Chapter 13
RETURN RISK AND THE
SECURITY MARKET LINE
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Chapter Outline
Expected Returns and Variances of a
portfolio
 Announcements, Surprises, and Expected
Returns
 Risk: Systematic and Unsystematic
 Diversification and Portfolio Risk
 Systematic Risk and Beta
 The Security Market Line (SML)

Expected Returns (1)
Expected return = return on a risky asset
expected in the future

Expected returns are based on the probabilities
of possible outcomes

Expected means average if the process is
repeated many timesn
E ( R)   pi Ri
i 1
2
Expected Returns (2)
Probability
Boom
Normal
Recession
•R
A
= 6%
•RB = 7%
0.2
0.4
0.4
Expected return
Stock A Stock B
20%
15%
10%
8%
-5%
2%
risk premium = 6 – 3.2=2.8%
risk premium = 7 – 3.2=3.8%
•If the risk-free rate = 3.2%, what is the risk premium
for each stock?
3
Variance and Standard Deviation (1)

Unequal probabilities can be used for the
entire range of possibilities

Weighted average of squared deviations
n
σ 2   pi ( Ri  E ( R)) 2
i 1
4
Variance and Standard Deviation (2)
Consider the previous example. What is
the variance and standard deviation for
each stock?
0.2
20
4
39.2
 Stock A
0.4
10
4
6.4


Stock B
0.4
-5
mean
-2
6
48.4
94 Var
9.6954 std
0.2
0.4
0.4
15
8
2
mean
3
3.2
0.8
7
12.8
0.4
10
23.2 Var
4.8166 std
5
Portfolios
Portfolio = a group of assets held by an
investor

The risk-return trade-off for a portfolio is
measured by the portfolio expected return and
standard deviation, just as with individual assets
Portfolio weights = Percentage of a
portfolio’s total value in a particular asset
6
Portfolio Weights

Suppose you have $ 20,000 to invest and you
have purchased securities in the following
amounts. What are your portfolio weights in
each security?
◦ $5,000 of A
WA=.25
◦ $9,000 of B WB=.45
◦ $5,000 of C
WC=.25
◦ $1,000 of D WD=.05
7
Portfolio Expected Returns (1)

The expected return of a portfolio is the
weighted average of the expected returns for
each asset in the portfolio
m
E ( RP )   w j E ( R j )
j 1

You can also find the expected return by finding
the portfolio return in each possible state and
computing the expected value
8
Expected Portfolio Returns (2)

Consider the portfolio weights computed
previously. If the individual stocks have the
following expected returns, what is the
expected return for the portfolio?
◦
◦
◦
◦

A: 19.65%
B: 8.96%
C: 9.67%
D: 8.13%
0.25 19.65 4.913
0.45 8.96 4.032
0.25 9.67 2.418
0.05 8.13 0.407
11.77
E(RP) = 11.77
9
Portfolio Variance (1)
Steps:
1. Compute the portfolio return for each
state:
RP = w1R1 + w2R2 + … + wnRn
2. Compute the expected portfolio return
using the same formula as for an
individual asset
3. Compute the portfolio variance and
standard deviation using the same
formulas as for an individual asset
10
Portfolio Variance (2)
Consider the following information
Invest 60% of your money in Asset A

◦ State Probability A
◦ Boom
.5
70%
◦ Recession .5
-20%
1.
2.
B
10%
30%
What is the expected return and
standard deviation for each asset?
What is the expected return and
standard deviation for the portfolio?
11
Solution:
prob

1.
0.5
0.5
E(R)
Var( R )
std ( R )
A (R)
B(R)
70
-20
25
2025
45
10
30
20
100
10
prob
A (R) B(R) Port( R )
0.5
70
10
46
0.5
-20
30
0
E(R)
25
20
23
Var( R ) 2025 100
529
std ( R )
45
10
23
12
Another Way to Calculate Portfolio
Variance

Portfolio variance can also be calculated
using the following formula:
  x   x   2 xL xU CORRL,U L U
2
P
2
L
2
L
2
U
2
U

Correlation is a statistical measure of how 2 assets
move in relation to each other

If the correlation between stocks A and B = -1,
what is the standard deviation of the portfolio?
13
Solution:

Var=1058.1 (1458+32-431.87)
14
Different Correlation Coefficients (1)
15
Different Correlation Coefficients (2)
1316
Different Correlation Coefficients(3)
17
Possible Relationships between Two Stocks
18
Diversification (1)

There are benefits to diversification
whenever the correlation between two
stocks is less than perfect (p < 1.0)

If two stocks are perfectly positively
correlated, then there is simply a riskreturn trade-off between the two
securities.
19
Diversification (2)
20
Expected vs. Unexpected Returns
Total return = Expected return + Unexpected return

Expected return from a stock is the part of
return that shareholders in the market predict
(expect)

The unexpected return (uncertain, risky part):
◦ At any point in time, the unexpected return
can be either positive or negative
◦ Over time, the average of the unexpected
component is zero
21
Announcements and News

Announcements and news contain both
an expected component and a surprise
component

It is the surprise component that affects a
stock’s price and therefore its return
Announcement = Expected part + Surprise
22
Systematic Risk

Risk factors that affect a large number of
assets

Also known as non-diversifiable risk or
market risk

Examples: changes in GDP, inflation,
interest rates, general economic
conditions
23
Unsystematic Risk

Risk factors that affect a limited number
of assets

Also known as diversifiable risk and
asset-specific risk

Includes such events as labor strikes,
shortages.
24
Returns

Unexpected return = systematic portion
+ unsystematic portion
Total return can be expressed as follows:
Total Return = expected return +
systematic portion + unsystematic
portion

25
Effect of Diversification

Portfolio diversification is the investment
in several different asset classes or
sectors
Principle of diversification = spreading
an investment across a number of assets
eliminates some, but not all of the risk
Diversification
assets
is not just holding a lot of
26
The Principle of Diversification

Diversification can substantially reduce the
variability of returns without an equivalent
reduction in expected returns

Reduction in risk arises because worse than
expected returns from one asset are offset by
better than expected returns from another
There
is a minimum level of risk that cannot be
diversified away and that is the systematic
portion
27
Portfolio Diversification (1)
28
Portfolio Diversification (2)
29
Diversifiable (Unsystematic) Risk

The risk that can be eliminated by
combining assets into a portfolio

If we hold only one asset, or assets in the
same industry, then we are exposing
ourselves to risk that we could diversify
away
The
market will not compensate
investors for assuming unnecessary risk
30
Total Risk

The standard deviation of returns is a
measure of total risk

For well diversified portfolios,
unsystematic risk is very small

Consequently, the total risk for a
diversified portfolio is essentially
equivalent to the systematic risk
31
Systematic Risk Principle

There is a reward for bearing risk

There is no reward for bearing risk
unnecessarily

The expected return (and the risk
premium) on a risky asset depends only
on that asset’s systematic risk since
unsystematic risk can be diversified away
32
Measuring Systematic Risk

Beta (β) is a measure of systematic risk

Interpreting beta:
◦ β = 1 implies the asset has the same
systematic risk as the overall market
◦ β < 1 implies the asset has less systematic
risk than the overall market
◦ β > 1 implies the asset has more systematic
risk than the overall market
33
High and Low Betas
34
Portfolio Betas

Consider the previous example with the
following four securities
◦ Security
◦A
◦B
◦C
◦D

Weight
.133
.2
.267
.4
Beta
3.69
0.64
1.64
1.79
What is the portfolio beta? 1.773
35
Beta and the Risk Premium

The higher the beta, the greater the risk
premium should be

The relationship between the risk
premium and beta can be graphically
interpreted and allows to estimate the
expected return
36

Consider a portfolio consisting of asset A
and a risk-free asset. Expected return on
asset A is 20%, it has a beta = 1.6. Riskfree rate = 8%.
37
Portfolio Expected Returns and Betas
Rf
38
Reward-to-Risk Ratio:

The reward-to-risk ratio is the slope of
the line illustrated in the previous slide
◦ Slope = (E(RA) – Rf) / (A – 0)
◦ Reward-to-risk ratio =

If an asset has a reward-to-risk ratio = 8?

If an asset has a reward-to-risk ratio = 7?
39
The Fundamental Result

The reward-to-risk ratio must be the
same for all assets in the market
E ( RA )  R f
A


E ( RM  R f )
M
If one asset has twice as much systematic
risk as another asset, its risk premium is
twice as large
40
Security Market Line (1)
The security market line (SML) is the
representation of market equilibrium
 The slope of the SML is the reward-torisk ratio: (E(RM) – Rf) / M
 The beta for the market is always equal to
one, the slope can be rewritten
Slope = E(RM) – Rf = market risk premium

41
Security Market Line (2)
42
The Capital Asset Pricing Model
(CAPM)

The capital asset pricing model defines
the relationship between risk and return

E(RA) = Rf + A(E(RM) – Rf)

If we know an asset’s systematic risk, we
can use the CAPM to determine its
expected return
43
CAPM

Consider the betas for each of the assets given
earlier. If the risk-free rate is 4.5% and the market
risk premium is 8.5%, what is the expected return
for each?
Security
Beta
Expected Return
A
3.6
35.1%
B
.7
10.45%
C
1.7
18.85%
D
1.9
20.65%
44
Factors Affecting Expected Return

Time value of money – measured by the
risk-free rate

Reward for bearing systematic risk –
measured by the market risk premium

Amount of systematic risk – measured by
beta
45