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# Mean Variance Optimisation in the Market

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Mean-variance theory was developed in the 50’s and 60’s by Markowitz, Tobin, Sharpe, and Lintner, among others. It is still called Modern Portfolio Theory (MPT) by some people. Mean-variance theory continues to be the main workhorse on which analytical portfolio management is based. The equilibrium version of mean-variance theory is called the Capital Asset Pricing Model (CAPM). The goal of the theory is to optimally invest funds in wide variety of assets. It is a quantitative tool, allowing making investment decisions by considering the trade-off between risk and return.

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“Mean Variance Optimisation in the Market”

There are single and multi period mean variance optimisers. Single-Period MVO considers designing of portfolio for single upcoming period and maximising return considering presumed level of risk. Multi-Period MVO is a strategy, rebalancing portfolio at the end of each period. 5.1 Single Period Problem [11]Inputs: o The expected return for each asset o The standard deviation of each asset (a measure of risk) o The correlation matrix between these assets Output: o The efficient frontier, i.e. the set of portfolios with expected return greater than any other with the same or lesser risk, and lesser risk than any other with the same or greater return. 5.2 Multi-period Problem Input: o The full historical data set Desired output: o The Geometric Mean Frontier; i.e. the set of rebalanced portfolios with greater geometric mean return than any other with the same or lesser standard deviation, and lesser standard deviation than any other with the same or greater geometric mean return 5.3 Benefits of MV optimiser [12]Satisfaction of client objectives and constraints. It is possible to integrate client constraints and objectives in MVO, which enables individual approach to each client, dependant on his risk acceptance level. Control of portfolio risk exposure. It is possible to control risk exposure of portfolio with the help of MVO. Implementation of style objectives and market outlook. It is possible to reflect company s investment style, philosophy and market outlook in MVO. Efficient use of investment information. MVO is designed to reflect all the relevant information available for the optimisation of the portfolio. Timely portfolio changes. MVO is capable of processing large volumes of data very quickly, which is useful for large organisation, that need to incorporate this information in there portfolios in order to reflect the changes in environment. Portfolio management is the process of managing assets of a mutual fund, including choosing and monitoring appropriate investments and allocating funds accordingly. A mutual fund is an open-ended fund managed by an investment company which raises money from shareholders and invests in a group of assets. Mutual funds raise money by selling shares to the public.

[1] In return for the money, shareholders receive an equity position in the fund and, in effecting each of its underlying securities. Investors often hold more shares in more than one company and there are two approaches to portfolio management. The two types are: Active Passive Maximising the expected utility of the excess return over a chosen benchmark is known as active portfolio management, whilst passive portfolio management just tracks the benchmark. The simplest example of passive management is the index fund that is designed to replicate exactly a well-defined index of common stock, such as S&P 500.The fund buys each stock in the index in exactly the proportion it represents the index. If J.P. Morgan Chase represents 3% of the index, the fund places 3% of its money in J.P. Morgan Chase stock. Every portfolio manager needs to decide if s/he will approach an active or passive approach to portfolio management. Factors that needs to be considered when choosing a portfolio strategy are the following: Cost/ Fees It s much cheaper to run an indexed portfolio than an actively managed one. Every time a portfolio manager is trading on behalf of an investor and is adjusting the portfolio the investor is charged a fee, which is often a substantial amount of money, and sometimes can prove to be too costly if the returns are not high. Taxes – Index funds sell stock only when companies enter or leave the index they re mirroring, which is infrequent, so shareholders rarely incur capital-tax gains tax. In contrast to active management, every time a sale has made a profit shareholders are liable for capital gains tax. State of the economy some firms may be very sensitive to the state of the economy, some not so. The index decreases during a recession, so investors need to be aware of which companies are very susceptible to the state of the economy.

For example during a recession share value of Tesco may not decrease as much as share price of a manufacturing firm during a recession. The reason being is that people still need to eat and so Tesco will have sales no matter what the state of the economy. Time Horizon Investors that want to invest in the long term, go for index, and will track the market.

For example the S&P 500 is higher today (even with the looming depression) at 1360.54 compared to 3 years ago at 1171.36 open on 31/1/05 (FTSE charting) Source: www.ft.com Investors investing over a short time period, such as day traders will actively manage a portfolio and be selling and buying shares very quickly and following the market very closely to make as much money as possible as quickly as possible. The time horizon can coincide with the goals they are trying to achieve, be it long, medium or short term. If the investor is faced with a short term goal, then it is possible that the investor will approach an active management strategy to make the most money as quickly as possible. Having an active approach over a long period of time can prove to be costly. Attitude towards risk an investor s attitude towards risk will play a integral part in the decision of managing a portfolio passively or actively. If I was an investor and my attitude was risk loving I would actively manage a portfolio, as this involves picking stock, based on forecasts of the stock being under priced and market timing. The believe, is the higher the risk the higher the return. However if I were a risk averse investor I would not like to run the risk of individually picking stock and would prefer to spread the risk by investing in a well-defined Index. Both strategies have its positive and negative points and the approach taken by the investor is depended on personal preferences to risk and the situation the investor is facing. I, personally, would take the passive approach to portfolio management.

The reasons behind this choice is that I am risk averse, so I would not like to run the risk of picking stock in the believe that the market has not been able to price the stock correctly and instantaneously to its true value. The costs of managing an active portfolio are high. These costs include transaction costs and taxes. So to reduce my costs of managing a portfolio I would choose a passive approach so that I am not paying out to brokers and to the government. Lastly I believe that the market is efficient and so this does not allow big amounts of profits to be generated as the market corrects any inefficiency. Q1b) The CAPM provides the relationship between an investments systematic risk and its expected return. The treatment of risk in the CAPM refines the notions of systematic and unsystematic risk developed by Harry. M. Markowitz

[3] Unsystematic risk is the risk to an assets value caused by factors that are specific to an organisation, such as quality of management, R&D, marketing effectiveness. A fundamental principle of portfolio theory is that unsystematic risk can be diversified away. Systematic risk is risk that cannot be diversified away.

This risk represents the variation in an assets value caused by economic fluctuations. The market only provides a return for risk that cannot be diversified away. The CAPM assumptions can be divided into two groups. First set of assumptions are for the investors and the second is for the financial market environment within which investments are bought and sold. The assumptions behind CAPM for investors are 1. Investors are rational, risk averse and utility maximisers 2. Investors perceive utility in terms of return. 3. Investors measure risk by the standard deviation of return 4. Investors have a single period investment time horizon 5. Investors have the same expectations about what the future holds. The assumptions for the financial market are perfect and that 1. there are no taxes 2. there are no transaction cost 3. investors can both lend and borrow at the risk free rate of return. As one can see these assumptions are highly restrictive and not a real representation of reality, as we live in complex world. Yet with these set of assumptions the model still gives an accurate picture, to a certain extent, of what investors experience when investing. I will go through all the assumptions in turn and explain what the result will be if they are untrue. Firstly in reality investors may be rational but not all are risk-averse, some are risk-loving and will be high risk investors in the hope to reap in high returns.

This may not be the most efficient portfolio or share and as a result the investor will not be on the Capital Market Line. Secondly if the assumption of investors perceive utility was untrue then it would not be possible to measure the efficient portfolio set. Whilst return is a dominant factor in determining utility other factors come into play. An investor will consider utility factors such as: will this investment complicate my tax calculations Would I be better off consuming this wealth instead? Could the investor gain any lifestyle benefits from this investment (ie. Art) Thirdly, not all investors understand statistics and hence do not know what standard deviation is so investors do not look at the standard deviation of return. Alternatively they can see risk in terms of the state of the economy, recent business conditions and trading information. As a result the CAPM should not solely rely on standard deviation of returns, as the measurement of risk is more complex and certain aspects are difficult to measure. In addition, investors rarely have a single time period of investment time horizon. All investors are different and have different time horizons of investment. For this reason this assumption makes the model static, but as we very well know the world is dynamic in which investors are free to manipulate their portfolio. One of the other assumptions of the model is that all investors are homogeneous and have the same expectations of the future. Clearly this is untrue in the real world.

Not all investors share the same view; some are optimistic others are not so. As a result this would lead to different investors to hold different portfolio sets. We have seen what the results are if the assumptions of the investors are untrue. Now I will move on to discuss what the results will be if the assumptions of the financial market are untrue. The figure below describes portfolio opportunities. The assumptions under CAPM The CAPM is important as it quantifies and prices systematic risk and expresses it relative to the market portfolio.

Thus CAPM provides us with the expected return of any asset or portfolio based on its risk as measured by beta, the risk premium of the market, and the risk free rate. This model is constructed in a hypothetical world based on several rigorous assumptions, but like in any economic model, these simplifications are essential in developing a workable model in a complex and diverse financial world. The assumptions of CAPM and deriving the CAPM through a simple proof, as pointed out in Elton and Gruber

[1] are as follows. The first of all assumption to take is there are no transaction costs.

This assumption implies frictionless markets; however in reality transaction costs play a varied part in investment decisions. If transaction costs were present, the return from any asset would be a function of whether or not the investor owned it before the decision period. Thus transaction costs play an important role in reality as in most cases as it is not instantaneous shifting one portfolio of assets to another, and also the delivery costs may dissuade an investor even though it might be an included to form an optimum portfolio. Portfolio theory is also called modern portfolio theory which was developed by Harry Markowitz in his famous paper Portfolio Selection in 1952. Before he developed this theory, investors had no specific measurement of risk. It is the theory that explains why the investors should choose several investments, rather than put all of their funds in just one. In his paper, he used some scientific methods and logical reasoning to demonstrate how the efficient portfolio theory works. It includes some assumptions to reinforce the theoretical model and applying the deductive and inductive reasoning to develop the theory. Also he used some mathematical ways to explain how the model works underlying the assumptions. Under these methods, it displays how the investors should select their efficient portfolio of investments. Before analyzing the portfolio theory, I should explain and illustrate some terminology which relates to the scientific and logical reasoning, including theory, model, assumption, inductive and deductive reasoning. One definition of assumption from the Oxford English Dictionary is That which is assumed or taken from the granted; a supposition, postulate . From the definition, we can see the term of assumption is a statement or a fact to consider as the truth. In other words, it is an idea to be accepted without the experimental proof as the basis of argument. Assumption is a basic condition as the part of logical analysis and comes into being before the theory formed.

Moreover, the development of theoretical model is underlying the assumptions. So it is a limited statement that cannot be treated as the principle and theory. As the example I mention above, before Newton developed the Gravity Theory, he used the existence of earth gravity as the assumption and then proved it through the lots of experiment. There are lots of explanations of model in the Oxford English Dictionary. One of them relates to the topic I discuss by explaining model is A simplified or idealized description or conception of a particular system, situation, or in the mathematical terms, that is put forward as a basis for theoretical or empirical understanding, or for calculation, prediction, etc . Although model also can be seen as a system, it is not a universal idea like theory. A model is a scientific statement through some experimental proof and is only accurate in the limited situations. In addition, a model also can be seen as a part of proof in the process of inference in the theory and can be described as a mathematical way. For instance, Helium atom model shows nucleus with two protons and two neutrons orbited by two electrons. There is no doubt that the effective argument needs the premises by logical steps to construct the valid conclusion. For a logical argument can take two shapes: Deductive and Inductive reasoning. In deductive reasoning, if the premises are true, the conclusion is impossible to be false.

Although it is valid from the proof, it can also lead a false conclusion, the reason is that the premise is incorrect. So the validity of the deductive reasoning depends on the truth of the premises. For example, the sum of the angles for any triangles is 180 degree, so we can conclude that a triangle whatever the shape is, its angle must be 180 degree. In inductive reasoning, the premise with some degree of the probability supports the conclusion, but do not sure the validity of the conclusion. In fact, it is not logically valid and can never be used to provide proofs, because the conclusion is drawing under the experience and observations. In some logical terms, it defines as from general to special . For instance, one day you see lots of people who do the presentations wearing the suits in the university. Then you conclude the people who wear the suits in university must have the presentation today. Obviously, the conclusion does not make sense. If people who have the presentation or not, it can not depend on wearing the suits. Instead, inductive reasoning need collect lots of information and evidences, in order to reach an appropriate conclusion. So the development of theory, it needs to construct propriety assumptions and using some theoretical models and logical reasoning to demonstrate them.

From the definition of theory, we can conclude the modern portfolio theory is an acceptable general principle to explain the investment behavior and investors can choose the efficient portfolio base on both the expect return and variance of return. In the paper Portfolio Selection , Markowitz also applied these logical and scientific methods to develop his theory, I will discuss below. Markowitz noted that there are two stages in the process of selecting a portfolio. The first stage starts with observation and experience and ends with beliefs about the future performances of available securities . Stage two starts with the relevant beliefs about future performances and ends with the choice of portfolio. In his paper, he considered the process of second stage. Firstly, he rejected the hypothesis that the investor does (or should) maximize discounted expected, or anticipated returns because it never implies that there is a diversified portfolio which is preferable to all non-diversified . Then he applied the reductive reasoning to explain the rejection of the hypothesis in terms of observed experience and logical reasoning.

From the observed experience he referred to diversification is both observed and sensible . However the rule of behaviors does not imply the superiority of diversification . Then he concluded this hypothesis should be rejected. In other words, the argument gives a premise that any acceptable theory attempts to account for investment behaviors must not contradict observed experience. However, the hypothesis contradicts the observed experience. According to the deductive reasoning, the conclusion is the hypothesis is not an acceptable theory to explain the investment behaviors. Markowitz noted that there are two stages in the process of selecting a portfolio. The first stage starts with observation and experience and ends with beliefs about the future performances of available securities . Stage two starts with the relevant beliefs about future performances and ends with the choice of portfolio. In his paper, he considered the process of second stage. Firstly, he rejected the hypothesis that the investor does (or should) maximize discounted expected, or anticipated returns because it never implies that there is a diversified portfolio which is preferable to all non-diversified . Then he applied the reductive reasoning to explain the rejection of the hypothesis in terms of observed experience and logical reasoning.

From the observed experience he referred to diversification is both observed and sensible . However the rule of behaviors does not imply the superiority of diversification . Then he concluded this hypothesis should be rejected. In other words, the argument gives a premise that any acceptable theory attempts to account for investment behaviors must not contradict observed experience. However, the hypothesis contradicts the observed experience. According to the deductive reasoning, the conclusion is the hypothesis is not an acceptable theory to explain the investment behaviors. From the logical reasoning, the rejection of the hypothesis also can be proved. Markowitz said the hypothesis implies that the investor places all his funds in the security with the greatest discounted value . Then he used the mathematical way to analyze the discounted return of two or more securities and give the conclusion that there is no evidence to show a diversified portfolio preferred to all non-diversified portfolios. So we rejected the hypothesis. It is also an example of deductive reasoning. After the rejection of the hypothesis, Markowitz indicates one of variation of hypothesis that the investor should diversify and that he should maximize expected return . He argued that adjusted hypothesis depended on the assumption that there is a portfolio which gives both maximum expected return and minimum variance . From the observational evidence, the assumption is not unreasonable, because the return on securities are too inter-correlated so that this kind of portfolio is difficult to exist. Then he concluded that the portfolio with maximum expected return is not necessarily the one with minimum variance . It can be seen as the example of reductive reasoning, Due to the hypothesis is inadequate to explain the investors behaviors.

Markowitz developed a new hypothesis, which is investor does (or should) consider expected return a desirable thing and variance of return an undesirable thing and used Expected Return-Variance of Return Model (E-V rule) to explain it, Under the E-V model, Markowitz used a number of assumptions to reinforce the theoretical model. Firstly, the investors consider the return on investment alternative as being represented by a probability distribution of rates of the investment outcome. Secondly, investors measure the risk using the variance of return. Also in a portfolio of shares, they calculate the risk by using the covariance computations. In addition, investors are willing to basing on decision on the two parameters the expected return and variance of return in their investment.

Finally, it assumed the risk-averse behaviors; an investor will select the efficient portfolio. For a given level of risk, investors prefer maximum expected return. Similarly, for a given expected return, investors prefer minimum risk. (Richard Dobbins, Stephen Witt and John Fielding, 1994) Markowitz argued that the E-V rule is used both as a maxim to guide the investment behaviors and as a hypothesis to explain it. Also using mathematical analysis to show the investors can select the efficient portfolio from the combinations of expected return and variance. One reason of rejecting the expected return rule that is never implies the superiority of diversification. So Markowitz used the deductive logic to conclude it does not mean that the E-V rule never implies the superiority of an undiversified portfolio . He indicated the one particular undeversified portfolio would give maximum expected return and minimum variance . However, most of efficient portfolios are diversified which is underlying the E-V rule. In the end, he concluded not only does the E-V hypothesis imply diversification, it implies the right kind of diversification for the right reason . He also indicated that the adequacy of diversification is not thought by investors to depend solely on the number of different securities held , because in the real world, the same industries have the similar economic characteristics. It is likely for the same industries to performance badly at the same time. Using the point as the premise, according to the inductive reasoning, he concluded that diversification depends on the investment on the different kind of industries. Moreover, the firms in different industry have lower covariance than the firms in the same industry. In order to avoid the high covariance in the portfolio of securities, the investor should select the securities in the different industries. Portfolio theory that was established in the early 1950s has five basic assumptions, which are: Agents prefer more wealth to less. Agents are risk-averse and require a higher expected rate or return for taking on more risk. The rates of return follow a normal distribution.

This means that the risk of an individual security can be measured by its standard deviation. The wealth-holder cannot affect the probability distribution of the security held. The theory considers only existing assets not new issues. (Keith Pilbean, finance & financial markets, published 1998, page 130). The return of assets on a portfolio can be observed by the next expression: N Rp=SwiE(Ri) I=1 Where we can say, that Rp is the price weighted return on a portfolio of risky assets, wi is the price-weighted proportion of a portfolio spent in asset i, Ri is the return on the asset i in the portfolio and N is the amount of securities in the investors portfolio. Furthermore, the average expected return on the portfolio is shown by the following expression, where E (Ri) shows the average expected return on asset I, and the E (Rp) is the average expected return on the portfolio. N E(Rp)=SwiE(Ri) i=1 isk/standard deviation When a portfolio dominates another, it will happen because either it has the same standard deviation, or it has a lower standard deviation and the same expected rate of return as another portfolio, and this is called the dominance principle. We can read from the graph that portfolio A dominates B and B dominates C and D. But not all combinations are of interest since some of them dominate others. In this case, we can see that A dominates C and E dominates portfolio F, so our attention focuses only in the thick black part of the efficiency frontier.

The portfolios that dominates the others portfolios is the so-called efficiency frontier. Furthermore, all dominated portfolios are called inefficient portfolios and the portfolio that for a given level of risk offers the highest possible return is called efficient portfolio. A technique, which is known as quadratic programming, and comes from the Markowitz method, can give us the mathematical solution for the portfolio efficiency frontier. The basic ingredient in this technique is to calculate a reachable target rate of return X% and then find a set of weights wi, which attains this target with the minimum of variance, which is illustrated next: N N N s p=Sw i s i +S S wi wj sij I=1 I=1 I=1 i =j Then by using the diversification, the risk can be reduced until the minimum risk of a given return is reached. This can be achieved by minimizing the correlation among the rates of return on two or more securities.

However, this method has a major problem, which has to do with the amount of covariances that have to be calculated. In N securities the formula that is used is (N – N)/2 and we think of 500 securities then the covariances goes over the number of 120.000. but even in N securities the risk of every combination will be less than the weighted average of the securities that make up the portfolio. The MVO Scenarios Commodities Scenario -1 Bonds and Equity Scenario (2) Commodities Scenario -3 Forex Scenario -4 Minerals Scenario -5 Security scenario – 6 Securities Forex Scenario 7 Insurance Scenario 8 Mutual Funds Scenario -9 Real Estate Scenario -10 Elton and Gruber Modern portfolio Theory and Investment Analysis p285 l CAPM Empirical Problems and the Distribution of Returns p. 1 l Eugene F. Fama and Kenneth R. French, The Capital Asset Pricing Model: Theory and Evidence l Cheol S. Eun The Benchmark Beta, CAPM, and Pricing Anomalies Oxford Economics Papers 46(1994)

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[1] Elton and Gruber Modern portfolio Theory and Investment Analysis p285

[2] CAPM empirical problems and the distribution of returns p. 1

[3] Elton, Gruber, Das and Hlavka, 1993

[4] Eugene F. Fama and Kenneth R. French, The Capital Asset Pricing Model: Theory and Evidence

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