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CHAPTER 5Risk and Return Portfolio Theory and

Asset Pricing Models

- Portfolio Theory
- Capital Asset Pricing Model (CAPM)
- Efficient frontier
- Capital Market Line (CML)
- Security Market Line (SML)
- Beta calculation
- Arbitrage pricing theory
- Fama-French 3-factor model

Portfolio TheoryCh 5, in-class, portfolio theory

exercise

- Suppose Asset A has an expected return of 10

percent and a standard deviation of 20 percent.

Asset B has an expected return of 16 percent and

a standard deviation of 40 percent. If the

correlation between A and B is 0.4, what are the

expected return and standard deviation for a

portfolio comprised of 30 percent Asset A and 70

percent Asset B?

n.b. this is .4, not .6, as in binder

Portfolio Expected Return

Portfolio Standard Deviation

Attainable Portfolios rAB 0.4

Attainable Portfolios rAB 1

Attainable Portfolios rAB -1

Attainable Portfolios with Risk-Free Asset

(Expected risk-free return 5)

Expected Portfolio Return, rp

Efficient Set

Feasible Set

Risk, ?p

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.

Investors preferences

- Add to this diagram information about the risk

aversion of investors

Expected Return, rp

IB2

IB1

Optimal Portfolio Investor B

IA2

IA1

Optimal Portfolio Investor A

Risk ?p

Optimal Portfolios

- Indifference curves reflect an investors

attitude toward risk as reflected in his or her

risk/return tradeoff function. They differ among

investors because of differences in risk

aversion. - An investors optimal portfolio is defined by the

tangency point between the efficient set and the

investors indifference curve. - Different indifference curves reflect different

investors risk aversion

WHAT is CAPM?

- An equilibrium model specifying the relationship

between risk and required return on assets held

in diversified portfolios. - RISK RETURN
- (Systematic) RISK RETURN
- If , then
- Only one factor affects risk, and it is?

ß

What are the assumptions of the CAPM?

- Investors all think in terms ofa single holding

period. - All investors have identical expectations.
- Investors can borrow or lend unlimited amounts at

the risk-free rate.

(More...)

- All assets are perfectly divisible.
- There are no taxes and no transactions costs.
- All investors are price takers, that is,

investors buying and selling wont influence

stock prices. - Quantities of all assets are given and fixed.

What impact does rRF have on the 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. - The straight line connecting rRF with M, the

tangency point between the line and the old

efficient set, becomes the new efficient frontier.

Portfolios Combining the Risk Free Asset and the

Market Portfolio

Expected Portfolio Return

Z

.

B

M

.

rM

The Capital Market Line (CML)

.

A

rRF

?M

Risk, ?p

Slope ?

- Portfolio M must contain every asset in exact

proportion to that assets fraction of total

market value of all securities

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

rM - rRF

?p.

rp

rRF

?M

Slope

Intercept

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

investors indifference curves.

Expected Return, rp

CML

I2

I1

.

M

.

rM

R

rR

R Optimal Portfolio

rRF

R0original portfolio

Risk, ?p

?M

?R

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 SML Equation

- The measure of risk used in the SML is the beta

coefficient of company i, ßi. - The SML equation
- ri rRF (RPM) ßi

Recall 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.

Illustration of beta calculation

.

20 15 10 5

.

Year rM ri 1 15 18 2 -5 -10

3 12 16

_

-5 0 5 10 15 20

rM

-5 -10

.

ri -2.59 1.44 kM

Method of Calculation

- Analysts use a computer with statistical or

spreadsheet software to perform the regression. - At least 3 years of monthly returns or 1 years

of weekly returns are used. - Many analysts use 5 years of monthly returns.

(More...)

- If beta 1.0, stock is average risk.
- If beta gt 1.0, stock is riskier than average.
- If beta lt 1.0, stock is less risky than average.
- Most stocks have betas in the range of 0.5 to 1.5.

Interpreting Regression Results

- The R2 measures the percent of a stocks variance

that is explained by the market. The typical R2

is - 0.3 for an individual stock
- over 0.9 for a well diversified portfolio

Interpreting Regression Results (Continued)

- The 95 confidence interval shows the range in

which we are 95 sure that the true value of beta

lies. The typical range is - from about 0.5 to 1.5 for an individual stock
- from about .92 to 1.08 for a well diversified

portfolio

What is the relationship between stand-alone,

market, and diversifiable risk.

?2 b2 ?2 ?e2. ?2 variance

stand-alone risk of Stock j. b2 ?2 market risk

of Stock j. ?e2 variance of error term

diversifiable risk of Stock j.

j

j

M

j

j

j

M

j

What are two potential tests that can be

conducted to verify the CAPM?

- Beta stability tests
- Tests based on the slope of the SML

Tests of the SML indicate

- A more-or-less linear relationship between

realized returns and market risk. - Slope is less than predicted.
- Irrelevance of diversifiable risk specified in

the CAPM model can be questioned.

(More...)

- Betas of individual securities are not good

estimators of future risk. - Betas of portfolios of 10 or more randomly

selected stocks are reasonably stable. - Past portfolio betas are good estimates of future

portfolio volatility.

Are there problems with the CAPM tests?

- Yes.
- Richard Roll questioned whether it was even

conceptually possible to test the CAPM. - Roll showed that it is virtually impossible to

prove investors behave in accordance with CAPM

theory.

What are our conclusions regarding the CAPM?

- It is impossible to verify.
- Recent studies have questioned its validity.
- Investors seem to be concerned with both market

risk and stand-alone risk. Therefore, the SML

may not produce a correct estimate of ri.

(More...)

- CAPM/SML concepts are based on expectations, yet

betas are calculated using historical data. A

companys historical data may not reflect

investors expectations about future riskiness. - Other models are being developed that will one

day replace the CAPM, but it still provides a

good framework for thinking about risk and return.

When using the CAPM, there are choices that have

to be made

CAPM COMMENTS ON ESTIMATING Required rates of

return

- Under the CAPM, required rate of return (or cost

of equity) depends on the risk free rate, the

market risk premium, and the Beta, as shown in

the Security market Line - ki krf (km -krf) ?i

- Risk free rate short term or long term?
- Long term used because (1)equities are long term

instruments, and (2)there is less volatility in

long term rate. - Market risk premium ex-post or ex-ante?
- Ex-post from Ibbotson
- Ex-ante from IBES (Will vary over time).

- Estimating Betas historical or anticipated?
- Historical from characteristic line assumes that

future will reflect the past. - Adjusted Betas (Marshall Blume) Betas tend to

move towards 1 over time. Math. procedure used

to estimate this. - Fundamental Betas Adjustment process to include

(1)financial leverage (2)sales volatility, etc.

Adjusted continuously.

Choices in estimating Betas

- Historical periods of different length one, two,

...five,... years? - Different holding periods daily, weekly,

monthly, annual returns? - More frequent--more data, but also more noise.
- What is used to represent km?
- SP500, NYSE index, Wilshire index or what?

TWO CALCULATIONS OF BETAS

- Merrill Lynch ( Yahoo!Finance)
- km SP500
- B Pure Historical
- Data Monthly(60 observations)

- Value Line
- km NYSE index
- B Adjusted
- Data Weekly(260 observations)

CONCLUSION

- One can calculate Betas in many ways depending

on the method used, different betas may result

from SML ki krf (km -krf)B - With luck, different Betas from different sources

will be close.

What is the difference between the CAPM and the

Arbitrage Pricing Theory (APT)?

- The CAPM is a single factor model.
- The APT proposes that the relationship between

risk and return is more complex and may be due to

multiple factors such as GDP growth, expected

inflation, tax rate changes, and dividend yield.

Required Return for Stock i under the APT

ri rRF (r1 - rRF)?1 (r2 - rRF) ? 2

... (rj - rRF) ? j.

rj required rate of return on a portfolio

sensitive only to economic Factor j.

? j sensitivity of Stock i to economic

Factor j.

What is the status of the APT?

- The APT is being used for some real world

applications. - Its acceptance has been slow because the model

does not specify what factors influence stock

returns. - More research on risk and return models is needed

to find a model that is theoretically sound,

empirically verified, and easy to use.

Fama-French 3-Factor Model

- Fama and French propose three factors
- The excess market return, rM-rRF.
- the return on, S, a portfolio of small firms

(where size is based on the market value of

equity) minus the return on B, a portfolio of big

firms. This return is called rSMB, for S minus B.

Fama-French 3-Factor Model (Continued)

- the return on, H, a portfolio of firms with high

book-to-market ratios (using market equity and

book equity) minus the return on L, a portfolio

of firms with low book-to-market ratios. This

return is called rHML, for H minus L.

Required Return for Stock i under the

Fama-French 3-Factor Model

ri rRF (rM - rRF)bi (rSMB)ci (rHMB)di bi

sensitivity of Stock i to the market return. cj

sensitivity of Stock i to the size factor. dj

sensitivity of Stock i to the book-to-market

factor.

Required Return for Stock i bi0.9, rRF6.8,

the market risk premium is 6.3, ci-0.5, the

expected value for the size factor is 4,

di-0.3, and the expected value for the

book-to-market factor is 5.

ri rRF (rM - rRF)bi (rSMB)ci (rHMB)di ri

6.8 (6.3)(0.9) (4)(-0.5) (5)(-0.3)

8.97

CAPM Required Return for Stock i

CAPM ri rRF (rM - rRF)bi ri 6.8

(6.3)(0.9) 12.47 Fama-French

(previous slide) ri 8.97

THE END, thank god.

507

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