Portfolio Management PDF

CHAPTER 6 5 PORTFOLIO MANAGEMENT LEARNING OUTCOMES After going through the chapter student shall be able to understand ❑ Activities in Portfolio Management ❑ Objectives of Portfolio M...

CHAPTER 6 5 PORTFOLIO MANAGEMENT LEARNING OUTCOMES After going through the chapter student shall be able to understand ❑ Activities in Portfolio Management ❑ Objectives of Portfolio Management ❑ Phases of Portfolio Management (1) Security Analysis (2) Portfolio Analysis (3) Portfolio Selection (4) Portfolio Revision (5) Portfolio Evaluation ❑ Portfolio Theories (1) Traditional Approach (2) Modern Approach (Markowitz Model or Risk-Return Optimization) ❑ Risk Analysis (1) Types of Risk (2) Diversion of Risk © The Institute of Chartered Accountants of India  2. 6.2 ADVANCED FINANCIAL MANAGEMENT 2 (3) Risk & Return (4) Portfolio Analysis ❑ Markowitz Model of Risk-Return Optimization ❑ Capital Market Theory ❑ Sharpe Index Model (Single Index Model) ❑ Capital Asset Pricing Model (CAPM) ❑ Arbitrage Pricing Theory Model (APT) ❑ Portfolio Evaluation Methods ❑ Sharpe’s Optimal Portfolio ❑ Formulation of Portfolio Strategy ❑ Portfolio Revision and Rebalancing ❑ Asset Allocation Strategies ❑ Fixed Income Portfolio ❑ Alternative Investment Avenues 1. INTRODUCTION Investment in the securities such as bonds, debentures and shares etc. is lucrative as well as exciting for the investors. Though investment in these securities may be rewarding, it is also fraught with risk. Therefore, investment in these securities requires a good amount of scientific and analytical skills. As per the famous principle of not putting all eggs in the same basket, an investor never invests his entire investable funds in one security. He invests in a well diversified portfolio of a number of securities which will optimize the overall risk-return profile. Investment in a portfolio can reduce risk without diluting the returns. An investor, who is expert in portfolio analysis, may be able to generate trading profits on a sustained basis. Every investment is characterized by return and risk. The concept of risk is intuitively understood by investors. In general, it refers to the possibility of the rate of return from a security or a portfolio of securities deviating from the corresponding expected/average rate and can be measured by the standard deviation/variance of the rate of return. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.3 2.3 How different type of Investors react in different situations Source: www.missiassaugahsale.com 1.1 Activities in Portfolio Management The following three major activities are involved in the formation of an Optimal Portfolio suitable for any given investor: (a) Selection of securities. (b) Construction of all Feasible Portfolios with the help of the selected securities. (c) Deciding the weights/proportions of the different constituent securities in the portfolio so that it is an Optimal Portfolio for the concerned investor. The activities are directed to achieve an Optimal Portfolio of investments commensurate with the risk appetite of the investor. 1.2 Objectives of Portfolio Management Some of the important objectives of portfolio management are: (i) Security/Safety of Principal Amount: Security not only involves keeping the principal sum intact but also its purchasing power. © The Institute of Chartered Accountants of India  2. 6.4 ADVANCED FINANCIAL MANAGEMENT 4 (ii) Stability of Income: To facilitate planning more accurately and systematically the reinvestment or consumption of income. (iii) Capital Growth: It can be attained by reinvesting in growth securities or through purchase of growth securities. (iv) Marketability i.e. the case with which a security can be bought or sold: This is essential for providing flexibility to investment portfolio. (v) Liquidity i.e. nearness to money: It is desirable for the investor so as to take advantage of attractive opportunities upcoming in the market. (vi) Diversification: The basic objective of building a portfolio is to reduce the risk of loss of capital and/or income by investing in various types of securities and over a wide range of industries. (vii) Favourable Tax Status: The effective yield an investor gets from his investment depends on tax to which it is subjected to. By minimising the tax burden, yield can be effectively improved. 2. PHASES OF PORTFOLIO MANAGEMENT Portfolio management is a process and broadly it involves following five phases and each phase is an integral part of the whole process and the success of portfolio management depends upon the efficiency in carrying out each of these phases. 2.1 Security Analysis  The securities available to an investor for investment are numerous in number and of various types. The securities are normally classified on the basis of ownership of securities such as equity shares, preference shares, debentures and bonds, In recent times a number of new securities with innovative features are available in the market e.g. Convertible Debentures, Deep Discount Bonds, Zero Coupon Bonds, Flexi Bonds, Floating Rate Bonds, Global Depository Receipts, Euro-currency Bonds, etc. are some examples of these new securities. Among this vast group of securities, an investor has to choose those ones which he considers worthwhile to be included in his investment portfolio. This requires a detailed analysis of the all securities available for making investment. Security analysis constitutes the initial phase of the portfolio formation process and consists of examining the risk-return characteristics of individual securities and also the correlation among them. A simple strategy in securities investment is to buy under-priced securities and sell overpriced securities. But the basic problem is how to identify under-priced and overpriced securities and this is what security analysis is all about. As discussed in the chapter of Security Analysis, there are two alternative approaches to analyse any security viz. Fundamental Analysis and Technical Analysis. They are based on different premises and follow different techniques. Fundamental analysis, the older of the two approaches, concentrates on the fundamental factors affecting the company such as © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.5 2.5 ❑ the EPS of the company, ❑ the dividend pay-out ratio, ❑ the competition faced by the company, ❑ the market share, quality of management, etc. ❑ fundamental factors affecting the industry to which the company belongs. The Fundamental Analyst compares this intrinsic value (true worth of a security based on its fundamentals) with the current market price. If the current market price is higher than the intrinsic value, the share is said to be overpriced and vice versa. This mispricing of securities gives an opportunity to the investor to acquire the share or sell off the share profitably. An intelligent investor would buy those securities which are under-priced and sell those securities which are overpriced. Thus, it can be said that fundamental analysis helps to identify fundamentally strong companies whose shares are worthy to be included in the investor's portfolio. The second approach to security analysis is ‘Technical Analysis’. As per this approach the share price movements are systematic and exhibit certain consistent patterns. Therefore, properly studied past movements in the prices of shares help to identify trends and patterns in security prices and efforts are made to predict the future price movements by looking at the patterns of the immediate past. Thus, Technical Analyst concentrates more on price movements and ignores the fundamentals of the shares. In order to construct well diversified portfolios, so that Unsystematic Risk can be eliminated or substantially mitigated, an investor would like to select securities across diverse industry sectors which should not have strong positive correlation among themselves. The Random Walk Theory holds that the share price movements are random and not systematic. Consequently, neither Fundamental Analysis nor Technical Analysis is of value in generating trading gains on a sustained basis. The Efficient Market Hypothesis (EMH) does not subscribe to the belief that it is possible to book gains in the long term on a sustained basis from trading in the stock market. Markets, though becoming increasingly efficient everywhere with the passage of time, are never perfectly efficient. So, there are opportunities all the time although their durations are decreasing and only the smart investors can look forward to booking gains consistently out of stock market deals. 2.2 Portfolio Analysis  Once the securities for investment have been identified, the next step is to combine these to form a suitable portfolio. Each such portfolio has its own specific risk-return characteristics which are not just the aggregates of the characteristics of the individual securities constituting it. The return and risk of each portfolio can be computed mathematically based on the Risk-Return profiles for the constituent securities and the pair-wise correlations among them. © The Institute of Chartered Accountants of India  2. 6.6 ADVANCED FINANCIAL MANAGEMENT 6 From any chosen set of securities, an indefinitely large number of portfolios can be constructed by varying the fractions of the total investable resources allocated to each one of them. All such portfolios that can be constructed out of the set of chosen securities are termed as Feasible Portfolios. Detailed discussion on Risk- Return concept has been covered later in this chapter. 2.3 Portfolio Selection  The goal of a rational investor is to identify the Efficient Portfolios out of the whole set of Feasible Portfolios mentioned above and then to zero in on the Optimal Portfolio suiting his risk appetite. An Efficient Portfolio has the highest return among all Feasible Portfolios having same or lower Risk or has the lowest Risk among all Feasible Portfolios having same or higher Return. Harry Markowitz’s portfolio theory (Modern Portfolio Theory) outlines the methodology for locating the Optimal Portfolio for an investor out of efficient portfolios. Detailed discussion on Markowitz’s Portfolio Theory has been covered later in this chapter. 2.4 Portfolio Revision  Once an optimal portfolio has been constructed, it becomes necessary for the investor to constantly monitor the portfolio to ensure that it does not lose it optimality. Since the economy and financial markets are dynamic in nature, changes take place in these variables almost on a daily basis and securities which were once attractive may cease to be so with the passage of time. New securities with expectations of high returns and low risk may emerge. In light of these developments in the market, the investor now has to revise his portfolio. This revision leads to addition (purchase) of some new securities and deletion (sale) of some of the existing securities from the portfolio. The nature of securities and their proportion in the portfolio changes as a result of the revision. This portfolio revision may also be necessitated by some investor-related changes such as availability of additional funds for investment, change in risk appetite, need of cash for other alternative use, etc. Portfolio revision is not a casual process to be taken lightly and needs to be carried out with care, scientifically and objectively so as to ensure the optimality of the revised portfolio. Hence, in the entire process of portfolio management, portfolio revision is as important as portfolio analysis and selection. 2.5 Portfolio Evaluation  This process is concerned with assessing the performance of the portfolio over a selected period of time in terms of return and risk and it involves quantitative measurement of actual return realized and the risk borne by the portfolio over the period of investment. The objective of constructing a portfolio and revising it periodically is to maintain its optimal risk-return characteristics. Various types of alternative measures of performance evaluation have been developed for use by investors and portfolio managers. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.7 2.7 This step provides a mechanism for identifying weaknesses in the investment process and for improving these deficient areas. It should however be noted that the portfolio management process is an ongoing process. It starts with security analysis, proceeds to portfolio construction, and continues with portfolio -revision and ends with portfolio evaluation. Superior performance is achieved through continual refinement of portfolio management skill. Detailed discussion on the techniques of Portfolio Evaluation has been covered later in this chapter. 3. PORTFOLIO THEORIES Portfolio theory forms the basis for portfolio management. Portfolio management deals with the selection of securities and their continuous shifting in the portfolio to optimise re turns to suit the objectives of an investor. This, however, requires financial expertise in selecting the right mix of securities in changing market conditions to get the best out of the stock market. In India as well as in a number of Western countries, portfolio management service has assumed the role of a specialised service and a number of professional investment bankers/fund managers compete aggressively to provide the best options to high net-worth clients, who have little time to manage their own investments. The idea is catching on with the growth of the capital market and an increasing number of people want to earn profits by investing their hard-earned savings in a planned manner. A portfolio theory guides investors about the method of selecting and combining securities that will provide the highest expected rate of return for any given degree of risk or that will expose the investor to the lowest degree of risk for a given expected rate of return. Portfolio theory can be discussed under the following heads: 3.1 Traditional Approach  The traditional approach to portfolio management concerns itself with the investor, definition of portfolio objectives, investment strategy, diversification and selection of individual investment as detailed below: (i) Investor's study includes an insight into his – (a) age, health, responsibilities, other assets, portfolio needs; (b) need for income, capital maintenance, liquidity; (c) attitude towards risk; and (d) taxation status; (ii) Portfolio objectives are defined with reference to maximising the investors' wealth which is subject to risk. The higher the level of risk borne, the more the expected returns. (iii) Investment strategy covers examining a number of aspects including: © The Institute of Chartered Accountants of India  2. 6.8 ADVANCED FINANCIAL MANAGEMENT 8 (a) Balancing fixed interest securities against equities; (b) Balancing high dividend payout companies against high earning growth companies as required by investor; (c) Finding the income of the growth portfolio; (d) Balancing income tax payable against capital gains tax; (e) Balancing transaction cost against capital gains from rapid switching; and (f) Retaining some liquidity to seize upon bargains. (iv) Diversification reduces volatility of returns and risks and thus adequate equity diversification is sought. Balancing of equities against fixed interest bearing securities is also sought. (v) Selection of individual investments is made on the basis of the following principles: (a) Methods for selecting sound investments by calculating the true or intrinsic value of a share and comparing that value with the current market value (i.e. by following the fundamental analysis) or trying to predict future share prices from past price movements (i.e., following the technical analysis); (b) Expert advice is sought besides study of published accounts to predict intrinsic value; (c) Inside information is sought and relied upon to move to diversified growth companies, switch quickly to winners than loser companies; (d) Newspaper tipsters about good track record of companies are followed closely; (e) Companies with good asset backing, dividend growth, good earning record, high quality management with appropriate dividend paying policies and leverage policies are traced out constantly for making selection of portfolio holdings. In India, most of the share and stock brokers follow the above traditional approach for selecting a portfolio for their clients. The Traditional Approach suggests that one should not put all money in one basket, instead an investor should diversify by investing in different securities and assets. As long as an investor invests in different assets and securities, he shall get the advantage of diversification. Markowitz questioned this wisdom of the Traditional Approach and proved that putting money in particular kinds of securities or assets will give the investor advantage of diversification. Therefore, one should not go blindly picking up securities and assets to make portfolio. 3.2 Modern Approach (Markowitz Model or Risk-Return Optimization) Originally developed by Harry Markowitz in the early 1950's, Portfolio Theory - sometimes referred to as Modern Portfolio Theory - provides a logical/mathematical framework in which investors can optimise their risk and return. The central plank of the theory is that diversification through portfolio © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.9 2.9 formation by adding securities whose returns are having low correlation or negative correlation into portfolio can reduce risk, and second, one can get higher return by taking higher risk. Harry Markowitz is regarded as the father of Modern Portfolio Theory. According to him, investors are mainly concerned with two properties of an asset: risk and return. The essence of his theory is that risk of an individual asset hardly matters to an investor. What really matters is the contribution it makes to the investor's overall risk. By turning his principle into a useful technique for selecting the right portfolio from a range of different assets, he developed the 'Mean Variance Analysis' in 1952. We shall discuss this theory in detail later in this chapter. 4. RISK ANALYSIS Before proceeding further it will be better if the concept of risk and return is discu ssed. A person makes an investment in the expectation of getting some return in the future, but the future is uncertain and so is the uncertain future return. It is this uncertainty associated with the returns from an investment that introduces risk for an investor. It is important here to distinguish between the expected return and the realized return from an investment. The expected future return is what an investor expects to get from his investment. On the other hand, the realized return is what an investor actually obtains from his investment at the end of the investment period. The investor makes the investment decision based on the expected return from the investment. However, the actual return realized from the investment may not correspond to the expected return. This possible variation of the actual return from the expected return is termed as risk. If actual realizations correspond to expectations exactly, there would be no risk. Risk arises where there is a probability of variation between expectations and realizations with regard to an investment. Thus, risk arises from the variability in returns. An investment whose returns are fairly stable is considered to be a low-risk investment, whereas an investment whose returns fluctuate significantly is considered to be a highly risky investment. Government securities whose returns are fairly stable and which are free from default are considered to possess low risk whereas equity shares whose returns are likely to fluctuate widely around their mean are considered risky investments. The essence of risk in an investment is the variation in its returns. This variation in returns is caused by a number of factors. These factors which produce variations in the returns from an investment constitute different types of risk. 4.1 dLJƉĞs of Risk Let us consider the risk in holding equity securities. The elements of risk may be broadly classified into two groups as shown in the following diagram. © The Institute of Chartered Accountants of India  2. 6.10 ADVANCED FINANCIAL MANAGEMENT 10 The first group i.e. systematic risk comprises factors that are external to a company (macro in nature) and affect a large number of securities simultaneously. These are mostly uncontrollable in nature. The second group i.e. unsystematic risk includes those factors which are internal to companies (micro in nature) and affect only those particular companies. These are controllable to a great extent. The total variability in returns of a security is due to the total risk of that security. Hence, Total risk = Total Systematic risk + Total Unsystematic risk 4.1.1 Systematic Risk Due to dynamic nature of an economy, the changes occur in the economic, political and social conditions constantly. These changes have an influence on the performance of companies and thereby on their stock prices but in varying degrees. For example, economic and political instability adversely affects all industries and companies. When an economy moves into recession, corporate profits will shift downwards, and stock prices of most companies may decline. Thus, the impact of economic, political and social changes is system-wide and that portion of total variability in security returns caused by such macro level factors is referred to as systematic risk. Systematic risk can be further subdivided into interest rate risk, market risk and purchasing power risk. (i) Interest Rate Risk: This arises due to variability in the interest rates from time to time and its impact on security prices. A change in the interest rates establishes an inverse relationship with the price of security i.e. price of securities tends to move inversely with change in rate of interest, long term securities show greater variability in the price with respect to interest rate changes than short term securities. (ii) Purchasing Power Risk: It is also known as inflation risk, as it also emanates from the very fact that inflation affects the purchasing power adversely. Nominal return contains both the real return component and an inflation premium in a transaction involving risk of the above type to compensate for inflation over an investment holding period. Inflation rates vary over time and investors are caught unaware when rate of inflation changes unexpectedly causing erosion in the value of realised rate of return and expected return. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.11 2.11 Purchasing power risk is more in inflationary conditions especially in respect of bonds and fixed income securities. It is not desirable to invest in such securities during inflationary periods. Purchasing power risk is however, less in flexible income securities like equity shares or common stock where rise in dividend income off-sets increase in the rate of inflation and provides advantage of capital gains. (iii) Market risk: This is a type of systematic risk that affects prices of a share that moves up or down consistently for some time periods in line with other shares in the market. A general rise in share prices is referred to as a bullish trend, whereas a general fall in share prices is referred to as a bearish trend. In other words, the share market moves between the bullish phase and the bearish phase. The market movements can be easily seen in the movement of share price indices such as the BSE Sensitive Index, BSE National Index, NSE Index etc. 4.1.2 Unsystematic Risk Also called ‘idiosyncratic risk’ the return from a security of any company may vary because of certain factors particular to that company. Variability in returns of the security on account of these factors (micro in nature), is known as unsystematic risk. It should be noted that this risk is in addition to the systematic risk affecting all the companies. Unsystematic risk can be further subdivided into business risk and financial risk. (i) Business Risk: Business risk emanates variability in the operating profits of a company – higher the variability in the operating profits of a company, higher is the business risk. Such a risk can be measured using operating leverage. (ii) Financial Risk: It arises due to presence of debt in the capital structure of the company. It is also known as leveraged risk and expressed in terms of debt-equity ratio. Excess of debt vis-à-vis equity in the capital structure indicates that the company is highly geared and hence, has higher financial risk. Although a leveraged company's earnings per share may be more but dependence on borrowings exposes it to the risk of winding-up for its inability to honour its commitments towards lenders/creditors. This risk is known as leveraged or financial risk of which investors should be aware of and portfolio managers should be very careful. 4.2 Diversion of Risk  As discussed above, the total risk of an individual security consists of two risks systematic risk and unsystematic risk. It should be noted that by combining many securities in a portfolio the unsystematic risk can be avoided or diversified which is attached to any particular security. The following diagram depicts how the risk can be reduced with the increase in the number of securities. © The Institute of Chartered Accountants of India  2. 6.12 ADVANCED FINANCIAL MANAGEMENT 12 From the above diagram it can be seen that total risk is reducing with the increase in the number of securities in the portfolio. However, ultimately when the size of the portfolio reaches certain limit, it will contain only the systematic risk. 4.3 Risk & Return It is very common that an intelligent investor would attempt to anticipate the kind of risk that he/she is likely to face and would also attempt to estimate the extent of risk associated with different investment proposals. In other words an attempt is made by him/her to measure or quantif y the risk of each investment under consideration before making the final selection. Thus quantification of risk is necessary for analysis of any investment. As risk is attached with return its risk cannot be measured without reference to return. The return, in turn, depends on the cash inflows to be received from the investment. Let us take an example of purchase of a share. With an investment in an equity share, an investor expects to receive future dividends declared by the company. In addition, he expects to receive capital gain in the form of difference between the selling price and purchase price, when the share is finally sold. Suppose a share of X Ltd. is currently selling at ` 12.00. An investor who is interested in the share anticipates that the company will pay a dividend of ` 0.50 in the next year. Moreover, he expects to sell the share at ` 17.50 after one year. The expected return from the investment in share will be as follows: Forecasted dividend + Forecasted end of the period stock price R= -1 Initial investment © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.13 2.13 ` 0.50 + ` 17.50 R= - 1 = 0.5 or 50 per cent ` 12.00 It is important to note that here the investor expects to get a return of 50 per cent in the future, which is uncertain. It might be possible that the dividend declared by the company may turn out to be either more or less than the figure anticipated by the investor. Similarly, the selling price of the share may be less than the price expected by the investor at the time of investment. It may sometimes be even more. Hence, there is a possibility that the future return may be more than 50 per cent or less than 50 per cent. Since the future is uncertain the investor has to consider the probability of several other possible returns. The expected returns may be 20 per cent, 30 per cent, 50 per cent, 60 per cent or 70 per cent. The investor now has to assign the probability of occurrence of these possible alternative returns as given below: Possible returns (in per cent) Probability of occurrence Xi p(Xi) 20 0.20 30 0.20 50 0.40 60 0.10 70 0.10 The above table gives the probability distribution of possible returns from an investment in shares. Such distribution can be developed by the investor with the help of analysis of past data and modifying it appropriately for the changes he expects to occur in a future period of time. With the help of available probability distribution two statistical measures one expected return and the other risk of the investment can be calculated. 4.3.1 Expected Return  a. In case the data is given with probabilities: The expected return of the investment is the probability weighted average of all the possible returns. If the possible returns are denoted by Xi and the related probabilities are p(Xi) the expected return may be represented as X and can be calculated as: n X= ∑x i p(X i ) i=1 It is the sum of the products of possible returns with their respective probabilities. The expected return of the share in the example given above can be calculated as shown below: © The Institute of Chartered Accountants of India  2. 6.14 ADVANCED FINANCIAL MANAGEMENT 14 Example Calculation of Expected Return Possible returns Probability Xi p(Xi) Xi p(Xi) 0.20 0.20 0.04 0.30 0.20 0.06 0.40 0.40 0.16 0.50 0.10 0.05 0.60 0.10 0.06 n ∑x i p(X i) 0.37 i=1 Hence the expected return is 0.37 i.e. 37% b. In case the historical data is given: n x1 +x 2 +x 3........xi x= i=1 n i.e. Simple average or Arithmetic Mean of the historical data series Example Suppose the total return from shares of X Ltd. for a period of last five years is as follows: Year Total Return % (x) 1 9.50 2 7.00 3 11.00 4 -6.00 5 2.50  x = 24.00 i 24.00 The Average Return of the share will be = = 4.80 5 © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.15 2.15 4.3.2 Risk  As risk is attached with every return hence calculation of only expected return is not sufficient for decision making. Therefore risk aspect should also be considered along with the expected return. The most popular measure of risk is the variance or standard deviation of the probability distribution of possible returns. Risk can be defined as deviation of actual/possible return from expected return i.e. Mean return. Variance of each security is generally denoted by σ 2 and is calculated by using the following formula: a. In case the data is given with probabilities: n σ2 = ∑ [(X - X) i 2 p(X i )] i=1 Continuing our earlier example the following table provides calculations required to calculate the risk i.e. Variance or Standard Deviation (SD). Possible Probability Deviation Deviation squared Product returns p(Xi) (Xi - X ) (Xi - X )2 (Xi - X )2 p(Xj) Xi 0.20 0.20 -0.17 0.0289 0.00578 0.30 0.20 -0.07 0.0049 0.00098 0.40 0.40 0.03 0.0009 0.00036 0.50 0.10 0.13 0.0169 0.00169 0.60 0.10 0.23 0.0529 0.00529 Var (σ2) 0.0141 Variance = 0.0141 Standard Deviation of the return will be the positive square root of the variance and is generally represented by σ. Accordingly, the standard deviation of return in the above example will be 0.0141 = 0.1187 i.e. 11.87%. b. In case the historical data is given: n (xi - x)2 σ 2 = i=1 n Thus, keeping the above formulae same, instead of multiplying with probabilities, we need to divide it by no of observations. Continuing our earlier example the following table provides calculations required to calculate the risk i.e. Variance or Standard Deviation (SD). © The Institute of Chartered Accountants of India  2. 6.16 ADVANCED FINANCIAL MANAGEMENT 16 Year Total Return Deviation Deviation squared % (x) (x i - x ) (x i - x )2 1 9.50 4.70 22.09 2 7.00 2.20 4.84 3 11.00 6.20 38.44 4 -6.00 -10.80 116.64 5 2.50 -2.30 5.29  x = 24.00 i 187.30 Mean 4.80 187.30 Variance = = 37.46 5 Standard Deviation of the return will be the positive square root of the variance and is generally represented by σ. Accordingly, the standard deviation of return in the above example will be 37.46 = 6.12 i.e. 6.12%. The basic purpose to calculate the variance and standard deviation is to measure the extent of variability of possible returns from the expected return. Several other measures such as range, semi - variance and mean absolute deviation can also be used to measure risk, but standard deviation has been the most popularly accepted measure. For example, if a security has a return of 11% and Standard Deviation i.e. risk of 3%, then it can be concluded that as per the given conditions, the return’s range can be expected from 8% to 14% i.e. 11% - 3% would be the lower range and 11% + 3% would be the upper range. The method described above is widely used for assessing risk. The standard deviation or variance, however, provides a measure of the total risk associated with a security. As we know, the total risk comprises two components, namely systematic risk and unsystematic risk. Unsystematic risk is the risk specific or unique to a company. Unsystematic risk associated with the security of a particular company can be eliminated/reduced by combining it with another security having negative correlation. This process is known as diversification of unsystematic risk. As a means of diversification the investment is spread over a group of securities with different characteristics. This collection of diverse securities is called a Portfolio. As unsystematic risk can be reduced or eliminated through diversification, it is not very important for an investor to consider. The risk that is relevant in investment decisions is the systematic risk because it is not diversifiable. Hence, the main interest of the investor lies in the measurement of systematic risk of a security. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.17 2.17 4.3.3 Measurement of Systematic Risk  As discussed earlier, systematic risk is the variability in security returns caused by changes in the economy or the market and all securities are affected by such changes to some extent. Some securities exhibit greater variability in response to market changes and some may exhibit less response. Securities that are more sensitive to changes in factors are said to have higher systematic risk. The average effect of a change in the economy can be represented by the change in the stock market index. The systematic risk of a security can be measured by relating that security’s variability vis-à-vis variability in the stock market index. A higher variability would indicate higher systematic risk and vice versa. The systematic risk of a security is measured by a statistical measure which is called Beta (β). The main input data required for the calculation of beta of any security are the historical data of returns of the individual security and corresponding return of a representative market return (stock market index). There are two statistical methods i.e. correlation method and the regression method, which can be used for the calculation of Beta. 4.3.3.1 Correlation Method : Correlation measures the extent to which two variables are related. In respect to Beta, it means how market returns and security returns are related. Using this method beta (β) can be calculated from the historical data of returns by the following formula: r σ βi = im i σm Where rim = Correlation coefficient between the returns of the stock i and the returns of the market index. σ i = Standard deviation of returns of stock i σ m = Standard deviation of returns of the market index. 4.3.3.2 Regression Method : The regression model is based on the postulation that there exists a linear relationship between a dependent variable and an independent variable. The model helps to calculate the values of two constants, namely alfa (α) and beta (β). β measures the change in the dependent variable in response to unit change in the independent variable, while α measures the value of the dependent variable even when the independent variable has zero value. The form ula of the regression equation is as follows: Y = α + βX where Y = Dependent variable X = Independent variable α and β are constants. © The Institute of Chartered Accountants of India  2. 6.18 ADVANCED FINANCIAL MANAGEMENT 18 α = Y - βX The formula used for the calculation of α and β are given below. n ∑ XY - (∑ X)( ∑ Y) β= n∑ X ( ∑ X) 2 2 where n = Number of items. Y = Dependent variable scores. X = Independent variable scores. For the purpose of calculation of β, the return of the individual security is taken as the dependent variable and the return of the market index is taken as the independent variable. Here it is very important to note that a security can have betas that are positive, negative or zero. Positive Beta- indicates that security’s return is dependent on the market return and moves in the direction in which market moves. Negative Beta- indicates that security’s return is dependent on the market return but moves in the opposite direction in which market moves. Zero Beta- indicates that security’s return is independent of the market return. Further as beta measures the volatility of a security’s returns relative to the market, the larger the beta, the more volatile the security. A beta of 1.0 indicates that a security has risk as that of the market. A stock with beta greater than 1.0 has above average risk i.e. its returns would be more volatile than the market returns. For example, when market returns move up by 6%, a stock with beta of 2 would find its returns moving up by 12% (i.e. 6% x 2). Similarly, decline in market returns by 6% would produce a decline of 12% (i.e. 6% x 2) in the return of that security. A stock with beta less than 1.0 would have below market risk. Variability in its returns would be less than the market variability. Beta is calculated from historical data of returns to measure the systematic risk of a security. It is a historical measure of systematic risk. In using this beta for investment decision making, the investor is assuming that the relationship between the security variability and market variability will continue to remain the same in future also. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.19 2.19 4.4 Portfolio Analysis Till now we have discussed the risk and return of a single security. Let us now discuss the return and risk of a portfolio of securities. 4.4.1 Portfolio Return  For a portfolio analysis an investor first needs to specify the list of securities eligible for selection or inclusion in the portfolio. Then he has to generate the risk-return expectations for these securities. The expected return for the portfolio is expressed as the mean of its rates of return over the time horizon under consideration and risk for the portfolio is the variance or standard deviation of these rates of return around the mean return. The expected return of a portfolio of assets is simply the weighted average of the returns of the individual securities constituting the portfolio. The weights to be applied for calculation of the portfolio return are the fractions of the portfolio invested in such securities. Let us consider a portfolio of two equity shares A and B with expected returns of 16 per cent and 22 per cent respectively. The formula for the calculation of expected portfolio return may be expressed as shown below: n rp = ∑x i r i i=1 r p = Expected return of the portfolio. Xi = Proportion of funds invested in security i r i = Expected return of security i. n = Number of securities in the portfolio. If 40 per cent of the total funds is invested in share A and the remaining 60 per cent in share B, then the expected portfolio return will be: Return on the portfolio = (0.40 x 16) + (0.60 x 22) = 19.6 per cent 4.4.2 Portfolio Risk  As discussed earlier, the variance of return and standard deviation of return are statistical measures that are used for measuring risk in investment. The variance of a portfolio can be written down as the sum of 2 terms, one containing the aggregate of the weighted variances of the constituent securities and the other containing the weighted co-variances among different pairs of securities. Covariance (a statistical measure) between two securities or two portfolios or a security and a portfolio indicates how the rates of return for the two concerned entities behave relative to each other. The covariance between two securities A and B may be calculated using the following formula: © The Institute of Chartered Accountants of India  2. 6.20 ADVANCED FINANCIAL MANAGEMENT 20 COVAB =  [R A - R A ][RB - RB ] N where COVAB = Covariance between A and B RA = Return of security A RB = Return of security B R A = Expected or mean return of security A RB = Expected or mean return of security B N = Number of observations. The calculation of covariance can be understood with the help of following table: Calculation of Covariance Deviation Deviation Year RX Rx - R x RY RY - R Y [R x - R x ][R y - R y ] 1 11 -4 18 5 -20 2 13 -2 14 1 -2 3 17 2 11 -2 -4 4 19 4 9 -4 -16 Rx = 15 Ry =13 -42 n ∑[R x - R x ] [R y - R y ] - 42 i=1 Cov xy = = = -10.5 n 4 From the above table it can be seen that the covariance is a measure of how returns of two securities move together. In case the returns of the two securities move in the same direction consistently the covariance is said to be positive (+). Contrarily, if the returns of the two securities move in opposite directions consistently the covariance would be negative (-). If the movements of returns are independent of each other, covariance would be close to zero (0). © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.21 2.21 The coefficient of correlation is expressed as: Cov AB rAB =  A B where rAB = Coefficient of correlation between A and B CovAB = Covariance between A and B σA= Standard deviation of A σB = Standard deviation of B It may be noted on the basis of above formula the covariance can be expressed as the product of correlation between the securities and the standard deviation of each of the s ecurities as shown below: CovAB = σA σB rAB It is very important to note that the correlation coefficients may range from -1 to 1. A value of -1 indicates perfect negative correlation between the two securities’ returns. A value of +1 indicates a perfect positive correlation between them. A value of zero indicates that the returns are independent. The calculation of the variance (or risk) of a portfolio is not simply a weighted average of the variances of the individual securities in the portfolio as in the calculation of the return of portfolio. The variance of a portfolio with only two securities in it can be calculated with the following formula.  p2 = x 12 12 + x 22  22 + 2x 1 x 2 (r12 1 2 ) where p2 = Portfolio variance. x1 = Proportion of funds invested in the first security. x2 = Proportion of funds invested in the second security (x1+x2 = 1).  12 = Variance of first security.  22 = Variance of second security.  1 = Standard deviation of first security.  2 = Standard deviation of second security. r12 = Correlation coefficient between the returns of the first and second securities. © The Institute of Chartered Accountants of India  2. 6.22 ADVANCED FINANCIAL MANAGEMENT 22 As the standard deviation is the square root of the variance the portfolio standard deviation can be obtained by taking the square root of portfolio variance. Let us take an example to understand the calculation of portfolio variance and portfolio standard deviation. Two securities A and B generate the following sets of expected returns, standard deviations and correlation coefficient: A B r= 20% 25% σ= 50% 30% rab= -0.60 Now suppose a portfolio is constructed with 40 per cent of funds invested in A and the remaining 60 per cent of funds in B (i.e. P = 0.4A + 0.6B). Using the formula of portfolio return the expected return of the portfolio will be: RP= (0.40 x 20) + (0.60 x 25) = 23% And the Variance and Standard Deviation of the portfolio will be: Variance σp 2 = (0.40)2 (50)2 + (0.60) 2 (30)2 + 2(0.40)(0.60)(- 0.60)(50)(30) = 400 + 324 - 432 = 292 Standard deviation σp = 292 = 17.09 per cent. The return and risk of a portfolio depends on following two sets of factors: (a) Returns and risks of individual securities and the covariance between securities forming the portfolio (b) Proportion of investment in each of securities. As the first set of factors is parametric in nature for the investor in the sense that he has no control over the returns, risks and co-variances of individual securities. The second set of factors is choice factor or variable for the investors in the sense that they can choose the proportions of each security in the portfolio. 4.4.3 Reduction or Dilution of Portfolio Risk through Diversification The process of combining more than one security in to a portfolio is known as diversification. The main purpose of this diversification is to reduce the total risk by eliminating or substantially mitigating the unsystematic risk, without sacrificing portfolio return. As shown in the example mentioned above, diversification has helped to reduce risk. The portfolio standard deviation of 17.09% is lower than the standard deviation of either of the two securities taken separately which were 50% and 30% respectively. Incidentally, such risk reduction is possible even when the two constituent securities © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.23 2.23 are uncorrelated. In case, however, these have the maximum positive correlation between them, no reduction of risk can be achieved. In order to understand the mechanism and power of diversification, it is necessary to consider the impact of covariance or correlation on portfolio risk more closely. We shall discuss following three cases taking two securities in the portfolio: (a) Securities’ returns are perfectly positively correlated, (b) Securities’ returns are perfectly negatively correlated, and (c) Securities’ returns are not correlated i.e. they are independent. 4.4.3.1 Perfectly Positively Correlated : In case two securities returns are perfectly positively correlated the correlation coefficient between these securities will be +1 and the returns of these securities then move up or down together. The variance of such portfolio can be calculated by using the following formula:  p2 = x 12 12 + x 22  22 + 2x 1 x 2 r12 1 2 As r12 = 1, this may be rewritten as: 2 2 2 2 2  = x  + x  + 2x x   p 1 1 2 2 1 2 1 2 or  = 2 p (x 11 + x 2  2 )2 Hence Standard Deviation will become  p = x 1 1 + x 2  2 In other words this is simply the weighted average of the standard deviations of the individual securities. Taking the above example we shall now calculate the portfolio standard deviation when correlation coefficient is +1. Standard deviation of security A = 40% Standard deviation of security B = 25% Proportion of investment in A = 0.4 Proportion of investment in B = 0.6 Correlation coefficient = +1.0 Portfolio standard deviation maybe calculated as: σp = (0.4) (40) + (0.6) (25) = 31 © The Institute of Chartered Accountants of India  2. 6.24 ADVANCED FINANCIAL MANAGEMENT 24 Thus it can be seen that the portfolio standard deviation will lie between the standard deviations of the two individual securities. It will vary between 40 and 25 as the proportion of investment in each security changes. Now suppose, if the proportion of investment in A and B are changed to 0.75 and 0.25 respectively; portfolio standard deviation of the portfolio will become: σp = (0.75) (40)+ (0.25) (25) = 36.25 It is important to note that when the security returns are perfectly positively correlated, diversification provides only risk averaging and no risk reduction because the portfolio risk cannot be reduced below the individual security risk. Hence, reduction of risk is not achieved when the constituent securities’ returns are perfectly positively correlated. 4.4.3.2 Perfectly Negatively Correlated : When two securities’ returns are perfectly negatively correlated, two returns always move in exactly opposite directions and correlation coefficient between them becomes -1. The variance of such negatively correlated portfolio may be calculated as: p2 = x12 12 + x 2222 − 2x1x 2 ( r12 12 ) As r12 = -1, this may be rewritten as: σ p2 = (x1σ1 - x 2σ 2 )2 Hence Standard Deviation will become σ p = x1σ1 - x 2σ 2 Taking the above example we shall now calculate the portfolio standard deviation when correlation coefficient is -1. σp = (0.4)(40) - (0.6)(25) =1 Thus, from above it can be seen that the portfolio risk has become very low in comparison of risk of individual securities. By changing the weights it can even be reduced to zero. For example, if the proportion of investment in A and B are 0.3846 and 0.6154 respectively, portfolio standard deviation becomes: = (0.3846)(40) - (0.6154)(25) = 0 Although in above example the portfolio contains two risky assets, the portfolio has no risk at all. Thus, the portfolio may become entirely risk-free when security returns are perfectly negatively correlated. Therefore, diversification can substantially reduce or even eliminate risk when securities are perfectly negatively correlated,. However, in real life it is very rare to find securities that are perfectly negatively correlated. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.25 2.25 4.4.3.3 Returns are uncorrelated or independent : When the returns of two securities are entirely uncorrelated, the coefficient of correlation of these two securities would be zero and the formula for portfolio variance will be as follows:  p2 = x 12 12 + x 22  22 + 2x 1 x 2 r12 1 2 As r12 = 0, this may be rewritten as: σ p2 = x12σ12 + x 22σ 22 Hence Standard Deviation will become σ p = x12 σ12 + x 22 σ 22 Taking the above example we shall now calculate the portfolio standard deviation when correlation coefficient is 0. (0.4) 2 (40)2 + (0.6) 2 (25)2 σp = σp = 256 + 225 σp =21.93 Thus, it can be observed that the portfolio standard deviation is less than the standard deviations of individual securities in the portfolio. Therefore, when security returns are uncorrelated, diversification can reduce risk. We may now tabulate the portfolio standard deviations of our illustrative portfolio having two securities A and B, for different values of correlation coefficients between them. The proportion of investments in A and B are 0.4 and 0.6 respectively. The individual standard deviations of A and B are 40 and 25 respectively. Portfolio Standard Deviations Correlation Portfolio coefficient Standard Deviation 1.00 31 0.60 27.73 0 21.93 -0.60 13.89 -1.00 1.00 © The Institute of Chartered Accountants of India  2. 6.26 ADVANCED FINANCIAL MANAGEMENT 26 Summarily it can be concluded that diversification reduces risk in all cases except when the security returns are perfectly positively correlated. With the decline of correlation coefficient from +1 to -1, the portfolio standard deviation also declines. But the risk reduction is greater when the security returns are negatively correlated. 4.4.4 Portfolio with more than two securities  So far we have considered a portfolio with only two securities. The benefits from diversification increase as more and more securities with less than perfectly positively correlated returns are included in the portfolio. As the number of securities added to a portfolio increases, the standard deviation of the portfolio becomes smaller and smaller. Hence, an investor can make the portfolio risk arbitrarily small by including a large number of securities with negative or zero correlation in the portfolio. But, in reality, securities rarely show negative or even zero correlation. Typically, securities show some positive correlation, that is above zero but less than the perfectly positive value (+1). As a result, diversification (that is, adding securities to a portfolio) results in some reduction in total portfolio risk but not in complete elimination of risk. Moreover, the eff ects of diversification are exhausted fairly rapidly. That is, most of the reduction in portfolio standard deviation occurs by the time the portfolio size increases to 25 or 30 securities. Adding securities beyond this size brings about only marginal reduction in portfolio standard deviation. Adding securities to a portfolio reduces risk because securities are not perfectly positively correlated. But the effects of diversification are exhausted rapidly because the securities are still positively correlated to each other though not perfectly correlated. Had they been negatively correlated, the portfolio risk would have continued to decline as portfolio size increased. Thus, in practice, the benefits of diversification are limited. 4.4.5 Calculation of Return and Risk of Portfolio with more than two securities  The expected return of a portfolio is the weighted average of the returns of individual securities in the portfolio, the weights being the proportion of investment in each security. The formula for calculation of expected portfolio return is the same for a portfolio with two securities and for portfolios with more than two securities. The formula is: n rp = ∑ x i ri i=1 Where rp = Expected return of portfolio. xi = Proportion of funds invested in each security. ri = Expected return of each security. n = Number of securities in the portfolio. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.27 2.27 Let us consider a portfolio with four securities having the following characteristics: Security Returns (per cent) Proportion of investment P 11 0.3 Q 16 0.2 R 22 0.1 S 20 0.4 The expected return of this portfolio may be calculated using the formula: rp = (0.3)(11) + (0.2)(16) + (0.1)(22) + (0.4)(20) = 16.7 per cent The portfolio variance and standard deviation depend on the proportion of investment in each security as also the variance and covariance of each security included in the portfolio. Let us take the following example to understand how we can compute th e risk of multiple asset portfolio. Security xi σi Correlation Coefficient X 0.25 16 X and Y = 0.7 Y 0.35 7 X and Z = 0.3 Z 0.40 9 Y and Z = 0.4 It may be noted that correlation coefficient between X and X, Y and Y, Z and Z is 1. A convenient way to obtain the result is to set up the data required for calculation in the form of a variance-covariance matrix. As per data given in the example, the first cell in the first row of the matrix represents X and X the second cell in the first row represents securities X and Y, and so on. The variance or covariance in each cell has to be multiplied by the weights of the respective securities represented by that cell. These weights are available in the matrix at the left side of the row and the top of the column containing the cell. This process may be started from the first cell in the first row and continued for all the cells till the last cell of the last row is reached as shown below: Weights 0.25 0.35 0.40 X Y Z 0.25 X 1 x 16 x 16 0.7 x 16 x 7 0.3 x 16 x 9 0.35 Y 0.7 x 7 x 16 1x7x7 0.4 x 7 x 9 0.40 Z 0.3 x 9 x 16 0.4 x 9 x 7 1x9x9 © The Institute of Chartered Accountants of India  2. 6.28 ADVANCED FINANCIAL MANAGEMENT 28 Once the variance-covariance matrix is set up, the computation of portfolio variance is a comparatively simple operation. Each cell in the matrix represents a pair of two securities. When all these products are summed up, the resulting figure is the portfolio variance. The square root of this figure gives the portfolio standard deviation. Thus the variance of the portfolio given in the example above can now be calculated. σp2 = (0.25 x 0.25 x 1 x 16 x 16) + (0.25 x 0.35 x 0.7 x 16 x 7) + (0.25 x 0.40 x 0.3 x 16 x 9) + (0.35 x 0.25 x 0.7 x 7 x 16) + (0.35 x 0.35 x 1 x 7 x 7) + (0.35 x 0.40 x 0.4 x7 x 9) + (0.40 x 0.25 x 0.3 x 9 x 16) + (0.40 x 0.35 x 0.4 x 9 x 7) + (0.40 x 0.40 x 1 x 9 x 9) = 16 + 6.86 + 4.32 + 6.86 + 6.0025 + 3.528 + 4.32 + 3.528 + 12.96 = 64.3785 The portfolio standard deviation is: σ p= 64.3785 =8.0236 Hence, the formula for computing Portfolio Variance may also be stated as follows: n n  x i x jri j i  j  p2 = =i 1 =j 1 Or p2 n n = ∑∑ x i x jσ i j i=1 j=1 where σp2 = Portfolio variance. xi = Proportion of funds invested in security i (the first of a pair of securities). xj = Proportion of funds invested in security j (the second of a pair of securities). σi = Standard Deviation of security i σj = Standard Deviation of security j rij = The co-efficient of correlation between the pair of securities i and j σij = The covariance between the pair of securities i and j n = Total number of securities in the portfolio. Thus from above discussion it can be said that a portfolio is a combination of assets. From a given set of 'n' securities, any number of portfolios can be created. These portfolios may comprise of two securities, three securities, all the way up to 'n' securities. A portfolio may contain the same securities as another portfolio but with different weights. A new portfolios can be created either by changing the securities in the portfolio or by changing the proportion of investment in the existing securities. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.29 2.29 Thus summarily it can be concluded that each portfolio is characterized by its expected return and risk. Determination of expected return and risk (variance or standard deviation) of each portfolio that can be used to create a set of selected securities which is the first step in portfolio management and called portfolio analysis. 5. MARKOWITZ MODEL OF RISK-RETURN OPTIMIZATION Unlike the CAPM, the Optimal Portfolio as per Markowitz Theory is investor specific. The portfolio selection problem can be divided into two stages: (1) finding the mean-variance efficient portfolios and (2) selecting one such portfolio. Investors do not like risk and the greater the riskiness of returns on an investment, the greater will be the returns expected by investors. There is a trade-off between risk and return which must be reflected in the required rates of return on investment opportunities. The standard deviation (or variance) of return measures the total risk of an investment. It is not necessary for an investor to accept the total risk of an individual security. Investors can diversify to reduce risk. As number of holdings approach larger, a good deal of total risk is removed by diversification. 5.1  Assumptions of the Model It is a common phenomenon that the diversification of investments in the portfolio leads to reduction in variance of the return, even for the same level of expected return. This model has taken into account risks associated with investments - using variance or standard deviation of the return. This model is based on the following assumptions. : (i) The return on an investment adequately summarises the outcome of the investment. (ii) The investors can visualise a probability distribution of rates of return. (iii) The investors' risk estimates are proportional to the variance of return they perceive for a security or portfolio. (iv) Investors base their investment decisions on two criteria i.e. expected return and variance of return. (v) Investors are assumed to be rational in so far as they would prefer greater returns to lesser ones given equal or smaller risk and are risk averse. Risk aversion in this context means merely that, as between two investments with equal expected returns, the investment with the smaller risk would be preferred. (vi) Though ‘Expected Return’ could be a suitable measure of monetary inflows like NPV but yield in different scenarios is the most commonly used measure of return. The standard deviation of return is referred to as deviation of yields from its expected value or return. © The Institute of Chartered Accountants of India  2. 6.30 ADVANCED FINANCIAL MANAGEMENT 30 5.2 Efficient Frontier Markowitz has formalised the risk return relationship and developed the concept of efficient frontier using the Mean-Variance Dominance Principle. For selection of a portfolio, comparison between combinations of portfolios is essential. As a rule, a portfolio is dominating another portfolio in terms of mean and variance if there is another portfolio with: (a) A lower expected value of return and same or higher standard deviation (risk). (b) The same or higher standard deviation (risk) but a lower expected return. Markowitz has defined the diversification as the process of combining assets that are less than perfectly positively correlated in order to reduce portfolio risk without sacrificing any portfolio returns. If an investors’ portfolio is not efficient he may: (i) Increase the expected value of return without increasing the risk. (ii) Decrease the risk without decreasing the expected value of return, or (iii) Obtain some combination of increase of expected return and decrease risk. This is possible by switching to a portfolio on the efficient frontier. Fig. 1: Markowitz Efficient Frontier If all the investments are plotted on the risk-return space, individual securities would be dominated by portfolios, and the efficient frontier would be containing all Efficient Portfolios (An Efficient Portfolio has the highest return among all portfolios with identical risk and the lowest risk among all portfolios with identical return). Fig – 1 depicts the boundary of possible investments in securities, A, B, C, D, E and F; and B, C, D, are lying on the efficient frontier. The best combination of expected value of return and risk (standard deviation) depends upon the investors’ utility function. The individual investor will want to hold that portfolio of securities which places him on the highest indifference curve, choosing from the set of available portfolios. The dark line at the top of the set is the line of efficient combinations, or the efficient frontier. The optimal portfolio for an investor lies at the point where the indifference curve for the concerned investor touches the efficient frontier. This point reflects the risk level acceptable to the investor in order to achieve a desired return and provide maximum return for the bearable level of risk. The concept of efficient frontier and the location of the optimal portfolio are explained with help of Fig-2. © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.31 2.31 Fig. 2 : Optimal Investment under Markowitz Model In Fig-2 A, B, C, D, E and F define the boundary of all possible investments out of which investments in B, C and D are the efficient portfolios lying on the efficient frontier. The attractiveness of the investment proposals lying on the efficient frontier depends on the investors’ attitude to risk. At point B, the level of risk and return is at optimum level. The returns are highest at point D, but simultaneously it carries higher risk than any other investment. Fig.3 : Selection of Portfolios The shaded area in Fig-3 represents all attainable or feasible portfolios, that is all the combinations of risk and expected return which may be achieved with the available securities. The efficient frontier contains all possible efficient portfolios and any point on the frontier dominates any point to the right of it or below it. Consider the portfolios represented by points B and E. B and E promise the same expected return E (R1) but the risk associated with B is  (R1) whereas the associated with E is  (R2). Investors, therefore, prefer portfolios on the efficient frontier rather than interior portfolios given the assumption of risk aversion; obviously, point A on the frontier represents the portfolio with the least possible risk, whilst D represents the portfolio with the highest possible rate of return with highest risk. The investor has to select a portfolio from the set of efficient portfolios lying on the efficient frontier. This will depend upon his risk-return preference. As different investors have different preferences , the optimal portfolio of securities will vary from one investor to another. © The Institute of Chartered Accountants of India  2. 6.32 ADVANCED FINANCIAL MANAGEMENT 32 6. CAPITAL MARKET THEORY The above figure 2 portrays the relationship between risk and return for efficient portfolio graphically. Point B represents the market portfolio and if a line tangent to this point is drawn and extended upto y-axis the point at which it will touch will be the riskless rate of interest. This is shown in Fig 4. Fig.4 : Selection of Portfolios Preferred investment strategies plot along line RfBZ, representing alternative combinations of risk and return obtainable by combining the market portfolio with borrowing or lending. This is known as the Capital Market Line (CML). Portfolio lying on line from R f to B shall be lending portfolio as it will involve some investment in risk-free securities and some investment in market portfolio. Portfolios lying from B to Z will be borrowing portfolio as it will be an investment in market portfolio by borrowing the same amount. The slope of the capital market line can be regarded as the reward per unit of risk borne and it is computed as follows: RM - R f Slope = M Where RM = Market Return Rf= Risk Free Rate of Return σM = Standard Deviation of Market From the Capital Market Line the expected return of a portfolio can be found as follows: RM - R f E(R) = Rf + × σP M Where σP = Standard Deviation of Portfolio © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.33 2.33 7. SINGLE INDEX MODEL (SHARPE INDEX MODEL) This model assumes that co-movement between stocks is due to change or movement in the market index. Casual observation of the stock prices over a period of time reveals that most of the stock prices move with the market index. When the Sensex increases, stock prices also tend to increase and vice-versa. This indicates that some underlying factors affect the market index as well as the stock prices. Stock prices are related to the market index and this relationship could be used to estimate the return on stock. Towards this purpose, the following equation can be used: R=i  i + i R m + i Where, Ri = expected return on security i  i = intercept of the straight line or alpha co-efficient  i = slope of straight line or beta co-efficient R m = the rate of return on market index i = error term or return expected on account of unsystematic risk According to the equation, the return of a stock can be divided into two components, the return due to the market and the return independent of the market. βi indicates the sensitiveness of the stock return to the changes in the market return. For example, βi of 1.5 means that the stock return is expected to increase by 1.5% when the market index return increases by 1% and vice-versa. Likewise, βi of 0.5 expresses that the individual stock return would change by 0.5 per cent when there is a change of 1 per cent in the market return. βi of 1 indicates that the market return and the security return are moving in tandem. The estimates of βi and αi are obtained from regression analysis. The single index model is based on the assumption that stocks vary together because of the common movement in the stock market and there are no effects beyond the market (i.e. any fundamental factor effects) that accounts for the stocks co-movement. The expected return, standard deviation and co- variance of the single index model represent the joint movement of securities. The mean return is: R=i  i +  i R m +  i The variance of security’s return: = 2 2 i  2 m +  2 i The covariance of returns between securities i and j is: © The Institute of Chartered Accountants of India  2. 6.34 ADVANCED FINANCIAL MANAGEMENT 34  ij =  i j  2m The variance of the security has two components namely, systematic risk or market risk and unsystematic risk or unique risk. The variance explained by the index is referred to as systematic risk. The unexplained variance is called Residual Variance or Unsystematic Risk. The systematic risk can be calculated by using following formula: = Systematic risk  2 i  variance of market index 2 2 =  i m Unsystematic risk = Total variance - Systematic risk. i 2 =  i 2 - Systematic risk. Thus, the total risk = Systematic risk + Unsystematic risk. = 2i  2m +2i. From this, the portfolio variance can be derived  N   2  +  N 2 2   2    X i i = 2  X   m    p i i   i=1    i =1    Where,  p2 = variance of portfolio  m 2 = expected variance of index 2 i = variation in security’s return not related to the market index x i = the portion of stock i in the portfolio. β i = Beta of stock i in the portfolio Likewise expected return on the portfolio also can be estimated. For each security α i and βi should be estimated. N =RP  x ( i =1 i i + i R m ) β i = Value of the beta for security i xi = Proportion of the investment on security i © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.35 2.35 αi = Value of alpha for security i N = The number of securities in the portfolio Portfolio return is the weighted average of the estimated return for each security in the portfolio. The weights are the respective stocks’ proportions in the portfolio. A portfolio’s alpha value is the weighted average of the alpha values for its component securities using the proportion of the investment in a security as weight. N = P  xi i i=1 α P = Value of the alpha for the portfolio Similarly, a portfolio’s beta value is the weighted average of the beta values of its component stocks using relative share of them in the portfolio as weights. N P = ∑ x i i i=1 Where, β P = Value of the beta for the portfolio. Illustration 1 The following details are given for X and Y companies’ stocks and the Bombay Sensex for a period of one year. Calculate the systematic and unsystematic risk for the companies’ stocks. If equal amount of money is allocated for the stocks what would be the portfolio risk? X Stock Y Stock Sensex Average return 0.15 0.25 0.06 Variance of return 6.30 5.86 2.25 β 0.71 0.685 Correlation Co-efficient 0.424 Co-efficient of determination (r 2) 0.18 Solution The co-efficient of determination (r2) i.e. square of Coefficient of Correlation gives the percentage of the variation in the security’s return that is explained by the variation of the market index return. In the X company stock return, 18 per cent of variation is explained by the variation of the index and 82 per cent is not explained by the index. © The Institute of Chartered Accountants of India  2. 6.36 ADVANCED FINANCIAL MANAGEMENT 36 According to Sharpe, the variance explained by the index is the systematic risk. The unexplained variance or the residual variance is the unsystematic risk. Company X: Systematic risk = β i2  Variance of market index = (0.71)2  2.25 = 1.134 Unsystematic risk( i ) 2 = Total variance of security return - Systematic Risk = 6.30 – 1.134 = 5.166 or = Variance of Security Return (1-r2) = 6.30 X (1-0.18) = 6.3 X 0.82 = 5.166 =   x  m +  2 2 2 Total risk = Systematic Risk + Unsystematic Risk = 1.134 + 5.166 = 6.30 Company Y: Systematic risk =  i2 x  m2 = (0.685)2 x 2.25 = 1.056 Unsystematic risk = Total variance of the security return - systematic risk. = 5.86-1.056 = 4.804 Portfolio Risk  N   2  +  N 2 2   2 =   i i  m     X i i   2 p X  i=1    i=1   = [(0.5 x 0.71 + 0.5 x 0.685) 2 2.25] + [(0.5) 2(5.166)+(0.5) 2(4.804)] = [(0.355 + 0.3425) 2 2.25] + [(1.292 + 1.201)] = 1.0946 + 2.493 = 3.5876 © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.37 2.37 8. CAPITAL ASSET PRICING MODEL (CAPM) The CAPM distinguishes between risk of holding a single asset and holding a portfolio of assets. There is a trade-off between risk and return. Modern portfolio theory concentrates on risk and stresses on risk management rather than on return management. Risk may be security risk involving danger of loss of return from an investment in a single financial or capital asset. Security risk differs from portfolio risk, which is the probability of loss from investment in a portfolio of assets. Portfolio risk is comprised of unsystematic risk and systematic risk. Unsystematic risks can be averted through diversification and is related to random variables. Systematic risk is market related component of portfolio risk. It is commonly measured by regression coefficient Beta or the Beta coefficient. Low Beta reflects low risk and high Beta reflects high risk. As the unsystematic risk can be diversified by building a portfolio, the relevant risk for determining the prices of securities is the non-diversifiable component of the total risk. As mentioned earlier, it can be measured by using Beta (β) a statistical parameter which measures the market sensitivity of returns. The beta for the market is equal to 1.0. Beta explains the systematic relationship between the return on a security and the return on the market by using a simple linear regression equation. The return on a security is taken as a dependent variable and the return on market is taken as independent variable then R j = Rf + β (Rm – Rf). The beta parameter (β) in this William Sharpe model represents the slope of the above regression relationship and measures the sensitivity or responsiveness of the security returns to the general market returns. The portfolio beta is merely the weighted average of the betas of individual securities included in the portfolio. Portfolio Beta (βP) =  Proportion of Security × Beta for Security. CAPM provides a conceptual framework for evaluating any investment decision where capital is committed with a goal of producing future returns. CAPM is based on certain assumptions to provide conceptual framework for evaluating risk and return. Some of the important assumptions are discussed below: (i) Efficient market: It is the first assumption of CAPM. Efficient market refers to the existence of competitive market where financial securities and capital assets are bought and sold with full information of risk and return available to all participants. In an efficient market, the price of individual assets will reflect a real or intrinsic value of a share as the market prices will adjust quickly to any new situation, John J. Hampton has remarked in “Financial decision making” that although efficient capital market is not much relevant to capital budgeting decisions, but CAPM would be useful to evaluate capital budgeting proposal because the company can compare risk and return to be obtained by investment in machinery with risk and return from investment in securities. (ii) Rational investment goals: Investors desire higher return for any acceptable level of risk or the lowest risk for any desired level of return. Such a rational choice is made on logical and consistent ranking of proposals in order of preference for higher good to lower good and this © The Institute of Chartered Accountants of India  2. 6.38 ADVANCED FINANCIAL MANAGEMENT 38 is the scale of the marginal efficiency of capital. Beside, transactive preferences and certainty equivalents are other parameters of rational choice. (iii) Risk aversion in efficient market is adhered to although at times risk seeking behaviour is adopted for gains. (iv) CAPM assumes that all assets are divisible and liquid assets. (v) Investors are able to borrow freely at a risk less rate of interest i.e. borrowings can fetch equal return by investing in safe Government securities. (vi) Securities can be exchanged without payment of brokerage, commissions or taxes and without any transaction cost. (vii) Securities or capital assets face no bankruptcy or insolvency. (viii) CAPM assumes that the Capital Market is in equilibrium. Based on above assumptions the CAPM is developed with the main goal to formulate the return required by investors from a single investment or a portfolio of assets. The required rate of return is defined as the minimum expected return needed so that investors will purchase and hold an asset. Risk and return relationship in this model stipulates higher return for higher level of risk and vice versa. However, there may be exception to this general rule where markets are not efficient. Three aspects are worth consideration: (a) Stock market is not concerned with diversifiable risk (b) It is not concerned with an investor having a diversified portfolio (c) Compensation paid is restricted to non-diversifiable risk. Thus, an investor has to look into the non-diversifiable portion of risk on one side and returns on the other side. To establish a link between the two, the required return one expec ts to get for a given level of risk has been mandated by the Capital Asset Pricing Model. If the risk free investment R f is 5%, an investor can earn this return of 5% by investing in risk free investment. Again if the stock market earns a rate of return R m which is 15% then an investor investing in stocks constituting the stock market index will earn also 15%. Thus, the excess return earned over and above the risk free return is called the market risk premium (Rm – Rf) i.e. (15% - 5%) = 10% which is the reward for undertaking risk. So, if an investment is as risky as the stock market, the risk premium to be earned is 10%. If an investment is 30% riskier than the stock market, it would carry risk premium i.e. 30% more than the risk premium of the stock market i.e. 10% + 30% of 10% = 10% + 3% = 13%. β identifies how much more risky is an investment with reference to the stock market. Hence the risk premium that a stock should earn is β times the risk premium from the market [β × (Rm – Rf)]. The total return from © The Institute of Chartered Accountants of India  PORTFOLIO MANAGEMENT 6.39 2.39 an investment is the risk free rate of return plus the risk premium. So the required return from a stock would be Rj = Rf + [β × (Rm – Rf)]. In the above example 5% + 1.3 × (15% - 5%) = 18% The risk premium on a stock varies in direct proportion to its Beta. If the market risk premium is 6% and β of a stock is 1.2 then the risk premium for that stock shall be 7.2% (6% × 1.2) Illustration 2 A company’s beta is 1.40. The market return is 14% and the risk free rate is 10%. (i) What is the expected return of the company’s stock based on CAPM. (ii) If the risk premium on the market goes up by 2.50% points, what would be the revised expected return on this stock? Solution (i) Computation of expected return based on CAPM Rj = Rf + β (Rm – Rf) = 10% + 1.40 (14% - 10%) = 10% + 5.6% = 15.6% (ii) Computation of expected return if the market risk premium goes up by 2.50% points The return from the market goes up by 2.50% i.e. 14% + 2.50% = 16.50% Expected Return based on CAPM is given by Rj = 10% + 1.40 (16.5% - 10%) = 10% + 1.40 × 6.5% = 10% + 9.10% = 19.10% A graphical representation of CAPM is the Security Market Line, (SML). This line indicates the rate of return required to compensate at a given level of risk. Plotting required return on Y axis and Beta on the X-axis we get an upward sloping line which is given by (R m – Rf), the risk premium. The higher the Beta value of a security, higher would be the risk premium relative to the market. This upward sloping line is called the Security Market Line. It measures the relationship between systematic risk and return. © The Institute of Chartered Accountants of India  2. 6.40 ADVANCED FINANCIAL MANAGEMENT 40 Illustration 3 The risk premium for the market is 10%. Assuming Beta values of Security K are 0, 0.25, 0.42, 1.00 and 1.67. Compute the risk premium on Security K. Solution Market Risk Premium is 10%  Value of K Risk Premium of K 0.00 0% 0.25 2.50% 0.42 4.20% 1.00 10.00% 1.67 16.70% Illustration 4 Treasury Bills give a return of 5%. Market Return is 13% (i) What is the market risk premium (ii) Compute the  Value and required returns for the following combination of investments. Treasury Bill 100 70 30 0 Market 0 30 70 100 Solution Risk Premium Rm – Rf = 13% - 5% = 8%