LESSON 28: CAPITAL ASSET PRICING MODEL (CAPM)psnacet.edu.in/courses/MBA/sapm/lecture-28.pdf · LESSON 28: CAPITAL ASSET PRICING MODEL (CAPM) The CAPM was developed to explain how

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LESSON 28:CAPITAL ASSET PRICING MODEL (CAPM)

The CAPM was developed to explain how risky securities arepriced in market and this was attributed to experts like Sharpeand Lintner. Markowitz theory being more theoretical, CAPMaims at a more practical approach to stock valuation.It is no doubt based on the mean-variance approach to risk forassessment of investment as developed by Markowitz. Itexplains the behavioural pattern of investors in building upportfolios.

CAPM-AssumptionsThe CAPM is based on certain assumptions some of which arecommon to CAPM and MPT. CAPM is it1 L1Ct developed aspart of MPT (Modern portfolio Theory). The assumptions arefirst set out below:1 The investor aims at maximising the utility of his wealth,

rather than the wealth or rerun. The difference betweenthem is that individual preferences are taken into account inthe utility concept. While some have preference for largerrisk who will have increasing marginal Utility for wealth, forothers, with less preference for risk the incremental wealthwill be less attractive if it is attached with more risk.Thus, the preference of investors for risk return will betaken into account in this model.

2 Investors have similar expectations of Risk and Return.Without these consensus standards, the estimates of meanand variance may lead to different forecasts with the resultthat the ct1icient portfolio of each will be different fromthat of the others. There will be innumerable efficientfrontiers, each dependent on the set of preferences ofindividuals for risk and return. If investors do not havesimilar expectations there will be no homogeneity in theirconception and no single efficient frontier line will apply toall.This in [Urn will imply that the price of an asset, which isthe best estimate ‘of the present value of future returns willbe different for different investors. This assumption istherefore 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 ismeasured by two factors, mean and variance. In the CAPMwe assume that rational investors diversify away theirdiversifiable risk, namely, unsystematic risk and onlysystematic risk remains which varies with the Beta of thesecurity. While some use the beta only, as a measure of risk,others use both Beta and variance of returns (total Risk) asthe sources of reward or expected return. As theseperceptions of risk and reward vary from individual toindividual, under CAPM we get a series of efficient frontierlines while in the case of MPT, there will be a single efficientfrontier line as the conception of risk and return expectationis assumed to be homogeneous in the latter.

4 Investors will have free access to all available information atno cost and no loss of time. If the information is not thesame for all, no common efficient frontier line can bedrawn. Besides even if the information is not available atthe same time different conclusions can be drawn regardingexpected return and risk and no single price of the capitalasset CU.1 be conceived.

5 Investors should have identical time horizons which againis highly unre-alistic. Investors have different time horizonsand their estimates of stock value will therefore differ, evenas the estimated earnings are the same per year. Continuoustime models are sometimes used to get over the abovedifficulty or again one ~ approximate a single period modelas a proxy to multiperiod model on the assump-tion thatreturns are the same over time and time has no relevance toexpected returns and that expected returns are againindependent of the past and current information.

While the above assumptions are common to both CAPM andMPT, some assumptions are specific to CAPM. Thus, there is arisk free asset, which gives risk free return. Investors can borrowand lend unlimited amounts at the same price. 1rus assump-tion of risk free asset transforms the curved efficient frontierline to a linear one. Risk can be reduced by adding a risk freeasset, or borrowing at the risk free rate.Besides, it is also necessary to assume in CAPM that total assetquantity is fixed and all assets are marketable and divisible. Thisassumption implies that the liquidity requirement of investorsis ignored and there will no new issues, which are bothunrealistic.After the brief review of the above assumptions we cansummarise the requirements for CAPM as follows:Risk is measured by variance of expected returns. There are twocomponents of Risk - systematic (non-diversifiable) andunsystematic (diversifiable). For diversifiable risk, the investormakes a proper diversification to reduce the risk and for thenon-diversifiable portion, he uses the relevant Beta measure toadjust to his requirement or preferences. Due to the possibilityof risk free asset and lending and borrowing at the free rate, theinvestor has two components of the portfolio - risk free assetsand the risky market assets. His total return is summation fromthe above two components.Under CAPM, the equilibrium situation arises when allfrictions, like taxes, divisibility transaction costs and differentrisk-free borrowing and lending rates are assumed away.Equilibrium will be brought about by changes in prices due tochanges in demand and supply.

CMLFigure below depicts the capital market line with risk less rate ofreturn. Point P is the riskless interest rate. Preferred investmentsare plotted along the line PMZ, by combination of both riskyassets and risk free asset along with the borrowing and lending.

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The slope of the PMZ is the measure of the reward for Risktaking. P is the risk free return, Em - T is the measure of therisk premium - a return for the risk taking. The reward forwaiting is the risk less interest rate, OP, and second reward is thereturn per unit of risk borne measured by the slope ‘a’ of thePMZ line. The internal rate can be considered as price of timeand the slope of capital market line as price of the risk.

If the borrowing and lending rates are different, then OQbecomes the borrowing rate and OP will be the lending rate, asshown below. The efficient frontier line with differentialborrowing and lending rates will be as shown below:

QMN is with the borrowing rate of OQ and PMN is with thelending rate of OP and ABC is the efficiency frontier linewithout borrowing and lending. The curved line will becomelinear; if once the riskless asset of borrowing and lending atfixed riskless rate is introduced.The CML as described above reflects the relationship of totalrisk and expected return. Total risk includes both systematic andunsystematic risks. It may also include the risk free assets toreduce the total risk. The CAPM has two components of thecapital market return, which are reward for waiting or risklessreturn, and the reward per unit of risk borne as measured bythe slope of the CML line.The investors will have their choice of efficient portfoliosomewhere along the line of CML, as all efficient portfolioswould be on it. Those which are less efficient will be below theline PMN, in the chart above. The risk free rate can be thoughtthe price of time and the slope of the capital market line as thePrice of Risk.

Security Market Line (SML)Unlike the CML, which considers the total risk as a measure ofvariability of returns, SML takes into account only the system-atic risk, which is market related and is not possible to reduce oreliminate by diversification. Beta is the measure of risk of asecurity relative to the whole market, and is used in the SML.Since the unsystematic risk is already taken care of by diversifica-tion in the construction of an efficient portfolio, it is desirableto develop an alternative to CML which will use Beta as the

independent variable and can be adopted for use in portfoliomanagement and in purchase of individual securities. Such aline is called Security Market Line, which depicts a linearrelationship between expected return and the systematic risk.The SML curve drawn below shows a positive slope, indicatingthat the return and risk are positively related. The higher the riskthe higher is the return. The Beta of Market portfolio, asrepresented by the BSE or NSE index is always one. But thecompany scrip can have Betas higher than 1 or lower than 1.Those with Betas less than 1 are defensive securities and thosewith Betas above 1 are called aggressive scrip’s. The graph belowshows these types of scrip’s and the relationship of Beta toexpected return.

SML is Security Market Line, OS is the risk-free return, OP is thereturn of the market, whose Beta is 1; those below Beta 1 aredefensive 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 returnRm is Market return.Bi is Beta of Scrip i related to Market RiskIf Rf = 10, Rm = 20 Bi = 1.5, which is more risky than themarket average,thenRi = 10 + 1.5 (20 - 10)= 10 + 15 = 25.0% which is higher than the market returnSuppose the Bi is less risky than the market at 0.75 thenRi = 10 + 0.75 (20 - 10)= 10 + 7.5 = 17.5%, which is lower than the market returnFrom the above equation, we can estimate the expected returnon a security. It is represented as something like a premium ordiscount on the market return and can be compared. It can be areturn on a security as distinguished from a portfolio. If thesecurity is correctly priced it will haveRi = Rf = 0 and SML curve goes through the origin (see chartbelow) Ri-- Rf measures the excess return which varies with therisk taken. Within the chart it is seen that Rm - P, f’ excessreturn if market Beta = 1. The security market line implies thatthe individual assets and port-folio should be on SML, If theyare cor-rectly priced. Beta values should then cor-rectly representthe contribution to the risk of the security to the portfolio. All


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assets lying above the SML are undervalued and those belowthe SML are overvalued.

If we buy undervalued securities, the returns will be more andvice versa. It will thus be seen that SML curve assumes a criticalimportance in portfolio selection and individual investmentdecision.

CAPM AnalysisThe expected return of a portfolio in equilibrium is equal to riskfree rate Rf, plus risk premium which is related to its Beta.Thus, Rp - Rf = risk premium and this is equal to BPM (Rm -Rf) where Rp is expected rate of return and Rf the risk freereturn.

These symbols are the same as explained above. This leads usto the market model, which relates the expected excess return ofthe portfolio to the excess return of the market. This is anexplanation of the risk premium which gives excess returns.The chart below presents the relation of E [(Rm)- (Rf)], to theE (RP-Rf) Viz., in words, excess returns on a portfolio’s excessrisk over the market risk. This chart presents CAPM in a generalform with expected excess market risk related to expected excessreturn.

Market ModelRisk Premium form is the one shown above and the equationfor this is

Beta relates the portfolio premium to market risk premium. IfBeta is one they are the same.Market Model can be presented in the form of a regressionequation. Taking the above equation for Risk premium of theportfolio, let us introduce a new concept of risk adjusted excessreturn which is generated by the expertise of the portfolioManager, represented by the aop. This can be graphicallyrepresented as the y-axis intercept for the regression line. aop canbe zero, positive or negative, depending on the performance ofthe portfolio.When this model is presented in the Risk premium form., theequation is

Rp - Rf = αp - Rf (1 - βpm) + β pm (Rm - Rf) + rp

αpo = αp – Rf (1- βpm)

As the risk of the portfolio remains unaffected, Bpm of thecharacteristic line remains unchanged but ap changes to apo asfollows:αpo = ap - Rf (1 - Bpm)αpo is risk adjusted excess returnαp is return on portfolio when the market return is zero. Inother words, αp represents the excess return of the portfolio,when the market return is equal to riskless and rp is the error orthe residual.αp0 can be the same as αp when Rf (1- Bpm) is zero whichhappens when Bpm is 1 ; then the excess market returndisappears. 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 returnon a portfolio is positive αpo >o, then the portfolio return isgreater than what is normally expected, indicated superiorreturn. If apo is positive te portfolio Manager has shown hisexpertise in beating the market by showing higher than themarket return.We have seen that the slope of SML curve is Beta. If we have aperfectly diversified portfolio in the CAPM, the error termdisappears. If at the same time expected risk adjusted excessreturn αo of the portfolio is zero which is assured under MarketEfficiency Theory, then the slope of the characteristic line is Betaof the portfolio. Thus, under condition of perfect markets, theslope of the Regression line is Beta and excess returns disappear(αp0 = 0)

Uses and LimitationsIn real world, investors get higher return for higher risk andthey are more concerned with company related risks than withthe market related risks, except in the case of trained investmentanalysts. Companies are found to use CAPM to determine thecost of equity for the firm, to estimate the required return fordivisions or lines of business and to determine the hurdle ratesfor corporate investments and to evaluate the performance ofinvestment Division in terms of costs and returns. Thesehurdle rates of return are in general the required rates of returnand the


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corporate assess the past performance of the costs and relatedreturns for each of the Division. In the case of public utilities,the CAPM can be used to estimate the costs and rates to becharged to cover the costs. The CAPM is used to regulate thepublic utilities from the point of view of costs.Historical return and Betas are used to select the proper risk ininvestments in the portfolio. CAPM is used to select securities,construct portfolios and evaluate the performance of theportfolio. It is thus a useful tool for investment analysis andportfolio management.The limitations of the theory are also pointed out by manycritics. This theory is unrealistic for any average investor, whogoes by the fundamental factors influencing the company, itsearning, dividend and bonus record. Empirical tests of theModel have not proved very useful. The model is built ex-antefactors while in reality the expectations of the future vary fromperson to person. Data and analysis is to be based on ex-postfactors while anticipations of future risk and returns are ex-anteand both may not be related. The CAPM is in fact not testableexactly as the exact composition of the market is known and isused in testing. The empirical tests conducted by Richard Rolland others were only tests on samples whether me proxymarket portfolio was efficient or not. The use of surrogativesand proxies have not proved the theory as really useful andpractical. .CAPM theory is thus a nice theoretical exposition but in actualworld, it does not conform to the real world risk-return trendsand empirical tests have not given unequivocal support to thetheory. It is also found mat there - are many non-Beta factorsinfluencing me returns. The calculation of Beta is itself ofdoubtful validity as me historical Betas may not reflect thefuture risks or returns. In the short-run in particular, projectionson the basis of Betas on returns and risk have been found to beunreliable and results contrary to C’APM Theory were noticed.Thus, CAPM is a good theoretical tool but with its ownlimitations in practical applications.

Capital Asset Pricing ModelThe assumptions of CAPM are that the market is in equilib-rium and the expected rate of return is equal to the required rateof return for a given level of risk. or Beta. CAPM presents alinear relationship between the required rate of return of asecurity and relates it to market related risk or Beta, whichcannot be avoided. The equation for the CAPM Theory isRj = Rf + Bj (RM- Rf)Rj is expected rate of return on security ‘j’ and Rf is risk freeturn.Bj is Beta coefficient - a risk measure for the non –diversifiablepart of total Risk. Rm is return on Market Portfolio and RM –Rf is the excess return for the extra risk.

Limitation of CAPMIt is not realistic in the real world. This assumes that allinvestors are risk averse and higher the risk, me higher is thereturn. Investors ignore the transactions cost information costs,brokerage” taxes etc., and make decisions on the basis of singleperiod horizon. The investors are given a choice on the basis ofrisk-return charac-teristics of an investment and they can buy atthe going rate in me market. There are many buyers and sellers

and the market is competitive and free forces of supply anddemand determine the prices.CAPM establishes a measure of risk premium and is measuredby BJ (RM-Rf) Beta coefficient is the non-diversifiable risk of theasset, relative to the risk of the asset.Suppose Cipla Company has a Beta equal to 1.5 and me risk freerate is say 6%. The required rate of return on the market (RM) is15%. Then, adopting the above equation, we haveRJ = Rf + (BJ) (RM - Rf)= 6 + 1.5 (15 - 6)= 6 + 13.5 = 19.5%If the market rate is 15% then the rc:tUn1 on Cipla should be19.5%, because the larger risk on Cipla than on the market.

SML Security Market LineSML plots the relationship between the Required rate of returnRJ and non- diversified risk, Beta, as expressed above in CAPM.

Example

Market Expected Return = 12%Market Risk Premium (RM - Rf) = 7

Risk free Return = 5 (Rf)RM - Rf = (12 - 5) = 7If security x has Beta of 1.20Then RJ = Rf + BJ (RM- Rf)- = 5 + 1.2 (7)= 5 + 8.4 = 13.4 (Aggressive Scrip)

If the security has Beta as 0.80, then it is a defensive scrip. ThusRj = 5+ 0.8 (7) =5+ 5.6 = 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 theabove chart.When we estimate the expected return after an year, in theabsence: of historic data on returns and probabilities thefallowing formula which is derived from the basic formulagiven above is useful.

Do(l +g)Expected return RJ = Po + gWhere, Do = last paid dividend

Po = current market priceg = Growth rate.


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If the above return is higher than the equilibrium rate effectingthe equilibrium price - a position on the SML - both the stocks,above the SML and below the SML have undergone somechanges: The expected rate: has to be equated to the requiredrate: of return, when the point of equilibrium is reached on theSML. Which the expected return is higher than the requiredrate:, the demand for that security will rise and the price: willalso rise, bringing down its return to the equilibrium level. Ifthe expected return is lower than the required return thedemand will fall leading to a fall in its price:, bringing it to theequilibrium level In the former case, the investors will buysecurities and in the letter case, they will scl1 securities.Thus, the CAPM is useful to provide insights for the financeManage~ to maximise the value of the firm. Following theprinciple of the higher the risk, the higher is the return, thefinance Manager will keep the risk level at the optimum level inperforming the investment function or financing function bykeeping in mind the return that shareholder expects to take at agiven level of risk at the company.The Finance Manager has to keep in mind the expected returnsof the share-holders and the returns he provides should becommensurate: with the risk. This risk is reflected in hisinvestment and financing decisions. The SML provides a benchmark reflecting the equilibrium position in the relationshipbetween the risk and return. The risk that is reflected in the non-diversifiable or systematic risk is that the company is exposed toin its operations, financing and investment decisions.

ProblemsQ. 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 riskwhat is the price he should pay.

B. If the bond is selling at a price of Rs. 97.59 what is its YTM.YTM is to be estimated by Trial and error method. Consider12% as YTM.Then

Hence, YTM is 12%What is the duration of Bond if the YTM is 12% and expectedreturn is 10.06%.Duration Value %of age Value WeightedDuration1. Year 9.82 = 11 10.06 0.1006=1x0.1006

1.122. Year 8.77 = 11 8.99 0.1798=2x0.0899

(1.12)23. Year 79.01 111 80.95 2.4285=3x0.8095

(1.12)3

Total 97.60 100.00 2.7089

The duration in the case of value bond of YTM of 12% andexcepted return 10.06% was shown above. Then the weightedduration of the portfolio is 2.7087.The Market Model and CAPM may not give similar results asexpected excess returns and risk premiums, in the two modelsare not identical.Say ∝p = 4%, Rf = 6% (riskless return )β Beta = 2, Rm (expected market return) = 10%Expected Risk Premium (Rp- Rf)Market Model = Pa + βpm (Rm-Rf)(Rp-Rf) = 4-6 (1-2) +2 (10-6)

4+6+8 =18%Where, αPα = αP-Rf (1-βm)

4-6 (1-2)

CAPM(Rp-Rf) = βPm (Rm- Rf)

= 2 (10-6) 20-12 = 8%

In the exceptional case when the portfolio is in equilibrium, theexpected risk premium of the portfolio under both modelsshould be identical. That happens when αPo =0= αP-Rf (1-Bpm)

Then under Market ModelThe excess Premium = 0+2 (10-6)

= 2 x4 = 8%and under CAPMRp-Rf= 2 (10-6)20-12= 8%

ProblemsIf the risk free rate is 10%, expected return on NSE Index is18%, standard deviation is 5% (SD). Construct an efficientportfolio to secure 16% return. The efficient portfolio consistsof market securities and risk free securities invested in theproportion of W and 1- W respectively, then

Rp = W (Rm) + (1-W) Rf16% = W (18%) + (1-W) 10%16 = 18w +10 –10 W16-10 =6= 18W-10W= 8 W. Thus, 8W =6

68

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


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If now an amount of Rs. 10,000 is to be invested and expectedreturn is 20% construct a portfolio and calculate its risk

Since it is more than 1, borrow 0.25 or 25% of Rs. 10,000 toinvest in market portfolio. Thus own funds are Rs. 10,000 andborrowed funds are Rs. 2,500, with a total of Rs. 12, 500.Risk on the portfolio is calculated as follows :

Diversification (Random and Naive )The traditional theory laid down diversification as a techniqueof selection of securities in a portfolio. This is called Randomdiversification or simple diversification on the basis of straightrule of two is a better than one simple diversification onRandom basis was found to be more remunerative by research-ers and these number of scraps in a portfolio of individuals isto be around 10-15 securities.Rational basis of why diversification and how to achieveoptimal diversification were studied by later researchers ofwhich Marko wiz is reputed to be the pioneer Naive diversifica-tions or superfluous diversifications may result from randomand indiscriminate selection of securities which does not lead to‘any reduction of risk Thus cycle and tubes are related industriesand one invests in those two types of industries. which arehighly correlated in a positive manner, the risk will be increased

by diversifications rather than reducing. Naive diversifications inname only, which does not reduce the risk. Thus an investormay have 10 scrip’s in steel, mini steel and ferrous metals, whichwill only increase risk. But an investor having 10 scrip spread incycle electronics sugar steel auto etc. will have less risk as theseindustries are not auto correlated and their risks are independentof each other or even negatively related.It is thus left to markoiz and later research to show thatdiversification is a tool to reduce unsystematic risk and how itcan be reduce by a study of variances and covariance of securitiesreturn, as against the market returns.

Why DiversificationIt is never prudent to put all ones eggs in one basket as it maylead to total ruin if the basket itself is broken or lost. Thehuman behavior is normally risk averse which means that forpsychological reason. he distributes his assets in a variety of riskclasses some in cash some in bank deposit, Insurance, Provi-dent fund, pension fund etc. These are all examples of thenormal human behavior of diversifying the asset holdings toreduce risk, provided for contingency and take all precautionsagainst total loss.Thus, the average investor never puts all his savings in oneform or in one security for self-production and for and forpsychological reason.Money kept idle or in some investments, which do not giveadequate return, will be a loss to investor, as he loses the valuethe value of money over time. By logic of common senseinvestors try to satisfy most of their objectives of saving byputting money in various avenues and that means diversifica-tions. The various objectives are income, capital appreciationsafety, marketability, contingency, and liquidity and hedge againstinflection and for future provision or larger incomes. His choiceof investments will cater to this requirement, which lead todiversification of investments.Even without the theoretical basis of covering or reducing theunsystematic diversifiable risk, the investor in the traditionalTheory used to adopt some methods of Diversification.

Example on Measurement of Risk

Calculation of Standard Deviation and VarianceAs diversification is meant for reduction of risk, its measure-ment is relevant here.


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Average return is the same in both cases but risk is different,DCM has a range of variation from 8% to 12% while theescorts has a variation from 9 to 11.

Calculation of Standard Deviation

Example DCM

Example Escorts

Risk on the Escorts is lower in the above example referred toabove.Example of Simple DiversificationTake two securities Diversification

The return on the portfolio by combining them would be Rp=W1 R1 + W2R2 Rs. are returns and W 1 and W 2 are weights oftwo investments.RP = 0.25 (.25) + 0.25 (0.75)

= 5% + 18.75%= 23.75%Here no consideration is given for risk and covariance of riskamong the securities invested in the portfolio. This can bedemonstrated by taking an example o Markowitz diversifica-tion.

Morkowitz DiversificationBefore discussing the Markowitz diversification what theresearches of investors and investment analysts have found isto be set out briefly. Firstly they found that putting all eggs inone basket is bad and most risky. Secondly, there should beadequate diversification of investment into various securities asthat will spread the risk and reduce it; if the number of themsay 10 to 15 it is adequate to enjoy the economies of time, scaleof operations and expertise utilised by the investors in hisanalysis.Reference was already made to Naive diversification, which is aspread of investments into many securities but will not reducethe risk, like buying ten securities all in the shipping industrywhich is risky industry in itself. Besides some researchers havefound that there is an optimisation process for diversification toreduce the unsystematic or company related risk by choosingsuch companies which are not closely related or not owned bythe same family group in the same industry group. Thisoptimisation process also leads to an investment in 10-15companies well chosen for differences in their characteristics,nature of the product market, pattern of production etc.

Example :What is the expected return of a portfolio, comprising of thefollowing securities :

RP = W1 R1 + W2 R2 + W3 R3

Rs are expected returns and Ws are weightsRp = (10 x 0.25) + 15 (0.25) + 20 (0.50)

= 2.5 + 3.75 + 10.0= 16.25%

ExampleGiven the following data on two stocks Tisco and Reliance andtheir returns calculate the required measures of expected returnsetc.


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= 5.2 + 9.0= 14.2%Calculation of Standard DeviationCalculation of CovarianceGiven, Or = 3 Os = 3Rx = Expected Return on Security xRy = Expected

Return on Security yN = number of observations

If the coefficient of correlation is high as 0.893, nearer to one,then the degree of risk in the portfolio is also highSML Example – Security Returns

Equation for SML is as follows :Calculate the SMLRi = Rf + Beta i (Rm- Rf)Scrip A = .08 + 1.70 (.12-0.08)

= .08 + .068 = 0.148 = 14.8%Scrip B = .08 + 1.40 (.12- .08)

= .08 + 0.056 = .136 = 13.6%Scrip C = .08 +1.10 (.12- 0.08)

= .08 +0.044 = .124 = 12.4%

CMLAll investors have to choose for a combination of two compo-nents of the portfolio : (1) Market Risky Portfolio. (2) RisklessSecurities. The straight line tangent to the efficient frontier line iscalled the Capital Market Line. On this line all the efficientportfolios would be lying. The CML choose the most efficientportfolio and this indicates the market price of risk representedby the formula.


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The term Rm – Rf/Omis the extra return over the risk free rateby increasing the level of risk (Sd) of the efficient portfolio byone unit.

SMLTaking non- systematic risk as zero for a well diversifiedportfolio, the only relevant risk is systematic risk measured byBeta. If Beta is zero, it is riskless security. For market Portfolio,the Beta is one. The straight line called SML is represented bythe equation Ri = α+ bβi.......... Equationif Bi = O, then the first point of the line isRf = α + b (O)Rf = αthe second point of the line isRM = α +b (1) where Beta m = 1As α = Rf, RM - α = bthen RM – Rf = b ......... EquationCombining the above two results, we have the equation ofSMLRi = Rf + Bo (RM- Rf)ExampleElecon stock is expected to sell at Rs. 70 a year hence and pay adividend of Rs. 4 per share. If the stock’s correlation withportfolio is-0.3

Example @Given the following data, answer the question below :Correlation Matrix

Portfolio Mix is 50 : 30: 20.What is its S.D. Following the formula used earlier we have,

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 1.5Portfolio (1)

S.D. = 0.7 (10.0%) + 0.3 (0) == 7.0 +0 = 7%

Portfolio (2)S.D.2

2 = B22 + 0

S.D.22 = (1.5)2 (10)2 = 2.25 x 100 = 225

S.D.22 = 225

S.D.2 = 225 = 15%

ExampleIndfund has three Investment strategies as given below. Findout the plan with greater risk

Strategy A has the largest Risk, as the portfolio Beta is highest at1.0, as compared to B and C.

ProblemsRepublic Forge has a beta 1.45, the risk free rate Rf is 10% andrp expected return on market portfolio is 16%. This company


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pays a dividend of Rs. 2 a share and expected growth individends 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?AnswerRP = 10 + 1.4 (16-10) = 10+ 8.7 = 18.7%Following the perptual Dividend Growth model, we have

= 2.20= 25.29The Numerator is the present value of Dividend stream anddenominator is the required rate of return or expected rateminus the growth rate of divided.

Asset Pricing Implication of SMLOne of the major assumptions of the CAPM is that the marketis in equilibrium and that the expected rate of return is equal tothe required rate of return for a given level of market risk orbeta. In other words, the SML provide a framework forevaluating whether high risk stocks are offering more or less inproportion to their risk and vice versa.

Do = Last dividend paidPo = Current purchase/ market priceg = Growth rate.To reach equilibrium and their required rate of return points onthe SML, both stocks have to go through a temporary priceadjustment.In practice, how does the price of say stock “X” get pushed upits equilibrium price ? Say Investors will be interested inpurchasing security “x” if it offers more than proportionatereturns in comparison to the risk. This demand will push upthe price of x as more of it is purchased and correspondinglybring down the returns. This process will continue till it reachesthe equilibrium price and expected returns are the same asrequired returns.

Equilibrium Price : (SLM)

ProblemShort term Government securities yield 7% and the expectedmarket 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 ‘sequilibrium price.Answer :Rx = RF + bx (RM- RF) and Px = Do (1+ g) [Rx-g]Rx = Rate of return required on stock xRF = Risk free rate of interest.Rm = Return on Market Portfolio.Bx = beta co-efficient of stock x.

g = Growth rate.Do = Last dividend per shareWe are given Rm 12% ; Bx = 0.8 ; Do= 2.00 ; g = 4% ; RF = 7%Rx = 7 + 0.8 (12-7) = 7+ 0.8x5 = 7+4 = 11%Equilibrium price can be worked out as given below.

ProblemThe beta co-efficient of standard company is 1.2. The companyis maintaining a 5% rate of growth in earnings or dividends.The last dividend paid was Rs. 2 per share. The risk free rate ofreturn 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 standardcompany ?

AnswerAccording to CAPM, the expected Rate of Return E (R) is equalto RF + Beta (Rm- RF)Given RF = 10% ; B= 1.2 ; RM = 155E (R) = 10 + 1.2 (15-10)

= 16%


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Problem on CAPM

Problem

Given the following variablesD1 = Dividend per share = 3.5%g= growth rate = 6%Kc = 18%Kc = Required rate of return.Rf = Risk –free Return = 9%B = Beta (Systematic risk measure) 1.3Ri = Rf + Beta (Rm –Rf)Where, Rf = 9%, RM = 16% and Beta = 1.3Thus, Ri = 9+ 1.3 916-9) = 18.1Applying the value of Ri as the required rate of return Kc. Wehave the formula

Return Ri =

Given Rf as 9% and Rm = 15%Calculate the Return on the securities 1 to 3 and compare themwith the expected Returns. Security line or SML is given by theformula.R1 = Rf = β (Rm- Rf)

= 9+ 1.20 (15-9)= 9+ 7.2 = 16.2%

R2 = 9+ 0.75 (15-9)= 9+ 4.5 = 13.5%

R3 = 9+ 1.5 (15-9)= 9+ 9 = 18%

Security 1 only could secure 16.25 which is higher than theexpected return E(R) : 14%

CAPMQuestion 1Given the following Security Market Line Ri =0.07 +0.09 βi.What are the returns on the two stocks whose βs are 1.2 and0.9.Question 2If the CAPM line is as shown belowRi = 0.04 + 0.10 βi.What is the excess return of the market over the risk free rateand what is the risk free rate ?Question 3Using the formula for SML, in equilibrium namely.Ri = Rf + bi (RM-RF) and GivenR1 =6% βi = 0.5R2 = 12% β2 = 1.5


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What is the exceed return on an asset with a Beta of 2 ?

Problems1. The following characteristic lines are given for three mutualfunds namely Kothari pioneer Templeton and Morgan Stanley.Kothari pioneer r1 = -0.5% + 1.25 rm, p= 0.8Templeton r2 = 1.25% + 0.95 rm, P = 0.75Morgan Stanley r3 =0.75% + 135 rm, P= 07Which fund has the most systematic riskWhat percentage of each Fund’s risk is systematic and unsys-tematic.AnswerThe Fund which has the highest Beta o 1.35 viz., Morganstanley has the most systematic Risk.The percentage of systematic and unsystematic risk in the totalrisk can be worked by taking the correlation coefficient squaredP2 which gives the systematic risk (p) and 1- systematic risk is theunsystematic risk.Kothari Pioneer :- Systematic Risk = (0.8)2 = 64%Unsystematic Risk :- 1-0.64 = 0.36 or 36%Templeton :- Systematic Risk = (0.75)2 = 56.25%Unsystematic Risk = 1- 0.5625 = 0.44 or 44%Morgan Stanley :- Systematic Risk = (0.7)2 =0.49 = 49%Unsystematic Risk :- 1- 0.49 = 0.51 or 51%2. If Risk free rate (Rf) is 5% and market return 14% and beta is1.5 for the security.Determine the expected return for the security.What happens to expected return if Rm or market returnincreases 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.Answera. Ri = Rf + (Rm- Rf) β

= 0.05 + (.14- .05) 1.5= 0.05 + 0.135 = 0.185 or 18.5%

b. Ri = 0.05 + (0.16 –0.05) 1.5= 0.05 + 0.165 = 0.215 or 21.5%

c. Ri = 0.05 + (0.14- 0.05) 0.75 == 0.05 + 0.0675 = .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 0.75SolutionRA= Rf+ (Rm –Rf) β18 = 18% + (15-8) β18= 8+7 β7β =10β = 1.43b. RA = Rf +b (Rm-Rf)

= 8+ 0.75 (15-8)= 8+5.25 = 13.25

= 13.255Given Rf = 6%, E (Rm) = 155 and expected returns andexpected Betas are as follows :Stock Expected Returns Expected BetaA 14% 1.20B 15% 0.75C 13% 1.50D 20% 1.60E 10% 0.80Which stock is overvalued and which is undervalued relative toexpected return.SolutionE (ri) = Rf + β (E (Rm) – Rf)E(RA) = 6 + 1.20 (15-6) = 6 + 10.8

= 16.8% (overvalued)E (RB) = 6+ 0.75 (15-6) = 6+ 6.75

= 12.75% (under valued )E (Rc) = 6+ 1.50 (15-6) 6+ 13.5

= 19.5% (over valued)E (Rd) = 6+ 1.60 (15-6) = 6 + 14.4

= 20.4% (over valued)E (RE) = 6+ 0.80 (15-6) = 6+ 7.2

= 13.2% (over valued)An investment company manages an Equity Fund consistingof five stock with the following market values and Betas.Stock Market Value BetasA Rs. 1,00,000 1.10B Rs. 25,000 1.20C Rs. 50,000 0.75D Rs. 125,000 0.60E Rs. 1,65,000 1.30

Total 465,000If Rf = 7% E (Rm) = 14%What is the portfolio’s expected return.SolutionWs are the weights given to five stock’s


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Notes

Documents

LESSON 28: CAPITAL ASSET PRICING MODEL (CAPM)psnacet.edu.in/courses/MBA/sapm/lecture-28.pdf · LESSON 28: CAPITAL ASSET PRICING MODEL (CAPM) The CAPM was developed to explain how