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Risk and Rates of Return
New words
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stand-alone risk:独立风险
portfolio risk：组合风险
Expected Rate of Return：期望报酬率
Probability Distributions：概率分布
Discrete Probability Distribution：离散概率分布
Continuous Probability Distribution连续概率分布
Variance:方差
Standard deviation: 标准偏差
Coefficient of Variation:变异系数
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%.
Defining and Measuring Risk
Risk is the chance that an outcome other than
expected will occur.
stand-alone risk: the risk associated with an
investment when it is held by itself or in
isolation, not in combination with other assets.
portfolio risk: the risk associated with an
investment when it is held in combination with
other assets ,not by itself.
Expected Rate of Return
 The rate of return expected to be
realized from an investment
The mean value of the probability
distribution of possible returns
The weighted average of the
outcomes, where the weights are
the probabilities
Expected Rate of Return
Probability of
This State
State of the
Economy Occurring (Pr i)
(1)
Boom
Normal
Recession
(2)
0.2
0.5
0.3
1.0
Martin Products
Return if This State Product:
Occurs (ki)
(2) x (3)
(3)
110%
22%
-60%
^ =
km
= (4)
22%
11%
-18%
15%
U. S. Electric
Return if This Product:
State Occurs (ki) (2) x (5)
(5)
20%
16%
10%
^ =
km
= (6)
4%
8%
3%
15%
Expected Rate of Return
k̂  Pr1k1  Pr2 k 2  ...  Prn k n
n
  Pri k i
i 1
Probability Distributions
Probability distribution is a listing of
all possible outcomes,or events,
with a probability(chance of
occurrence) assigned to each
outcome.
[must sum to 1.0 (100%)]
Probability Distributions
It either will rain, or it will not – only two
possible outcomes
Outcome (1)
Probability (2)
Rain
0.40 = 40%
No Rain
0.60 = 60%
1.00 100%
Probability Distributions
Martin Products and U. S. Electric
State of the
Economy
Boom
Normal
Recession
Probability of This State
Occurring
0.2
0.5
0.3
1.0
Rate of Return on Stock if
This State Occurs
Martin Products
U.S. Electric
110%
22%
-60%
20%
16%
10%
Continuous versus Discrete Probability
Distributions
 Discrete Probability
Distribution:
the number of possible outcomes is limited,
or finite
 Continuous Probability
Distribution:
the number of possible outcomes is
unlimited, or infinite
Discrete Probability Distributions
a. Martin Products
Probability of
Occurrence
b. U. S. Electric
Probability of
Occurrence
0.5 -
0.5 -
0.4 -
0.4 -
0.3 -
0.3 -
0.2 -
0.2 -
0.1 -
0.1 -
-60 -45 -30 -15 0 15 22 30 45 60 75 90 110
Rate of
Expected Rate
Return (%)
of Return (15%)
-10
-5
0
5
10
16 20
Expected Rate
of Return (15%)
25 Rate of
Return (%)
Continuous
Probability Distributions
Probability Density
U. S. Electric
Martin Products
-60
0
15
110
Rate of Return
(%)
Expected Rate of
Return
Measuring Risk:
The Standard Deviation
Calculating Martin Products’ Standard Deviation
Expected
Payoff Return
ki
k^
(2)
(1)
15%
110%
15%
22%
15%
-60%
ki - k^
(1) - (2) = (3)
95
7
-75
^2
(ki - k)
(4)
9,025
49
5,625
Probability
(5)
^ 2Pr
(ki - k)
i
(4) x (5) = (6)
1,805.0
0.2
24.5
0.5
1,687.5
0.3
Variance   2  3,517.0
Standard Deviation   m   m2  3,517  59.3%
Measuring Risk:
The Standard Deviation
n
Expected rate of return  k̂   Pri k i
i 1
n


Variance     k i - k̂ Pri
2
i 1
2
Standard deviation     
2
 k - k̂  Pr
n
i1
2
i
i
Measuring Risk:
The Standard Deviation
• Standard deviation (σ) measures total, or stand-alone,
risk.
• The larger σ is, the lower the probability that actual
returns will be closer to expected returns.
• Larger σ is associated with a wider probability
distribution of returns.
• Difficult to compare standard deviations, because return
has not been accounted for.
Measuring Risk:
Coefficient of Variation
 Standardized measure of risk per unit of
return
 Calculated as the standard deviation divided
by the expected return
 Useful where investments differ in risk and
expected returns

Risk
Coefficient of variation  CV 

Return
k̂
Portfolio Returns
k̂ p  w 1k̂ 1  w 2 k̂ 2  ...  w N k̂ N

^
N
w
j1
j
k̂ j
 Expected return on a portfolio, kp
The weighted average expected
return on the stocks held in the
portfolio
^
k p  0.10 (3.0%)  0.20 (6.4%)  0.40 (10.0%)
 0.20 (12.5%)  0.10 (15.0%)  9.6%
Amount
in each
stock is
equal
Economy
Prob.
HT
Coll
Port.
Recession
0.1
-22.0%
28.0%
3.0%
Below avg
0.2
-2.0%
14.7%
6.4%
Average
0.4
20.0%
0.0%
10.0%
Above avg
0.2
35.0%
-10.0%
12.5%
Boom
0.1
50.0%
-20.0%
15.0%
20%
13.4%
σ
Calculating portfolio standard deviation and CV
 0.10 (3.0 - 9.6)
 0.20 (6.4 - 9.6)2

2
 p   0.40 (10.0 - 9.6)
 0.20 (12.5 - 9.6)2

2
9.6)
(15.0
0.10


2
3.3%
 0.34
CVp 
9.6%







1
2
 3.3%
Comments on portfolio risk measures
• σp = 3.3% is much lower than the σi of
either stock (σHT = 20.0%; σColl. =
13.4%).
• σp = 3.3% is lower than the weighted
average of HT and Coll.’s σ (16.7%).
• Portfolio provides average return of
component stocks, but lower than average
risk.
• Why? Negative correlation between
stocks.
Investment 1 million
on A and B, fifty-tofity
case
A
B
portfolio
year
return
Rate of
return
return
Rate of
return
return
Rate of
return
20×6
20
40%
-5
-10%
15
15%
20×7
-5
-10%
20
40%
15
15 %
20×8
17.5
35%
-2.5
-5%
15
15 %
20×9
-2.5
-5%
17.5
35%
15
15 %
mean
7.5
15%
7.5
15%
15
15 %
σ
case
A
B
portfolio
year
return
Rate of
return
return
Rate
of
return
return
Rate
of
return
20×6
20
40%
20
40%
40
40%
20×7
-5
-10%
-5
-10%
-10
-10%
20×8
17.5
35%
17.5
35%
35
35%
20×9
-2.5
-5%
-2.5
-5%
-5
-5%
mean
7.5
15
7.5
15
15
15
σ
Returns Distribution for Two Perfectly Negatively
Correlated Stocks (r = -1.0) and for Portfolio WM:
Stock W
Stock M
Portfolio WM
25
25
25
15
15
15
0
0
0
-10
-10
-10
Returns Distributions for Two Perfectly Positively
Correlated Stocks (r = +1.0) and for Portfolio MM:
Stock MM’
Stock MM’
Stock M
25
25
25
15
15
15
0
0
0
-10
-10
-10
Portfolio Risk
Correlation Coefficient, r
A measure of the degree of relationship
between two variables
Positively correlated stocks rates of
return move together in the same
direction
Negatively correlated stocks rates of
return move in opposite directions
Portfolio Risk
Risk Reduction
Combining stocks that are not perfectly
positive correlated will reduce the
portfolio risk by diversification
The riskiness of a portfolio is reduced as
the number of stocks in the portfolio
increases
The smaller the positive correlation, the
lower the risk
Portfolio Risk
• Most stocks are positively correlated with the
market (rk,m  0.65).
• σ  35% for an average stock.
• Combining stocks in a portfolio generally lowers
risk. because they would not be perfectly
correlated with the existing portfolio.
• Eventually the diversification benefits of adding
more stocks (after about 10 stocks), and for
large stock portfolios, σp tends to converge to 
20%.
Firm-Specific Risk versus Market
Risk
 Firm-Specific Risk

That part of a security’s risk
associated with random outcomes
generated by events, or behaviors,
specific to the firm
It can be eliminated by proper
diversification
Firm-Specific Risk
versus Market Risk
 Market Risk
 That part of a security’s risk that cannot be
eliminated by diversification because it is
associated with economic, or market factors that
systematically affect most firms
Firm-Specific Risk
versus Market Risk
 Relevant Risk
 The risk of a security that cannot
be diversified away, or its market
risk
 This reflects a security’s
contribution to the risk of a
portfolio
Risk Aversion and
Required Returns
•
Risk aversion – assumes investors dislike
risk and require higher rates of return to
encourage them to hold riskier securities.
Risk Aversion and
Required Returns
 Risk Premium (RP)


The portion of the expected return that
can be attributed to the additional risk of
an investment
The difference between the expected
rate of return on a given risky asset and
that on a less risky asset, which serves
as compensation for investors to hold
riskier securities.
Market Risk Premium
RPM is the additional return over the
risk-free rate needed to
compensate investors for
assuming an average amount of
risk
Risk Premium for a Stock
Risk Premium for Stock j
= RPj = RPM x bj
The Concept of Beta
Beta Coefficient, b
•
•
Measures a stock’s market risk, and
shows a stock’s volatility relative to the
market.
Indicates how risky a stock is if the
stock is held in a well-diversified
portfolio.
Firm-Specific Risk
versus Market Risk
CAPM
A model based on the proposition that
any 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.
Portfolio Beta Coefficients
 The beta of any set of securities is
the weighted average of the
individual securities’ betas
b p  w1b1  w 2 b 2  ...  w n b n
N
  w jb j
j1
The Relationship Between
Risk and Rates of Return
k̂ j  expected rate of return on the jth stock
The Relationship Between
Risk and Rates of Return
k̂ j  expected rate of return on the j stock
th
k j  required rate of return on the jth stock
The Relationship Between
Risk and Rates of Return
k̂ j  expected rate of return on the j stock
th
k j  required rate of return on the jth stock
k RF  risk  free rate of return
The Relationship Between
Risk and Rates of Return
k̂ j  expected rate of return on the j stock
th
k j  required rate of return on the jth stock
k RF  risk  free rate of return
RPM  k M - k RF   market risk premium
The Relationship Between
Risk and Rates of Return
k̂ j  expected rate of return on the j stock
th
k j  required rate of return on the j stock
th
k RF  risk  free rate of return
RPM  k M - k RF   market risk premium
RP j  k M - k RF b j  risk premium
th
on the j stock
The Required Rate of Return for
a Stock
k j  required rate of return for stock j
k j  k RF  RPM b j
 k RF  k M  k RF b j
Security Market Line (SML)
 The line that shows the relationship
between risk as measured by beta and
the required rate of return for individual
securities
Security Market Line
SML : k j  k RF  k M  k RF b j
Required Rate
of Return (%)
khigh = 22
kM = kA = 14
kLOW = 10
Safe Stock Risk
Premium: 4%
kRF = 6
Market (Average
Stock) Risk Premium:
8%
Relatively
Risky
Stock’s
Risk
Premium:
16%
Risk-Free
Rate: 6%
0
0.5
1.0
1.5
2.0 Risk, bj
The Impact of Inflation
 kRF is the price of money to a
riskless borrower
 The nominal rate consists of
 a real (inflation-free) rate of return
 an inflation premium (IP)
 An increase in expected inflation
would increase the risk-free rate
Changes in Risk Aversion
 The slope of the SML reflects the
extent to which investors are averse to
risk
An increase in risk aversion increases
the risk premium and increases the
slope
Changes in a Stock’s
Beta Coefficient
 The Beta risk of a stock is affected by




composition of its assets
use of debt financing
increased competition
expiration of patents
 Any change in the required return
(from change in beta or in expected
inflation) affects the stock price
Stock Market Equilibrium
 The condition under which the
expected return on a security is just
equal to its required return
 Actual market price equals its
intrinsic value as estimated by the
marginal investor, leading to price
stability
Changes in Equilibrium Stock Prices
Stock prices are not constant due to
changes in:
 Risk-free rate, kRF
 Market risk premium, kM - kRF
 Stock X’s beta coefficient, bx
 Stock X’s expected growth rate, gX
 Changes in expected dividends, D0