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Let’s start with a review of what we did in the class!!! In the last class… We discussed Markowitz Model … a model helping us to create an optimum portfolio Markowitz Model of Portfolio •First, we discussed ----- calculation of returns and risk for each portfolio as they have to be evaluated in this two parametric framework. Markowitz Model of Portfolio •We were trying to find an optimum portfolio!!!!! •One step towards that is ----- Reduce choice set by using Mean-Variance Dominance Principle and thus, obtain EFFICIENT FRONTIER. Efficient Frontier OPTIMUM SELECTION OF A PORTFOLIO DEPENDS UPON RISK - RETURN TRADE - OFF!!! Expected Return F P OPTIMUM PORTFOLIO E Standard Deviation What are the most important contribution of Markowitz model? ???????? !!!!!!!!!! What are the most important contributions of Markowitz model? It has two important contributions: FIRST, it has provided tools of ‘quantification of ‘Risk and Return ’!!! What are the most important contributions of Markowitz model? Second is the concept of ‘Efficient Portfolio’!!! Is there anything in the Markowitz Model at which you would like to ‘ATTACK’? FIRST... Are you comfortable with TwoParametric model to evaluate a security/portfolio? Are Mean and Variance sufficient to evaluate a security or a portfolio? Look at the following two shares… SHARE - A Return (%) Probability 2 0.05 9 0.29 12 0.24 16 0.17 19 0.12 23 0.07 28 0.03 30 0.04 1.00 Expected 14.10% Return Standard 6.40 Deviation Skewness 0.80 SHARE - B Return (%) Probability 1 0.02 4 0.08 7 0.10 9 0.13 12 0.16 16 0.18 21 0.31 30 0.02 1.00 Expected 14.10% Return Standard 6.40 Deviation Skewness -1.07 Now, look at their distribution … PROBABILITY DISTRIBUTION OF RETURNS 0.35 SHARE A SHARE B 0.30 PROBABILITY 0.25 0.20 0.15 What do you think in which shares should you invest? 0.10 0.05 0.00 0 5 10 15 20 RETURN 25 30 35 Now, look at again the following two shares… SHARE - A Return (%) Probability 2 0.05 9 0.29 12 0.24 16 0.17 19 0.12 23 0.07 28 0.03 30 0.04 1.00 Expected 14.10% Return Standard 6.40 Deviation Skewness 0.80 SHARE - B Return (%) Probability 1 0.02 4 0.08 7 0.10 9 0.13 12 0.16 16 0.18 21 0.31 30 0.02 1.00 Expected 14.10% Return Standard 6.40 Deviation Skewness -1.07 It shows that Skewnwss is also that may matter in making a choice!!!! SECOND... Why is Markowitz Model not working? I want to invest in RISK-FREE ASSET and the Markowtiz model does not allow is this!!!!!! I do not know how to help her!!!! THIRD... Large Volume of data required. I would become mad!!! I really do not know how many pieces of input data I need to generate my best portfolio? Too much information required!!! • This model requirement of information is huge and it increases exponentially with increase in the number of securities. • Markowitz model requires (n (n+3))/2 pieces of input data. FOURTH... Have you ever wondered why returns of shares of companies from various industries are correlated? Scatter Diagram 40 R = 0.2814 30 ONGC (Return%) 20 10 0 -10 -5 0 5 -10 -20 ACC (Return%) 10 15 SCATTER DIAGRAM OF RETURNS 4 R = 0.2674 RANBAXY LABORATORIES LTD.(%) 3 2 1 0 -3 -2 -1 0 1 -1 -2 -3 INFOSYS TECHNOLOGIES LTD.(%) 2 3 4 Scatter Diagram 6 R = 0.289 4 RIL (Return%) 2 0 -10 -5 0 5 -2 -4 -6 ACC (Return%) 10 15 SCATTER DIAGRAM OF RETURNS 10 R = 0.3027 8 STATE BANK OF INDIA(%) 6 4 2 0 -3 -2 -1 0 1 -2 -4 RANBAXY LABORATORIES LTD.(%) 2 3 4 What makes shares’ return to have correlation across the companies from the different industries? THINK!!! Is there some underlying FACTOR which makes these correlations to exist? But, are we in a position to identify that factor? If that factor exists, then your data requirement will also be considerably reduced!!!! Yes!!!! We can identify that factor... And, this takes us to ... And, now… ?????????????????????????……… Ri Rm SHARPE’S SINGLE FACTOR/INDEX MODEL It is a linear relation between the return of a security and the underlying factor which is the MARKET INDEX. Ri Rm • It is ex-post relationship. • It shows how a factor leads to generation of returns in a security. • Its intercept represents unique return of security which is independent of Market Index. a • The slope of the Single Index Model represents which is a measure of SYSTEMATIC RISK. Systematic Risk Vs. Unsystematic Risk • Systematic Risk: Return on an asset is systemically influenced by return on market portfolio; hence if any variation in the return of an asset is explained by the variation in the market return, then such a variation is called SYSTEMATIC RISK. Such a risk is caused mainly by the macro factors; and it is non-diversifiable risk. • Unsystematic Risk: Any variation in the return of an asset that is not explained by the variation in the market return and is independent of the market risk, or that resides within the asset itself is called UNSYSTEMATIC RISK. Such a risk is caused mainly by the micro factors; and it is diversifiable risk. CHARACTERISTIC LINE • A regression line fitted to the scatter plot of returns from the market portfolio and a security is called CHARACTERISTIC LINE. • This is also a line that gives us the estimates of the parameters of the Single Factor Model. • The slope of the characteristic line is called that represents SYSTEMATIC RISK. • It is called a characteristic line as its slope showing the risk characteristics of a security which is different for different securities. CHARACTERISTICS LINE 3 y = 0.4619x - 0.2251 R2 = 0.1813 2 1 0 -1.5 -1 -0.5 0 -1 -2 -3 0.5 1 1.5 2 2.5 3 COMPONENTS OF TOTAL RISK OF A SECURITY • Total Risk of a security is determined by the variance of the returns. • It is equal to Unsystematic Risk and Systematic Risk. That is--- TOTAL RISK = SYSTEMATIC RISK. UNSYSTEMATIC RISK + Risk - – Where Total Risk of ith security = si2; Systematic Risk = i2 sm2 ; and Unsystematic Risk = Total Systematic Risk = si2 - i2 sm2. Is there any statistical measure that can tell us - out of total variation, how much per cent variation is due to systematic part and how much is due to unsystematic part? • YES!!! • It is R2. It represents proportion of total risk which is SYSTEMATIC. • In what way, the information of R2 is useful for an investment manager? What’s the difference between … • Total Systematic Risk? • β? • R2 ? ESTIMATION OF • The estimation of of a security needs the following steps: – First, identify a suitable MARKET INDEX. – Collect information about the prices of the security and the Index. – Fit the regression equation on the returns of the security and the Index where the security return will be taken as a dependent variable and the return on the Index will be taken as an independent variable. ESTIMATION [EXCEL output] SUMMARY OUTPUT Dr. Reddy'S Laboratories Ltd. Regression Statistics Multiple R 0.423823119 R Square 0.179626036 Adjusted R Square 0.178161082 Standard Error 6.876151354 Observations 562 ANOVA df Regression Residual Total Intercept X Variable 1 1 560 561 SS MS F Significance F 5797.440487 5797.440487 122.6155199 6.61196E-26 26477.61617 47.28145745 32275.05666 Coefficients Standard Error t Stat P-value 0.841961864 0.290330094 2.900015814 0.003877893 0.753268111 0.068026302 11.07318924 6.61196E-26 ESTIMATION [EXCEL output] SUMMARY OUTPUT Oil & Natural Gas Corpn. Ltd. Regression Statistics Multiple R 0.339940172 R Square 0.11555932 Adjusted R Square 0.113237954 Standard Error 7.369805289 Observations 383 ANOVA df Regression Residual Total Intercept X Variable 1 1 381 382 SS MS F Significance F 2703.791967 2703.791967 49.78072823 8.16384E-12 20693.64543 54.31403 23397.4374 Coefficients Standard Error t Stat P-value 0.275169466 0.376636098 0.730597698 0.465473937 0.696256196 0.098682115 7.05554592 8.16384E-12 ESTIMATION [EXCEL output] SUMMARY OUTPUT Reliance Industries Ltd. Regression Statistics Multiple R 0.714636907 R Square 0.510705909 Adjusted R Square 0.509833727 Standard Error 5.35907977 Observations 563 ANOVA df Regression Residual Total Intercept X Variable 1 SS MS F Significance F 1 16816.83319 16816.83319 585.5497139 3.95203E-89 561 16111.77189 28.71973599 562 32928.60508 Coefficients Standard Error t Stat P-value 0.214380529 0.226065807 0.94831028 0.343379833 1.282653728 0.053006306 24.19813451 3.95203E-89 Any comment?? Source: BSE Site Beta of a Portfolio … • Beta of a portfolio is the weighted average of individual securities betas. n β P = ∑X i β i i =1 What next…? And, now… something exciting… Concept of Beta from Single Index Model What a cocktail!!!! All these take us to … Concept of Systematic Risk - βeta Markowitz Efficient Frontier ????? Risk-Free Asset WHAT’S THE WORTH OF A CAPITAL ASSETS??? Dr. C. P. Gupta CAPITAL ASSET PRICING MODEL It is a model that tries to answer the following questions: What is the relevant CHOICE SET OF SECURITIES/PORTFOLIOS given the risk free asset and risky assets? How investors select the final OPTIMAL PORTFOLIO? What risk is considered by the market in pricing a security? What should be the equilibrium return and price? It makes use of the foundations built by the Markowitz Model and the Single Factor Model of Sharpe. Its main contribution is LINEARITY and SIMPLICITY. Assumptions of Capital Assets Pricing Model Investments are judged on the basis of risk and return associated with them. Returns are visualized in stochastic manner by investors. Investors maximise their expected utility function which is determined by return and risk. Investors are rational investors. Investors are risk averse. Market is perfectly competitive. Market is frictionless i.e. it has no transaction cost and information is also cost free. Assumptions of Capital Assets Pricing Model(continued…) Capital assets are perfectly divisible. Investors can have unlimited borrowing and lending at risk free rate. All investors have homogenous probability distributions and expected returns for future returns. All investors have same one holding period time horizon. All investors are Markowitz efficient. None is expecting any unanticipated inflation. All assets are available in fixed quantities. Capital market is in equilibrium. WHAT HAPPENS TO EFFICIENT FRONTIER WHEN A RISK FREE ASSET IS INTRODUCED INTO CAPITAL MARKET??? Will it be a non-linear or linear ??? EFFICIENT FRONTIER becomes a straight line that is tangent to Markowitz Efficient Frontier and it is called CAPITAL MARKET LINE. Capital Market Line (CML) CML is a line rising from the risk free rate, Rf, on the vertical axis and tangential to the Markowitz Efficient Frontier at M, which is market portfolio. It consists of efficient portfolios constructed by combing risk free security and market portfolio. It represents equilibrium in the capital market. Borrowing Lending M Rf sM Risk Capital Market Line (CML) (continued…) All risky assets are included in the market portfolio to extent of their supply. All portfolios on CML are perfectly correlated with the market portfolio and it implies that they are completely diversified and hence, possesses no unsystematic risk. CML relates the expected rate of return of an efficient portfolio to its standard deviation. The equation of CML is - E (RP ) RF E (RM ) - RF sM sP Capital Market Line (CML) (continued…) The slope of CML represents the price per unit of risk. It does not show how the expected rate of return of an asset relates to its individual risk. Therefore, in an equilibrium situation, the market will price only systematic risk and eta measures the systematic risk. This is known as the ‘SYSTEMATIC RISK PRINCIPLE’ which states that the expected return on an asset depends only on its systematic risk. ONE - FUND THEOREM It says that “one can generate an Efficient Portfolio by taking only ONE FUND and that is, the Market Portfolio and combine it with a risk free asset. WHICH PORTFOLIO FROM CML SHOULD BE SELECTED BY AN INVESTOR…??? Depending upon an investor’s return - risk trade-off which is reflected in his indifference map, he selects an optimum portfolio for himself. B M A Rf sM Risk Does the idea of Capital Market Line ensure better risk-return tradeoff for me??? Yes!!! It will improve risk-return trade-off for our Topiwalla. M Rf Do you see this? sM Risk ARE Investment Decisions and Financing Decisions independent ??? TOBIN’S SEPARATION THEOREM * Decision to invest in a capital asset has two stages: » “How to find the proportion of optimal portfolio of risky assets?” [Investment Decision ] and » “How to finance the portfolio of risky assets?” [Financing Decision ] TOBIN’S SEPARATION THEOREM (continued…) * Investment Decision is same for all investors as every one selects the market portfolio of risky assets. * Financing Decision is left for the individual investor. He/she can decide how much to borrow or to lend at risk free rate depending upon his/her degree of risk averseness. * Thus, investment decision and financing decision of each investor are totally independent investment decisions are same for all; and financing decisions are different and independent of investment decisions. SECURITY MARKET LINE (SML) SML is a line drawn in E(R) and space. It shows a linear relation between a security’s expected return and its . Security lying above SML is under-priced while security below SML is over-priced. Security lying to the right of = 1 is aggressive while security on the left of = 1 is defensive. The equation of SML is: E(Ri) = RF + ( E(RM) - RF ) i SECURITY MARKET LINE (SML) The equation of SML is called CAPITAL ASSET PRICING MODEL. E(RM) - RF is called risk premium per unit of systematic risk. SML M Aggressive Security Rf Defensive Security 1 What should be the price of a security in an equilibrium capital market …??? CAPM directly does not provide price of a security. However, indirectly through expected return, it provides price as return and price are inversely related. Let P1 and P0 represent price of a security at time 1 and time 0 respectively. Also, if P1 is the expected price, then by definition, the expected return, E(R), would be: E (R ) P1 - P0 P0 RF ( E ( RM ) - RF ) P0 1 {RF P1 - P0 P0 P1 ( E ( RM ) - RF ) } CAPM and its IMPLICATIONS CAPM makes investment decision simple. Just buy market portfolio. CAPM helps in identifying over - and under - priced securities. CAPM helps in the performance evaluation of an investment portfolio. A number of measures are developed to evaluate a portfolio. They are: Jensen’s Index Sharpe’s Index Treynor’s Index CAPM says “ Simplified diversification works “. CAPM is very useful in capital budgeting decisions. It helps in finding: Certainty Equivalent; and Risk Adjusted Discount Rate CAPM vs. SINGLE FACTOR MODEL SFM - a linear relation between the return of a security and the underlying factor. CAPM - a linear relation between the return of a security and its . SFM - represents ex-post relationship while CAPM represents ex-ante relationship. SFM - shows how a factor leads to generation of returns in a security, i.e. it shows return generating process while CAPM shows how the market price a security and how much risk premium, the market is willing to pay for one unit of systematic risk. SFM - its intercept represents unique return of a security when the return on the factor is zero while the intercept of CAPM represents risk free rate. The slope of SFM represents while the slope of CAPM represents the risk premium. What’s Next…??? ????? Dr. C. P. Gupta MEASURING PORTFOLIO PERFORMANCE …??? Portfolio performance MEASUREMENT AND EVALUATION is the last step in the process of portfolio management. The basic objective of measuring performance is - to judge the return of a portfolio vis-à-vis with the risk involved in it. That is to say, ASSOCIATE A MEASURE OF RISK WITH THE RETURN and then, determine whether the portfolio manager is able to generate more returns than expected. Portfolio Evaluation is concerned with the evaluation of the PORTFOLIO AS A WHOLE without examining the performance of individual securities in the portfolio. Before, we proceed further... We should also evaluate to what extent a portfolio is diversified. For that we must use - R2. WHY? Measures of Portfolio Evaluation THE SHARPE INDEX S s Pt THE TREYNOR’S INDEX T RPt - RF RPt - RF Pt THE JENSEN INDEX (ALSO KNOWN AS THE JENSEN’S ) J RPt - {RF ( RMt - RF ) P } Measures of Portfolio Evaluation (continued…) APPRAISAL RATIO - P/s(eP): It divides the alpha of the portfolio by the non-systematic risk of the portfolio. It measures abnormal return per unit of risk that in principle could be diversified away by holding a market index portfolio. Measures of Portfolio Evaluation (continued…) The M2 Measure of Performance: This measure wad made popular by Leah Modigliani, grand daughter of Franco Modigiliani. To compute M2, an imaginary portfolio is constructed by mixing the managed portfolio(say, P*) with a position in risk free assets in such a manner that the variance of such a portfolio matched with the variance of the market portfolio. Then, M2 = RP* - RM Looking for the exact source of success/failure! ! Why a portfolio manager is able to perform better or worse? FAMA’S DECOMPOSITION OF TOTAL RETURN ... E. Fama has provided an analytical framework that allows a detailed breakdown of a fund’s performance into the source or components of performance. Such a decomposition of total return is useful in identifying the different skills in portfolio management and to what extent the portfolio manager is capable of managing each one of them. This may suggest the areas of strength and those of weakness in the ability of a portfolio manager. FAMA SUGGESTED THE FOLLOWING DECOMPOSITION OF THE TOTAL RETURN FROM A PORTFOLIO... TOTAL RETURN EXCESS RETURN RISK FREE RETURN RISK PREMIUM DUE TO SYSTEMATIC RISK RETURN FROM SHARE SELECTION DUE TO UNSYSTEMATIC RISK FAMA’S DECOMPOSITION... Using the decomposition scheme discussed, Fama suggested the following: RP = RF + R1 + R2 + R3 where RP = Return on the managed portfolio; RF = Return on a risk free asset; R1 = Return from SYSTEMATIC RISK and is equal to (RM - RF)P; R2 = Return from UNSYSTEMATIC RISK & is equal to (RM - RF)(sP/sM - P); and R3 = Residual Return and Fama named as NET SELECTIVITY MEASURE. PORTFOLIO PERFORMANCE EVALUTION - AN ILLUSTRATION Consider the following information about Portfolio A, Portfolio Z and the Market Portfolio -M: PORTFOLIOS RETURN A Z M (MARKET INDEX) Risk - Free Return = 7% 12% 19% 15% STANDARD DEVIATION 18% 25% 20% BETA 0.7 1.3 1.0 SHARPE RATIOS A Z M 0.28 0.48 0.40 TREYNOR RATIOS A Z M 7.14 9.23 8.00 JENSEN RATIOS A Z M -0.60 1.60 0.00 APPRAISAL RATIOS Unsystematic Risk A Z M 11.31% 25.00% 20.00% APPRAISAL RATIOS -5.30 6.40 0.00 2 A Z M M - MEASURE Proportion of Investment in Fund 1.11 0.80 1.00 FAMA'S DECOMPOISTION Risk - Free Rate A Z M 7% 7% 7% M-SQUARE MEASURE -0.017 0.002 0.000 Risk Premium Due to Due to Unsystematic Systematic Risk Risk 5.60% 1.60% 10.40% -0.40% 8.00% 0.00% Net Selectivity Measure -2.20% 2.00% 0.00% Total 12% 19% 15% Other Measures... Expense Ratio: It is a ratio of the total expenses of a fund to the average net assets of a fund. Portfolio Turnover Ratio: It is defined as minimum of assets bought or assets sold during a year divided by average assets of a fund. Other Measures…(continued) Tracking Error: It is defined as deviation of the difference in returns portfolio under consideration and benchmark or target; that is to say, DEVIATION OF (Rp-Rb) where Rp is the portfolio under consideration while Rb is the benchmark portfolio. the standard between the a specified STANDARD return on the the return on Portfolio Evaluation completes the cycle of activities comprising portfolio management. And, thus, we come to an end of the course. But, before that - the last words At, the end of the Course, I feel that we have enough light about Investment Management !!!!!