Risk & Return

Chapter 12

RISK & RETURN:
PORTFOLIO
APPROACH

Alex Tajirian

Risk & Return: Portfolio Approach 12-2

1. OBJECTIVE

! What type of risk do investors care about? Is it "volatility"?...
! What is the risk premium on any asset, assuming that investors are
well diversified?

! As a byproduct: Why should investors diversify?

2. OUTLINE
# Statistical Background

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

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

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

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

© morevalue.com, 1997
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 given
portfolio?

Alex Tajirian


RISK & RETURN

Statistical Backgound

Stand-Alone Porfolio Risk
Risk

Variance Aggregate Assets within
of individual Asset Portfolio Aggroach A Portfolio

Beta Risk

Financial Projects,
Assets Divisions

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3.0 STATISTICAL
BACKGROUND
3.1 RANDOM VARIABLE
Examples: temperature, stock prices
Return Return

6% 6%
time time

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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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.

expected return ' p1k1 % p2k2 % ... % p NkN
N
' E pi × k i
i' 1

pi = probability of outcome i occurring
ki = value of outcome i
N = number of observations

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

Alex Tajirian


Case 2a: past observations are available (sample)
1
each Pi '
N

k % k2 % ... % kN
average ' k '
N
sum of actual rates of return
'
number of observation

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Example 1: Calculating Average Returns

k ' ? ; Probability Is Given

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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Example 2: Calculating Average Return

k ZZZ ' ? ; Only Past Observations Given

Given:

Year kZZZ, t
1985 10%2
1986 -5%
1987 10%

Solution:

10 % (& 5) % 10 15
k ZZZ ' ' ' 5%
3 3

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1.2 Measure of volatility: variance

L Motivations:
Simple example: You toss a coin, you win $1 if head, or
lose $1 if tail.
1 1
average payoff ' x ' p1x1 % p2x2 '
¯ (& 1) % (1) '
2 2

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.

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Case 1b: Probability of outcomes known

F2 ' variance ' p1(k1 & k)2 % p2(k2 & k)2 % ... % p N(kN & k)2
¯ ¯ ¯

' j [ p i × (k i & k)2 ]
N

i' 1

' j [p i × (i th deviation from average)2]
N

i' 1

' sum of weighted squared deviations from average

F ' standard deviation ' F2

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Example 3: Calculating Variance
(Data used in Example 1)

observation ¯
ki & k ¯
(ki & k)2 ¯
pi × (k i & 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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Case 2b: sample is available

j (k t & k)
N
2

F2 ' t' 1
N & 1
(k1 & k)2 % (k2 & k)2 % ... % (k N & k)2
'
N & 1
sum of squared deviations from mean
'
N & 1
' Average of squared deviations from the mean
F ' standard deviation ' F2

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Example 4: Calculate Variance of A Stock
Using data From Example 2,

2 (10% & 5%)2 % (& 5% & 5%)2 % (10% & 5%)2
FZZZ '
3 & 1
.0025 % .01 % .0025 .015
' ' ' .0075
2 2
FZZZ ' .0075 ' 8.68%

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 volatile
than ZZZ.

Alex Tajirian


Historical Returns & Standard Deviations

Series Average Annual Standard
Return Deviation
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, small
firms historically have achieved higher returns!

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

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2.2 SUMMARY MEASURES: SEVERAL VARIABLES
A portfolio of stocks
2.1 Central tendency (mean/average/expected value)

k p ' average return on a portfolio
' w1k1 % w2k2% .....% wnk N
(9)

' j ( wi × k i )
N

i' 1

where,
"i" represents assets (not observations), and p
represents ?portfolio," which is also an asset.

N = number of assets in the portfolio

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

Alex Tajirian


Example: Calculating Portfolio Returns
You have $100 to invest. Choose 50% in IBM, 50% in RCA.

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

Solution:
The weights are (½) each. Therefore,

1 1 1 1
kp ' × k IBM % × k RCA ' × 15% % × 20%
2 2 2 2
' 7.5% % 10% ' 17.5%

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

Alex Tajirian


2.2 Measures of Co-Movement (no causality)
(a) Absolute measure : Co-variance of x and y / cov(x,y)

(x1 & x)(y1 & y) % (x2 & x)(y2 & y) % ...% (xN & x)(y N & y)
cov(x,y) '
N & 1

Example: Calculating Covariance

X(%) Y(%)
i=1 1 6
i=2 3 2
i=3 2 4
Average 4

Solution:
(1%& 2%)(6%& 4%)% (3%& 2%)(2%& 4%)% (2%& 2%)(4%& 4%)
cov(x,y) '
3& 1
& .0002& .0002% 0
' ' & .0002 ' & .02%
2

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 an
asset’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

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DIVERSIFICATION

Return Stock AA Return Stock BB Return Portfolio AA + BB

2% 2% 2%

Portfolio AA + CC
Return Stock AA Return Stock CC Return

6% 6% 6%

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3.2 PORTFOLIO SIZE AND RISK

Portfolio Size & Risk
Naive Diversification

Portfolio Standard
Deviation

Diversifiable
risk

Market
Portfolio
Systematic
Risk

Total Risk 40 Number of Stocks in
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Alex Tajirian


Alex Tajirian


3.3 DECOMPOSING TOTAL RISK
From "portfolio size and risk" relationship,

F2 ' variance ' Total Risk
' Stand& Alone Risk
' Systematic Risk % Diversifiable Risk

L systematic risk / non-diversifiable risk / market risk
L diversifiable risk / idiosyncratic risk / unsystematic risk

Therefore,

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

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

Alex Tajirian


Examples of factors contributing to risk:
definition: Factors are sources of risk, which are outside the control
of 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 context
of a portfolio) is market risk. Thus, it would make sense to measure the
risk of a stock in terms of how it moves with the market, i.e., in terms of
how sensitive is a stock’s actual return (ki, t ) to changes in the actual
return of the market (km, t).

Implication:
A company's systematic component of actual return would depend
only 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 an
estimate 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 MEASURE
OF RISK
Actual return
on stock (%)
Stock A
high beta

Stock C
Low beta negative beta
Stock B

10 15 KM

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?

?5 What 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 19.8
Capital Holding 1.11 26.4
Digital Equipment 1.30 38.4
Exxon .67 19.8
Ford Motor Co. 1.30 28.7
Genentech 1.40 51.8
McDonald's 1.02 21.7
McGraw-Hill 1.32 29.3
Tandem Computer 1.69 50.7

† based on 1984-89.

? Which is riskier: Genentech or Tandem?

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

Alex Tajirian


4. CAPITAL ASSET PRICING MODEL (CAPM)
4.1 RISK/RETURN TRADEOFF

required return ' ks ' kRF % Risk Premium

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

4.2 MOTIVATION

Alternatively,

Alex Tajirian


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

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

Specifically,

k s ' kRF % RPs
' kRF % (kM & kRF ) × $s
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's
exposure to the market: $. F2 of stock, F2 of residuals,
industry, size of firm, and inflation are not part of the
equation; they are irrelevant in determining the
compensation.

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

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


4.4 PORTFOLIO RISK IMPLICATIONS

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

' weighted average of betas in the portfolio

where,
wi = weight of each asset i in the portfolio
= proportion of total assets invested asset i.

Alex Tajirian


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:

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

k p ' k RF % (k M & kRF)$p

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

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 ] The
tradeoff that an average investor should get Y All positions on the
SML are equally "good".

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

# Applications:
! Corporate finance: determining k project, kdivision, kcompany
! Investments: Given the SML, an investor then determines
which of these “correct” combinations of
risk-return she wants to accept based on her
individual appetite for risk.

Alex Tajirian


# Graphical representation:
The CAPM can be written as an equation of a straight line, namely

k s ' k RF % (kM & k RF)$s
y ' a % (slope)x

where,
a = y-intercept

Alex Tajirian


SECURITY MARKET LINE (SML)
required return

SML

kM
compensation for systematic risk
kRF
compensation for “time value of money”

1 beta Risk

SML: Ki = KRF + ( kM - KRF) * β i



UNDER/OVER
REWARDED STOCKS
rate of return

SML
11
9
B
A
6
4
KRF
.8 1.8 beta

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

Stock B is UNDER-REWARDED, since actual
return (9%) < required return (11%), for a level of
1.8 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 great
buy. However, if people start buying it, then its price _.

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

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



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

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

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



Simple Application8 1
Given two mutual funds GoGo and SoSo, with respective average
historical returns of 20% and 15%. Which is a better mutual fund
to hold?

Simple Application 2

? If FIBM _ Y IBM

Solution:
F2 = market risk + idiosyncratic risk
Thus,

Market Idiosyncratic Total Required
Risk ($) Risk Risk (F2) Return (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?



5. HOW TO ESTIMATE BETA?
# Alternative 1: Pure play as discussed earlier. This is what I
am emphasizing in this course.

# Alternative 2: Run the following regression, only if you have the
statistical background.
ki,t ' a i % $i × km,t % ei,t

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



# Alternative 3: use following formula9:

cov(ki , kM)
$i '
ˆ
2
FM

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 stocks
as a portfolio. Thus, you are combining portfolios that
do not necessarily have high positive correlation.
Therefore, the concept of diversification is still
applicable.



7. SUMMARY

T vocabulary
CAPM, SML, $, diversifiableidiosyncraticnon-systematic
risk, marketnon-diversifiablesystematic risk, variance,
covariance, volatility, "best fit line"

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

T Theoretically, according to the CAPM, the only source of equity
risk is Beta. Thus, company size, idiosyncratic risk, stock
volatility, and industry are irrelevant. The only risk investors care
about 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



cov(ki , kM)
$i '
ˆ
2
FM

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 January
higher than any other month?

! day of the week effect



9. ENDNOTES
1. This is what financial markets determine as a tradeoff between how much risk an investor has
to accept for a specific level of desired average return. Moreover, if these markets are fair, so
would the tradeoff be. There would not exist securities that are under- or over-rewarded for their
inherent “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 the
tradeoff might be.

2.

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

3. Actual annual risk premium for an average stock is by definition = (actual average annual return
on 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 the
slope 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 tend
to increase their consumption of electricity only modestly. Thus, company returns would be
increase significantly. Conversely, if the market is not doing very well, consumers would cut their
electricity consumption by only a small out. Thus, the performance, or return, of these companies
would 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 the
Market such as the S&P 500.

RPm ' (km & kRF) ' 10 & 5 ' 5%
6.



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

7.

8. Cannot tell, since we do not know the betas. Moreover, these funds could be overvalued given
their 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 expected
return, a rational investor should choose the one that has highest variance as it offers higher
chance 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 are
20%, 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 stock
with 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 another
possible 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. Numerical
1. Given the following information:

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

February 1991 -9% 10%
March 1991 1 -2
April 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 their
covariance 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 or
disagree? Explain.
(c) If you substitute Y by stock Z in your portfolio, which has a variance of 20% and is
negatively 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 beta
A $5,000 9% .8
B 5,000 10% 1.0
C 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 of
this portfolio is 1.2. Suppose you sell one of the companies, which has a beta of .4, and
invest 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 not
provided 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. See
Simple Application 2 p. 44 .

6. Disagree. They care about total risk (variance of returns), since their life depends on how
well 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 to
buy 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 should
be 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 way
around.



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, by
larger proportions.

13. Agree. One of the factors that affects beta is the D/A ratio. Moreover, they are directly
related. The higher debt makes the firm more sensitive to interest rate, which is a
systematic 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 benchmark
such as the S&P500.



II. NUMERICAL
1.

¯ & 9% (& 1)% 1 9
k ' ' & ' &3
3 3

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

(& 6%)2 % (4%)2 % (2%)2 .0056
' ' ' .0028
2 2

F' F2 ' .0028 ' 5.29%

b) STEP 1. Calculate average returns

¯ (& 9) % 1 % (& 1)
k Potatoe ' ' & 3%
3

¯ 10 % (& 2) % 4
k SP500 ' ' 4%
3

STEP 2 Calculate beta of Potato

Thus,



cov(kpotato,kS&P500)
$potatoe '
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
& 60
' ' & .83
72

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



2.
(a) Note that $100,000 is irrelevant (extraneous information).
2 2 2 2 2
Fp ' w X FX % w Y FY % 2w Xw Ycov(X,Y)

' (.25)2(.15) % (.75)2(.2) % 2(.25)(.75)(20)
' .0094 % .1125 % 7.5 ' 7.62

(b) By _ wX you would ` variance of portfolio. However, you also need
to 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) are
positively correlated, Then Z has to be negatively correlated with
both.



3.a)
¯ 1% 3% 2
X ' ' 2%
3

¯ 6% 2% 4
Y ' ' 4%
3

2 (1%& 2%)2 % (3%& 2%)2 % (2%& 2%)2 .0001% .0001
Fx ' ' ' .01%
3& 1 2

2 (6%& 4%)2 % (2%& 4%)2 % (4%& 4%)2 .0004% .0004
Fy ' ' ' .04%
3& 1 2
& .0002
ˆ rxy ' ' &1
.0001 .0004

b) Yes, since these stocks are negatively correlated.



4. Given: Portfolio of three securities A, B, & C, with:
Security Amount invested Average k beta
A $5,000 9% 0.8
B 5,000 10% 1.0
C 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:
5,000 5,000
(a) wA ' ' ' .25
5,000% 5,000% 10,000 20,000
5,000
wB ' ' .25
20,000
10,000
wc ' ' .5
20,000



¯ ¯ ¯ ¯
(b) k p ' wAk A % wBk B % wCk C
' .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

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

Since (required return' 12.45) > (average actual return' 10.25)
Y under& rewarded



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:
(a) k p ' w Ak A % wT& BillkT& Bill

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

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

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

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

.5 5 5 2
Y wA ' ' and wT& Bill ' 1& '
.7 7 7 7



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

k p ' 10% ' k RF % (k M& k RF)$p

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

.1 ' .09 % .06$p

.1& .09 1
Y $p ' '
.06 6

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

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

1.5
Y wA ' ' 2.14 > 1 and wT& bill ' & 1.14
.7

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.



6.

Given: from equation for $ of portfolio,
.5$A % .5$B ' 1.2

suppose you sell A. Thus,
1.2& .5(.4) 1
$A ' .4 Y $B ' ' ' 2
.5 .5
ˆ $new portfolio ' .5($new) % .5($B) ' .5(1.4) % .5(2) ' 1.7



ELIMINATIONS
(b) Relative co-movement: more intuitive than cov(x,y)
correlation between x and y = rxy
cov(x,y)
rxy '
Fx × Fy

such that,

On average: & 1 # rxy # 1
if rxy = 0; then x
and y have no
systematic co-movement
if 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 Correlation
Given data used above (in calculation of covariance)

(1%& 2%)2 % (3%& 2%)2 % (2%& 2%)2 .0001% .0001
Fx '
2
' ' .0001
3& 1 2

(6%& 4%)2 % (2%& 4%)2 % (4%& 4%)2 .0004% .0004
Fy
2
' ' ' .0004
3& 1 2
& .0002
ˆ r xy ' ' &1
.0001 .0004
L Compare rxy = -1 with cov(x,y) = -.02%. Former more intuitive.



.1 Variance of a portfolio1 (volatility):
Special case: only two stocks/assets x & y
Effect of covariance contribution on variance
2 2 2 2 2
w x Fx % w y Fy % 2w xw ycov(x,y) ' Fp

variance effect covariance effect total
+ + 0 No Effect
+ + + _

+ + - `

L Thus, variance of a portfolio of assets depends on:
1. # of assets included
2. Weight of each asset in portfolio
3. Variance of each asset
4. Covariance of each pair of assets

Note. In practice, diversification works as long as there are many
assets in the portfolio which are not highly positively
correlated.



Example: Calculating Variance of a Portfolio
Given: 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 of covariance Portfolio Effect on
weighted contribution Variance Variance
variances
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 that
are negatively related) < (sum of weighted variance
contribution).



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.

Y we need to make some assumptions (restrictions) about how stocks
move.



# More realistic description of historical stock returns

kit ' $ik Mt % e it
where e it is firm specific return at time t

Y
2
Fi ' systematic risk % firm specific risk
2
' $2FM % firm specific risk



i.1 More on firm specific return

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

! Illustration

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

Q Stock3 has the highest variance. In the third
period, it actually went down while the market
was up.



30.0%

20.0%

10.0%

0.0%

-10.0%

-20.0%

-30.0%
1 2 3 4 5 6

market portfolio stock1 with beta =2

stock2 with beta = 2 stock3 with beta = 2



i.2 More Factors Influencing Actual Returns

k t ' a % $k Mt % $1F1t % $2F2t% ... % et



a.ii ASSUMPTIONS
G There exists a risk-free asset (kRF)
G Investors are risk averse
G 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. The
higher 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 variability
means higher uncertainty about the firm's ability to sell its product. Thus, the higher the
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.

11. International diversification cannot decrease portfolio variance since an investor is stuck
with 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 do
not have perfectly correlated systematic risks. Diversification of international systematic
risk 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 by
100%, that of Zed tends to increase by 1%.

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

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 stock
market 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 company
changes, 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 are
better 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 in
addition an idiosyncratic components of risk.

F2 = variance = total risk = systematic risk + idiosyncratic risk
If 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 return
and 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 higher
idiosyncratic 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 low
beta 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), it
must 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 associated
with a company's debt level. The higher the debt to asset
ratio, the higher the beta. Thus, the higher the systematic
risk.

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

Agree. Demand variability is one of the sources of
Business risk. The higher the variability means higher
uncertainty about the firm's ability to sell its
product. 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 portfolio
variance since an investor is stuck with a country's non-
diversifiable risk.

Disagree. International diversification can lower
systematic risk as different countries do not have perfectly
correlated systematic risks. Diversification of
international systematic risk works in the same way as the
diversification of domestic firm-specific risk.

40. If the correlation between stocks Zart and Zed is one, then
if return on Zart increases by 100%, that of Zed tends to
increase by 1%.

Disagree. Zed tends to increase by 100%.

41. If two variables are highly correlated, then a movement in
one 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 and
summing 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 a
similar firm(s) ( same line of business and size as your
division).

44. If an asset has a beta of 1, then it must have the same
variance as the market.
Disagree. The market portfolio has only systematic risk. A
stock with beta of one, has in addition an
idiosyncratic components of risk.

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

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

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

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. Low
beta stocks would have low systematic risk. However,
such a low beta stock could have a much higher
idiosyncratic 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 higher
beta. Y must have higher idiosyncratic risk.



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

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

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

In terms of an equation, then the above "best fit line" would look like:
constant (sensitivity of asset)) )) to market) × (return on the market period
i
intercept % (slope of line ) × (return on the market period t )) )
% $ik Mt

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

; Rest (



1. In general,
2 2 2 2 2
Fp ' w1 F1% w2 F2% 2w1w2cov(k1,k2) %
2 2
w3 F3% 2w1w3cov(k1,k3)% 2w2w3cov(k2,k3)
' j wi Fi % j j 2wiwjcov(k i,kj)
2 2

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


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