2201afe vw week 9 return, risk and the sml

7/27/2019 2201AFE VW Week 9 Return, Risk and the SML

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2201AFE Corporate FinanceWeek 9:

Return, Risk and the Security Market Line

Readings: Chapter 11


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Agenda

Last Lecture

Return, Risk and the Security Market Line Key Concepts and Skills

Real World Application

Estimating Microsofts Beta

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Last Lecture

Returns

Holding Period Returns Averages: Arithmetic Mean & Geometric Mean

Risk

Variance

Standard Deviation

There is a reward for bearing risk

Positive risk-return relationship

Risk Premium

Efficient Market Hypothesis: weak, semi-strong, strong

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Return, Risk and the Security Market Line

Chapter 11

4


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1. Introduction & Financial

Statements

2. Time Value of Money

3. Valuing Shares & Bonds

7. Mid-semester Exam

8. Some Lessons from Capital

Market History

11. Financial Leverage & Capital

Structure Policy

13. Options & Revision

9. Return, Risk & the Security

Market Line

5. Making Capital Investment

Decisions & Project Analysis

12. Dividends & Dividend Policy

6. Revision for Mid-sem Exam

4. Net Present Value & Other

Investment Criteria

10. Cost of Capital


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Key Concepts and Skills

Expected Returns and Variances

Probabilities

Portfolios

Risk and Returns

The principle of diversification

Risk: Systematic and Unsystematic

The Security Market Line (SML)

Capital Asset Pricing Model (CAPM)

Reward to Risk Ratio

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Expected Returns

Consider an asset which has many possible future returns,

returns that are not equally likely. What is the average

return? What is the expected return?

Average or Expected returns is based on the average of all

possible future returns weighted by their probabilities.

Suppose there are Tpossible returns, and that R1 hasprobability p1 of occurring, R2 has probability p2, , and RT

has probability pT . Then:

7

TT2211

Ti

1i ii

RpRpRpE(R)

RpE(R)


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Example: Expected Returns

Suppose you have predicted the following returns for

stocks C and T in three possible states of nature. What are

the expected returns?

RC = 0.3(0.15) + 0.5(0.10) + 0.2(0.02) = 9.99%

RT = 0.3(0.25) + 0.5(0.20) + 0.2(0.01) = 17.7%

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State Probability Stock C Stock T

Boom 0.3 15% 25%

Normal 0.5 10% 20%Recession ??? 2% 1%


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Variance and Standard Deviation

Variance and standard deviation still measure the volatility

of returns.

Using unequal probabilities for the entire range of

possibilities.

Weighted average of squared deviations.

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2

2

ii

2

22

2

11

2

T

1i

2

ii

2

VARSD

E(R)][Rp...E(R)][RpE(R)][RpVAR

E(R)][RpVAR


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Example: Variance and Standard Deviation

Consider the previous example. What are the variance and

standard deviation for each stock?

E(R)C = 9.9% and E(R)T = 17.7%

Stock C:

2 = 0.3(0.15-0.099)2 + 0.5(0.10-0.099)2 + 0.2(0.02-0.099)2

= 0.3(0.051)2 + 0.5(0.001)2 + 0.2(-0.079)2

= 0.3(0.002601) + 0.5(0.000001) + 0.2(0.006241)

= 0.0007803 + 0.0000005 + 0.0012482 = 0.002029

Stock T:

2 = 0.3(0.25-0.177)2 + 0.5(0.20-0.177)2 + 0.2(0.01-0.177)2

= 0.3(0.073)2 + 0.5(0.023)2 + 0.2(-0.167)2

= 0.0015987 + 0.0002645 + 0.0055778 = 0.007441

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%5.4045044.0002029.02

%63.8086261.0007441.02


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Portfolios

A portfolio is a collection of assets.

An assets risk and return are important in how they affect

the risk and return of the portfolio.

The risk-return trade-off for a portfolio is measured by the

portfolio expected return and standard deviation, just aswith individual assets.

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Example: Portfolio Weights

Suppose you have $15,000 to invest and you have

purchased securities in the following amounts. What are

your portfolio weights in each security?

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Companies Amount invested Weights

CBA $2,000

WOW $3,000TLS $4,000

BHP $6,000


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Portfolio Expected Returns

The expected return of a portfolio is the weighted averageof the expected returns for each asset in the portfolio (see

example 11.3) Method 1:

Step 1: calculate E(Rasset) based on probability of state

Step 2: calculate E(RP) based on weights of assets

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nn2211asset Rp...RpRp)E(R

)E(Rw...)E(Rw)E(Rw)E(R nn2211P


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Portfolio Expected Returns

You can also find the expected return by finding theportfolio return in each possible state and computing the

expected value as we did with individual securities (seeexample 11.4)

Method 2:

Step 1: calculate E(RP) in each state, eg. boom or bust

Step 2: add the state returns weighted by each probability

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nn2211state,P Rw...RwRw)E(R

)E(Rp...)E(Rp)E(Rp)E(R state nn2state21state1P


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Example: E(RP)

Consider the following information

What are the expected return for a portfolio with an

investment of $6,000 in asset X and $4,000 in asset Z?

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State Probability Asset X Asset Z

Boom 0.25 15% 10%

Normal 0.60 10% 9%

Recession 0.15 5% 10%


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Example: E(RP) Method 1

Weight X = 0.6, Weight Z = 0.4

First way of calculating E(RP):

Step 1: Calculate the expected return of each asset based on

each probability of state occurring:

E(RX) = (0.250.15) + (0.60.10) + (0.150.05) = 0.105

E(RZ) = (0.250.10) + (0.60.09) + (0.150.10) = 0.094

Boom Normal Recession

Step 2: Calculate the E(RP) based on the weights of each

asset:E(RP) = 0.60.105 + 0.40.094 = 0.1006 (10.06%)

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Example: E(RP) Method 2

Weight X = 0.6, Weight Z = 0.4

Second way to calculate E(RP):

Step 1: Calculate the E(RP) in each state based on each asset

weight:

E(RP)Boom = (0.60.15) + (0.40.10) = 0.13

E(RP

)Normal

= (0.60.10) + (0.40.09) = 0.096

E(RP)Recession = (0.60.05) + (0.40.10) = 0.07

Step 2: Calculate total E(RP) using probabilities as weights:

E(RP) = pB E(RBoom) + pN E(RNormal) + pR E(RRecession)

E(RP) = (0.250.13) + (0.60.096) + (0.150.07)

= 0.0325 + 0.05756 + 0.0105 = 0.1006 (10.06%)

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Portfolio Variance with Probabilities

1. Compute the portfolio return for each state (step 1):

E(RP,state) = w1R1 + w2R2

2. Compute the expected portfolio return using probabilities as for asingle asset (step 2):

E(RP) = p1 E(Rstate1) + p2 E(Rstate2) + p3 E(Rstate3)

3. This E(RP) becomes the mean

4. Compute the deviations of each state from the mean, then square thedeviation:

[E(RP,state) E(RP)]2

5. Multiply the squared deviation with probability of each state, then

sum: (pstate [E(RP,state) E(RP)]

2)

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Example: Variance & SD

Variance: (pstate [E(RP,state) E(RP)]2)

Portfolio return in each state (boom, normal, recession) and two-

asset (X and Z) total portfolio return.

E(Rp)boom = 13%

E(Rp)normal = 9.6%

E(Rp)recession = 7%

E(Rp) = 10.06%

Variance:

= 0.25(0.13-0.1006)2 + 0.6(0.096-0.1006)2 + 0.15(0.07-0.1006)2

= 0.00021609 + 0.000012696 + 0.000140454

= 0.00036924

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%92.1019215619.000036924.0SD


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Example 2: E(R), Variance & SD

Consider the following information:

Invest 50% of your money in Asset A

What are the expected return and standard deviation for

each asset?

What are the expected return and standard deviation forthe portfolio?

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State Probability Asset A Asset B Portfolio

Boom 0.40 30% -5% 12.5%

Bust 0.60 -10% 25% 7.5%


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Example 2: E(R), Variance & SD Asset

E(Rasset) = (pstate1 Rasset) + (pstate2 Rasset)

VAR = [pstate (Rasset E(Rasset)2]

Asset A: E(RA) = 0.4(0.30) + 0.6(-0.10) = 6%

Variance(A) = 0.4(0.30-0.06)2 + 0.6(-0.10-0.06)2

= 0.02304 + 0.01536 = 0.0384

Asset B: E(RB) = 0.4(-0.05) + 0.6(0.25) = 13%

Variance(B) = 0.4(-0.05-0.13)2 + 0.6(0.25-0.13)2= 0.01296+0.00864 = 0.0216

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%7.14147.00216.0)B(SD

%6.19196.00384.0)B(SD

2VARSD


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Example 2: E(R), Variance & SD Portfolio

Calculate the Expected return of portfolio in each state:

E(Rp)state = (wasset A RA state) + (wasset B RB state)

E(Rp)boom = 0.5(0.30) + 0.5(-0.05) = 0.125

E(Rp)bust= 0.5(-0.10) + 0.5(0.25) = 0.075

Then the overall Expected portfolio return:

E(Rp) = (pstate1 E(Rp)state1) + (pstate2 E(Rp)state2)

E(Rp) = 0.4(0.125) + 0.6(0.075) = 0.095 Then the Variance of the portfolio:

VARp = [pstate (E(Rp)state E(Rp))2]

VARp = 0.4(0.125 - 0.095)2 + 0.6(0.075 - 0.095)2

= 0.00036 + 0.00024 = 0.0006 Then the Standard Deviation of the portfolio:

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%45.202449.00006.0SD

k d f l h


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Risk and Portfolio Theory

Risk Averse Investors: Require a higher average return to

take on a higher risk.

Portfolio theory assumptions:

Investors prefer the portfolio with the highest expected

return for a given variance or the lowest variance for a given

expected return.

Expected returns and variances of portfolios derived from

historical returns, variances, and co-variances of individual

assets in portfolio.

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C i d C l i C ffi i


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Covariance and Correlation Coefficient

Covariance is an absolute measure of the degree to which

two variables move together over time relative to their

individual mean (average).

Correlation Coefficient, , is a standardised measure of the

relationship between the two variables, ranging between

-1.00 to +1.00

25

T

)R)(RR(RCov

2211

2,1

Var

T

)R)(RR(RCov

1111

1,1

212,12,1

21

2,1

1,22,1

Cov

Coventn CoefficiCorrelatio

P tf li VAR & SD f 2 A t P tf li


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Portfolio VAR & SD for a 2-Asset Portfolio

In order to reduce the overall risk, it is best to have assetswith low positive or negative correlation (covariance).

The smaller is the covariance between the assets, the

smaller will be the portfolios variance.

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)nCorrelatio(ww2wwVAR

)iancevarCo(Covww2wwVAR

2,1212122

22

21

21

2p

2,121

2

2

2

2

2

1

2

1

2

p

2

PP SD

E l Ri k f 2 A t P tf li


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Example: Risk of 2-Asset Portfolio

Consider these two assets that have equal weights of 0.50

in the portfolio, and with the following returns and

standard deviation:E(R1) = 30% and 1 = 0.20

E(R2) = 15% and 2 = 0.12

Correlation Coefficient = 0.10

p2 = (0.5)2(0.2)2 + (0.5)2(0.12)2 + 2(0.5)(0.5)(0.2)(0.12)(0.1)

= 0.01 + 0.0036 + 0.0012 = 0.0148

= 12.16% (lower risk for 22.5% portfolio return)0.50.3 + 0.50.15 = 0.225 or 22.5%

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2,12121

2

2

2

2

2

1

2

1

2

p ww2ww

1216.00148.0P


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Diversification


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Diversification

The Principle of Diversification:states that spreading an

investment across many types of assets will eliminate some

but not all of the risk.

Diversification can substantially reduce the variability of

returns without an equivalent reduction in expected

returns.

Size of risk reduction depends on co-variances between

assets in the portfolio.

However, there is a minimum level of risk that cannot bediversified away and that is the systematic portion.

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Two Types of Risk


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Two Types of Risk

Systematic or Non-Diversifiable Risk:

That portion of an assets risk attributed to the market

factors that affect all firms and cannot be eliminated through

the process of diversification.

Unsystematic or Diversifiable Risk:

That portion of an assets risk which is firm specific and can

be eliminated through the process of diversification.

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Total Risk


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Total Risk

Total risk = Systematic risk + Unsystematic risk

The standard deviation of returns is a measure of total risk.

For well-diversified portfolios, unsystematic risk is very

small.

Consequently, the total risk for a diversified portfolio is

essentially equivalent to the systematic risk.

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Systematic Risk Principle


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Systematic Risk Principle

There is a reward for bearing risk.

There is not a reward for bearing risk unnecessarily.

The expected return on a risky asset depends only on that

assets systematic risk since unsystematic risk can bediversified away.

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Measuring Systematic Risk =


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Measuring Systematic Risk =

We use the beta coefficient to measure systematic risk.

Beta measures the responsiveness of a security to

movements in the market.

Market beta m = 1

Therefore if:

A= 1, the asset has the same systematic risk as the overallmarket.

A < 1 implies the asset has less systematic risk than the

overall market.

A > 1 implies the asset has more systematic risk than theoverall market.

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2

m

A,m

A

Cov


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CAPM


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CAPM

E(RA) = Rf+ A[E(RM) Rf]

Where:

E(RA) = expected return on asset A

Rf= risk free rate

A= beta of asset A

E(RM) = expected return on the market

Note: E(RM

) Rf= Market Risk Premium

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Example CAPM


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Example CAPM

If the beta for IBM is 1.15, the risk-free rate is 5%, and the

expected return on market is 12%, what is the required rate

of return for IBM?

Applying the CAPM:

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%05.13or1305.0

0805.005.0

]07.0[15.105.0

]05.012.0[15.105.0]R)[E(RR)E(R fMIBMfIBM

Example CAPM


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Example CAPM

Consider the betas for each of the assets given earlier. If

the risk-free rate is 2.13% and the market risk premium is

8.6%, what is the expected return for each?


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Security Beta Expected Return

CBA 2.685

WOW 0.195

TLS 2.161

BHP 2.434

Security Market Line (SML)


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The security market line (SML) is the graphical

representation of CAPM.

Shows the relationship between systematic risk and

expected return.

Positive slope.

The higher the risk, the higher the return.

According to the CAPM, all stocks must lie on the SML,

otherwise they would be under or over-priced.

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y ( )

40

Asset expected

return = E(ri)

E(RM)

SML

Asset beta =iM= 1.0

Market

A undervalued

B overvalued

RF

A B

E(RB)

E(RA) = E(RM) RF

Reward-to-Risk Ratio


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SML slope = Reward-to-Risk Ratio of the Market = Market

Risk Premium

In equilibrium, all assets and portfolios must have the samereward-to-risk ratio and they all must equal the reward-to-

risk ratio for the market.

If not, assets are undervalued or overvalued.

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)RE(R

)RE(R

R)E(R

R)E(RfM

M

fM

B

fB

A

fA

Reward-to-Risk Ratio: Example


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p

If RM = 12%, Rf= 6%

If Asset A has E(RA) = 15%, A = 1.3, and

Asset B has E(RB) = 10%, B = 0.8 then:

Asset B offers insufficient reward for its level of risk, so B is relatively

overvalued compared to A, or A is relatively undervalued.42

%58.0

%6%10

R)E(R

%9.63.1

%6%15

R)E(R

B

fB

A

fA

%61

%)6%12(Slope

)RE(R

)RE(R

Slope fMM

fM

CAPM and Beta of Portfolio


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Ifw1, w2, , wn are the proportions of the portfolio

invested in nassets 1, 2, , n, the beta of a portfolio (P)

can be written:

Example: If 30% of a portfolio is invested in asset 1 and thebalance in asset 2, and asset 1s beta = 1.7 while asset 2s

beta = 1.2, what is the beta of the portfolio (P).

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nj

1j

jjnn2211P ww...ww

35.1

)2.1(7.0)7.1(3.0

ww 2211P

Example: Portfolio Beta


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Consider our previous four securities and their betas:

What is the portfolio beta?

= 0.133(2.685) + 0.2(0.195) + 0.167(2.161) + 0.4(2.434)

= 1.731

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Security Weight Beta

CBA 0.133 2.685

WOW 0.200 0.195

TLS 0.167 2.161

BHP 0.400 2.434

Factors Affecting E(R)


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Pure time value of money

measured by the risk-free rate, Rf

Reward for bearing systematic risk measured by the market risk premium, E(RM) Rf

Amount of systematic risk

measured by beta,

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Quick Quiz


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What is the difference between systematic and

unsystematic risk?

What type of risk is relevant for determining the expected

return?

Consider an asset with a beta of 1.2, a risk-free rate of 5%and a market return of 13%.

What is the reward-to-risk ratio in equilibrium?

What is the expected return on the asset?

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Real World Application


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To calculate the parameters for an asset, say a share in

Microsoft, we perform a regression of the returns on

Microsoft with the returns on the market.

Historical data for Microsoft and the Market index (S&P

500) are collected and a time series of both returns are

calculated.

An OLS regression is then performed.

The regression provides values for the parameters and a

plot of the observations may be made.

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Example: Microsoft vs S&P500


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Monthly Returns: Microsoft and S&P 500 (n = 60)

-20

-10

0

10

20

30

40

-20 -15 -10 -5 0 5 10

S&P 500 Return

M

icrosoftReturn

Coefficients Standard error t-statistics

2.14 1.21 1.77 1.30 0.27 4.76

Adjusted R2= 0.27

n = 60

Next Week


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We move to calculating a firms cost of capital, termed the

weighted average cost of capital (WACC).

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2201afe vw week 9 return, risk and the sml

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