risk & return

Chapter 12

RISK & RETURN:PORTFOLIOAPPROACH

Alex Tajirian

Risk & Return: Portfolio Approach 12-2

© morevalue.com, 1997

Intuitively develop a model (theory) that tells us "whatshould happen to an asset's required return (price) if‘risk’ changes.”

]

What is the risk premium (RP) required (by an averageinvestor) to hold the asset?

1. OBJECTIVE

! What type of risk do investors care about? Is it "volatility"?...

! What is the risk premium on any asset, assuming that investors arewell diversified?

! As a byproduct: Why should investors diversify?

2. OUTLINE

# Statistical Background

# Portfolio "Risk" Diversification: Why not put all your eggs in onebasket?

# Optimal risk-reward tradeoff: a market-based1 approach

Alex Tajirian



Objective from a financial manager's perspective:

! Company Valuation: Is Company over/under-valued?

! What return do shareholders require for new projects? (Ch 14)

! How risky is a division, project, or the company? (Ch 14)

Objective from an investor's view:

! Which stock(s) is under/over-valued, i.e. mis-priced?

! Why do some portfolios make sense while others do not?

! Why ?putting all your eggs in one basket” does not make sense.

! How "risky" is your portfolio?

! How much return should an investor require form a givenportfolio?

Alex Tajirian



RISK & RETURN

Varianceof individual Asset

Stand-AloneRisk

AggregatePortfolio Aggroach

FinancialAssets

Projects,Divisions

Beta Risk

Assets withinA Portfolio

Porfolio Risk

Statistical Backgound

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3.0 STATISTICALBACKGROUND

3.1 RANDOM VARIABLEExamples: temperature, stock prices

ReturnReturn

time time6% 6%

Stock A Stock B

Stocks A & B have same center (average) of 6%

B is more VOLATILE than A

3.2 PROBABILITY DISTRIBUTION

probability = likelihood =frequency of occurrence

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expected return ' p1k1 % p2k2 % ... % pNkN

' EN

i'1pi × ki

2.1 SUMMARY MEASURES: ONE VARIABLE

(Center & Volatility)

Motivation: Need to summarize the data into few indicators

1.1 Measure of center of data: expected value

Case 1a: probability of outcome is known.

pi = probability of outcome i occurring

ki = value of outcome i

N = number of observations

Remember. We are interested in average return not averageprice, since price level is not very informative.

Alex Tajirian



each Pi '1N

average ' k 'k % k2 % ... % kN

N

'sum of actual rates of return

number of observation

Case 2a: past observations are available (sample)

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k ' ? ; Probability Is Given

Example 1: Calculating Average Returns

Given:

Data state of economy pi ki

i = 1 + 1% change in GNP .25 -5%

i = 2 +2% change in GNP .50 15%

i = 3 +3% change in GNP .25 35%

Solution:

Observations (pi x ki)

i = 1 -1.25%

i = 2 7.50%

i = 3 8.75%

ˆ Expected return = (-1.25 + 7.5 + 8.75 ) = 15%

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kZZZ ' ? ; Only Past Observations Given

k ZZZ '10 % (&5) % 10

3'

153

' 5%

Example 2: Calculating Average Return

Given:

Year kZZZ, t

1985 10%2

1986 -5%

1987 10%

Solution:

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average payoff ' x ' p1x1 % p2x2 '12

(&1) %12

(1) '

1.2 Measure of volatility: variance

L Motivations:Simple example: You toss a coin, you win $1 if head, or

lose $1 if tail.

What about the dispersion (deviations)? !

Sum of Deviations = (-1 + 0 ) + (1 - 0) = 0

We obviously have dispersion: -1 and 1.

Thus, one solution is to square deviations before you take sum.

Alex Tajirian



FF2 ' variance ' p1(k1 & k)2 % p2(k2 & k)2 % ... %pN(kN & k)2

' jN

i'1

[ pi × (ki & k)2 ]

' jN

i'1

[pi × (i th deviation from average)2]

' sum of weighted squared deviations from average

F ' standard deviation ' F2

Case 1b: Probability of outcomes known

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Example 3: Calculating Variance

(Data used in Example 1)

observation ki & k (ki & k)2 pi × (ki & k)2

i=1 (-.05 - .15) .04 .25 x.04 = .01

i=2 (.15 - .15) 0 .5 x 0 = 0

i=3 (.35 - .15) .04 .25 x .04 = .01

ˆ Variance = .01 + 0 + .01 = .02 = 2%

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FF2 '

jN

t'1

(kt & k)2

N & 1

'(k1 & k)2 % (k2 & k)2 % ... % (kN & k)2

N & 1

'sum of squared deviations from mean

N & 1' Average of squared deviations from the mean

FF ' standard deviation ' F2

Case 2b: sample is available

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FF2ZZZ '

(10% & 5%)2 % (&5% & 5%)2 % (10% & 5%)2

3 & 1

'.0025 % .01 % .0025

2'

.0152

' .0075

FZZZ ' .0075 ' 8.68%

Example 4: Calculate Variance of A Stock

Using data From Example 2,

Note. To make any intuitive sense out of the variance number(.0075) is to compare it to another stock, say that of Xerox =.01. Here, you can say that Xerox stock is more volatilethan ZZZ.

Alex Tajirian



Historical Returns & Standard Deviations

Series Average AnnualReturn

StandardDeviation

common stocks 12.1% 20.9%

small stocks 17.8 35.6

long-term corporate bonds 5.3 8.4

long-term government bonds 4.7 8.5

U.S. T-bills 3.6 3.3

Inflation 3.2 4.8

Source. R.G. Ibbotson and R.A. Sinquefield, Stocks, Bonds, Bills and Inflation.

?3 How much is the historical risk premium on stocks?

Puzzle 1: (Size Effect) Even when "risk" is taken into account, smallfirms historically have achieved higher returns!

Puzzle 2: (January Effect) Return in January have historically beenhigher than any other month!

Alex Tajirian



k p ' average return on a portfolio

' w1k1 % w2k2% .....%wnkN

' jN

i'1

( wi × k i )

(9)

wi ' weight of asset i in the portfolio' proportion of total invested in stock i

'amount invested in asset itotal value of investment

2.2 SUMMARY MEASURES: SEVERAL VARIABLES

A portfolio of stocks

2.1 Central tendency (mean/average/expected value)

where,"i" represents assets (not observations), and p

represents ?portfolio," which is also an asset.

N = number of assets in the portfolio

Alex Tajirian



average returns (k) are 15% and 20% respectively

k p '12

× k IBM %12

× kRCA '12

× 15% %12

× 20%

' 7.5% % 10% ' 17.5%

Example: Calculating Portfolio Returns

You have $100 to invest. Choose 50% in IBM, 50% in RCA.

Solution:

The weights are (½) each. Therefore,

Obviously, if you invest 75% in IBM, then the weights will be(3/4) and (1/4) respectively.

Alex Tajirian



cov(x,y) '(x1 & x)(y1 & y) % (x2 & x)(y2 & y) % ...% (xN & x)(yN & y)

N & 1

cov(x,y) '(1%&2%)(6%&4%)% (3%&2%)(2%&4%)% (2%&2%)(4%&4%)

3&1

'& .0002& .0002%0

2' & .0002 ' & .02%

2.2 Measures of Co-Movement (no causality)

(a) Absolute measure : Co-variance of x and y / cov(x,y)

Example: Calculating Covariance

X(%) Y(%)

i = 1 1 6

i = 2 3 2

i = 3 2 4

Average 4

Solution:

Alex Tajirian



Thus,

If cov(x,y) is < 0, then x and y move in opposite direction

If cov(x,y) is > 0, then x and y move in same direction

If cov(x,y) is = 0, then x and y have no systematic co-movement

;; I. 1-16, II. 1 ((

Alex Tajirian



3. Modern Portfolio Theory (MPT)

3.1 OUTLINE:Step 1: Diversification based on:

# stocks with negative co-movement (correlation)# stock within a large portfolio

Step 2: Develop a new measure of risk -- $: sensitivity of an

asset to movements in “the market". This measures anasset’s risk relative to a benchmark or the “market.”

Step 3: Develop a market-based risk/return tradeoff model

Step 4: How to measure “the market" in practice

Step 5: How to obtain estimates of $ in practice

Step 6: International diversification

Alex Tajirian



DIVERSIFICATION

Return

6 %

Stock AA

Return

2 %

Stock AA Return

2 %

Stock BB

Return

6 %

Stock CC

Return

2 %

Portfolio AA + BB

Return

6 %

Portfolio AA + CC

Alex Tajirian



Portfolio Size & RiskNaive Diversification

Portfolio StandardDeviation

MarketPortfolio

Diversifiablerisk

SystematicRisk

Total Risk 40 Number of Stocks inthe Portfolio

3.2 PORTFOLIO SIZE AND RISK

Alex Tajirian



Alex Tajirian



F2 ' variance ' Total Risk' Stand&Alone Risk

' Systematic Risk % Diversifiable Risk

In a large portfolio, unsystematic risk is essentially eliminated bydiversification. But in practice, total risk cannot be completelyeliminated by increasing the number of stock in a portfolio.

Thus, the only relevant risk for investors who hold a well diversifiedportfolio is systematic risk, not total variance.

3.3 DECOMPOSING TOTAL RISK

From "portfolio size and risk" relationship,

L systematic risk / non-diversifiable risk / market risk

L diversifiable risk / idiosyncratic risk / unsystematic risk

Therefore,

Alex Tajirian



Examples of factors contributing to risk:

definition: Factors are sources of risk, which are outside the controlof management:

!! systematic factors: GNP, inflation, interest rates, oil shocks

!! diversifiable factors: law suits, labor strikes, management luck

Alex Tajirian



3.4 SYSTEMATIC COMPONENT OF ACTUAL RETURNS

In the previous section, we saw that the only relevant risk (in the contextof a portfolio) is market risk. Thus, it would make sense to measure therisk of a stock in terms of how it moves with the market, i.e., in terms ofhow sensitive is a stock’s actual return (ki, t ) to changes in the actualreturn of the market (km, t).

Implication:A company's systematic component of actual return would dependonly on: (a) the company’s exposure to the market, denoted by $,and (b) the actual return on the market.

Thus, to get a better feel for this relationship, you can:! plot the historical data

t-period ki, t km, t

6/82 5% 5%

7/82 9% 10%

... ... ...

12/94 7% 3%

! The slope of the "best fit line" through the data gives you anestimate of $$.

! Thus, $ is a measure of relative risk.

Therefore, Graphically we will have:

Alex Tajirian


Alex Tajirian



4.1 $$ AS A MEASURE OF RISK:

$$ value Implication

$i =1 Y if market _ by 100%, kit _ 100% on average

$i =1.5 Y if market _ by 100%, kit _ 150% on average

$i =.5 Y if market _ by 100%, kit _ 50% on average

$i =-.5 Y if market _ by 100%, kit ` 50% on average

! Graphical Representation

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BETA AS A MEASUREOF RISK

Actual returnon stock (%)

Stock Ahigh beta

Stock Cnegative betaLow beta

Stock B

KM1510

The higher the BETA, the higher the RISK

For same level of increase in market return(15-10)

* Stock A increase by 100% (16-8)

* Stock B increase by less

* Stock C decreases

Alex Tajirian



?4 What are possible theoretical and actual value of beta?

?5What industry stocks tend to have high/low $?

! Factors influencing $ and their direction:

Amount of debt, Earnings Variability, . . .

+ +

Alex Tajirian



Sample of Betas & Their Standard Deviations

Company Beta† St. Deviation AT&T .76 24.2%Bristol Myers Squibb .81 19.8Capital Holding 1.11 26.4Digital Equipment 1.30 38.4Exxon .67 19.8Ford Motor Co. 1.30 28.7Genentech 1.40 51.8McDonald's 1.02 21.7McGraw-Hill 1.32 29.3Tandem Computer 1.69 50.7

† based on 1984-89.

? Which is riskier: Genentech or Tandem?

;; I.6-13, II.3, 4 ((

Alex Tajirian



required return ' ks ' kRF % Risk Premium

4. CAPITAL ASSET PRICING MODEL (CAPM)

4.1 RISK/RETURN TRADEOFF

What is the risk premium, RP, for asset i? ] Required Return onasset "s" = ?

4.2 MOTIVATION

Alternatively,

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Investors get rewarded only for non-diversifiable risk. Obviously theywould not require a risk premium for a "bad" that they can themselveseliminate through diversification

ks ' kRF % RPs

' kRF % (kM & kRF ) × $s

The only reason two assets would have a different required return is adifference in their $.

4.3 RESULT: Capital Asset Pricing ModelCAPM [ pronounced "CAP-M"]

Specifically,

where,

ks = required return on asset s

km = required return on the market

kRF = risk-free rate = return on a T-bill

! Compensation (required return) depends only on an asset'sexposure to the market: $. F2 of stock, F2 of residuals,industry, size of firm, and inflation are not part of theequation; they are irrelevant in determining thecompensation.

Alex Tajirian



! linear (proportional) relationship between risk and return:investors require ( kM - kRF) % compensation for each unit of$-risk. ] RPs = { (kM -kRF) $s } is proportional to stock's $.

Illustration:Suppose ( kM - kRF) = 8.5%.If $s _ from 1 to 2 Y RPs increases 2 times.

! RPM is independent of the security.

Alex Tajirian



Example: Calculating Required Return

Given: kRF = 5%, kM = 10%, bxyz = 2

kxyz = ?

Solution:

kxyz = 5% + (10%-5%)(2) = 15%

?6 What is risk premium of market (RPM) in this example?

?7 What is risk premium of XYZ Inc.?

Alex Tajirian



$$p '' w1$$1 %% w2$$2%% ...%%wn$$n

' weighted average of betas in the portfolio

4.4 PORTFOLIO RISK IMPLICATIONS

where,

wi = weight of each asset i in the portfolio

= proportion of total assets invested asset i.

Alex Tajirian



$p ' (.25)(1) % (.5)(1.5) % (.25)(.5) ' 1.125

kp ' kRF % (kM & kRF)$p

' 3% % (10% & 3%)(1.125) ' 10.875%

Example: Calculating $p and kp

Given: kRF = 3%, kM = 10%, and

Asset $$i wi

X 1 25%

Y 1.5 50%

Z .5 25%

Solution:

Alex Tajirian



4.5 WHAT IS THE OPTIMAL $$p FOR AN INVESTOR?

# SML / Security Market Line

/ Relationship between $$ of any asset and ks

# SML provides the "correct" tradeoff between risk & return ] Thetradeoff that an average investor should get Y All positions on theSML are equally "good".

YY The portfolio that an individual should choose, out of all the"good" ones on the SML, depends only on the individual'sappetite for risk.

# Applications:

! Corporate finance: determining k project, kdivision, kcompany

! Investments: Given the SML, an investor then determineswhich of these “correct” combinations ofrisk-return she wants to accept based on herindividual appetite for risk.

Alex Tajirian



ks ' kRF % (kM & kRF)$s

y ' a % (slope)x

# Graphical representation:

The CAPM can be written as an equation of a straight line, namely

where,

a = y-intercept

Alex Tajirian



SECURITY MARKET LINE (SML)

βSML: Ki = KRF + ( kM - KRF) * i

compensation for systematic risk

compensation for “time value of money”

SML

beta Risk

required return

1

kM

kRF



UNDER/OVERREWARDED STOCKS

rate of return

11 9

6

4

KRF

A B

SML

beta1.8.8

Stock A is OVER-REWARDED, since actual return(6%) > required return (4%), for a level of .8 betarisk.

Stock B is UNDER-REWARDED, since actualreturn (9%) < required return (11%), for a level of1.8 beta risk.



? What is the slope of the above SML?

? If km = 8%, what is kRF?

Dynamic Mechanism: Consider stock "A"

Step 1: Suppose that stock "A" has had an average return of 6%,which is > required return.

Step 2: Suppose now people discover this stock; it looks like a greatbuy. However, if people start buying it, then its price _.

Step 3: If price _, then its actual return` until it becomes equal torequired return.

ˆ Historical average return will end up = required return. otherwiseEMH would not hold.



? What can you say about a financial market where you observea number of securities like A Inc. and B Inc.?

? Suppose "A" and "B" represent projects. What can you say aboutthem?

? So what might happen to the industry that "A" belongs to, i.e. whatwill A's competitors do?



? If FIBM _ Y IBM

Simple Application8 1

Given two mutual funds GoGo and SoSo, with respective averagehistorical returns of 20% and 15%. Which is a better mutual fundto hold?

Simple Application 2

Solution:

F2 = market risk + idiosyncratic risk

Thus,

MarketRisk ($)

IdiosyncraticRisk

Total Risk (F2)

RequiredReturn (ks)

_ _ __

_ _

_ _ _ __



Application 3

? Only relevant risk is ß?

? For investor or manager?

? What kind of investor are we assuming?

Application 4

? How are the values of kRF and km determined?

? Current market values?

? Historical?

? Other?



ki,t ' ai % $i × km,t % ei,t

5. HOW TO ESTIMATE BETA?

# Alternative 1: Pure play as discussed earlier. This is what Iam emphasizing in this course.

# Alternative 2: Run the following regression, only if you have thestatistical background.

Where,

ki,t = actual return on stock i at time t

km,t = actual return on the "market" portfolio at time t

ei,t = error in return specific to stock i

$i = slope of ?best fit” line

! S&P500 is usually used for the market

! Regression analysis is used to estimate beta (b); the best line

that fits the data (observation of returns over time)

! Beta measures co-movement of stock i with the "market." ]

Beta measures the sensitivity of an asset to the market



$i 'cov(ki , kM)

F2M

## Alternative 3: use following formula9:

Note. ?i” stands for any asset. This includes individual stocks,also portfolios (p), division, . . .

# Alternative 4: Obtain from $ service (Merrill Lynch, BARRA,...)



6. INTERNATIONAL DIVERSIFICATION

Motivation: Easiest way to see it is to look at each country's stocksas a portfolio. Thus, you are combining portfolios thatdo not necessarily have high positive correlation. Therefore, the concept of diversification is stillapplicable.



7. SUMMARY

T vocabulary

CAPM, SML, $, diversifiable\idiosyncratic\non-systematicrisk, market\non-diversifiable\systematic risk, variance,covariance, volatility, "best fit line"

T Stock variance is not a good measure of equity risk since most ofstock variance (80%) is firm specific (diversifiable).

T Theoretically, according to the CAPM, the only source of equityrisk is Beta. Thus, company size, idiosyncratic risk, stockvolatility, and industry are irrelevant. The only risk investors careabout is if it contributes to portfolio risk.

T To obtain estimates of beta,

! Pure play method; free-hand drawing of "best fit line".

! Use beta-services: Merrill Lynch, BARRA

! Run regression yourself using standard software

! Or use following formula



$i 'cov(ki , kM)

F2M

T Dynamic Mechanism:

Stocks; projects

T The SML provides the correct risk-return tradeoff.



' 8. IN FUTURE CLASSES '

O Tests of CAPM

O Alternatives to CAPM

O Limitations of CAPM

O More on portfolio selection and diversification

O More complicated ways to estimate beta

## Anomalies! Size effect: Why do small companies have had higher

returns?

! January effect: Why are the historical returns in Januaryhigher than any other month?

! day of the week effect



1. This is what financial markets determine as a tradeoff between how much risk an investor hasto accept for a specific level of desired average return. Moreover, if these markets are fair, sowould the tradeoff be. There would not exist securities that are under- or over-rewarded for theirinherent “risk.” Note, that an investor might have her own view as to what the correct tradeoff is.The issues of market fairness and its implications on investment decision fall under the rubric of“Efficient Market Hypothesis (EMH).”

Thus, countries without any developed financial markets would have no clue as to what thetradeoff might be.

kZZZ,1985'PZZZ,Dec.)85&PZZZ,Jan.2,)85%DividendZZZ,)85

PZZZ,Jan.2,)85

2.

3. Actual annual risk premium for an average stock is by definition = (actual average annual returnon common stocks) - (actual average annual return on U.S. T-bills) = 12.1% - 3.6% = 8.5%

4. Theoretically, they can be any number between (- infinity) and (+ infinity), as they represent theslope of a line. However, in the U.S., they tend to be between .1 and 3.

5. Utility companies tend to have low betas, i.e., when the market is doing very well, people tendto increase their consumption of electricity only modestly. Thus, company returns would beincrease significantly. Conversely, if the market is not doing very well, consumers would cut theirelectricity consumption by only a small out. Thus, the performance, or return, of these companieswould not suffer much.

In a similar argument, entertainment stock tend to have high betas.

Make sure that you distinguish between a stock’s volatility and its performance relative to theMarket such as the S&P 500.

6.

RPm ' (km & kRF) ' 10 & 5 ' 5%

9. ENDNOTES



7.

RPxyz ' (km & kRF)×$xyz ' (10&5) × 2 ' 10%

8. Cannot tell, since we do not know the betas. Moreover, these funds could be overvalued giventheir betas.

9. Formula comes from OLS regression of ki on kM.



10. QUESTIONS

I. Agree/Disagree- Explain

1. If stocks Chombi Inc. and Xygot Inc. have the same required return, or market expectedreturn, a rational investor should choose the one that has highest variance as it offers higherchance of attaining high returns.

2. A good measure of volatility (dispersion) is: sum of deviations from the mean.

3. Variance of a stock is a good measure of risk to investors.

4. If the historical returns on mutual funds Saddam Inc., Whoopi Inc., and the market are20%, 10%, and 15% respectively, then Saddam Inc. is the better buy.

5. If Kumquat Inc.'s variance increases, then its required return must increase.

6. Firm managers only care about beta risk, as the rest is diversifiable.

7.H, I No one will invest in an asset that has a negative beta.

8.H, I If you (personally) believe that the stock market will rally, then you would buy the stockwith the highest beta.

9. CAPM is used in determining an appropriate rate of return for regulated utility companies.[ Note, this is not discussed in the notes or the book. I put it here just to indicate anotherpossible application of CAPM]

10. If variance of a stock increases, its beta must increase too.

11. If the beta of a stock increases, its variance must increase too.

12. The higher the beta of a stock, the riskier the returns.

13. The higher the proportion of debt to total assets, the higher the firm's beta.

14. The higher the earnings variability, the higher the beta of a firm.

15. Investors prefer to have low beta portfolios.



16. Beta is a measure of variance.



II. Numerical1. Given the following information:

Observation Return on Potato Inc. Return on S&P 500

February 1991 -9% 10%March 1991 1 -2April 1991 -1 4

(a) Calculate the variance and standard deviation of Potato Inc.(b) Calculate the beta of Potato Inc.(c) Interpret your result in (b).

2.H Given the variances of stocks X and Y are 15% and 20% respectively, with theircovariance equal to 20. (a) You are investing $100,000 of which 25% is in X. What is the variance of this

portfolio?(b) Since the variance of X < variance of Y, a rational investor would increase the

proportion invested in X so as to reduce the variance of the portfolio. Agree ordisagree? Explain.

(c) If you substitute Y by stock Z in your portfolio, which has a variance of 20% and isnegatively correlated with X, what happens to your answer in (a)?

(d) Can the stock Z be positively correlated with Y?

3.H Given the following rate of return (%) information on companies X and Y:

i = 1 i = 2 i = 3

X 1 3 2

Y 6 2 4

(a) Calculate, FX, FY, cov(X,Y), rXY.(b) Is it possible to obtain a portfolio of X and Y that has a zero variance?



4. Given: A portfolio of three securities A, B, & C, with:

Security Amount invested Average k betaA $5,000 9% .8B 5,000 10% 1.0C 10,000 11% 1.2

(a) What are the portfolio weights?(b) What is the average return on the portfolio?(c) What is the portfolio's beta?(d) If kRF = 3%, km = 12%, what is the required return on the portfolio? Is this portfolio

under or over-rewarded? Explain.

5. Given: kT-Bills = 9% , ßA = .7, kA = 13.5%, and kM =15%.(a) What is k of a portfolio with equal investments in A and T-Bills ?

(b) If ßp = .5, what are the portfolio weights?(c) If kp = 10%, what is its ß ?(d) if ßp = 1.5, what are the portfolio weights?

6. You have a portfolio of equally valued investments in two companies A & B. The beta ofthis portfolio is 1.2. Suppose you sell one of the companies, which has a beta of .4, andinvest the proceeds in a new stock with a beta of 1.4.What is the beta of your new portfolio?



13. ANSWERS TO QUESTIONS I. Agree/Disagree Explain

1. Disagree. Other things equal, you choose the one with smallest variance. Variance is"bad". Thus, you do not want to accept it if it offers you the same return (compensation),as a less risky asset.

2. Disagree. Calculation results in 0 variation, as in a coin-toss example shown below.

Sum of Deviations = (-1 - 0) + (1 - 0) = 0

3. Disagree. Most of the variance is diversifiable.

4. Disagree. We cannot tell. It depends on the betas of the mutual funds, which are notprovided in the question. It also depends on the risk preferences of the investor. Also see p.38 where stock B has higher returns but also is under-rewarded.

5. Disagree. Only if the increase in variance is due to an increase in the stock's beta. SeeSimple Application 2 p. 44 .

6. Disagree. They care about total risk (variance of returns), since their life depends on howwell the company does.

7. Disagree. Such an asset can be great when times are bad.

8. Disagree. Remember that most of a company's variance is diversifiable. Thus, you need tobuy a portfolio of stocks with high betas to diversify some of the firm-specific risk.

9. Agree. The CAPM is used by regulatory agencies to figure out what a fair return shouldbe for the utilities. This is one way to decide on how much you pay for their services.

10. Disagree. Theory tells us what happens to variance if beta changes and not the other wayaround.



11. Agree. Remember there are two sources of variance risk: market (beta) and firm-specific. So if beta increases, then variance must increase too, other things equal.

12. Agree. Higher beta means that stock prices go up and down, in relation to the market, bylarger proportions.

13. Agree. One of the factors that affects beta is the D/A ratio. Moreover, they are directlyrelated. The higher debt makes the firm more sensitive to interest rate, which is asystematic factor.

14. Agree. Earnings variability and beta are directly proportional. High earnings variability suggests that the firm's earnings move with the market. Good times bring in high earnings,while bad times have an adverse effect on them.

15. Disagree. It depends on how risk averse the individual is. Remember the higher the beta,the higher the required return.

16. Disagree. Beta measures an asset's return (price) fluctuations with respect to a benchmarksuch as the S&P500.



kPotatoe '(&9) % 1 % (&1)

3' &3%

kSP500 '10 % (&2) % 4

3' 4%

II. NUMERICAL1.

k '&9% (&1)%1

3' &

93

' &3

F2 '(&9%& (&3%))2 % (1%& (&3%))2 % (&1%& (&3%))2

3&1

'(&6%)2 % (4%)2 % (2%)2

2'

.00562

' .0028

F ' F2 ' .0028 ' 5.29%

b) STEP 1. Calculate average returns

STEP 2 Calculate beta of Potato

Thus,



$potatoe 'cov(kpotato,kS&P500)

variance(kS&P500)

'[(&9& (&3))(10&4) % (1& (&3))(&2&4) % (&1& (&3))(4&4)] / N&1

[(10&4)2 % (&2&4)2 % (4&4)2] / N&1

'&6072

' & .83

C) since beta is -.83, if the market (S&P500) _ 100%, then Potato Inc.tends to ` by 83%.



F2p ' w 2

XF2X % w 2

YF2Y % 2wXwYcov(X,Y)

' (.25)2(.15) % (.75)2(.2) % 2(.25)(.75)(20)

' .0094 % .1125 % 7.5 ' 7.62

2.(a) Note that $100,000 is irrelevant (extraneous information).

(b) By _ wX you would ` variance of portfolio. However, you also needto look at RETURN too. Return on the portfolio could _ or `.

(c) It would ` variance of portfolio.

(d) No. Since Z is negatively correlated with X, and (X,Y) arepositively correlated, Then Z has to be negatively correlated withboth.



X '1%3%2

3' 2%

Y '6%2%4

3' 4%

3.a)

F2x '

(1%&2%)2 % (3%&2%)2 % (2%&2%)2

3&1'

.0001% .00012

' .01%

F2y '

(6%&4%)2 % (2%&4%)2 % (4%&4%)2

3&1'

.0004% .00042

' .04%

ˆ rxy '& .0002

.0001 .0004' &1

b) Yes, since these stocks are negatively correlated.



(a) wA '5,000

5,000%5,000%10,000'

5,00020,000

' .25

wB '5,00020,000

' .25

wc '10,00020,000

' .5

4. Given: Portfolio of three securities A, B, & C, with:Security Amount invested Average k betaA $5,000 9% 0.8B 5,000 10% 1.0C 10,000 11% 1.2

(a) What are the portfolio weights?(b) What is the average return on the portfolio?(c) What is the portfolio's beta?(d) If kRF = 3%, km = 12%, what is the required return on the

portfolio? Is this portfolio under or over-rewarded? Explain.

Solution:



(d) using CAPM,kp ' 3% % (9%)(1.05) ' 12.45%

Since (required return'12.45) > (average actual return'10.25)

Y under& rewarded

(b) kp ' wAkA % wBkB % wCkC

' .25(9%) % .25(10%) % .5(11%)' 10.25%

(c) $p ' wA$A % wB$B % wC$C

' (.25)(.8) % (.25)(1) % (.5)(1.2)' 1.05



(a) kp ' wAkA % wT&BillkT&Bill

' (.5)(13.2%) % (.5)(9%)

(b) $p' .5 ' wA$A % wT&Bill$T&Bill

But wA % wT&Bill ' 1 and $T&bill ' 0

Y .5 ' wA(.7) % (1&wA)(0)

Y wA '.5.7

'57

and wT&Bill ' 1& 57

'27

5. Given: kT-Bills = 9% , ßA = .7 , kA = 13.2% , and kM =15%(a) What is k of portfolio, with equal investment in A and T-Bills ? (b) If ßp = .5, what are the portfolio weights?(c) If kp = 10%, what is its ß ?(d) if ßp = 1.5, what are the portfolio weights?

Solution:



kp ' 10% ' kRF % (kM&kRF)$p

.1 ' 9% % (15%&9%)$p

.1 ' .09 % .06$p

Y $p '.1& .09

.06'

16

(d) $p ' 1.5 ' wA$A % wT&bill$T&bill

1.5 ' wA(.7) % (1&wA)$T&bill

Y wA '1.5.7

' 2.14 > 1 and wT&bill ' &1.14

c) Since we do not know the weights of the assets in the portfolio, wecannot use the "formula" in (b). We need to use CAPM.

Since wA > 1, then you are borrowing 114% of your investment at the T-bill rate and investing your capital + borrowed amount in asset A. Thus,the negative weight of T-bill reflects borrowing the asset.



Given: from equation for $ of portfolio,

.5$A % .5$B ' 1.2

suppose you sell A. Thus,

$A ' .4 Y $B '1.2& .5(.4)

.5'

1.5

' 2

ˆ $new portfolio ' .5($new) % .5($B) ' .5(1.4) % .5(2) ' 1.7

6.



rxy ''cov(x,y)FFx × FFy

such that,

&1 # rxy # 1

ELIMINATIONS(b) Relative co-movement: more intuitive than cov(x,y)

correlation between x and y = rxy

On average:if rxy = 0; then xand y have nosystematic co-movementif rxy = 1; then if one _ by 100%, the other _ by 100%if rxy = -1; if one _ by 100%, the other ` by 100%if rxy = .5; if one _ by 100%, the other _ by 50%



Example: Calculating CorrelationGiven data used above (in calculation of covariance)

F2x '

(1%&2%)2 % (3%&2%)2 % (2%&2%)2

3&1'

.0001% .00012

' .0001

F2y '

(6%&4%)2 % (2%&4%)2 % (4%&4%)2

3&1'

.0004% .00042

' .0004

ˆ rxy '& .0002

.0001 .0004' &1

L Compare rxy = -1 with cov(x,y) = -.02%. Former more intuitive.



w 2x F

2x % w 2

y F2y % 2wxwycov(x,y) ' F2

p

.1 Variance of a portfolio1 (volatility): Special case: only two stocks/assets x & y

Effect of covariance contribution on variance

variance effect covariance effect total

+ + 0 No Effect

+ + + __

+ + - `

L Thus, variance of a portfolio of assets depends on:1. # of assets included2. Weight of each asset in portfolio3. Variance of each asset4. Covariance of each pair of assets

Note. In practice, diversification works as long as there are manyassets in the portfolio which are not highly positivelycorrelated.



Example: Calculating Variance of a PortfolioGiven: Two firms X and Y, such that variances of X and Y are 10%

and 20% respectively. What is the variance of an equal-weighted portfolio if cov(x,y) is 10%, 0, and -10%?

Solution:sum of weighted variances = (.5)2(10%) + (.5)2(20%) = 7.5%covariance contribution:

sum ofweightedvariances

covariancecontribution

PortfolioVariance

Effect onVariance

7.5% 2(.5)(.5)(0%) = 0 7.5% no effect

7.5% 2(.5)(.5)(10%) = 5% 12.5% _

7.5% 2(.5)(.5)(-10%) = - 5% 2.5% `

L Thus, (the variance of the portfolio that includes assets thatare negatively related) < (sum of weighted variancecontribution).



a.i LIMITATION OF PORTFOLIO VARIANCE To use Fp formula:

! too many items to calculate N variances and {(N2 - N)/2} co-variances if N = 100, we need over 4,000 co-variances to calculate

! Does not tell us the riskiness of individual stock/asset, i.e.cannot measure risk premium (RP) of individual stock.

YY we need to make some assumptions (restrictions) about how stocksmove.



kit ' $ikMt % eit

where eit is firm specific return at time t

F2i ' systematic risk % firm specific risk

' $2F2M % firm specific risk

# More realistic description of historical stock returns

Y



i.1 More on firm specific return

! In U.S., a company's systematic risk is on averageless than 20% of its total risk. Thus, most of anindividual company's total risk is firm specific.

! Illustration

Q Three stocks each with same beta = 2, butdifferent idiosyncratic variances.

Q Stock3 has the highest variance. In the thirdperiod, it actually went down while the marketwas up.



-30.0%

-20.0%

-10.0%

0.0%

10.0%

20.0%

30.0%

market portfolio stock1 with beta =2

stock2 with beta = 2 stock3 with beta = 2

1 2 3 4 5 6



kt ' a % $kMt % $1F1t % $2F2t% ... % et

i.2 More Factors Influencing Actual Returns



a.ii ASSUMPTIONS G There exists a risk-free asset (kRF)G Investors are risk averseG Investors maximize satisfaction (utility)G All non-diversifiable factors are aggregated (incorporated) in

kM

G Investors hold portfolios and not individual stocks2.

Note. (compare to limitations of Fp)! we only need to calculate N betas (simpler than variance)! we have risk measure for each stock (beta)



7. Portfolio variance, as a measure of equity risk, has a number of shortcomings.True. See notes p. 44, 74 .

8. Financial risk is diversifiable.

9. The higher a company's product demand variability, the higher its Business Risk.

10. The higher the fixed costs, the lower the Business Risk.

Disagree. Financial risk is defined as the risk associated with a company's debt level. Thehigher the debt to asset ratio, the higher the beta. Thus, the higher the systematic risk.

True. Demand variability is one of the sources of Business risk. The higher the variabilitymeans higher uncertainty about the firm's ability to sell its product. Thus, the higher therisk.

Disagree. High fixed costs put stress on a company's CFs, as they are unavoidable cashoutflows in the short-run. Thus, the higher the Business Risk.

11. International diversification cannot decrease portfolio variance since an investor is stuckwith a country's non-diversifiable risk.

12. International diversification increases risk. Therefore it should be avoided.Disagree. International diversification can lower systematic risk as different countries donot have perfectly correlated systematic risks. Diversification of international systematicrisk works in the same way as the diversification of domestic firm-specific risk.

13. Disagree. Although there is an additional component, foreign exchange risk,diversification principles still hold.

14. If the correlation between stocks Zart and Zed is one, then if return on Zart increases by100%, that of Zed tends to increase by 1%.

15. If two variables are highly correlated, then a movement in one causes a movement in theother.

16. The variance of an asset can be less than 0.



17. Disagree. Zed tends to increase by 100%.

18. Disagree. Correlation does not imply causality. An example would be football and stockmarket correlations as in "Football and Seesaw Finance."

19. Disagree. It has to be š 0, since you are squaring and summing the deviations.

20. You cannot obtain a beta estimate for a division.

21. A stock's required return (ks) tends to change daily, just as stock prices do.

22. Actual returns (kit) and required return (ks) tend to move in the same direction.23. Disagree. You can look at a division as a separate entity. Then try to obtain a beta

estimate based on that of a similar firm(s) ( same line of business and size as your division).

24. Disagree. Required return does not change every day. If the beta of the companychanges, then it would. Remember the actual and required returns are rarely equal.

25. Disagree. See #17 above.

26. If an asset has a beta of 1, then it must have the same variance as the market.

27. If systematic risk of a stock increases, then required return increases too. Thus, you arebetter off because you would be necessarily receiving higher returns.

28. Disagree. The market portfolio has only systematic risk. A stock with beta of one, has inaddition an idiosyncratic components of risk.

F2 = variance = total risk = systematic risk + idiosyncratic riskIf stock's beta= 1, then company systematic risk = market risk = market variance. But,since company idiosyncratic risk > 0, then company variance > market variance.

29. Disagree. See Application 2, p. 44, 74. You need to distinguish between required returnand realized/actual return.

30. Low beta stocks are less volatile than high beta stocks.

31. Two stocks X&Y have the same variance but X has a higher beta. Y must have higheridiosyncratic risk.



32. If the variance of the market increased, then required return on an asset increases too.

33. Disagree. Volatility, measured in terms of variance, has two components: systematic risk +idiosyncratic risk. Low beta stocks would have low systematic risk. However, such a lowbeta stock could have a much higher idiosyncratic risk than a high beta portfolio. Thus,low beta does not imply low volatility. Also see p. ?.

34. Agree. Since total risk is the same and X has a higher beta (i.e. higher systematic risk), itmust also have a lower idiosyncratic risk than Y.

35. Disagree. Look at CAPM. There is no compensation for the variance of the market.



More Questions

Agree/Disagree-Explain

36. Financial risk is diversifiable.

Disagree. Financial risk is defined as the risk associatedwith a company's debt level. The higher the debt to assetratio, the higher the beta. Thus, the higher the systematicrisk.

37. The higher a company's product demand variability, thehigher its Business Risk.

Agree. Demand variability is one of the sources ofBusiness risk. The higher the variability means higheruncertainty about the firm's ability to sell itsproduct. Thus, the higher the risk.

38. The higher the fixed costs, the lower the Business Risk.

Disagree. High fixed costs put stress on a company's CFs,as they are unavoidable cash outflows in the short-run. Thus, the higher the Business Risk.

39. International diversification cannot decrease portfoliovariance since an investor is stuck with a country's non-diversifiable risk.

Disagree. International diversification can lowersystematic risk as different countries do not have perfectlycorrelated systematic risks. Diversification ofinternational systematic risk works in the same way as thediversification of domestic firm-specific risk.

40. If the correlation between stocks Zart and Zed is one, thenif return on Zart increases by 100%, that of Zed tends toincrease by 1%.

Disagree. Zed tends to increase by 100%.

41. If two variables are highly correlated, then a movement inone causes a movement in the other.

Disagree. Correlation does not imply causality. An example



would be football and stock market correlations

42. The variance of an asset can be less than 0.

Disagree. It has to be š 0, since you are squaring andsumming the deviations.

43. You cannot obtain a beta estimate for a division.

Disagree. You can look at a division as a separate entity.Then try to obtain a beta estimate based on that of asimilar firm(s) ( same line of business and size as yourdivision).

44. If an asset has a beta of 1, then it must have the samevariance as the market.Disagree. The market portfolio has only systematic risk. A

stock with beta of one, has in addition anidiosyncratic components of risk.

F2 = variance = total risk = systematic risk + idiosyncraticrisk

If stock's beta= 1, then company systematic risk = marketrisk = market variance. But, since company idiosyncraticrisk > 0, then company variance > market variance.

45. If systematic risk of a stock increases, then its requiredreturn increases too. Thus, you are better off because youwould necessarily be receiving higher returns.

Disagree. See Application 2, p. 44, 74. You need todistinguish between required return and realized/actualreturn.

46. Low beta stocks are less volatile than high beta stocks.Disagree. Volatility, measured in terms of variance, has two

components: systematic risk + idiosyncratic risk. Lowbeta stocks would have low systematic risk. However, such a low beta stock could have a much higheridiosyncratic risk than a high beta portfolio. Thus,low beta does not imply low volatility.

47. Two stocks X&Y have the same variance but X has a higherbeta. Y must have higher idiosyncratic risk.



constant (sensitivity of asset))i )) to market) × (return on the market period

intercept % (slope of line ) × (return on the market period t )))

% $ikMt

Agree. Since total risk is the same and X has a higher beta(i.e., higher systematic risk), it must also have alower idiosyncratic risk than Y.

48. If the variance of the market increased, then requiredreturn on an asset increases too.

Disagree. Look at CAPM. There is no compensation for thevariance of the market.

In terms of an equation, then the above "best fit line" would look like:

where,kit / actual return observations on asset over period "t"ai / y-intercept$i/ sensitivity (exposure) of asset i to the marketkMt / actual return observations on the market over period "t"M /market portfolio, typically S&P500

;; Rest ((



1. In general,

F2p ' w 2

1 F21%w 2

2 F22%2w1w2cov(k1,k2) %

w 23 F

23%2w1w3cov(k1,k3)%2w2w3cov(k2,k3)

' j w 2i F

2i%j j 2wiwjcov(ki,kj)

2. Why diversify? See LAT 4/19/93 p. E83.