The modern portfolio theory

The term ‘portfolio’ is usually applied to combinations of securities, but the principles underlying security portfolio formation can be applied to combinations of any type of assets, including investment projects. Most firms diffuse their efforts across a range of products, market segments and customers in order to spread more thinly the risks of declining trade and profitability. If a firm can reduce its reliance on particular products or markets, then it can withstand more comfortably the impact of a major reverse in any single market. Diversification can generate some major strategic advantage, for example, the wider spread of activities, the greater the potential access to high performing market sectors.

The modern portfolio theory was developed by Harry Markowitz, presenting it in 1952 in an article entitled ‘Portfolio Selection’. Markowitz was the first to show the important benefits from diversification that arise from combining individual securities into portfolios and to demonstrate that the portfolio decision problem of an investor is equivalent to the maximisation of his or her expected utility. Modern Portfolio Theory explores how risk averse investors construct portfolio in order to optimise market risks against the expected return. The theory suggested that we could reduce the standard deviation of returns on asset portfolio by choosing assets, which do not move together.

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Allocating funds to a single security can be an extremely risky investment. The primary reason for investing in portfolios is diversification, that is, the allocation of funds to a variety of securities in order to reduce risk. As the number of securities held in the portfolio increases, the overall variability of the portfolio’s return, measured by its standard deviation, diminishes very sharply for small portfolios, but falls more gradually for larger combinations. This decline in risk is achieved because the exposure to the risk of volatile securities can be offset by the inclusion of low-risk securities or even high-risk ones, so long as their returns are not closely correlated. The key point here is that not all the risk of individual securities is relevant for assessing the risk of a portfolio of risky shares. The total risk of a portfolio is composed of two components:

1. Specific risk. The variability of a security’s rate of return due to factors unique to the individual firm.

2. Systematic risk. The variability of a security’s rate of return due to dependence on factors which influence the return on all securities.

Specific risk refers to the expected impact on sales and earnings of largely random events like industrial relations problems, equipment failure, R achievements. In a portfolio of shares , such factors tend to cancel out as the number of component securities increases. Systematic risk refers to the impact of movements in the macro-economy, such as fiscal changes, swings in exchange rates and interest rate movements, all of which cause reactions in security markets, captured in the movement of an index reflecting securities prices in general. No firm is entirely isolated from these factors, and even portfolio diversification cannot provide total protection against this form of risk. For this reason, it is often called market risk.

Reduction in the total risk of a portfolio is achieved by gradual elimination of the risks unique to individual companies, leaving an irreducible, undiversifiable, risk floor.

Substantial reduction in specific risk can be achieved with quite small portfolios, and the main scope for risk reduction is achieved with a portfolio of around thirty securities. To eliminate unique risk totally would involve holding a vast portfolio comprising all the securities traded in the market. This is called market portfolio and has a pivotal role in the CAPM, but for the individual investor it is neither practicable nor cost-effective, in view of the likely scale of the dealing fees required to construct and manage it. Since relatively small portfolios can capture the lion’s share of diversification benefits, it is only a minor simplification to use a well-diversified portfolio as a proxy for the overall market such as one of the well-known market indices

1. It is clear that risk-verse investors should diversify

2. Investors should not expect rewards for bearing specific risk

3. Securities have varying degree of systematic risk

The rate of return of a portfolio can be described by a probabilty distribution. The assumption is that such a probability distribution can only be characterized by its expected return and the variances of rates of return. The rate of return on a two-security portfolio is a weighted average of the rates of return on the two individual securities in the portfolio, where the weight associated to a security is the proportion of portfolio funds invested in the security.

The expected return on a portfolio E(R) comprising 2 assets a and b, whose individual expected returns are E(Ra) and E(Rb) and a and 1- a are respective weightings.

The riskiness of the portfolio expresses the extent to which the actual return may deviate from the expected return. This may be expressed in terms of the variance of the return s2 or in terms of its standard deviation s.

Portfolio analysis deals with the calculation of the efficient frontier. The outputs will be an efficient frontier; a 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.

Portfolios lying along the efficient frontier dominate all other risk/return combinations lying to the right or below the efficient frontier. They are clearly better than any in the interior of the shaded area.

The individual securities can be combined into portfolios. All the possible combinations represent the set of available investment opportunities. Among these opportunities we prefer the portfolios with the higher expected returns and lower standard deviations. Once the efficient frontier is identified, the investor’s risk/return preferences are taken into consideration The final choice of an individual investor is dependent upon the following two factors: (a) his/her preferences regarding a particular risk/return combination; and (b) relevant investment opportunities on the efficient frontier. A combination of these two factors gives the investor’s optimal portfolio, i.e., the efficient portfolio that maximazes his/her expected utility with reference to the risk/return trade off.

Investors will prefer one of the portfolios on the efficient frontier and their selection depends upon personal preferences for a low portfolio expected return versus a larger and more risky portfolio expected return.

However we cannot specify an optimal portfolio, except for an outright risk-minimizer, who would select portfolio D or the maximizer of expected return who would settle at poit A. A risk averse investor might select any portfolio along AD, depending on his degree of risk aversion, i.e, what additional return he would require to compensate for a specified increase in risk. So, the most desirable combination of risky assets depends on the decision -maker’s attitude towards risk. If we know the extent of his or her risk-aversion that is, how large a premium he/she requires for a given increase in risk, we could specify the best portfolio.

The portflolio combination model, although having limited operational usefulness for many investment projects, provides the infrastructure of a more sophisticate d approach to investment decision-making under risk, The capital asset pricing model (CAPM). This is based on an examination of the risk-return characteristics and resulting portfolio opportunities of securitires. The CAPM explains how individual securities are valued, or priced, in efficient capital markets. Essentially, this involves discounting the future expected returns from holding a security at a rate which adequately reflects the degree of risk incurred in holding that security. A major contribution of the CAPM is the determination of the premium for risk demanded by the market from different securities. This provides a clue as to the appropriate discount rate to apply when evaluating risky projects.

The second part of optimization involves the risk-free asset.. Because the portfolio expected return is the weighted average of its component expected returns, whereas its standard deviation is less than the weighted average of the component standard deviation, portfolio is less than perfectly correlated assets always offer better risk-return opportunities than the individual securities on their own. The lower the correlation between the assets, the greater the gain in efficiency.

In the case of two risky assets, the solution for the weights of the optimal risky portfolio can be shown as follows

5 10 15 20 25

CAL is the capital allocation line. It depicts all the risk-return combinations of risky and risk-free assets available to investors. The slope of the CAL equals the increase in the expected return of the chosen portfolio per unit of additional standard deviation.

The CAL that is supported by the optimal portfolio P, is tangent to the efficient frontier. This CAL dominates all alternative feasible lines

Now I am going to discuss relevant risk measures for portfolios and models regarding the way capital assets are priced in relation to their risks. was originally erected by Sharpe (1964) to explain how the capital market sets prices. If the market, that vast impersonal mass of investors, sets a value on a security which implies a particular discount rate, it is reasonable to conclude that any further activity of similar risk to existing ones should offer at least the same rate. This argument depends critically on the market prices being unbiased indicators of the intrinsic value of firms, thus resting heavily on the validity of the Efficient Market Hypothesis. The CAPM postulates that when the capital market is in equilibrium, i. e. all securities are correctly priced, the relationship between risk and return is given by an expression known as the security market line (SML). In a competitive market, the expected risk premium varies in direct proportion to beta. The equation of the SML states that the required return on shares is made up of two components: the return on a risk-free asset, plus a market risk premium, which varies according to the Beta of the share in question. The CAPM formula consists of three elements: the risk-free rate, the risk premium on the market portfolio and the Beta coefficient.

E(Ri) = the expected return on security or portfolio i.

Rf = the return on the riskless security.

E(Rm) = the expected return on the market portfolio.

Bi = the beta coefficient of a security or portfolio i.

The CAPM indicates that an investor can obtain above the riskless return only by taking on additional risk. The Beta coefficient for security I can be expressed as

Bi = Qim/Q2 m the CAPM indicates that a portfolio’s return is

directly determined by a single risk factor, beta.

Where Qim = the covariance between the rate of return on the market portfolio and the rate of return on security I, Q2m = variance of rate of return on portfolio m.

If Beta is 1, then the required return is simply the average return for all securities, i.e. the return on the market portfolio. The higher the Beta, the higher the risk premium and the total return required. A relatively high beta does not guarantee a relatively high return. The actual return depends partly on the behaviour of the market, which acts as a proxy for general economic factors.. Similarly, expected returns for the individual security hinge on the expected return for the market.

By acquiring investments that are not subject to the same influences on market value, a Mr. can reduce covariance

within the portfolio and increase the safety of the trust capital. This is the principal benefit flowing from diversification of the

portfolio. It is the central pillar of modern portfolio theory.

The cornerstone of Modem Portfolio Theory, developed by Harry Markowitz, is the efficient frontier consisting of portfolios

with a maximum level of expected return, given some investor-selected level of risk. In his landmark Joumal of Finance paper,

“Portfolio Selection,” published in 1952, Markowitz legitimized the concept of risk diversification. He demonstrated that the

riskiness of a portfolio depends on the covariance of its holdings, not on the average riskiness of the separate investments.

Aggregate portfolio risk for all potential portfolios is determined by the sum of the covariances of its holdings. For a given level

of risk, only the portfolio with the maximum level of expected return belongs on the efficient frontier. Importantly, such a

portfolio could be determined through a new construct he developed, called portfolio optimization.

Pike, R. and Neale, B.,1996. Corporate Finance and investment. Decisions and Strategies. Second edition. Prentice Hall Europe. HG5436 P4

Diacogiannis, G., 1994. Financial Management. A modelling approach using spreadsheets. McGraw-Hill Book Company.

c. Portfolio analisys ; rateof return, efficient frontier

In the case of PLC, there are two important messages. First, it is not enough just to spread your activites. Different activities are subject to different types of risk, which are not always closely related. The factors affecting the profitability of packaging operations, such as…….. If changes in these influences are random and relatively uncorrelated, diversification may significantly reduce the variability of company earnings. Second, to generate an appreciable impact on overall returns, diversification must usually be substantial in relation to the whole enterprise. These are the key

Pike, R. and Neale, B.,1996. Corporate Finance and investment. Decisions and Strategies. Second edition. Prentice Hall Europe. HG5436 P4

Diacogiannis, G., 1994. Financial Management. A modelling approach using spreadsheets. McGraw-Hill Book Company.


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