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Mean Variance Optimization |
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Purpose of Mean Variance Optimization Mean Variance Optimization is a technique from the field of portfolio optimization. The Mean Variance Optimization technique is originally based on the work of finance professor Harry Markowitz. The goal of the Mean Variance Optimization technique is to construct a portfolio of assets that minimizes the variance of the total portfolio for a given level of return or equivalently, to maximize return of the portfolio for a given level of variance. The technique uses estimates of future standard deviations and returns from each asset and correlations between the individual assets, and a target return. For two assets the number of inputs would be two standard deviations, two returns and two correlations: six total inputs. For three assets the number of inputs would be three standard deviations, three returns and three correlations: nine inputs total. The outputs would be the weights of the individual assets that give the above portfolio so that the portfolio manager would know in what percentage to mix the individual assets. Mean Variance Optimization was a big step in finance because it illustrated that one must consider how assets move with respect to each other when considering the risk of an entire portfolio as opposed to just considering how assets move individually. Assumptions of Mean Variance Optimization Here are some of the assumptions behind the mathematical technique of Mean Variance Optimization: 1) Assume the measure of risk of an asset (stock, bond, exchange traded fund, etc) is standard deviation. 2) Assume that past returns and past standard deviations can be used as a proxy for future returns and standard deviations. 3) Assume assets have fixed correlations between them. 4) Assume asset returns are normally distributed. 5) Assume all investors (market participants) are rational so all assets are always correctly priced. The actual technique is something from mathematics called constrained optimization which involves differential calculus, Lagrange multipliers and possibly some linear algebra. So for the non-mathematician, the details of the technique can be difficult. It is very important to note that in all cases subjective future estimates of standard deviations and returns and correlations are being input into the Mean Variance Optimization algorithm. Risk of using Mean Variance Optimization Incorrectly The risk of using Mean Variance Optimization for commercial portfolio management is that it is a technique that is mathematically elegant but with very unrealistic assumptions. Also risky is that the model is very sensitive to changes in the inputs. Hence investors who might not be very strong in math or common sense might be impressed by the model because it is presented with mathematical rigor and its results are presented with precision, even though the technique uses very imprecise inputs and the results are very sensitive to changes in the inputs. But with weak or unrealistic assumptions, a mathematical model can only be true by coincidence. All mathematical models are only approximations of reality and the Mean Variance Optimization is no exception. And a model with poor assumptions could lead to very unpredicatable (poor results) if used in practice. Comments on the assumptions of Mean Variance Optimization Model 1) Standard deviation as a measure of risk is only one possibility as a measurement of risk. Most investors think that upside standard deviation would be a good thing but that downside standard deviation would be a bad thing. Also many investors consider the word "risk", to mean the possibility of a permanent loss of wealth due to some event happening. So it is very possible that a more useful measure of risk should include a definition that incorporates the concept of downside loss and a permanent loss of wealth. 2) In practice, choosing recent returns and standard deviations as an estimate for future returns could be very dangerous because of something called regression to the mean. Basically last year's winner might only have normal returns or losses this year. And last year's loser could have great returns this year. So using recent results could actually have the opposite effect of what you trying to achieve. 3) Correlations between asset classes have changed over the years and in a financial crisis assets become even more positively correlated. 4) Standard deviations in markets move widely, sometimes more than three to six standard deviations from the mean which is more than a normal distribution would predict. 5) The dotcom era in the United States and also the real estate bubble in the United States are examples from recent financial history that examples that investors are not always rational. General problem of poor or incorrect assumptions Here are some examples that illustrate the general problem of building arguments, theories, and plans of action on poor or incorrect assumptions. Example 1: Imagine an army preparing to defend a particular beach from invasion. The army takes the most stringent and detailed steps to defend the beach. But then the invaders invade a different beach. Example 2: Imagine a beautiful physics theory built on the assumption that the speed of light is different for observers moving with respect to one another. The theory exists for many years and is commonly accepted by the scientific community. But the theory cannot explain many different natural phenomenon.Then a physicist comes along and makes a different assumption. That the speed of light is the same for all observers no matter how they are moving with respect to one another. Because of this change in assumptions, the theory now works much better than before. Example 3: Imagine a country deciding on whether or not to go to war. The argument for going to war is based on the assupmtion that the potential enemy has weapons that are capable of severely damaging the first country. But the beginning assumption that the potential has these destructive weapons is unproven. The arguments created to support going to war are built on this unproven assumption and are colored in elogent rhetoric. People who question the assumption that the second country has such destructive weapons are labeled as traitors and ostracized by politicians and citizens in the firs country. Hence the first country goes to war against the second country and destroys it. And destroys part of itself and part of its own reputation. But no destructive weapons are found in the second country. Maybe they never existed. Hence the entire argument for going to war might have been based on incorrect assumptions and was based on unproven assumptions. Human History is full of groups of people following arguments based on poor assumptions. So finance is probably not immuned. Mean Variance Optimization as used in the commercial world The Mean Variance Optimization technique is incorporated into modern portfolio optimization software and is sold to money managers. You can even do Mean Variance Optimization on your laptop using Microsoft Excel and Excel's solver module. So what is Mean Variance Optimization good for? Mean variance optimization is a good learning tool to show how a portfolio could be constructed if the assumptions held. Mean Variance optimization also illustrates quantitatively how variance of a portfolio can be reduced by reducing correlations between assets. This was a huge leap forward in finance. Mean Variance Optimization could be useful in testing what-if scenarios using different combinations of inputs rather than just one set of inputs. For example given many different combinations of standard deviations, returns and correlations, what would the portfolio returns look like? Given a target return, how would the inputs of different standard deviations, correlations, and individual returns affect the weights of the individual assets? Mean Variance Optimization is also useful in showing how a portfolio looks in a worst case scenario. For example, what if there was a situation where all assets suddenly became correlated for a short time with large variances in the individual assets and large negative returns for each individual asset, like for instance what happened in the global financial crisis that began in 2008. Uses such as those above could be very useful to show to clients as far as making sure their expectations were properly managed and that they don't panic as much if/when another financial crisis repeats itself. Conclusion Mean Variance Optimization is a great pioneering academic work that motivated researchers to study the concept of portfolio management in a quantitative way. Mean Variance Optimization has useful academic teaching uses. Mean Variance Optimization has good commercial uses insofar as managing client expectations of how a client portfolio could look under various circumstances. Mean Variance Optimization may have better commercial use by mixing the technique with random combinations of inputs to show what portfolios can look like under different circumstances. Mean Variance Optimization also introduces the ideas of matching a portfolio with certain risk-return characteristics to investors with a certain risk tolerance. As a practical tool taken by itself and used with naive inputs, Mean Variance Optimization is not optimal for use in commercial portfolio management. If used and presented improperly, Mean Variance Optimization will not manage a client's expectations properly and could lead to disappointment and the loss of clients because the clients will be expecting a certain return and may not get it. Clients with more worldly experience and more common sense might laugh at you, if you, the portfolio manager present Mean Variance Optimization as your only portfolio management tool. Especially if these clients are older or who have experienced unanticipated losses or who have lived through a repressive government or financial situation. As a measure of risk, standard deviation (variance) as used in Mean Variance Optimization is very limited. Many clients will become confused if an advisor talks about risk as also having an upside component. In our opinion, Mean Variance Optimization has a hint of "focusing on the details" but not paying attention to the big picture. We ascribe this to the fact that Mean Variance Optimization was created in the world of academia and not created in the commercial investment world. This would be the difference between theories about business taught in the classroom Vs the practice of business in the real world. Or learning military tactics in the classroom Vs learning military tactics on the battlefield. Or learning about investing and wealth management in the classroom Vs learning about investing and wealth management in the real world. Also in our opinion, Mean Variance Optimization has a hocus pocus or pseudo scientific feel to it when presented as a portfolio selection tool. Like a magic black box that matches the user with the perfect portfolio. The world is not that simple and predictable. Just because something is formulated using mathematics does not mean it works or is scientifically grounded. In fact the use of mathematics could be being used to give a technique the appearance of scientific credibility in order to win non-believers and skeptics over. Final Recommendation Use Mean Variance Optimization in combination with other portfolio management techniques and know the limitations of Mean Variance Optimization. Don't use Mean Variance Optimization by itself, especially if you don't understand the assumptions. |
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