LESSON 28: CAPITAL ASSET PRICING MODEL (CAPM)

Transcription

1 LESSON 28: CAPITAL ASSET PRICING MODEL (CAPM) The CAPM was developed to explain how risky securities are priced in market and this was attributed to experts like Sharpe and Lintner. Markowitz theory being more theoretical, CAPM aims at a more practical approach to stock valuation. It is no doubt based on the mean-variance approach to risk for assessment of investment as developed by Markowitz. It explains the behavioural pattern of investors in building up portfolios. CAPM-Assumptions The CAPM is based on certain assumptions some of which are common to CAPM and MPT. CAPM is it1 L1Ct developed as part of MPT (Modern portfolio Theory). The assumptions are first set out below: 1 The investor aims at maximising the utility of his wealth, rather than the wealth or rerun. The difference between them is that individual preferences are taken into account in the utility concept. While some have preference for larger risk who will have increasing marginal Utility for wealth, for others, with less preference for risk the incremental wealth will be less attractive if it is attached with more risk. Thus, the preference of investors for risk return will be taken into account in this model. 2 Investors have similar expectations of Risk and Return. Without these consensus standards, the estimates of mean and variance may lead to different forecasts with the result that the ct1icient portfolio of each will be different from that of the others. There will be innumerable efficient frontiers, each dependent on the set of preferences of individuals for risk and return. If investors do not have similar expectations there will be no homogeneity in their conception and no single efficient frontier line will apply to all. This in [Urn will imply that the price of an asset, which is the best estimate of the present value of future returns will be different for different investors. This assumption is therefore unrealistic for application in the real world. 3 Investors make investment decision on a rational basis, depending on their assessment of risk and return. Risk is measured by two factors, mean and variance. In the CAPM we assume that rational investors diversify away their diversifiable risk, namely, unsystematic risk and only systematic risk remains which varies with the Beta of the security. While some use the beta only, as a measure of risk, others use both Beta and variance of returns (total Risk) as the sources of reward or expected return. As these perceptions of risk and reward vary from individual to individual, under CAPM we get a series of efficient frontier lines while in the case of MPT, there will be a single efficient frontier line as the conception of risk and return expectation is assumed to be homogeneous in the latter. 4 Investors will have free access to all available information at no cost and no loss of time. If the information is not the same for all, no common efficient frontier line can be drawn. Besides even if the information is not available at the same time different conclusions can be drawn regarding expected return and risk and no single price of the capital asset CU.1 be conceived. 5 Investors should have identical time horizons which again is highly unre-alistic. Investors have different time horizons and their estimates of stock value will therefore differ, even as the estimated earnings are the same per year. Continuous time models are sometimes used to get over the above difficulty or again one ~ approximate a single period model as a proxy to multiperiod model on the assump-tion that returns are the same over time and time has no relevance to expected returns and that expected returns are again independent of the past and current information. While the above assumptions are common to both CAPM and MPT, some assumptions are specific to CAPM. Thus, there is a risk free asset, which gives risk free return. Investors can borrow and lend unlimited amounts at the same price. 1rus assumption of risk free asset transforms the curved efficient frontier line to a linear one. Risk can be reduced by adding a risk free asset, or borrowing at the risk free rate. Besides, it is also necessary to assume in CAPM that total asset quantity is fixed and all assets are marketable and divisible. This assumption implies that the liquidity requirement of investors is ignored and there will no new issues, which are both unrealistic. After the brief review of the above assumptions we can summarise the requirements for CAPM as follows: Risk is measured by variance of expected returns. There are two components of Risk - systematic (non-diversifiable) and unsystematic (diversifiable). For diversifiable risk, the investor makes a proper diversification to reduce the risk and for the non-diversifiable portion, he uses the relevant Beta measure to adjust to his requirement or preferences. Due to the possibility of risk free asset and lending and borrowing at the free rate, the investor has two components of the portfolio - risk free assets and the risky market assets. His total return is summation from the above two components. Under CAPM, the equilibrium situation arises when all frictions, like taxes, divisibility transaction costs and different risk-free borrowing and lending rates are assumed away. Equilibrium will be brought about by changes in prices due to changes in demand and supply. CML Figure below depicts the capital market line with risk less rate of return. Point P is the riskless interest rate. Preferred investments are plotted along the line PMZ, by combination of both risky assets and risk free asset along with the borrowing and lending

2 The slope of the PMZ is the measure of the reward for Risk taking. P is the risk free return, Em - T is the measure of the risk premium - a return for the risk taking. The reward for waiting is the risk less interest rate, OP, and second reward is the return per unit of risk borne measured by the slope a of the PMZ line. The internal rate can be considered as price of time and the slope of capital market line as price of the risk. independent variable and can be adopted for use in portfolio management and in purchase of individual securities. Such a line is called Security Market Line, which depicts a linear relationship between expected return and the systematic risk. The SML curve drawn below shows a positive slope, indicating that the return and risk are positively related. The higher the risk the higher is the return. The Beta of Market portfolio, as represented by the BSE or NSE index is always one. But the company scrip can have Betas higher than 1 or lower than 1. Those with Betas less than 1 are defensive securities and those with Betas above 1 are called aggressive scrip s. The graph below shows these types of scrip s and the relationship of Beta to expected return. If the borrowing and lending rates are different, then OQ becomes the borrowing rate and OP will be the lending rate, as shown below. The efficient frontier line with differential borrowing and lending rates will be as shown below: QMN is with the borrowing rate of OQ and PMN is with the lending rate of OP and ABC is the efficiency frontier line without borrowing and lending. The curved line will become linear; if once the riskless asset of borrowing and lending at fixed riskless rate is introduced. The CML as described above reflects the relationship of total risk and expected return. Total risk includes both systematic and unsystematic risks. It may also include the risk free assets to reduce the total risk. The CAPM has two components of the capital market return, which are reward for waiting or riskless return, and the reward per unit of risk borne as measured by the slope of the CML line. The investors will have their choice of efficient portfolio somewhere along the line of CML, as all efficient portfolios would be on it. Those which are less efficient will be below the line PMN, in the chart above. The risk free rate can be thought the price of time and the slope of the capital market line as the Price of Risk. Security Market Line (SML) Unlike the CML, which considers the total risk as a measure of variability of returns, SML takes into account only the systematic risk, which is market related and is not possible to reduce or eliminate by diversification. Beta is the measure of risk of a security relative to the whole market, and is used in the SML. Since the unsystematic risk is already taken care of by diversification in the construction of an efficient portfolio, it is desirable to develop an alternative to CML which will use Beta as the SML is Security Market Line, OS is the risk-free return, OP is the return of the market, whose Beta is 1; those below Beta 1 are defensive and others are aggressive scrip s in the market. SML can be represented symbolically by an equation as Ri = Rf + Bi (Rm - Rf) Ri is the return on the security, i, Rf is Risk-free return Rm is Market return. Bi is Beta of Scrip i related to Market Risk If Rf = 10, Rm = 20 Bi = 1.5, which is more risky than the market average, then Ri = (20-10) = = 25.0% which is higher than the market return Suppose the Bi is less risky than the market at 0.75 then Ri = (20-10) = = 17.5%, which is lower than the market return From the above equation, we can estimate the expected return on a security. It is represented as something like a premium or discount on the market return and can be compared. It can be a return on a security as distinguished from a portfolio. If the security is correctly priced it will have Ri = Rf = 0 and SML curve goes through the origin (see chart below) Ri-- Rf measures the excess return which varies with the risk taken. Within the chart it is seen that Rm - P, f excess return if market Beta = 1. The security market line implies that the individual assets and port-folio should be on SML, If they are cor-rectly priced. Beta values should then cor-rectly represent the contribution to the risk of the security to the portfolio. All

3 assets lying above the SML are undervalued and those below the SML are overvalued. Beta relates the portfolio premium to market risk premium. If Beta is one they are the same. Market Model can be presented in the form of a regression equation. Taking the above equation for Risk premium of the portfolio, let us introduce a new concept of risk adjusted excess return which is generated by the expertise of the portfolio Manager, represented by the a o p. This can be graphically represented as the y-axis intercept for the regression line. a o p can be zero, positive or negative, depending on the performance of the portfolio. When this model is presented in the Risk premium form., the equation is Rp - Rf = αp - Rf (1 - βpm) + β pm (Rm - Rf) + rp αp o = αp Rf (1- βpm) If we buy undervalued securities, the returns will be more and vice versa. It will thus be seen that SML curve assumes a critical importance in portfolio selection and individual investment decision. CAPM Analysis The expected return of a portfolio in equilibrium is equal to risk free rate Rf, plus risk premium which is related to its Beta. Thus, Rp - Rf = risk premium and this is equal to B PM (Rm - Rf) where Rp is expected rate of return and Rf the risk free return. These symbols are the same as explained above. This leads us to the market model, which relates the expected excess return of the portfolio to the excess return of the market. This is an explanation of the risk premium which gives excess returns. The chart below presents the relation of E [(Rm)- (Rf)], to the E (RP-Rf) Viz., in words, excess returns on a portfolio s excess risk over the market risk. This chart presents CAPM in a general form with expected excess market risk related to expected excess return. Market Model Risk Premium form is the one shown above and the equation for this is As the risk of the portfolio remains unaffected, Bpm of the characteristic line remains unchanged but ap changes to ap o as follows: αp o = ap - Rf (1 - Bpm) αp o is risk adjusted excess return αp is return on portfolio when the market return is zero. In other words, αp represents the excess return of the portfolio, when the market return is equal to riskless and rp is the error or the residual. αp 0 can be the same as αp when Rf (1- Bpm) is zero which happens when Bpm is 1 ; then the excess market return disappears. The above equation now becomes ; Rp- Rf = αp + Bpm (Rm- Rf) +rp; putting it differently. If error term is dropped then the equation becomes Rp = Rf = αp + βpm (Rm Rf) The above equation states that if the risk adjusted excess return on a portfolio is positive αp o >o, then the portfolio return is greater than what is normally expected, indicated superior return. If apo is positive te portfolio Manager has shown his expertise in beating the market by showing higher than the market return. We have seen that the slope of SML curve is Beta. If we have a perfectly diversified portfolio in the CAPM, the error term disappears. If at the same time expected risk adjusted excess return α o of the portfolio is zero which is assured under Market Efficiency Theory, then the slope of the characteristic line is Beta of the portfolio. Thus, under condition of perfect markets, the slope of the Regression line is Beta and excess returns disappear (αp 0 = 0) Uses and Limitations In real world, investors get higher return for higher risk and they are more concerned with company related risks than with the market related risks, except in the case of trained investment analysts. Companies are found to use CAPM to determine the cost of equity for the firm, to estimate the required return for divisions or lines of business and to determine the hurdle rates for corporate investments and to evaluate the performance of investment Division in terms of costs and returns. These hurdle rates of return are in general the required rates of return and the

4 corporate assess the past performance of the costs and related returns for each of the Division. In the case of public utilities, the CAPM can be used to estimate the costs and rates to be charged to cover the costs. The CAPM is used to regulate the public utilities from the point of view of costs. Historical return and Betas are used to select the proper risk in investments in the portfolio. CAPM is used to select securities, construct portfolios and evaluate the performance of the portfolio. It is thus a useful tool for investment analysis and portfolio management. The limitations of the theory are also pointed out by many critics. This theory is unrealistic for any average investor, who goes by the fundamental factors influencing the company, its earning, dividend and bonus record. Empirical tests of the Model have not proved very useful. The model is built ex-ante factors while in reality the expectations of the future vary from person to person. Data and analysis is to be based on ex-post factors while anticipations of future risk and returns are ex-ante and both may not be related. The CAPM is in fact not testable exactly as the exact composition of the market is known and is used in testing. The empirical tests conducted by Richard Roll and others were only tests on samples whether me proxy market portfolio was efficient or not. The use of surrogatives and proxies have not proved the theory as really useful and practical.. CAPM theory is thus a nice theoretical exposition but in actual world, it does not conform to the real world risk-return trends and empirical tests have not given unequivocal support to the theory. It is also found mat there - are many non-beta factors influencing me returns. The calculation of Beta is itself of doubtful validity as me historical Betas may not reflect the future risks or returns. In the short-run in particular, projections on the basis of Betas on returns and risk have been found to be unreliable and results contrary to C APM Theory were noticed. Thus, CAPM is a good theoretical tool but with its own limitations in practical applications. Capital Asset Pricing Model The assumptions of CAPM are that the market is in equilibrium and the expected rate of return is equal to the required rate of return for a given level of risk. or Beta. CAPM presents a linear relationship between the required rate of return of a security and relates it to market related risk or Beta, which cannot be avoided. The equation for the CAPM Theory is R j = Rf + B j (R M - Rf) R j is expected rate of return on security j and Rf is risk free turn. Bj is Beta coefficient - a risk measure for the non diversifiable part of total Risk. R m is return on Market Portfolio and RM Rf is the excess return for the extra risk. Limitation of CAPM It is not realistic in the real world. This assumes that all investors are risk averse and higher the risk, me higher is the return. Investors ignore the transactions cost information costs, brokerage taxes etc., and make decisions on the basis of single period horizon. The investors are given a choice on the basis of risk-return charac-teristics of an investment and they can buy at the going rate in me market. There are many buyers and sellers and the market is competitive and free forces of supply and demand determine the prices. CAPM establishes a measure of risk premium and is measured by B J (R M -Rf) Beta coefficient is the non-diversifiable risk of the asset, relative to the risk of the asset. Suppose Cipla Company has a Beta equal to 1.5 and me risk free rate is say 6%. The required rate of return on the market (RM) is 15%. Then, adopting the above equation, we have R J = Rf + (B J ) (R M - Rf) = (15-6) = = 19.5% If the market rate is 15% then the rc:tun1 on Cipla should be 19.5%, because the larger risk on Cipla than on the market. SML Security Market Line SML plots the relationship between the Required rate of return R J and non- diversified risk, Beta, as expressed above in CAPM. Example Market Expected Return = 12% Market Risk Premium (R M - Rf) = 7 Risk free Return = 5 (Rf) RM - Rf = (12-5) = 7 If security x has Beta of 1.20 Then R J = Rf + B J (R M - Rf) - = (7) = = 13.4 (Aggressive Scrip) If the security has Beta as 0.80, then it is a defensive scrip. Thus R j = (7) = = 10.6 These can be represented as follows : The actual prices of securities may fall above or below the SML. The over- valuation and under-valuation can be seen from the above chart. When we estimate the expected return after an year, in the absence: of historic data on returns and probabilities the fallowing formula which is derived from the basic formula given above is useful. Do(l +g) Expected return RJ = Po + g Where, Do = last paid dividend Po = current market price g = Growth rate

5 If the above return is higher than the equilibrium rate effecting the equilibrium price - a position on the SML - both the stocks, above the SML and below the SML have undergone some changes: The expected rate: has to be equated to the required rate: of return, when the point of equilibrium is reached on the SML. Which the expected return is higher than the required rate:, the demand for that security will rise and the price: will also rise, bringing down its return to the equilibrium level. If the expected return is lower than the required return the demand will fall leading to a fall in its price:, bringing it to the equilibrium level In the former case, the investors will buy securities and in the letter case, they will scl1 securities. Thus, the CAPM is useful to provide insights for the finance Manage~ to maximise the value of the firm. Following the principle of the higher the risk, the higher is the return, the finance Manager will keep the risk level at the optimum level in performing the investment function or financing function by keeping in mind the return that shareholder expects to take at a given level of risk at the company. The Finance Manager has to keep in mind the expected returns of the share-holders and the returns he provides should be commensurate: with the risk. This risk is reflected in his investment and financing decisions. The SML provides a bench mark reflecting the equilibrium position in the relationship between the risk and return. The risk that is reflected in the nondiversifiable or systematic risk is that the company is exposed to in its operations, financing and investment decisions. Problems Q. An investor wants to purchase a Bond with maturity 3 years, coupon rate 11 % and par value of 100. A. If the investor is requiring YTM 15% of equivalent risk what is the price he should pay. B. If the bond is selling at a price of Rs what is its YTM. YTM is to be estimated by Trial and error method. Consider 12% as YTM. Then Hence, YTM is 12% What is the duration of Bond if the YTM is 12% and expected return is 10.06%. Duration Value %of age Value Weighted Duration 1. Year 9.82 = =1x Year 8.77 = =2x (1.12) 2 3. Year =3x The duration in the case of value bond of YTM of 12% and excepted return 10.06% was shown above. Then the weighted duration of the portfolio is The Market Model and CAPM may not give similar results as expected excess returns and risk premiums, in the two models are not identical. Say p = 4%, Rf = 6% (riskless return ) β Beta = 2, Rm (expected market return) = 10% Expected Risk Premium (Rp- Rf) Market Model = P a + βpm (Rm-Rf) (Rp-Rf) = 4-6 (1-2) +2 (10-6) Where, αp α CAPM =18% = αp-rf (1-βm) 4-6 (1-2) (Rp-Rf) = βpm (Rm- Rf) = 2 (10-6) = 8% In the exceptional case when the portfolio is in equilibrium, the expected risk premium of the portfolio under both models should be identical. That happens when αp o =0= αp-rf (1- Bpm) Then under Market Model The excess Premium = 0+2 (10-6) and under CAPM Rp-Rf= 2 (10-6) 20-12= 8% = 2 x4 = 8% Problems If the risk free rate is 10%, expected return on NSE Index is 18%, standard deviation is 5% (SD). Construct an efficient portfolio to secure 16% return. The efficient portfolio consists of market securities and risk free securities invested in the proportion of W and 1- W respectively, then Rp = W (Rm) + (1-W) Rf 16% = W (18%) + (1-W) 10% 16 = 18w W =6= 18W-10W= 8 W. Thus, 8W =6 6 8 and W = 75% 1-W = 25% Investment in market securities is 75% of the amount and 25% in risk free securities. The portfolio risk in the case is (1.12) 3 Total

6 by diversifications rather than reducing. Naive diversifications in name only, which does not reduce the risk. Thus an investor may have 10 scrip s in steel, mini steel and ferrous metals, which will only increase risk. But an investor having 10 scrip spread in cycle electronics sugar steel auto etc. will have less risk as these industries are not auto correlated and their risks are independent of each other or even negatively related. It is thus left to markoiz and later research to show that diversification is a tool to reduce unsystematic risk and how it can be reduce by a study of variances and covariance of securities return, as against the market returns. If now an amount of Rs. 10,000 is to be invested and expected return is 20% construct a portfolio and calculate its risk Since it is more than 1, borrow 0.25 or 25% of Rs. 10,000 to invest in market portfolio. Thus own funds are Rs. 10,000 and borrowed funds are Rs. 2,500, with a total of Rs. 12, 500. Risk on the portfolio is calculated as follows : Why Diversification It is never prudent to put all ones eggs in one basket as it may lead to total ruin if the basket itself is broken or lost. The human behavior is normally risk averse which means that for psychological reason. he distributes his assets in a variety of risk classes some in cash some in bank deposit, Insurance, Provident fund, pension fund etc. These are all examples of the normal human behavior of diversifying the asset holdings to reduce risk, provided for contingency and take all precautions against total loss. Thus, the average investor never puts all his savings in one form or in one security for self-production and for and for psychological reason. Money kept idle or in some investments, which do not give adequate return, will be a loss to investor, as he loses the value the value of money over time. By logic of common sense investors try to satisfy most of their objectives of saving by putting money in various avenues and that means diversifications. The various objectives are income, capital appreciation safety, marketability, contingency, and liquidity and hedge against inflection and for future provision or larger incomes. His choice of investments will cater to this requirement, which lead to diversification of investments. Even without the theoretical basis of covering or reducing the unsystematic diversifiable risk, the investor in the traditional Theory used to adopt some methods of Diversification. Example on Measurement of Risk Calculation of Standard Deviation and Variance As diversification is meant for reduction of risk, its measurement is relevant here. Diversification (Random and Naive ) The traditional theory laid down diversification as a technique of selection of securities in a portfolio. This is called Random diversification or simple diversification on the basis of straight rule of two is a better than one simple diversification on Random basis was found to be more remunerative by researchers and these number of scraps in a portfolio of individuals is to be around securities. Rational basis of why diversification and how to achieve optimal diversification were studied by later researchers of which Marko wiz is reputed to be the pioneer Naive diversifications or superfluous diversifications may result from random and indiscriminate selection of securities which does not lead to any reduction of risk Thus cycle and tubes are related industries and one invests in those two types of industries. which are highly correlated in a positive manner, the risk will be increased

7 Average return is the same in both cases but risk is different, DCM has a range of variation from 8% to 12% while the escorts has a variation from 9 to 11. Calculation of Standard Deviation Example DCM = 5% % = 23.75% Here no consideration is given for risk and covariance of risk among the securities invested in the portfolio. This can be demonstrated by taking an example o Markowitz diversification. Morkowitz Diversification Before discussing the Markowitz diversification what the researches of investors and investment analysts have found is to be set out briefly. Firstly they found that putting all eggs in one basket is bad and most risky. Secondly, there should be adequate diversification of investment into various securities as that will spread the risk and reduce it; if the number of them say 10 to 15 it is adequate to enjoy the economies of time, scale of operations and expertise utilised by the investors in his analysis. Reference was already made to Naive diversification, which is a spread of investments into many securities but will not reduce the risk, like buying ten securities all in the shipping industry which is risky industry in itself. Besides some researchers have found that there is an optimisation process for diversification to reduce the unsystematic or company related risk by choosing such companies which are not closely related or not owned by the same family group in the same industry group. This optimisation process also leads to an investment in companies well chosen for differences in their characteristics, nature of the product market, pattern of production etc. Example Escorts Example : What is the expected return of a portfolio, comprising of the following securities : Risk on the Escorts is lower in the above example referred to above. Example of Simple Diversification Take two securities Diversification RP = W 1 R 1 + W 2 R 2 + W 3 R 3 Rs are expected returns and Ws are weights Rp = (10 x 0.25) + 15 (0.25) + 20 (0.50) = = 16.25% Example Given the following data on two stocks Tisco and Reliance and their returns calculate the required measures of expected returns etc. The return on the portfolio by combining them would be Rp =W 1 R 1 + W 2 R 2 Rs. are returns and W 1 and W 2 are weights of two investments. RP = 0.25 (.25) (0.75)

8 = = 14.2% Calculation of Standard Deviation Calculation of Covariance Given, Or = 3 Os = 3 Rx = Expected Return on Security x Ry = Expected If the coefficient of correlation is high as 0.893, nearer to one, then the degree of risk in the portfolio is also high SML Example Security Returns Return on Security y N = number of observations Equation for SML is as follows : Calculate the SML Ri = Rf + Beta i (Rm- Rf) Scrip A = ( ) = = = 14.8% Scrip B = ( ) = =.136 = 13.6% Scrip C = ( ) = =.124 = 12.4% CML All investors have to choose for a combination of two components of the portfolio : (1) Market Risky Portfolio. (2) Riskless Securities. The straight line tangent to the efficient frontier line is called the Capital Market Line. On this line all the efficient portfolios would be lying. The CML choose the most efficient portfolio and this indicates the market price of risk represented by the formula

9 The term Rm Rf/Omis the extra return over the risk free rate by increasing the level of risk (Sd) of the efficient portfolio by one unit. Portfolio Mix is 50 : 30: 20. What is its S.D. Following the formula used earlier we have, SML Taking non- systematic risk as zero for a well diversified portfolio, the only relevant risk is systematic risk measured by Beta. If Beta is zero, it is riskless security. For market Portfolio, the Beta is one. The straight line called SML is represented by the equation Ri = α+ bβi... Equation if Bi = O, then the first point of the line is Rf = α + b (O) Rf = α the second point of the line is RM = α +b (1) where Beta m = 1 As α = Rf, RM - α = b then RM Rf = b... Equation Combining the above two results, we have the equation of SML Ri = Rf + Bo (RM- Rf) Example Elecon stock is expected to sell at Rs. 70 a year hence and pay a dividend of Rs. 4 per share. If the stock s correlation with portfolio is-0.3 CAPM Example If the risk free rate is 8% Market Risk premium 6%, Market SD is 10% Calculate the variance of two portfolios Risk free investment 30%, Risky Fund 70% Diversified portfolio with Beta of 1.5 Portfolio (1) S.D. = 0.7 (10.0%) (0) = = = 7% Portfolio (2) 2 2 S.D. 2 = B S.D. 2 = (1.5) 2 (10) 2 = 2.25 x 100 = S.D. 2 = 225 S.D. 2 = 225 = 15% Example Indfund has three Investment strategies as given below. Find out the plan with greater risk Given the following data, answer the question below : Correlation Matrix Strategy A has the largest Risk, as the portfolio Beta is highest at 1.0, as compared to B and C. Problems Republic Forge has a beta 1.45, the risk free rate Rf is 10% and r p expected return on market portfolio is 16%. This company

10 pays a dividend of Rs. 2 a share and expected growth in dividends is 10% per annum. What is the stock required rate of return according to CAPM? What is the stock s Market price, assuming the required return? Answer RP = (16-10) = = 18.7% Following the perptual Dividend Growth model, we have g = Growth rate. Do = Last dividend per share We are given R m 12% ; B x = 0.8 ; Do= 2.00 ; g = 4% ; RF = 7% R x = (12-7) = x5 = 7+4 = 11% Equilibrium price can be worked out as given below. = 2.20 = The Numerator is the present value of Dividend stream and denominator is the required rate of return or expected rate minus the growth rate of divided. Asset Pricing Implication of SML One of the major assumptions of the CAPM is that the market is in equilibrium and that the expected rate of return is equal to the required rate of return for a given level of market risk or beta. In other words, the SML provide a framework for evaluating whether high risk stocks are offering more or less in proportion to their risk and vice versa. Do = Last dividend paid Po = Current purchase/ market price g = Growth rate. To reach equilibrium and their required rate of return points on the SML, both stocks have to go through a temporary price adjustment. In practice, how does the price of say stock X get pushed up its equilibrium price? Say Investors will be interested in purchasing security x if it offers more than proportionate returns in comparison to the risk. This demand will push up the price of x as more of it is purchased and correspondingly bring down the returns. This process will continue till it reaches the equilibrium price and expected returns are the same as required returns. Problem The beta co-efficient of standard company is 1.2. The company is maintaining a 5% rate of growth in earnings or dividends. The last dividend paid was Rs. 2 per share. The risk free rate of return and the return on market portfolio are 10% and 15% respectively. The current market price of the company is Rs. 14. What will be the equilibrium price per share of standard company? Answer According to CAPM, the expected Rate of Return E (R) is equal to RF + Beta (R m - RF) Given RF = 10% ; B= 1.2 ; RM = 155 E (R) = (15-10) = 16% Equilibrium Price : (SLM) Problem Short term Government securities yield 7% and the expected market return is 12%. Stock X s beta is 0.8, its growth rate is 4% and its last dividend was Rs. 2.0 what would be the stock s equilibrium price. Answer : R x = R F + bx (R M - R F ) and P x = Do (1+ g) [Rx-g] R x = Rate of return required on stock x R F = Risk free rate of interest. R m = Return on Market Portfolio. B x = beta co-efficient of stock x

11 Return Ri = Problem on CAPM Problem Given the following variables D 1 = Dividend per share = 3.5% g= growth rate = 6% K c = 18% K c = Required rate of return. Rf = Risk free Return = 9% B = Beta (Systematic risk measure) 1.3 Ri = Rf + Beta (R m Rf) Where, Rf = 9%, RM = 16% and Beta = 1.3 Thus, Ri = ) = 18.1 Applying the value of Ri as the required rate of return Kc. We have the formula Given Rf as 9% and R m = 15% Calculate the Return on the securities 1 to 3 and compare them with the expected Returns. Security line or SML is given by the formula. R 1 = Rf = β (Rm- Rf) = (15-9) = = 16.2% R 2 = (15-9) R 3 = (15-9) = = 13.5% = 9+ 9 = 18% Security 1 only could secure which is higher than the expected return E(R) : 14% CAPM Question 1 Given the following Security Market Line R i = βi. What are the returns on the two stocks whose βs are 1.2 and 0.9. Question 2 If the CAPM line is as shown below R i = βi. What is the excess return of the market over the risk free rate and what is the risk free rate? Question 3 Using the formula for SML, in equilibrium namely. R i = R f + b i (R M -RF) and Given R 1 =6% β i = 0.5 R 2 = 12% β 2 =

12 What is the exceed return on an asset with a Beta of 2? Problems 1. The following characteristic lines are given for three mutual funds namely Kothari pioneer Templeton and Morgan Stanley. Kothari pioneer r 1 = -0.5% rm, p= 0.8 Templeton r 2 = 1.25% rm, P = 0.75 Morgan Stanley r 3 =0.75% rm, P= 07 Which fund has the most systematic risk What percentage of each Fund s risk is systematic and unsystematic. Answer The Fund which has the highest Beta o 1.35 viz., Morgan stanley has the most systematic Risk. The percentage of systematic and unsystematic risk in the total risk can be worked by taking the correlation coefficient squared P 2 which gives the systematic risk (p) and 1- systematic risk is the unsystematic risk. Kothari Pioneer :- Systematic Risk = (0.8) 2 = 64% Unsystematic Risk : = 0.36 or 36% Templeton :- Systematic Risk = (0.75) 2 = 56.25% Unsystematic Risk = = 0.44 or 44% Morgan Stanley :- Systematic Risk = (0.7) 2 =0.49 = 49% Unsystematic Risk : = 0.51 or 51% 2. If Risk free rate (Rf) is 5% and market return 14% and beta is 1.5 for the security. Determine the expected return for the security. What happens to expected return if Rm or market return increases to 16% assuming that other variables do not change. What happens to expected return if the beta falls to 0.75, assuming that other variables do not change. Answer a. Ri = Rf + (Rm- Rf) β = ( ) 1.5 = = or 18.5% b. Ri = ( ) 1.5 = = or 21.5% c. Ri = ( ) 0.75 = = =.1175 = 11.75% given Rf = 8% Rm = 15% and RA = 18% Find out the Beta for stock A. What is stock A s return of its Beta falls to 0.75 Solution RA= Rf+ (Rm Rf) β 18 = 18% + (15-8) β 18= 8+7 β 7β =10 β = 1.43 b. RA = Rf +b (Rm-Rf) = (15-8) = = = Given Rf = 6%, E (Rm) = 155 and expected returns and expected Betas are as follows : Stock Expected Returns Expected Beta A 14% 1.20 B 15% 0.75 C 13% 1.50 D 20% 1.60 E 10% 0.80 Which stock is overvalued and which is undervalued relative to expected return. Solution E (ri) = Rf + β (E (Rm) Rf) E(RA) = (15-6) = = 16.8% (overvalued) E (RB) = (15-6) = = 12.75% (under valued ) E (Rc) = (15-6) = 19.5% (over valued) E (Rd) = (15-6) = = 20.4% (over valued) E (RE) = (15-6) = = 13.2% (over valued) An investment company manages an Equity Fund consisting of five stock with the following market values and Betas. Stock Market Value Betas A Rs. 1,00, B Rs. 25, C Rs. 50, D Rs. 125, E Rs. 1,65, Total 465,000 If Rf = 7% E (Rm) = 14% What is the portfolio s expected return. Solution Ws are the weights given to five stock s

13 Notes