Sapm CHAPTER 8 Solution

8/2/2019 Sapm CHAPTER 8 Solution

http://slidepdf.com/reader/full/sapm-chapter-8-solution 1/25

Chapter-8

CAPITAL ASSET PRICING MODEL AND ARBITRAGE PRICING

THEORY

Q1) Mention the assumptions underlying the standard capital asset pricing model.

Ans: The CAPM is based on the following assumptions.

• Individuals are risk averse.

• Individuals seek to maximize the expected utility of their portfolio over a single period

planning horizon.

• Individuals have homogenous expectations-they have identical subjective estimates of themeans, variances and co-variances among returns.

• Individuals can borrow and lend freely at a riskless rate of interest.

• The market is perfect: there are no taxes: there are no transactions costs; securities are

divisible; the market is competitive.

• The quantity of risky securities in the market is given.

Q2) What is the relationship between risk and return for efficient portfolios?

Ans: The relationship between risk and return for efficient portfolios, as given by the capital

market line, is:

E ( Ri) = R f + λσi

Where λ = E (R M ) - R f

σM

Q3) What is the measure of risk for an individual security?

Ans: The risk of a security is expressed in terms of its covariance with the market portfolio, σiM.



The measure of risk is σiM.

We can find standardized measure of systematic risk, popularly known as β by using relationship,

βi = σiM/ σ2M

Q4) Discuss the relationship embodied in the security market line.

Ans: The Security Market Line (SML) is a part of the CAPM which gives the

Risk/return relationship for individual stocks.

There is a linear relationship between expected return and covariance of individual stocks with

market portfolio. This relationship, called the Security Market Line is as follows:

E (R i) = R f + [(E (R M) -R f ) / σ2M] σiM

Where E (R i) is the expected return on security i, R f is the risk –free return, E (R M) is the expected

return on the market portfolio, σ2M is the variance of return on market portfolio, and σiM is the

covariance of returns between security i and market portfolio.

In words, the SML relationship says:

Expected return on the security i =Risk free return + (Price per unit of risk) Risk

The price per unit of risk: [E(R M) -R f ] / σ

2

M

The risk of the security i expressed in terms of the covariance with the market portfolio, σ iM .

The standardized measure of systematic risk i.e, beta(β) can be found out by

βi = σiM / σ2M

βi reflects the slope of a linear regression relationship in which the return on security i is regressed

on the return on the market portfolio.

Thus, The SML is popularly expressed as

E (R i) = R f + [E (R M) -R f ] βi

In words, the SML relationship says:



Expected return in the security i =Risk-free return + Market risk Premium × Beta of security i

The SML which reflects the expected return-beta relationship is as shown below.

Note: The slope of the SML is the market risk premium

Assets which are fairly priced plot exactly on the SML. Underpriced securities(such as P) plot

above the SML, whereas overpriced securities( such as O) plot below the SML. The difference

between the actual expected return on a security and its fair return as per the SML is called the

security’s alpha, denoted by ɑ

Q5) What is the risk free rate? How would you measure it?

Ans: The risk-free return is the return on a security (or portfolios of securities) that is free from

default risk and is uncorrelated with returns from anything else in the economy.

The return on a zero-beta portfolio is the best estimate of the risk-free rate. The risk free

return is represented by the rate on a 364-day treasury bill or the rate on the long term government

bond.

Q6) Suggest an appropriate measure for the market risk premium and justify it.

Ans: The market risk premium is the difference between the expected return on the market

portfolio and risk-free return.



Arithmetic Mean is most appropriate measure for measuring market risk premium because

arithmetic mean is more consistent with the mean variance framework and can better predict the

premium in the next period.

And the longest possible historical period is used as the measurement period.

Q7) What factors influence market risk premium?

Ans: There are three main factors which influence market risk premium. They are

1. Variance in the Underlying Economy : If the Underlying economy is more Volatile, the

market risk premium is likely to be large. For e.g., the market risk premiums for emerging

markets, given their high growth and high-risk economies, are larger than the market risk

premiums for developed markets.

2. Political Risk : Market risk premiums are larger in the markets subject to higher political

instability. Remember that political instability causes economic uncertainty.

3. Market Structure : If the companies listed on the market are mostly large, stable and

diversified, the market risk premium is smaller. On the other hand, if the companies are

listed on the market are generally small and risky, the market premium is larger.

Q8) What is Beta and how is it measured?

Ans: Beta is the ratio of product of covariance of security i and market to the variance of market

βi = σiM / σ2M

βi =Σ(R i - R i bar)(R M - R M bar)

Σ((R M - R M bar)2

Beta is measured as follows• If Beta > 1, more risky than the market

Beta of more than 1 indicates an aggressive stock and the value of fund

is likely to rise or fall more than the benchmark.

• If Beta < 1, indicates less risky than the market.



Beta of less than 1 indicates that the stock will react less than the

market index. It is called as defensive stock.

• If Beta = 1, same risk as the market.

If the beta of a stock is 1 it is called neutral stock.

Q9) What adjustment is done to historical betas?

To overcome some of the limitations the following adjustments can be made:

A procedure that is sometimes recommended is to take a weighted average of the historical beta,

on the one hand and 1.0 (the value of market beta) on the other. The weighting scheme should take

in to account the degree of historical estimation error and the dispersion of individual firms around

the average is large, the weight assigned to the historical beta should be small. If the dispersion of

individual firms around the average is large, the weight assigned to 1.00 (the market beta) should

be small. By balancing these factors a suitable weighting scheme can be developed.

Q10) What fundamental factors drive beta? How are fundamental betas superior to

historical betas?

The following fundamental factors drive beta

• Industry affiliation

• Corporate growth

• Earnings variability

• Financial leverage

Fundamental betas have several advantages over historical betas:• Fundamental betas have a stronger intuitive appeal as the rules for predicting them are

consistent with over general understanding of what makes a company risky.

• Fundamental betas are ideal for the analysis of non trading assets like individual projects,

strategic business units and corporate divisions as he price behavior for such assets is not

available.



• Fundamental betas, in general, seem to outperform historical betas.

• Fundamental betas can be based on future descriptors, where as historical betas necessarily

require past data (for E.g., a firm’s growth orientation can be estimated using revenue

projections).

Q11) Discuss the limitation of betas based on accounting earnings.

Limitations of betas based on accounting earnings are:

• Accounting earnings are generally smoothed out, related to the value of the company. This

results in betas which have an upward bias or downward bias.

• Accounting earnings are influenced by non-operating factors like extraordinary gains or

losses and changes in accounting policies with respect to depreciation, inventory valuationand so on.

• Compare to stock price which are observed on daily basis, accounting earning are

measured at less frequent intervals. This means that regression analysis using accounting

data will have fewer observations and lesser power.

Q 12) How is beta estimated from cross-sectional regressions?

Beta from Cross – sectional Regression Yet another approach to estimate beta for unlisted

companies or divisions or projects calls for cross – sectional regression analysis. It involves two

steps:

Step 1: Estimate a cross – sectional regression relationship for publicly traded firms in which the

dependent variables is beta and the independent variables are fundamental firm factors like growth

rate, earnings variability, financial leverage , size , and dividend payout ratios, which seem to drive

betas.

Step 2: Plug the specific characteristics of the project, division, or unlisted company in the

regression relationship to arrive at an estimate of beta.



For example, the following is a regression relating the beta of NYSE and AMEX stocks in 1944 to

four variables: coefficient of variation in operating income, debt-equity ratio, earnings growth, and

total assets.

Beta = 0.6507 + 0.27 coefficient of variation in operating income + 0.09 Debt/ equity ratio + 0.54

Earnings growth – Total assets (millions of dollars)

Suppose an unlisted firm has the following characteristics:

Coefficient of variation in operating income = 1.85

Debt equity ratio = 0.09

Earnings growth rate = 12 %

Total assets = $ 150 million

Plugging these values in to the regression relation yields the following beta:

Beta = 0.6507 + 0.27 (1.85) + 0.09 (0.09) + 0.54 (0.12) – 0.00009 (150) = 1.2095

Q13) Discus the procedure commonly used in practice to test the CAPM.

In practice, researchers have tested the CAPM using ex post facto data, rather than ex ante data.

The commonly followed procedure involves three basic steps:

1. Set up the sample data.

2. Estimate the security characteristics lines (SCLs).

3. Estimate the security market line (SML).

Set up the sample data : Suppose you are looking at a sample of 75 securities over a

period of 60 monthly holding periods (5 years). For each of the 60 holding perios, you have



to collect the rates of returns on 75 securities, a market portfolio proxy, and one – month

(risk – free) Treasury bills. Your data set thus consists of:

R it :returns on 75 securities over the 60 month period (i = 1,2,…….75 and t = 1,2,…..60)

R mt : returns on a market portfolio proxy over the 60 month period

R ft : risk – free rates over the 60 month period

This constitutes a total of 77 * 60 = 4620 rates of return.

Estimate the security characteristics lines : you have to estimate the beta for each securities

in the sample. The beta for each security is simply the slope of its security characteristic

line. There are two ways in which security beta is estimated:

R it = ai + bi R mt + eit

R it – R ft = ai + bi (R mt – R ft) + eit

Note that in Eq.(9.12) the return on security I is regressed on the return on the market

portfolio, where in Eq. (9.13) the excess return on security I is regressed on the excess

return on market portfolio. It appears that Eq. (9.13) is used more commonly.

Estimate the security market line : once you have the beta estimates of various securities ,

you can estimate the security market line:

/R i = y0 + y1 bi + et i = 1,2,……..75



Comparing Eqs (9.13) and (9.14) you can infer that if the CAPM holds:

• The relationship should be linear . This means that terms like bi^2, if substituted

for bi , should not yield better explanatory power.

• Y0 , the intercept , should not be significantly different from the risk – free

rate, /R f .

• Y1 , the slope coefficient , should not be significantly different from /(R m – R f )

• No other factors, such as company size or total variance, should affect /R i.

• The model should explain a significant portion of variation in returns among

securities.

Q14) What is the empirical evidence on the CAPM?

The empirical evidence on the CAPM model are :

• The relation appears to be linear.

• In general y0 is greater than the risk-free rate and y1 is less than R m – R f (bar).

This means that the actual relationship between risk and return is flatter than what theCAPM says.

• In addition to beta, some other factors , such as standard deviation of returns and

company size, too have a bearing on return.



• Beta does not explain a very high percentage of the variance in return among

securities.

Q15) Despite its limitations, why is the CAPM widely used?

The CAPM widely used,

• Some objective estimate of risk premium is better than a completely subjective estimate or

no estimate.

• CAPM is a simple and intuitively appealing risk-return model. Its basic message that

diversifiable risk does not matter is accepted by nearly every one.

• While there are plausible alternative risk measures, no consensus has emerged on what

course to plot if beta is abandoned. As Brealey and Myers say: “So the capital asset pricing

model survives not from a lack of competition but from a surfeit”.

Q16) Show how the capital market line is a special case of the security market line?

Relationship between SML And CML

• SML

E(Ri) = Rf + {[E(RM ) – Rf]/ σ2M} σiM

SINCE σiM = iM σi σ M

E(Ri) = Rf + {[E(RM ) – Rf]/ σM} ʆiM σi



IF i AND M ARE PERFECTLY CORRELATED ʆiM = 1. SO

E(Ri) = Rf + {[E(RM ) – Rf]/ σM} σi

THUS CML IS A SPECIAL CASE OF SML

17. Define the return – generating process according to the APT ?

Return generating process – The APT assumes that the return on an asset is linearly related to a set

of risk factors as shown below :

R i = E(R i) + bi1I1 + bi2 I2.........bik Ik + ei

Where,

R i = actual return on asset i during a specified time period ( i = 1,2,...... n).

E (R i) = expected return from on asset i if all the risk factors have Zero changes.

bij = sensitivity of asset i’s return to the coppn risk factor j.

I j (1, 2, .....k) = deviation of a systematic risk factor j from its expected value.

ei = random error term , unique to asset i.

The expected value of each risk factor , I, is Zero. Hence , the I j’s in the above equation measure the

deviation of each risk factor from its expected value. Asset returns are related in a linear manner to a

limited number of systematic risk factors.

18. What is the equilibrium risk return relationship according to the APT ?

APT establishes the equilibrium risk and return relationship. The key idea that guides the development of

the equilibrium relationship is the law of one price which says that two identical things cannot sell at

different prices. Applied to portfolios, expected returns . If it were so, arbitrageurs will step in and their

actions will ensure that the law of one price is satisfied.

The equilibrium relationship according to the APT is as follows:



E(R i) = ƛ 0 + bi1ƛ1 + bi2ƛ2 + ....bijƛ j

Where E(R i) = expected return on the asset i, ƛ0 = expected return on asset with zero systematic risk, bij =

sensitivity of asset i’s treturn to the common risk factor j ,and ƛ j = risk premium related to the jth Risk

factor.

19. What is a multifactor model? Describe the types of multifactor models used in practice.

In this approach researchers chooses a priori the exact number and identity of risk factors and specifies

the multi factor model of the following kind.

R it = a1 + ( bit F1t+ bi2 F2t +.....+ bik Fkt ) + eit

Where, R it = return on security i in period t and F1t is the return associated with the jth risk factor in period t.

The advantage of factor model like this is that the researcher can specify the risk factors; the disadvantage

of such a model is that there is very little theory to guide it. Hence, developing a useful factor model is as

much an art as science. The variety of multifactor models employed in practice may be divided into two

broad categories: macroeconomic based risk factor models and micro-economic based risk factor models.

Macroeconomic based risk factor models: these models consider risk factors that are macroeconomic in

nature.

R it =ai + bil R mt + bi2 MPt +bi3 DEI t+ b i4UI t+b5 UPR t + bi6 UTSt +eit

Where R m is the return on a value weighted index of NYSE – listed stocks, MP is the monthly growth rate

in the US industrial production ,DEI is the change in inflation, measured by the Us consumer

price index, UI is the difference between actual and expected levels of inflation ,UPR is the

unanticipated term structure shift ( long term RFR – short term RFR ).

Microeconomic based risk factor models: These models consider risk factors that are

microeconomic in nature.

(R it – RFR t) =ὰ

i + bil (R mt –RFR t) + bi2 SMBt + bi3HMLt + eit.

In this model, in addition to ( R mt – RFR t), the excess return on a stock market portfolio, there are

two other micro economic risk factors. SMB,and HMLt.. SMBt (i.e., small minus big) is the return to a

portfolio of small capitalization stocks less the return to a portfolio of low book – to – market values less

the return to a portfolio of low book –to- market value stocks. In this model, Smb is intended to capture



the risk associated with firm size while HML is meant to reflect risk differentials associated with

“growth” (i.e., low book – to – market ratio) and “value” (i.e., book – to – market ratio).

20. Explain the theory of Stock market as a complex adaptive system.

Michael J. Mauboussin has suggested that the capital market may be regarded as a complex adaptive

system .This model appears to be more consistent with what is known in disciplines like physics and

biology.

The central characteristics and properties of a complex adaptive system are as follows:

Aggregation – The collective interactions of many less- complex agents produces complex, large – scale

behavior.

Adaptive decision rules – Agents in the adaptive system take information from the environment and

develop decision rules. The competition between various decisions rules ensures that eventually the most

effective decision rules survive.

Non – Linearity – unlike a linear system, wherein the value of the whole is equal to the sum of its parts, a

non – linear system is one wherein the aggregate behavior is very complex because of interaction effects.

Feedback Loops – In a system that has feedback loops the output of one interaction becomes the input of

the next. A positive feedback can magnify an effect, whereas a negative feedback can dampen an effect.

CAPITAL ASSET PRICING MODEL AND

ARBITRAGE PRICING THEORY

PROBLEMS:

Q1. The returns on the equity stock of Auto Electricals Ltd and the market portfolio over 11 years

period are given below:



Year Return on the stock Return

Auto Electricals (%) Market Portfolio (%)

1 15 12

2 -6 1

3 18 14

4 30 24

5 12 16

6 25 30

7 2 -3

8 20 24

9 18 15

10 24 22

11 8 12

(i) Calculate the beta for the stock of Auto Electricals Ltd?

(ii) Establish the characteristic line for the stock of Auto Electricals Ltd.

Solution:

Year R A R M R A - R A R M - R M (R A - R A) (R M - R M) (R M - R M)2

1 15 12 .09 -3.18 -.2862 10.1124

2 -6 1 -21.09 -14.18 299.05 201.0724



3 18 14 2.91 -1.18 -3.4338 1.3924

4 30 24 14.91 8.82 131.50 77.7924

5

6

7

8

9

10

11

12

25

2

20

18

24

8

16

30

-3

24

15

22

12

-3.09

9.91

-13.09

4.91

2.91

8.91

-7.09

.82

14.82

-18.18

8.82

-.18

6.82

-3.18

-2.5338

146.8662

237.9762

43.3062

-.5238

60.7662

22.5462

.6724

219.6324

330.5124

77.7924

0.0324

46.5124

10.1124

Sum 166 167 935.2334 975.6364

Mean 15.09 15.18

975.6364 935.2334

σ M 2 = = 97.56 Cov A,M = = 93.52

11 - 1 11 – 1

93.52

β A = = .9586

97.56

(ii) Alpha = R A – βA R M

= 15.09– (.9586x 15.18) = 0.538

Equation of the characteristic line is

R A = 0.538+ .9586 R M

Q2. Calculation of beta for stock B



Year R A (%) R M (%) R A-R Abar R M-R Mbar (R A-R A)

x(R M-R M)

(R M-R M)2

1 15 9 6 -1 -6 1

2 16 12 7 2 14 4

3 10 6 1 -4 -4 16

4 -15 4 -24 -6 144 36

5 -5 16 -14 6 -84 36

6 14 11 5 1 5 1

7 10 10 1 0 0 0

8 15 12 6 2 12 4

9 12 9 3 -1 -3 1

10 -4 8 -13 -2 26 4

11 -2 12 -11 2 -22 412 12 14 3 4 12 16

13 15 -6 6 -16 -96 256

14 12 2 3 -8 -24 64

15 10 8 1 -2 -2 4

16 9 7 0 -3 0 9

17 12 9 3 -1 -3 1

18 9 10 0 0 0 0

19 22 37 13 27 351 729

20 13 10 4 0 0 0

Σ R A = 180 ΣR M = 200

Cov A,M = 326/19= 17.15

R A bar =9 R M bar = 10

σ M2 = 1186/19= 62.42

326/19

βA = ------------------- = 0.27

1186/19



Q4. A (i) What is the beta for stock A?

Solution:

Cov ( A,M ) Cov ( A,M )

ρ AM = ; 0.7 = ⇒ Cov ( A,M ) = 336

σ A σ M 24 x 20

σ M 2

= 202

= 400

Cov ( A,M ) 336

β A = = = 0.84

σ M 2 400

(ii) What is the beta for stock B?

Cov ( B,M ) Cov ( B,M )

ρ BM = ; 0.8 = ⇒ Cov ( B,M ) = 512

σ B σ M 32 x 20

σ M 2

= 202 = 400

Cov ( B,M ) 512

β B = = = 1.28

σ M 2 400

B (i) What is the expected return for stock A?



Solution:

E ( R A) = R f + β A ( E ( R M ) - R f )

= 6% + 0.84 (8%) = 12.72%

(ii) What is the expected return for stock B?

Solution:

E ( R B) = R f + β B ( E ( R M ) - R f )

= 6% + 1.28 (8%) = 16.24%

Q6. The following table gives an analyst’s expected return on two stocks for particular market

returns.

Market Return Aggressive Stock Defensive Stock

8% 2% 10%

20% 32% 16%

(i) What are the betas of the two stocks?

Solution:

Beta= 40% - (-5%)/25% - 5% = 2.25



What is the beta of the defensive stock?

Beta= 18% -8%/25% - 5% = 0.5

MINICASE

a. Calculation of expected return and standard deviation

Solution:

For stock A:

Expected return = (0.2 x -15) + (0.5 x 20) + (0.3 x 40) = 19

Standard deviation = [0.2 (-15 -19)2 + 0.5 (20-19)2 + 0.3 (40 – 19)2 ] 1/2

= [231.2 + 0.5 + 132.3]1/2 = 19.07

For stock B:

Expected return = (0.2 x 30) + (0.5 x 5) + [0.3 x (-) 15] = 4

Standard deviation = [0.2 (30 – 4)2 + 0.5 (5 -4)2 + 0.3 (-15–4)2]1/2

= (135.2 + 0.5 + 108.3) ½ = 15.62

For stock C:

Expected return = [0.2 x (-5)] + (0.5 x 15) + (0.3 x 25)] = 14



Standard deviation = [0.2 (-5 – 14)2 + 0.5 (15 -14)2 + 0.3 (25-14)2] ½

= [72.2 + 0.5 + 36.3] ½ = 10.44

For market portfolio:

Expected return = [0.2 x (-) 10] + (0.5 x 16) + (0.3 x 30) = -2 + 8 + 9 = 15

Standard deviation = [0.2 (-10-15)2 + 0.5(16-15)2 + 0.3 (30 – 15)2] ½

= (125 + 0.5 + 67.5) ½ = 13.89

Solution:

b. calculation of covariance b/w the returns on A and B? Returns on A and C?

State of the

Economy

Prob-

ability (p)

Return on

A (%) (R A)

Return

B (%) (R B)

R A-E(R A) R B-E(R B) P

x [R A-E(R A)]

x[R B-E(R B)]

Recession 0.2 -15 30 -34 26 -176.8

Normal 0.5 20 5 1.0 1 .5

Boom 0.3 40 -15 21 -19 -119.7

total = -296

Covariance between the returns of A and B is (-) 296

State of the

Economy

Prob- Return on A

(%) (R A)

Return C

(%) (R C)

R A-E(R A) R C-E(R C) P



ability (p) x [R A-E(R A)]

x [R C-E(R C)]

Recession 0.2 -15 - 5.0 -34 -19 129.2

Normal 0.5 20 15.0 1.0 1 .5

Boom 0.3 40 25.0 21 11 69.3

total = 199

Covariance between the returns of A and C is 199

c. Coefficient of correlation between the returns of A and B = -296/ 19.07x 15.62 = (-) 1

Coefficient of correlation between the returns of A and C = 199/19.07x10.44 = 1

d. Portfolio in which stocks A and B are equally weighted:

Economic condition Probability Overall expected return

Recession 0.2 0.5 x (-) 15 + 0.5 x 30 = 7.5

Normal 0.5 0.5 x 20 + 0.5 x 5 = 12.5

Boom 0.3 0.5 x 40 + 0.5 x (-)15 = 12.5

Expected return of the portfolio = (0.2 x 7.5) + (0.5 x 12.5) + (0.3 x 12.5)

= 1.5 + 6.25 + 3.75 = 11.5



Standard deviation of the portfolio

= [ 0.2 (7.5 – 11.5)2 + 0.5 (12.5 – 11.5)2 + 0.3 (12.5 – 11.5)2]1/2

= [ 3.2 + 0.5 + 0.3] ½ = 2

Portfolio in which weights assigned to stocks A, B and C are 0.4, 0.4 and 0.2 respectively.

Expected return of the portfolio = (0.4 x 19.0) + (0.4 x 4) + 0.2 x 14)

= 7.6 + 1.6 + 2.8 = 12

For calculating the standard deviation of the portfolio we also need covariance between B

and C, which is calculated as under:

State of the

Economy

Prob-

ability (p)

Return on

B (%) (R B)

Return on

C (%)

(R C)

R B-E(R B) R C-

E(R C)

p

x[R B-E(R B)]

x[R C-E(R C)]

Recession 0.2 30 - 5.0 26 -19 (-) 98.8

Normal 0.5 5 15.0 1 1 0.50

Boom 0.3 (-)15 25.0 (-)19 11 (-) 62.7

total = (-)161

Covariance between the returns of B and C is (-)161

We have the following values:

WA = 0.4 WB = 0.4 WC = 0.2

σA = 19.07 σB = 15.62 σC = 10.44



σAB = (-)296 σAC = 199 σBC = (-) 161

Standard deviation

= [ (0.4 x 19.07)2 + (0.4 x 15.62)2 + (0.2 x 10.44)2 + [ 2 x 0.4 x 0.4 x (-) 296 ] +

+ [2 x 0.4 x 0.2 x 199] + [2 x 0.4 x 0.2 x (-) 161]1/2

= (58.18 + 39.03 + 4.35 – 94.72 + 31.84 – 25.76)1/2 = 3.59

e.

(i) Risk-free rate is 6% and market risk premium is 15 – 6 = 9%

The SML relationship is

Required return = 6% + β x 9%

(ii) For stock A:

Required return = 6 % + 1.2 x 9 % = 16.8 %; Expected return = 19 %

Alpha = 19 – 16.8 = 2.2%

For stock B:

Required return = 6 % - 0.7 x 9 % = -0.3%; Expected return = 4 %

Alpha = 4 –(- 0.3) = 4.3 %

For stock C:

Required return = 6% + 0.90 x 9 % = 14.1 %; Expected return = 14%

Alpha = 14 – 14.1 = (-) 0.1 %



f. Calculation of Beta for stock D?

Period R D (%) R M (%) R D-R D R M-R M (R M-R M )2 (R D-R D) (R M-R M)

1 -12 -5 -18.4 -11.2 125.44 206.08

2 6 4 -0.4 -2.2 4.84 0.88

3 12 8 5.6 1.8 3.24 10.08

4 20 15 13.6 8.8 77.44 119.68

5 6 9 -0.4 2.8 7.84 - 1.12

∑R D = 32 ∑ R M = 31 ∑(R M-R M)2 = 218.80 ∑ (R D-R D) (R M-R M) = 335.6

_ _

R D = 6.4 R M = 6.2 σ2m = 218.8/4 = 54.7 Cov (D,M) = 335.6/4 = 83.9

ß = 83.9 / 54.7 = 1.53

Interpretation: The change in return of D is expected to be 1.53 times the expected change in

return on the market portfolio.

g. What is the Capital Market Line (CML)? Security Market Line (SML)? How is CML

related to SML?

Capital Market Line (CML: 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 gives the risk/return relationship for efficient portfolios.



Security Market Line (SML): The Security Market Line (SML), is also a part of the CAPM, it

gives the risk/return relationship for individual stocks.

h. What is system risk? Unsystematic risk? Present the formula for them

• Systematic risk stems from economy wide factors also known as non‐diversifiable/market risk.

Factors like tax policy, interest rate, GDP growth, inflation etc

• Unsystematic risk stems from firm specific factors also known as diversifiable/unique risk.

Factors like strike, lockout, fire, raw material supply/price, competition, management decision etc.

Systematic Risk (SR) =Beta Square * Market Variance

Un-Systematic Risk (USR) =Security Variance – Systematic Risk

i. What is the basic difference b/w the CAPM and APT?

CAPM assumes that return on a stock/portfolio is solely influenced by the market factor

whereas the APT assumes that the return is influenced by a set of factors called risk factors.

• 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.

Sapm CHAPTER 8 Solution

Documents