FIN303
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Risk and return
(chapter 8)
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Investment
returns
The rate of return on an investment can be calculated as
follows:
Return =
(Amount received – Amount invested)
________________________
Amount invested
For example, if $1,000 is invested and $1,100 is returned
after one year, the rate of return for this investment is:
($1,100 - $1,000) / $1,000 = 10%.
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What is investment risk?


Investment risk is related to the probability of earning a low or negative
actual return.
The greater the chance of lower than expected or negative returns, the riskier
the investment.
Firm
X
Firm Y
-70
0
15
100
Rate of
Return (%)
Expected Rate of Return
Firm X (red) has a lower distribution of returns than firm Y (purple) though both have
the same average return. We say that firm X’s returns are less variable/volatile
(lower standard deviation ) and thus X is a less risky investment than Y
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Selected Realized Returns, 1926-2009
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Why is the T-bill return independent of the
economy? Do T-bills promise a completely
risk-free return?
Vicentiu Covrig
•
•
•
•
T-bills will return the promised 5.5%, regardless of the
economy.
No, T-bills do not provide a completely risk-free return,
as they are still exposed to inflation. Although, very
little unexpected inflation is likely to occur over such a
short period of time.
T-bills are also risky in terms of reinvestment risk.
T-bills are risk-free in the default sense of the word.
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Return: Calculating the expected return
^
r  expected rate of return
^
r  P1r1  ....  Pn rn
where n  number of outcomes
r  expected return for each outcome
p  probabilit y that each outcome occurs
Outcome
Prob. of outcome
1(recession)
.1
2 (normal growth)
.6
3 (boom)
.3
Return in
-15%
15%
25%
r^ =expected rate of return = (.1)(-15) + (.6)(15) +(.3)(25)=15%
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Risk: Calculating the standard deviation for
each alternative
Standard deviation (σ) measures total, or stand-alone,
risk. Greater the σ, greater the risk. Why?

n

i 1
^
(ri  r ) 2 Pi
(-15.0 - 15.0) 2 (0.1)  (15.0 - 15.0) 2 (0.6)
 
2
 (25.0 - 15.0) (0.3)
 10.96%
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


1
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Investor attitude towards risk





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Risk aversion – assumes investors dislike risk and require higher
rates of return to encourage them to hold riskier securities.
Risk premium – the difference between the return on a risky asset
and less risky asset, which serves as compensation for investors
to hold riskier securities
Very often risk premium refers to the difference between the
return on a risky asset and risk-free rate (ex. a treasury bond)
Standard deviation (σi) measures total, or stand-alone, 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.
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Coefficient of Variation (CV)


A standardized measure of dispersion about the expected
value, that shows the risk per unit of return.
You want to invest in a security with the highest expected
return per unit of risk, and thus the lowest CV
Standard deviation 
CV 

Expected return
r̂
Small-company stocks
Large-company stocks
L-T corporate bonds
Average Standard
Return
Deviation
17.3%
33.2%
12.7
20.2
6.1
8.6
9
CV
1.92
1.59
1.41
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Illustrating the CV as a Measure of
Relative Risk
Prob.
A
B
Rate of Return (%)
0
σA = σB , but A is riskier because of a larger
probability of losses. In other words, the
same amount of risk (as measured by σ) for
smaller returns.
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Portfolio returns
The rate of return on a portfolio is a weighted average of
the rates of return of each asset comprising the portfolio,
with the portfolio proportions as weights.
rp = W1r1 + W2r2
W1 = Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2
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Assume that you invested $3000 in Countrywide stock and $2,000
in Yahoo stock. The expected return of Countrywide stock is 15%
and the expected return of Yahoo is 20%.
What is the portfolio expected return?
Answer:
W1=3,000/(3,000+2,000)=0.6
W2=2,000/ (3,000+2,000)=0.4
Expected portfolio return = 0.6*15%+0.4*20%= 17%
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The benefits of diversification










Come from the correlation between asset returns
Correlation, : a measure of the strength of the linear relationship between two
variables
-1.0 < r < +1.0
If r = +1.0, securities 1 and 2 are perfectly positively correlated
If r = -1.0, 1 and 2 are perfectly negatively correlated
If r = 0, 1 and 2 are not correlated
The smaller the correlation, the greater the risk reduction potential  greater
the benefit of diversification
If  = +1.0, no risk reduction is possible
Most stocks are positively correlated with the market (ρ  0.65)
Combining stocks and bonds in a portfolio generally lowers risk.
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Partial Correlation, ρ = +0.35
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Illustrating diversification effects of a
stock portfolio
p (%)
35
Company-Specific Risk
Stand-Alone Risk, p
20
Market Risk
0
10
20
30
40
2,000+
# Stocks in Portfolio
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Creating a Portfolio: Beginning with One
Stock and Adding Randomly Selected Stocks
to Portfolio
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•
•
•
σp decreases as stocks are added, because they would not be
perfectly correlated with the existing portfolio.
Expected return of the portfolio would remain relatively constant.
Eventually the diversification benefits of adding more stocks
dissipates (after about 10 stocks), and for large stock portfolios,
σp tends to converge to  20%.
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Illustrating Diversification Effects of a Stock
Portfolio
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Breaking down sources of risk
Stand-alone risk = Market risk + Firm-specific risk



Market risk – portion of a security’s stand-alone risk that cannot be eliminated
through diversification. Measured by beta.
Firm-specific risk – portion of a security’s stand-alone risk that can be
eliminated through proper diversification.
If an investor chooses to hold a one-stock portfolio (exposed to more risk than a
diversified investor), would the investor be compensated for the risk they bear?
- NO!
- Stand-alone risk is not important to a well-diversified investor.
- Rational, risk-averse investors are concerned with σp, which is based upon
market risk.
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Capital Asset Pricing Model
(CAPM)



Model based upon concept that a stock’s required rate of return is
equal to the risk-free rate of return plus a risk premium that
reflects the riskiness of the stock after diversification.
Primary conclusion: The relevant riskiness of a stock is its
contribution to the riskiness of a well-diversified portfolio
Beta: measures a stock’s market risk
Indicates how risky a stock is if the stock is held in a welldiversified portfolio
Beta is calculated using regression analysis
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The Security Market Line (SML):
Calculating required rates of return
SML: ri = rRF + βi (rM – rRF)

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SML is the empirical part of CAPM
Assume rRF = 8% , rM = 15% and company’s BETA (βi ) is 1.2
The market (or equity) risk premium is RPM = rM – rRF =
15% – 8% = 7%.
ri = 8% + 1.2x(15% - 8%) = 16.4%
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What is the market risk premium?
•
•
•
Additional return over the risk-free rate needed to
compensate investors for assuming an average amount of
risk.
Its size depends on the perceived risk of the stock market
and investors’ degree of risk aversion.
Varies from year to year, but most estimates suggest that it
ranges between 4% and 8% per year.
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Expected vs. Required Returns
rˆ(exp ected ) r (required)
Stock A
12.4%
12.1%
Stock B
10.5
10.5
Stock C
9.8
9.9
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Undervalue (r̂  r)
d
(r̂  r)
Fairly
valued
Overvalued (r̂  r)
FIN303
Vicentiu Covrig
Illustrating the Security Market Line
SML: ri = 5.5% + (5.0%)bi
ri (%)
SML
.
..
HT
rM =
10.5
.r
=
RF
-1 Coll
0
5.5
.T-bills
USR
1
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2
Risk, bi
FIN303
Vicentiu Covrig
An Example:
Equally-Weighted Two-Stock Portfolio
•
•
Create a portfolio with 50% invested in High
Tech and 50% invested in Collections.
The beta of a portfolio is the weighted average of
each of the stock’s betas.
bP = wHTbHT + wCollbColl
bP = 0.5(1.32) + 0.5(-0.87)
bP = 0.225
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Calculating Portfolio Required Returns
•
•
The required return of a portfolio is the weighted
average of each of the stock’s required returns.
rP = wHTrHT + wCollrColl
rP = 0.5(12.10%) + 0.5(1.15%)
rP = 6.625%
Or, using the portfolio’s beta, CAPM can be used
to solve for expected return.
rP = rRF + (RPM)bP
rP = 5.5% + (5.0%)(0.225)
rP = 6.625%
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Factors That Change the SML
•
What if investors raise inflation
expectations by 3%, what would happen to
theri (%)
SML?
SML2
ΔI = 3%
13.5
SML1
10.5
8.5
5.5
Risk, bi
0
0.5
1.0
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1.5
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Factors That Change the SML
•
What if investors’ risk aversion increased,
causing the market risk premium to
increase by 3%, what would happen to the
SML?
ri (%)
SML
ΔRPM = 3%
2
13.5
SML1
10.5
5.5
Risk, bi
0
0.5
1.0
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1.5
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Comments on beta


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If beta = 1.0, the security is just as risky as the average stock (the
market)
If beta > 1.0, the security is riskier than average (the market)
If beta < 1.0, the security is less risky than average (the market)
CAPM/SML concepts are based upon expectations, but betas are
calculated using historical data. A company’s historical data may
not reflect investors’ expectations about future riskiness.
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Can the beta of a security be negative?
•
•
•
Yes, if the correlation between Stock i and
the market is negative (i.e., ρi,m < 0).
If the correlation is negative, the regression
line would slope downward, and the beta
would be negative.
However, a negative beta is highly
unlikely.
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Calculating Betas
•
•
•
Well-diversified investors are primarily concerned with how
a stock is expected to move relative to the market in the
future.
Without a crystal ball to predict the future, analysts are forced
to rely on historical data. A typical approach to estimate beta
is to run a regression of the security’s past returns against the
past returns of the market.
The slope of the regression line is defined as the beta
coefficient for the security.
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Learning objectives
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Know how to calculate a rate of return; historical rates of return 1926-2001
Discuss the investment risk; know that our risk measure will be the standard
deviation of returns (no calculations are necessary)
Know how to calculate expected return, standard deviation and coefficient of
variation given probabilities of each outcome
Know what is risk aversion and risk premium
Know how to calculate the portfolio return
Discuss the diversification effects of a portfolio; the role of correlation and its two
signs and the benefits of diversification
Know the two sources of risk; market and firm specific
Briefly discuss what is CAPM and beta
Know how to calculate required return using SML
Recommended problems: ST-1, Questions 8-4,8-7,8-8, problems 8-1 to 8-8
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