Modern Portfolio Theory and the Markowitz Model

Alex CarrNonlinear ProgrammingModern Portfolio Theory and the Markowitz ModelLouis BachelierFather of Financial MathematicsThe Theory of Speculation, 1900The first to model the stochastic process, Brownian MotionStock options act as elementary particlesBefore modern day portfolio theory there were many scholars who helped pave the way for Markowitz and his peers. Louis Bachelier was the first to unite the fields of economics and mathematics, and has, therefore, been dubbed the father of financial mathematics. Bachelier outlined his model in his thesis, The Theory of Speculation, in 1900. Bachelier believed that stocks acted as elementary particles, and followed the continuois-time stochastic process now called Brownian motion. Thus, Bachelier believe one could forecast the performance of stock options with his model of Brownian Motion.

2John Burr WilliamsTheory of Investment Value, 1938Present Value ModelDiscounted Cash Flow and Dividend based ValuationAssets have an intrinsic valuePresent value of its future net cash flowsDividend distributions and selling priceThe most significant player in the field of financial mathematics after Bachelier was John Burr Williams. He believed that in forecasting, investors must focus on a stocks intrinsic, long-term value. In Williams 1938 publication, The Theory of Investment Value, he opens with: "Separate and distinct things not to be confused, as every thoughtful investor knows, are real worth and market price..." He believed instead of simply looking at market price, one should focus on future corporate earnings and dividends to determine forecasts. He proposed the idea that the value should be calculated using present value of future cash flows in the form of dividends and selling price. Williams believed that the value of a stock should be calculated using discounted cash flow valuation. This aided in developing the dividend discount model.

3Harry MarkowitzMathematics and Economics at University of ChicagoEarlier Models Lacked Analysis of Risk Portfolio Selection in the Journal of Finance, 1952Primary theory of portfolio allocation under uncertaintyPortfolio Selection: Efficient Diversification of Investments, 1959Nobel PrizeMarkowitz Efficient Frontier and PortfolioAfter being encouraged by his thesis advisor at the University of Chicago, Harry Markowitz dedicated his career to studying the mathematical analysis of the stock market, specifically the effects of asset risk, return, correlation, and diversification on expected portfolio returns. He realized that his predecessors models had lacked any analysis of risk. From this he developed his basis theory of portfolio allocation under uncertainty. After his initial publication, Portfolio Selection, in 1952, Markowitz teamed up with George Dantzig to further research optimization techniques, which led to the development of the critical line algorithm for the identification of the optimal mean-variance portfolios, which fell on what would later be called the Markowitz frontier. A few years and much research later, Markowitz would publish a finalized version of the critical line algorithm. In 1959, Markowitz finally published a fully developed theory on portfolio allocation in his first book, for which he won the Nobel Prize among many other awards. The optimal portfolio, or Markowitz Efficient Portfolio, is defined, in his publications, as that where no added diversification can lower the portfolios risk for a given return expectation, or where no additional expected return can be gained without increasing the risk. Additionally, the Markowitz Efficient Frontier is the set of all the portfolios with the highest expected return at every given level of risk. His model is often called the Mean-Variance model because it analyzed the mean, expected return, and variance, standard deviation, of different portfolios.

4FoundationsExpected return of an asset is the meanRisk of an asset is the variability of an assets historical returnsReduce the risk of an individual asset by diversifying the portfolioSelect a portfolio of various investments Maximize expected return at fixed level of riskMinimize risk at a fixed amount of expected returnChoosing the right combination of stocksThe Modern Portfolio Theory is a financial theory that carefully selects a portfolio of various investments assets such that the expected return is maximized at a fixed amount of risk, or the level of risk is minimized at a fixed expected portfolio return. This is also generally known as the mean variance framework. The MPT follows a mathematical formulation that attempts to build a diversified portfolio that has collectively lower risk than one stand alone asset. Diversification is the investment practice of not putting all your eggs in one basket. That is, to reduce risk by investing in a variety of assets. In this model, the expected return of an investment is the weighted average of all possible values that this variable can take on. The risk of an individual asset is the standard deviation of the assets historical returns. Markowitz emphasized the importance of not just simply choosing stocks, but choosing the right combinations of stocks that dont have the same historical trends such that the stocks pay out and dont pay out at different times. If this is done correctly an investor can create a portfolio that will always pay out. 5Model AssumptionsRisk of a portfolio is based on the variability of returns from the said portfolio.An investor is risk averse.An investor prefers to increase consumption.The investor's utility function is concave and increasing.

The model assists in the selection of the most efficient by analyzing various possible portfolios of the given assets. Through diversification and selection of investment assets that do not have the same movement, the model shows investors how to reduce their risk. In his model, Markowitz made certain assumptions. First, he calculated the risk of a portfolio based on the variance of returns from the portfolio. He assumed that an investor is risk averse, prefers to increase consumption, or utility of their terminal wealth, and is rational in nature. Also, an investors utility function is concave and increasing, based on the assumptions stated previously about investors. In this case, the investors utility function based on consumption and the aversion to risk at different levels of consumption. He also assumed that an investor will either maximize expected return at a given level of risk, or minimize risk at a given expected return. Lastly, he used a single period model of investment for all his analysis.

6Model AssumptionsAnalysis is based on single period model of investment.An investor either maximizes his portfolio return for a given level of risk or maximum return for minimum risk.An investor is rational in nature.The model assists in the selection of the most efficient by analyzing various possible portfolios of the given assets. Through diversification and selection of investment assets that do not have the same movement, the model shows investors how to reduce their risk. In his model, Markowitz made certain assumptions. First, he calculated the risk of a portfolio based on the variance of returns from the portfolio. He assumed that an investor is risk averse, prefers to increase consumption, or utility of their terminal wealth, and is rational in nature. Also, an investors utility function is concave and increasing, based on the assumptions stated previously about investors. In this case, the investors utility function based on consumption and the aversion to risk at different levels of consumption. He also assumed that an investor will either maximize expected return at a given level of risk, or minimize risk at a given expected return. Lastly, he used a single period model of investment for all his analysis.

7Risk Standard deviation of the mean (or return)Systematic Risk: market risks that cannot be diversified away Interest rates, recessions and warsUnsystematic Risk: specific to individual stocks and can be diversified awayNot correlated with general market moves

In the Markowitz model, risk is defined as the standard deviation of an asset from its mean historical returns. The risk of an individual asset can be broken up in two types systematic, and unsystematic risk. Systematic risk are market risks, such as interest rates, recessions, and wars, that cannot be diversified away. Unsystematic risk, on the other hand, is risk that is specific only to an individual stock, based on the success of the corporation, that is not correlated to any general market trends. The unsystematic risk of an asset can be diversified. The overall risk of a portfolio, therefore, is based not on the individual risks of the assets, but on the covariance between the risks of the different assets. These covariances allow for an investor to select the optimal share of each of the assets. Investors will benefit from holding two risky stocks rather than one individual stock if they are selected properly. 8

Risk and DiversificationIn the Markowitz model, risk is defined as the standard deviation of an asset from its mean historical returns. The risk of an individual asset can be broken up in two types systematic, and unsystematic risk. Systematic risk are market risks, such as interest rates, recessions, and wars, that cannot be diversified away. Unsystematic risk, on the other hand, is risk that is specific only to an individual stock, based on the success of the corporation, that is not correlated to any general market trends. The unsystematic risk of an asset can be diversified. The overall risk of a portfolio, therefore, is based not on the individual risks of the assets, but on the covariance between the risks of the different assets. These covariances allow for an investor to select the optimal share of each of the assets. Investors will benefit from holding two risky stocks rather than one individual stock if they are selected properly9DiversificationOptimal: 25-30 stocks Smooth out unsystematic riskLess risk than any individual assetAssets that are not perfectly positively correlatedForeign and Domestic InvestmentsMutual FundsDiversification is a risk management technique that mixes a wide variety of investments within a portfolio. This is done because a portfolio ofdifferent kinds of investments will, on average, yield higher returns and pose a lower risk than any individual investment found within the portfolio. Diversification strives tosmooth out unsystematic risk eventsin a portfolio so that the positiveperformance ofsome investments willneutralizethe negative performance of others. Therefore, the benefits of diversification will hold only if the securities in the portfolio are not perfectly correlated. Studies and mathematical models have shown that maintainingawell-diversifiedportfolioof 25to 30 stocks will yield themost cost-effectivelevel of risk reduction. Investing in more securities will still yield further diversification benefits, albeit at a drastically smaller rate. Further diversification benefits can be gained by investing in foreign securities becausethey tendbe less closelycorrelated withdomestic investments. For example, aneconomic downturn in the U.S. economy may not affect Japan's economy in the same way; therefore, having Japanese investments wouldallow an investor to have a small cushion of protection against losses due to an American economic downturn. Most non-institutional investors have a limited investment budget, andmay find it difficult to create an adequately diversified portfolio. This fact alone can explain why mutual fundshave been increasing in popularity. Buying shares in a mutual fundcan provide investors with aninexpensive source ofdiversification. 10Correlation Negative PositiveNone0.09 to 0.0 0.0 to 0.09Small0.3 to 0.10.1 to 0.3Medium0.5 to 0.30.3 to 0.5Strong1.0 to 0.50.5 to 1.0

When choosing certain assets it is important to make sure that you pick stocks that do not have strong positive correlation. That is, you dont want a portfolio of stocks that move in the same direction. In order to maximize returns, a portfolio should be comprised of assets that move in different directions. To show the importance of this, look at a portfolio of all domestic stocks in the same industry. If something goes wrong with the U.S. economy or that specific industry than all your entire portfolio will decrease. But if you have a portfolio of stocks in different industries both foreign and domestic, then when one particular industry or economy is in downturn, your other assets will help balance out those negative effects. 11Expected ReturnIndividual AssetWeighted average of historical returns of that assetPortfolioProportion-weighted sum of the comprising assets returnsFor an individual asset, the expected return can be calculated as the weighted average of the historical returns of that asset. For a portfolio the expected return is the expected value, or mean, of all the likely returns of investments comprising a portfolio.The expected return of a portfolio helps explain a relationship between the risk and return trade-off of a certain portfolio. This assumes that investors will always want to maximize expected return at a given level of risk. To calculated the expected return of a portfolio you take the sum of the weighted proportions of each individual assets expected return. 12

Mathematical ModelEquations to calculate portfolio return, variance, and standard deviation. For I = 1 to n different assets comprising a portfolio. 13The ProcessFirst: Determine a set of Efficient PortfoliosSecond:Select best portfolio from the Efficient FrontierIn the mathematical model of Markowitzs theory one must plot the relationship between the portfolios return at different fixed levels of risk or the portfolios risk at given levels of return. To choose the best portfolio from a number of possible portfolios, each with different return and risk, there are two steps to do. First, the set of efficient portfolios, or the efficient frontier, must be determined. Secondly, the best portfolio on the efficient frontier will be chosen. This most efficient portfolio will be chosen depending on the risk-aversion of the investor, and whether or not a risk-free asset will be added on top of the portfolio.

14Risk and ReturnEither expected return or risk will be the fixed variablesFrom this the other variable can be determinedRisk, standard deviation, is on the Horizontal axisExpected return, mean, is on the Vertical axisBoth are percentages

Either the risk or return will be inputs, and you be able one from the other. The risk, standard deviation, is plotted on the horizontal axis, while the return, mean, is plotted on the vertical axis. The region is first plotted as all possible combinations of assets in the portfolio. As the investor is rational, they would like to have higher return. And as he is risk averse, he wants to have lower risk. The region includes all the possible portfolios an investor can invest in. The efficient portfolios are the ones that lie on the left boundary of the region, which makes a hyperbola. The portfolios on the boundary are most efficient because for the same risk they have a higher return than all others, or for the same expected return they have lower risk. For every level of return, there is one portfolio that offers the lowest possible risk, and for every level of risk, there is a portfolio that offers the highest return. The top half of the left boundary, the section with a positive slope, is called the Efficient Frontier. The Efficient Frontier is the same for all investors, as all investors want maximum return with the lowest possible risk. 15Plotting the GraphAll possible combinations of the assets form a region on the graphLeft Boundary forms a hyperbolaThis region is called the Markowitz Bullet

16Determining the Efficient FrontierThe left boundary makes up the set of most efficient portfoliosThe half of the hyperbola with positive slope makes up the efficient frontierThe bottom half is inefficient

18Indifference Curve

Each curve represents a certain level of satisfactionPoints on curve are all combinations of risk and return that correspond to that level of satisfactionInvestors are indifferent about points on the same curveEach curve to the left represents higher satisfactionFor selection of the optimal portfolio the investor must analyze what level of risk they are willing to take; this is done by creating indifference curves at certain levels of satisfaction. Every investor will have a set of unique indifference curves, because the relationship between risk and return is different for everyone. Every point on a particular curve shows different bundles of risk and return to which the investor is indifferent. Each next curve to the left represents higher utility or satisfaction. The goal of the investor would be to maximize his satisfaction by moving to a curve that is higher. 19Optimal Portfolio

The optimal portfolio is found at the point of tangency of the efficient frontier with the indifference curveThis point marks the highest level of satisfaction the investor can obtainThe point will be different for every investor because indifference curves are different for every investor The investor's optimal portfolio is found at the point of tangency of the efficient frontier with the indifference curve. This point marks the highest level of satisfaction the investor can obtain. The optimal portfolio is at the point where the investor can get the most satisfaction and the best risk-return relationship. 20Capital Market LineE(RP)= IRF + (RM - IRF)P/MSlope = (RM IRF)/MTangent linefrom intercept point on efficient frontierto point where expected return equals risk-free rate of returnRisk-return trade off in the Capital MarketShows combinations of different proportions of risk-free assets and efficient portfolios The option of adding a risk-free asset to the portfolio is now considered. The asset sits at a point on the vertical axis at the risk-free return rate. This is asset is usually a government security because government securities usually have no risk. A line from this point on the vertical axis tangent to the efficient frontier is drawn. This line is known as the Capital Market Line, which represents the risk-return trade off in a capital market. Any point on the line shows a combination of different proportions of risk-free securities and efficient portfolios.RP = Expected Return of PortfolioRM = Return on the Market PortfolioIRF = Risk-Free rateM = Standard Deviation (Risk) of the market portfolioP = Standard Deviation (Risk) of portfolio(RM - IRF)/M is the slope of CML. There, slope measures the reward per unit of market risk.

21Additional Use of Risk-Free AssetsInvest in Market PortfolioBut CML provides greatest utilityTwo more choices:Borrow Funds at risk-free rate to invest more in Market Portfolio Combinations to the right of the Market Portfolio on the CMLLend at the risk-free rate of interestCombinations to the left of the Market Portfolio on the CML

Any portfolios on this new line, except for the point of tangency, have greater utility than any portfolios on the efficient frontier. But the point of tangency is the most efficient portfolio because it lies on both the CML and the efficient frontier, thus it is also the most well-diversified portfolio because it consists of shares and risk-free securities in the capital market. This most efficient portfolio is called the Market Portfolio. All portfolio combinations to the left of P show combinations of investing in the Market Portfolio and lending at the risk-free rate of interest, and all those to the right of P represent purchases of risky assets made with funds borrowed at the risk-free rate of interest.

22Efficient Frontier with CML

23CriticismsThere are a very large number of possible portfolio combinations that can be madeLots of data needs to be includedCovariancesVarianceStandard DeviationsExpected ReturnsAsset returns are, in reality, not normally distributedLarge swings occur much more often3 to 6 standard deviations from the meanWhile the Markowitz model is one of the most lasting portfolio theories there have been many criticisms of it over the years. First the number of possible portfolio combinations is exponential especially when dealing with many assets. There is also a lot of data needed to model this problem, and finding all the different covariances, and standard deviations, etc. can be tedious. Thirdly, the model assumes that returns are normally distributed while, in reality, the market almost never follow a normal distribution. It is much more likely to have a large swing that is 3 to 6 standard deviations from the mean. 24Criticisms Investors are not rationalHerd BehaviorGamblersFractional shares of assets cannot usually be boughtInvestors have a credit limitCannot usually buy an unlimited amount of risk-free assetsAlso, the model assumes that investors are rational. But time and time again we have seen this is not true. Investors can be gamblers that are willing to spend big on high risk. It does not allow for "herd behavior" or investors who will accept lower returns for higher risk. Also, the model assumes that the combinations of assets can be fractional, when usually fractional shares of assets cannot be bought. Lastly, the model does not account for the fact that many investors may have a credit limit when looking at combinations on the CML. Most investors cannot buy an unlimited amount of risk-free assets. 25