. = commissions + market impact 　This concerns the size of orders and the excellence of execution. 　You could try using more than one T.C. to see sensitivity (if you have a simulator with a fast optimizer). 　You should simulate for a long period as well as taking a look at unusual periods. (2) What are E r g 　and the Sharpe ratio for a long period? (I will discuss later if time permits) (3) Ex-port vs. Ex-??? 　(4) What frequency of reopt has been assumed? 　Compare results with greater or less frequent reopt. 　Ask to see a plot of return vs. time; or perhaps (return - index) vs. time More generally in any historical simulation Eri 　Cov ri Optimizer Frontier (Pick a point) Constraints 　Ask yourself 'How Each': i.e. how each compares with the alternatives. Sharpe Ratio: 　How to pick out an investment company. (This ratio was worked out by portfolio theorists in the 1960s.) 　You might pick an investment company that made the portfolio grow most. Portfolio theorists would say no to this choice. You should do the following calculation: (This calculation assumes you will put all your yen into either yen or one investment company. How to pick one? ) (Table) 1 2 3 4 5 　Investment Augmented Subtract Standard Ratio of 　Company Return short-term Deviation 3:4* interest rate (or range) (*) adjusted for changes in management etc. 　Column 5 represents the Sharpe Ratio; you should choose the firm with the highest ratio. 　Why is this the case? You can see why if you imagine two investment companies A and B 　when B manages the same portfolio as A plus 10% cash. 　You want a way of measuring the two firms which considers these different holdings to be equally good. The two firms get exactly the same performance in every period except that A will hold more cash and B less (subtracting the interest rate in the calculation for return on cash.) 　Similarly if any two investment companies have the same Sharpe Ratio you can get the same risk and return on portfolio as a whole by holding one + more cash rather than the other + less cash. If one has a higher Sharpe Ratio you can have the same average and lower standard deviation (or the same standard deviation and a higher average than the other) by adjusting cash so that _________ 　So if you go to select one investment company you should compute the Sharpe Ratio. 　I have described some research into cov. and also research into how to estimate Eri. In fact you can't do a historical simulation without estimating Eri or something to serve in its stead. Various techniques have been tried in practice as well as simulation and more than one has been successful. 　Historical average returns: not bad (I'm a bit surprised about this) 　but see Hackanson and Trower's research. 　Use of fundamentals such as balance sheet information 　One of the conferences at which MPT people discuss their experiences is the semi-annual Berkeley Conference 　Other methods use analyst earnings estimates and estimate dbs So 　what you want to know of your back test is what method of estimating E and V they have used (insofar as they are willing to divulge this); what TC costs have been assumed; and what constraints are employed etc. Warnings: 　Watch out for data using some methods which allow various pay-out settings. If you explore enough you will find a security that did well in the past 　but typically won't in the future. Frequently 　elaborate methods work beautifully until you bet on them. Warning 2: 　Don't expect to get rich quick. U.S. pension funds are delighted to make an extra 2% more than the market average (200 basis points) for the same standard deviation as the market. (There is a big theoretical debate as to whether even this is possible: I think it is.) 　Having pointed out complexities and given warnings 　I should say a few words in defense of the method. 　Start with the notions of risk and return on portfolio. We need to estimate Eri Vri 　and Cov. We would like to know which stocks will rise 　but can only ask which did. The reason we can ask this is because of a data base and computer programs 　and having estimation methods which have worked and seem plausible enough to continue. We are asking how do we combine securities in a portfolio 　given the latest data and current prices. There must be an element of doubt about estimation methods. We want to constrain the portfolio 　not to invest in various ways. In principal if we could trust our estimates we would let the optimizer decide the amounts. But we can give instructions that if the estimation procedure wants more than say 2% or 3% in some security not to let this happen. 　I hope I haven't made it sound too complex or too worrisome. If you can 　try it - not as a black box but as a way of you and the computer and the data base together exploring portfolio selection methods.