Transcript Risk and Return

CHAPTER 4
Risk and Return: The Basics
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Basic return concepts
Basic risk concepts
Stand-alone risk
Portfolio (market) risk
Risk and return: CAPM/SML
HW CHAPTER 4
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ST1, pg 164 B&E
4-3, 4-7, 4-8, 4-9, 4-13 pg 166-167 B&E
What are investment returns?
Investment returns measure the
financial results of an investment.
Returns may be historical or
prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
What is the return on an investment that
costs $1,000 and is sold
after 1 year for $1,100?
Dollar return:
$ Received - $ Invested
$1,100
$1,000
= $100.
Percentage return:
$ Return/$ Invested
$100/$1,000
= 0.10 = 10%.
What is investment risk?
Typically, investment returns are not
known with certainty.
Investment risk pertains to the
probability of earning a return less
than that expected.
The greater the chance of a return far
below the expected return, the
greater the risk.
Selected Realized Returns,
1926 – 2001
Small-company stocks
Large-company stocks
L-T corporate bonds
L-T government bonds
U.S. Treasury bills
Average
Return
17.3%
12.7
6.1
5.7
3.9
Standard
Deviation
33.2%
20.2
8.6
9.4
3.2
Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2002
Yearbook (Chicago: Ibbotson Associates, 2002), 28.
Probability distribution
Stock X
Stock Y
-20
0
15
50
Rate of
return (%)
 Which stock is riskier? Why?
Assume the Following
Investment Alternatives
Economy
Prob.
T-Bill
Recession
0.10
8.0%
Below avg.
0.20
Average
Repo
Am F.
-22.0%
28.0%
10.0%
8.0
-2.0
14.7
-10.0
1.0
0.40
8.0
20.0
0.0
7.0
15.0
Above avg.
0.20
8.0
35.0
-10.0
45.0
29.0
Boom
0.10
8.0
50.0
-20.0
30.0
43.0
1.00
Alta
MP
-13.0%
What is unique about
the T-bill return?
The T-bill will return 8% regardless
of the state of the economy.
Is the T-bill riskless? Explain.
(nominal)
Do the returns of Alta Inds. and Repo Men
move with or counter to the economy?
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Alta Inds. moves with the economy, so it is
positively correlated with the economy. This is
the typical situation.
Repo Men moves counter to the economy. Such
negative correlation is unusual.
Calculate the expected rate of return
on each alternative.
^
r = expected rate of return.

r=
n
 rP .
i i
i=1
^
rAlta = 0.10(-22%) + 0.20(-2%)
+ 0.40(20%) + 0.20(35%)
+ 0.10(50%) = 17.4%.
^r
Alta
Market
Am. Foam
17.4%
15.0
13.8
T-bill
Repo Men
8.0
1.7
 Alta has the highest rate of return.
 Does that make it best?
What is the standard deviation
of returns for each alternative?
  Standarddeviation
  Variance  
 2


   ri  r  Pi .

i 1 
n
2
Standard Deviation: Another View

 
 ki kˆ
n
i 1
2
n 1
Why doesn’t this formula have “Probability” in it?
All states assumed equally likely.
What does the “-1” in the denominator tell us?
This is the calculation assuming a sample.
 2


    ri  r  Pi .

i 1 
n
Alta Inds:
 = ((-22 - 17.4)20.10 + (-2 - 17.4)20.20
+ (20 - 17.4)20.40 + (35 - 17.4)20.20
+ (50 - 17.4)20.10)1/2 = 20.0%.
T-bills = 0.0%.
Alta = 20.0%.
Repo = 13.4%.
Am Foam= 18.8%.
Market = 15.3%.
Prob.
T-bill
Am. F.
Alta
0
8
13.8
17.4
Rate of Return (%)
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Standard deviation measures the stand-alone
risk of an investment.
The larger the standard deviation, the higher
the probability that returns will be far below the
expected return.
Coefficient of variation is an alternative
measure of stand-alone risk.
Comments on standard deviation as a
measure of risk
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Standard deviation (σi) measures total, or standalone, risk.
The larger σi is, the lower the probability that actual
returns will be closer to expected returns.
Larger σi is associated with a wider probability
distribution of returns.
Difficult to compare standard deviations, because
expected return has not been accounted for comparing two risky propositions, without any idea of
the payoff.
Expected Return versus Risk
Security
Alta Inds.
Market
Am. Foam
T-bills
Repo Men
Expected
return
17.4%
15.0
13.8
8.0
1.7
Risk, 
20.0%
15.3
18.8
0.0
13.4
Standardized Risk
Coefficient of Variation is a measure of relative
variability.
CV 

kˆ
Shows risk per
unit of return.
(pain/gain ratio)
Should you take the investment with the lowest
coefficient of variation (small CV is generally
better)?
Coefficient of Variation:
CV = Standard deviation/expected return
CVT-BILLS
= 0.0%/8.0%
= 0.0.
CVAlta Inds
= 20.0%/17.4%
= 1.1.
CVRepo Men
= 13.4%/1.7%
= 7.9.
CVAm. Foam
= 18.8%/13.8%
= 1.4.
CVM
= 15.3%/15.0%
= 1.0.
***sigma(Portfolio) = [w1 sigma1 + w2 sigma2 ] , ie weighted average of individual
StDevs. If and only If -> ρ12 =1
CV has sigma in the numerator, hence for the reason above, taking a wt. avg. of
CV will involve taking a wt. avg . of sigma (or variance) - which is not allowed.
Expected Return versus Coefficient
of Variation
Expected
Risk:
Risk:
Security
return

CV
Alta Inds
17.4%
20.0%
1.1
Market
15.0
15.3
1.0
Am. Foam
13.8
18.8
1.4
T-bills
8.0
0.0
0.0
Repo Men
1.7
13.4
7.9
Return
Return vs. Risk (Std. Dev.):
Which investment is best?
20.0%
18.0%
16.0%
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
Alta
Mkt
Am. Foam
T-bills
0.0%
Repo
5.0%
10.0%
15.0%
Risk (Std. Dev.)
20.0%
25.0%
Portfolio Risk and Return
Assume a two-stock portfolio with
$50,000 in Alta Inds. and $50,000 in
Repo Men.
^
Calculate rp and p.
^
Portfolio Return, rp
^
rp is a weighted average:
n
^
^
rp =  wiri
i=1
^
rp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.
^
^
^
rp is between rAlta and rRepo.
Alternative Method
Estimated Return
Economy
Recession
Below avg.
Average
Above avg.
Boom
Prob.
0.10
0.20
0.40
0.20
0.10
Alta
Repo
Port.
-22.0%
-2.0
20.0
35.0
50.0
28.0%
14.7
0.0
-10.0
-20.0
3.0%
6.4
10.0
12.5
15.0
^
rp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40
+ (12.5%)0.20 + (15.0%)0.10 = 9.6%.
(More...)
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p = ((3.0 - 9.6)20.10 + (6.4 - 9.6)20.20 +
9.6)20.40 + (12.5 - 9.6)20.20 + (15.0 9.6)20.10)1/2 = 3.3%.
p is much lower than:
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(10.0 -
either stock (20% and 13.4%).
average of Alta and Repo (16.7%).
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The portfolio provides average return but much
lower risk. The key here is negative (or less than
perfect) correlation.
Var(P) = w1^2 Var1+w2 ^2 Var2+2w1w2*Cov(1,2)
And Cov(1,2) = corr(1,2)*(Var1Var2)^1/2
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If corr(1,2) = 1 then Var(P) = [w1 1 + w2 2 ]^2
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StDev(P) = [w1 1 + w2 2 ] , ie weighted average of
individual StDevs. IFF -> ρ12 =1
σ = w1 σ1 + w2 σ2 , IFF -> ρ12 = σ12/σ1σ2 = 1
If the correlation coefficient is -1 then portfolio
standard deviation is equal σ = w1 σ1 - w2 σ2 and
it is possible to achieve the zero portfolio standard
deviation by varying the proportion of assets
weights w1 and ;w2 in the portfolio. Practically
impossible since very few assets are perfectly
negatively correlated.
Correlation Coefficient
Correlation coefficients () range from …
-1 to +1
 = -1 implies
perfectly negative correlation
 = +1 implies
perfectly positive correlation
 = 0 implies
variables are not related
Do most stocks have positive, negative, or zero correlations with each other?
Positive, but not perfectly so
What is correlation of any security with riskless asset (T-bill – is it riskless?)?
Zero
Two-Stock Portfolios
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Two stocks can be combined to form a riskless
portfolio if  = -1.0.
Risk is not reduced at all if the two stocks have  =
+1.0.
In general, stocks have   0.65, so risk is lowered
but not eliminated.
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Investors typically hold many stocks.
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What happens when  = 0?
General comments about risk
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Most stocks are positively correlated with
the market (ρk,m  0.65).
σ  35% for an average stock. (what is
range in 2 of 3 years? E® = 12% from
market)
Combining stocks in a portfolio generally
lowers risk.
What would happen to the
risk of an average 1-stock
portfolio as more randomly
selected stocks were added?
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p would decrease because the added stocks
would not be perfectly correlated, but r^p would
remain relatively constant.
Prob.
Large
2
1
0
15
1  35% ; Large  20%.
Return
p (%)
Company Specific
(Diversifiable) Risk
35
Stand-Alone Risk, p
20
Market Risk
0
10
20
30
40
2,000+
# Stocks in Portfolio
Stand-alone
risk
=
Market
risk
+
Diversifiable
risk
.
Market risk is that part of a security’s
stand-alone risk that cannot be
eliminated by diversification.
Firm-specific, or diversifiable, risk is
that part of a security’s stand-alone risk
that can be eliminated by
diversification.
Two Components of Risk
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Company-specific (Diversifiable) risk
Unique to specific firms.
Results from random or uncontrollable events.
What are some examples?
Natural disasters, accidents, strikes, lawsuits, death of
CEO, etc.
Market (systematic) risk
Relates to forces affecting all investments.
What are some examples?
Inflation, recession, war, yield inversion etc.
Conclusions
As more stocks are added, each new
stock has a smaller risk-reducing
impact on the portfolio.
p falls very slowly after about 40
stocks are included. The lower limit
for p is about 20% = M .
By forming well-diversified portfolios,
investors can eliminate about half the
riskiness of owning a single stock.
Can an investor holding one stock earn
a return commensurate with its risk?
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
No
Stand-alone risk is not important to a welldiversified investor – because it vaporizes
Rational, risk-averse investors are concerned
with σp, which is based upon market risk.
There can be only one price (the market return)
for a given security.
No compensation should be earned for holding
unnecessary, diversifiable risk.
How is market risk measured for
individual securities?
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Market risk, which is relevant for stocks held in welldiversified portfolios, is defined as the contribution of
a security to the overall riskiness of the portfolio.
It is measured by a stock’s beta coefficient. For stock
i, its beta is:
bi = cov[i,m]/var[m]
= i m iM / m^2
= (iM i) / M
Betas….?
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In addition to measuring a stock’s contribution of
risk to a portfolio, beta also measures the stock’s
volatility relative to the market.
Shows how the price of a security responds to
changes in the overall stock market (not just its
variance)
Stock’s Beta is the only relevant measure of risk
from a portfolio standpoint: in a well diversified
portfolio there is no USR so beta gives the stock’s
response to a change in the market (syst. risk)
Using a Regression to Estimate Beta
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Run a regression with returns on the stock in
question plotted on the Y axis and returns on
the market portfolio plotted on the X axis.
The slope of the regression line, which
measures relative volatility, is defined as the
stock’s beta coefficient, or b.
Calculation of Beta
_
ki
20
.
15
.
10
Year
1
2
3
kM
15%
-5
12
ki
18%
-10
16
5
-5
.
0
-5
-10
5
10
15
_
20
kM
Regression line:
^
^
k = -2.59 + 1.44 k
i
M
Use the historical stock returns to
calculate the beta for PQU.
Year
1
2
3
4
5
6
7
8
9
10
Market
25.7%
8.0%
-11.0%
15.0%
32.5%
13.7%
40.0%
10.0%
-10.8%
-13.1%
PQU
40.0%
-15.0%
-15.0%
35.0%
10.0%
30.0%
42.0%
-10.0%
-25.0%
25.0%
Calculating Beta for PQU
40%
0.830833
0.396277
0.354616
4.395725
0.213482
r pqu
20%
rM
0%
-40%
-20%
0%
20%
40%
-20%
-40%
r PQU = 0.83r M + 0.03
2
R = 0.36
0.025608391
0.082199263
0.220376811
8
0.388527511
Show in
Excel,
and how
to get
return
series
on
stocks,
market
What is beta for PQU?
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The regression line, and hence beta, can be
found using a calculator with a regression
function or a spreadsheet program. In this
example, b = 0.83.
Calculating Beta in Practice
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Many analysts use the S&P 500 to find the
market return.
Analysts typically use four or five years’ of
monthly returns to establish the regression
line.
Some analysts use 52 weeks of weekly
returns.
How is beta interpreted?
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If b = 1.0, stock has average risk.
If b > 1.0, stock is riskier than average.
If b < 1.0, stock is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5.
Can a stock have a negative beta?
Beta of .5 If …
Market goes up 1%, …
stock only goes up .5%.
But if market goes down 1%,
stock drops just .5%.
Beta of 0
No correlation with market
Finding Beta Estimates on the Web
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http://finance.yahoo.com/q/ks?s=C
http://www.investor.reuters.com/StockEntry.aspx

http://new.quote.com/stocks/company.action?sym=C
[change the ticker in the URL Address bar, directly]

http://finance.google.com/finance?q=C
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** Changes in estimation window length, return
frequency, choice of market and statistical
adjustments will result in varying betas from
different sources. It may be best to calculate your
own – you can fix the variables and parameters.
Expected Return versus Market Risk
Security
Alta
Market
Am. Foam
T-bills
Repo Men
Expected
return
17.4%
15.0
13.8
8.0
1.7
Risk, b
1.29
1.00
0.68
0.00
-0.86
 Which of the alternatives is best?
Use the SML to calculate each
alternative’s required return.

The Security Market Line (SML) is part of the Capital
Asset Pricing Model (CAPM).
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How do we find a security’s required return ki?

SML: ri = rRF + (rM - rRF)bi .
Assume rRF = 8%; rM = rM = 15%.
RPM = (rM - rRF) = 15% - 8% = 7%.
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Higher beta means higher market risk and thus
higher expected or required [not realized] return.
Security Market Line
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What is the proxy for the risk-free rate?
U.S. Treasury Bills
What is the proxy for the return on the
market?
Typically use the S&P500 Index [ticker
is spx or ^spx or spy]
Required Rates of Return
rAlta = 8.0% + (7%)(1.29)
= 8.0% + 9.0%
rM
=
15.0%.
rAm. F. =
12.8%.
rT-bill =
rRepo =
= 17.0%.
8.0% + (7%)(1.00)
=
8.0% + (7%)(0.68)
=
8.0% + (7%)(0.00) =
8.0% + (7%)(-0.86) =
8.0%.
2.0%.
Expected versus Required Returns
^
Alta
r
17.4%
r
17.0%
Undervalued
Market
15.0
15.0
Fairly valued
Am. F.
13.8
12.8
Undervalued
T-bills
8.0
8.0
Fairly valued
Repo
1.7
2.0
Overvalued
Buy Alta
Sell Repo
ri (%) SML: ri = rRF + (RPM) bi
ri = 8% + (7%) bi
.
Alta
.
rM = 15
rRF = 8
.
-1
Market
. T-bills
Repo
0
1
2
Risk, bi
SML and Investment Alternatives
Portfolio Beta
N
bP   wi bi
i 1
Portfolio beta is just a weighted average of the
security betas
b1=.8, b2=1, b3=1.2 wi{.2,.3,.5}
bw =.16+.30+.60=1.06
Note that =wi =1
Calculate beta for a portfolio with 50%
Alta and 50% Repo
bp = Weighted average
= 0.5(bAlta) + 0.5(bRepo)
= 0.5(1.29) + 0.5(-0.86)
= 0.22.
What is the required rate of return
on the Alta/Repo portfolio?
rp = Weighted average r
= 0.5(17%) + 0.5(2%) = 9.5%.
Or use SML:
rp = rRF + (RPM) bp
= 8.0% + 7%(0.22) = 9.5%.
Impact of Inflation Change on SML
What if investors raise inflation expectations by 3%? (reqd return on
all risky assets increases by 3%, Prices drop)
Required Rate
of Return r (%)


If Expected Return increases ->Current Price drops
If Realized Return increases -> Final Price rises
 I = 3%
New SML
SML2
SML1
18
15
11
8
Original situation
0
0.5
1.0
1.5
2.0
Impact of Risk Aversion Change
What if investors’ risk aversion increased, causing the market risk premium to increase
by 3% - rise/run=3/1? (km rises from 15 to 18%, so ki for b=.5 rises by 1.5%, and for b=2 rises by 6%)
Required
Rate of
Return
(%)
rM = 18%
After increase
in risk aversion
SML2
rM = 15%
SML1
18
 RPM =
3%
15
8
Original situation
1.0
Risk, bi
Capital Asset Pricing Model & Security Market Line

Does a higher required return mean that the
actual return you get will be higher?
No, may lose all of your money on the stock.
If the expected return on a risky asset is <0,
what would your weight in the asset be? But
all assets with b>0 have a positive expected
return. So what would happen when the
CAPM is empirically tested?
CAPM Issues
Investors’ required returns are based on
future risk, but betas are calculated with
historical data. Will a company’s beta be
the same this year and next year?
 No – Non-Stationarity Problem
 Assumes markets are efficient
 There have been studies that both support
and dispute the CAPM
 Still used in practice [provides a conceptual framework
useful for linking risk and return in financial decisions]
Next Chapter (5):Portfolio Risk
ExpectedPortfolioReturn (x1 r1 )  (x 2 r2 )
PortfolioVariance  x12σ 12  x 22σ 22  2(x1x 2ρ 12σ 1σ 2 )