CAPM

• CAPM stands for Capital Asset Pricing Model.

The CAPM is an equilibrium model that specifies the relationship between risk and required rate of return for assets held in well-diversified portfolios. All investors have identical expectations.

• Investors can borrow or lend unlimited amounts at the risk-free rate.

• All assets are perfectly divisible.

• There are no taxes and no transactions costs.

• All investors are price takers, that is, investors’ buying and selling won’t influence stock prices.

• Quantities of all assets are given and fixed.

Portfolio Theory

Stock PQR Stock XYZ

Return % 11 or 17 20 or 8

Probability 0.5 each return 0.5 each return

Expected Return 14 14

Variance 9 36

Standard Deviation 3 6

• Expected Return (PQR)= .5x11+.5x17=14• Expected Return (XYZ)= .5x20 +.5x8=14• Variance (PQR)= .5(11-14)2 + .5( 17-14)2 = 9• Variance (XYZ)= .5(20-14)2 + .5)(8-14)2 = 36• Standard Deviation (PQR) =( 9) ½ = 3• Standard Deviation (XYZ) = (36)1/2 = 6In the above example both the companies have

same expected return of 14%. Here XYZ company’s stock is more risky than PQR’S stock because Standard deviation of

former is 6 and the latter is 3.

• Holding two securities may reduce the risk too. The portfolio risk can be calculated from the following formula:

Risk of a portfolio = W12S1

2+W22S2

2+2W1W2 ( r12 S1S2)

W1 = Weight of total portfolio value in stock I

W2 = Weight of total portfolio value in stock II

S1 = Standard deviation of stock I

S2 = Standard deviation of stock II

r12 = Correlation coefficient of stock I & II

r12 = Covariance of stock I & II

S1S2

• In this example Covariance of Stock I & II = Covariance = ½ (11-14)(20-14)+(17-14)(8-14) = ½(-18)+(-18) = -36/2 = -18

r12 = -18 3x6

r12 = - 1 The correlation coefficient indicates the similarity or

dissimilarity in the behavior of stock I & II .In our example as the relationship is, r = -1 ,it indicates that both the stocks are having negative relationship and the returns move in the opposite direction.

Suppose Investor holds 2/3 in Stock of ABC and 1/3 in stock of XYZ then :

Portfolio risk = (2/3)2(3)2+(1/3)2(6)2 + 2x(2/3)(1/3) (-1x3x6)

= (4/9)(9)+(1/9)(36)+2(2/9)(-18)

= 4 + 4 + 4/9 (- 18)

= 8-8 = 0

Therefore the portfolio risk is nil if the securities are negatively correlated.

Thus Markowitz diversification can lower the risk if the

securities in the portfolio have low correlation coefficients( r).

Example

Correlation Coefficient = .4Stocks % of Portfolio Avg ReturnABC Ltd. 28 60% 15%XYZ Ltd. 42 40% 21%

Standard Deviation = weighted avg = 33.6 Standard Deviation = Portfolio = 28.1 Return = weighted avg. = Portfolio = 17.4%

Let’s Add stock PQR Ltd. to the portfolio

Example Correlation Coefficient = .3Stocks % of Portfolio Avg ReturnPortfolio 28.1 50% 17.4%New CorpNew Corp 3030 50%50% 19% 19%

NEW Standard Deviation = weighted avg = 31.80 NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20%

NOTE: Higher return & Lower risk .How did we do that?

DIVERSIFICATION

Optimal Portfolio Concept

• The optimal portfolio among the given set of efficient portfolio is arrived at by using investors utility indifference curves.

• According to economists, utility define the relationship between Psychological satisfaction and wealth.

• The relationship between utility & wealth can be linear, concave and convex function depending upon the behavior of an individual.

• Linear Function:

Wealth

Utility

In case of Linear function for each unit change in wealth, there is an equal increase in satisfaction.

• Concave Function:

Wealth

Utility

In Case of Concave function for each unit change in wealth ,there is a less than proportional increase in satisfaction. Each successive increase in wealth adds less satisfaction as the level of wealth increase

• Convex Function:

Wealth

Utility

In case of convex functions, for each unit change in wealth, there is more than proportional increase in satisfaction.

Standard Deviation

Expected Return

I B I A

Here I B represent indifference curve of Investor B & IA represent the indifference curve of Investor A.Both the investors are risk averse that means both the investor would like to have higher amount of return than the corresponding amount of risk. However Investor A is less risk averse than Investor B as B wants a higher expected return for bearing a given amount of risk as compared to A. Thus we can conclude that steeper the slope of the curve the greater the degree of risk aversion.

Return

Risk

Goal is to move up and left. WHY?

ReturnLow Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

ExpectedPortfolio Return, kp

Risk, p

Efficient Set

Feasible Set

Feasible and Efficient Portfolios

• The feasible set of portfolios represents all portfolios that can be constructed from a given set of stocks.

• An efficient portfolio is one that offers:

– the most return for a given amount of risk, or

– the least risk for a give amount of return.

• The collection of efficient portfolios is called the efficient set or efficient frontier.

IB2 IB1

IA2IA1

Optimal PortfolioInvestor A

Optimal Portfolio

Investor B

Risk p

ExpectedReturn, kp

Optimal Portfolios

M

Z

.ArF

M Risk, p

Efficient Set with a Risk-Free Asset

The Capital MarketLine (CML):

New Efficient Set

..B

rM^

ExpectedReturn, rp

C

D

What impact does Rf have onthe efficient frontier

• When a risk-free asset is added to the feasible set, investors can create portfolios that combine this asset with a portfolio of risky assets.

• Risk free asset is the one whose return is fixed and standard deviation is nil. Government securities, treasury bills etc. are treated as risk free assets.

• The straight line connecting rRF with M, the tangency point between the line and the old efficient set, becomes the new efficient frontier.

• When a risk free asset is introduced in the construction of the optimal portfolio ,it is presumed that investor can invest part of his funds in the risk free assets and the remaining part in the optimal portfolio.

Continued….

Here Rf is the risk free asset and M is the optimal portfolio for any investor. If the investor put his entire funds in the optimal portfolio then the returns are C Rm and if the entire funds are invested in the risk free assets then the returns are C Rf.

If the investor invest part of his funds in the optimal portfolio and part of the funds in the risk free assets his returns will be in the range of C Rf to C Rm and the portfolio will be represented on Rf M.

Now it shall be observed than any point on the line DMZ will be higher than the point on the arc AMB. Thus investor would be tempting to use risk free asset along with the portfolio on efficient frontiers for investing their funds instead of investing only in efficient portfolio.

What is the Capital Market Line?

• The Capital Market Line (CML) is all linear combinations of the risk-free asset and Portfolio M.

• Portfolios below the CML are inferior.– The CML defines the new efficient set.– All investors will choose a portfolio on the CML.

The CML Equation

rp = rRF +

SlopeIntercept

^ p.rM - rRF^

M

Risk measure

What does the CML tell us?

• The expected rate of return on any efficient portfolio is equal to the risk-free rate plus a risk premium.

• The optimal portfolio for any investor is the point of tangency between the CML and the investor’s indifference curves.

rRF

MRisk, p

I1

I2

CML

R = Optimal Portfolio

.R .MrR

rM

R

^

^

ExpectedReturn, rp

What is the Security Market Line (SML)?

• The CML gives the risk/return relationship for efficient portfolios.

• The Security Market Line (SML), also part of the CAPM, gives the risk/return relationship for individual stocks.

• The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m)

)Cov(RFR-R

RFR)E(R Mi,2M

Mi

RFR)-R(Cov

RFR M2M

Mi,

2M

Mi,Cov

RFR)-(RRFR)E(R Mi i

)( i

The Security Market Line Equations

We then define as beta

)E(R i

)Beta(Cov 2Mim/0.1

mR

SML

0

RFR

Negative Beta

Graph of SML

Determining the Expected Return

RFR)-(RRFR)E(R Mi i

• The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset

• The risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (RM-RFR)

How are betas calculated?

• Run a regression line of past returns on Stock i versus returns on the market.

• The regression line is called the characteristic line.

• The slope coefficient of the characteristic line is defined as the beta coefficient.

• If beta = 1.0, stock is average risk.

• If beta > 1.0, stock is riskier than average.

• If beta < 1.0, stock is less risky than average.

• Most stocks have betas in the range of 0.5 to 1.5.