Return, Risk and the Security
Market Line
Expected Return and Variance
•
Expected return—the weighted average of the distribution of
possible returns in the future.
•
Variance of returns—a measure of the dispersion of the
distribution of possible returns.
•
Rational investors like return and dislike risk.
Example—Calculating Expected
Return
St at e o f
Eco no my
Boo m
No rma l
Recessio n
Expected return
Pi
Pro ba bilit y
o f St at e i
0.25
0.50
0.25
Ri
Ret urn in
St at e i
35%
15%
–5%
 0.25  35%  0.50  15%  0.25   5%
 15%
Example—Calculating Variance
State of
Economy
Boom
Normal
Recession
(Ri – R)
(Ri – R)2
Pi x (Ri – R)2
0.20
0
– 0.20
0.04
0
0.04
2=
0.01
0
0.01
0.02
  0.02
 0.1414 or 14.14%
Example—Expected Return and
Variance
State of
Economy
Boom
Bust
Pi
0.40
0.60
Return on
Asset A
30%
–10%
Return on
Asset B
–5%
25%
Expected Returns:
E RA   0.40  0.30  0.60   0.10  0.06  6%
E RB   0.40   0.05  0.60  0.25  0.13  13%
Example—Expected Return and
Variance
Variances:
Var RA   0.40  0.30  0.06  0.60   0.10  0.06
2
2
 0.0384
Var RB   0.40   0.05  0.13  0.60  0.25  0.13
2
 0.0216
Standard deviations:
σ RA   0.0384  0.196  19.6%
σ RB   0.0216  0.147  14.7%
2
Portfolios
• A portfolio is a collection of assets.
• An asset’s risk and return is important in how it
affects the risk and return of the portfolio.
• The risk–return trade-off for a portfolio is measured
by the portfolio’s expected return and standard
deviation, just as with individual assets.
Portfolio Expected Returns
• The expected return of a portfolio is the weighted
average of the expected returns for each asset in
the portfolio.
m
E(Rp) = ∑ wjE (Rj)
j =1
• You can also find the expected return by finding
the portfolio return in each possible state and
computing the expected value as we did with
individual securities.
Example—Portfolio Return and
Variance
Assume 50 per cent of portfolio in asset A and
50 per cent in asset B.
State of
Economy
Boom
Bust
Pi
RA
RB
0.40
0.60
30%
–10%
–5%
25%
E R p   0.40  0.125  0.60  0.075
 0.095 or 9.5%
R
12.
7.5
Example—Portfolio Return and
Variance
•
•
Var(Rp)  (0.50 x Var(RA)) + (0.50 x Var(RB)).
By combining assets in a portfolio, the risks faced by the
investor can significantly change.
Var R p   0.40  0.125  0.095  0.60  0.075  0.095
2
 0.0006
R p   0.0006
 0.0245 or 2.45%
2
Announcements, Surprises and
Expected Returns
•
Key Issues
– What are the components of the total return?
– What are the different types of risk?
•
Expected and Unexpected Returns
– Total return (R) = expected return (E(R))+ unexpected
return (U)
•
Announcements and News
– Announcement = expected part + surprise
– It is the surprise component that affects a stock’s price
and, therefore, its return.
Risk
•
•
Systematic risk: that component of total risk which is due to
economy-wide factors.
Non-systematic risk: that component of total risk which is
unique to an asset or firm.
Total return  Expected return  Unexpected return
R  E R   U
 E R   systematicportion  non - systematicportion
Standard Deviations of Monthly
Portfolio Returns
Number of
Shares
1
5
10
15
20
25
30
35
40
45
Average Standard
Deviation
11.49%
7.91%
6.61%
6.08%
5.71%
5.60%
5.50%
5.50%
5.26%
5.12%
Ratio of Standard
Deviations
1.00
0.69
0.58
0.53
0.50
0.49
0.48
0.48
0.46
0.45
Diversification
•
The process of spreading investments across different
assets, industries and countries to reduce risk.
•
Total risk = systematic risk + non-systematic risk
•
Non-systematic risk can be eliminated by diversification;
systematic risk affects all assets and cannot be diversified
away.
The Principle of Diversification
• Diversification can substantially reduce the
variability of returns without an equivalent reduction
in expected returns.
• This reduction in risk arises because worse than
expected returns from one asset are offset by
better than expected returns from another.
• However, there is a minimum level of risk that
cannot be diversified away and that is the
systematic portion.
Portfolio Diversification
Systematic Risk
•
•
The systematic risk principle states that the expected return
on a risky asset depends only on the asset’s systematic risk.
The amount of systematic risk in an asset relative to an
average risky asset is measured by the beta coefficient.
Security A
Security B
•
Std Deviation
30%
10%
Beta
0.60
1.20
Security A has greater total risk but less systematic risk (more
non-systematic risk) than Security B.
Measuring Systemic Risk
• What does beta tell us?
-
-
A beta of 1 implies the asset has the same
systematic risk as the overall market.
A beta < 1 implies the asset has less
systematic risk than the overall market.
A beta > 1 implies the asset has more
systematic risk than the overall market.
Beta Coefficients for Selected
Companies
Company
Amcor
BHP
Boral
Caltex Australia
CSR
Coles Myer
Mayne Nickless
NAB
Beta Coefficent
0.78
1.33
0.85
1.38
0.96
0.45
0.68
1.27
Example—Portfolio Beta Calculations
Amount
Invested
Portfolio
Weights
Beta
(2)
(3)
(4)
(3)  (4)
ABC Company
$6 000
50%
0.90
0.450
LMN Company
4 000
33%
1.10
0.367
XYZ Company
2 000
17%
1.30
0.217
$12 000
100%
Share
(1)
Portfolio
1.034
Example—Portfolio Expected
Returns and Betas
•
Assume you wish to hold a portfolio consisting of asset A and
a riskless asset. Given the following information, calculate
portfolio expected returns and portfolio betas, letting the
proportion of funds invested in asset A range from 0 to 125
per cent.
•
Asset A has a beta of 1.2 and an expected return of 18 per
cent.
The risk-free rate is 7 per cent.
Asset A weights: 0 per cent, 25 per cent, 50 per cent, 75 per
cent, 100 per cent and 125 per cent.
•
•
Example—Portfolio Expected
Returns and Betas
Proportion
Invested in
Asset A (%)
Proportion
Invested in
Risk-free Asset (%)
Portfolio
Expected
Return (%)
Portfolio
Beta
0
100
7.00
0.00
25
75
9.75
0.30
50
50
12.50
0.60
75
25
15.25
0.90
100
0
18.00
1.20
125
–25
20.75
1.50
Return, Risk and Equilibrium
•
Key issues:
– What is the relationship between risk and return?
– What does security market equilibrium look like?
•
The ratio of the risk premium to beta is the same for every
asset. In other words, the reward-to-risk ratio for the market
is constant and equal to:
Reward/ris k ratio 
E Ri   R f
i
Example—Asset Pricing
•
Asset A has an expected return of 12 per cent and a beta of
1.40. Asset B has an expected return of 8 per cent and a
beta of 0.80. Are these two assets valued correctly relative to
each other if the risk-free rate is 5 per cent?
0.12  0.05
A:
 0.05
1.40
0.08  0.05
B:
 0.0375
0.80
•
Asset B offers insufficient return for its level of risk, relative to
A. B’s price is too high; therefore, it is overvalued (or A is
undervalued).
Security Market Line
• The security market line (SML) is the
representation of market equilibrium.
• The slope of the SML is the reward-to-risk ratio:
(E(RM) – Rf)/ßM
• But since the beta for the market is ALWAYS equal
to one, the slope can be rewritten.
• Slope = E(RM) – Rf = market risk premium
Security Market Line (SML)
Asset expected
return (E (Ri))
= E (RM) – Rf
E (RM)
Rf
M
= 1.0
Asset
beta (i)
The Capital Asset Pricing
Model (CAPM)
•
•
An equilibrium model of the relationship between risk and
return.
What determines an asset’s expected return?
– The risk-free rate—the pure time value of money.
– The market risk premium—the reward for bearing
systematic risk.
– The beta coefficient—a measure of the amount of
systematic risk present in a particular asset.


CAPM  E Ri   R f  E RM  R f    i
11-27
Calculation of Systematic Risk


~ ~
i  Cov Ri , RM /M
Where: Cov = covariance
~
Ri
= random distribution of return for asset i
~
R M = random distribution of return for the
market
M = standard deviation of market return
Covariance and Correlation
•
•
•
The covariance term measures how returns change
together—measured in absolute terms.
The correlation coefficient measures how returns change
together—measured in relative terms.
Correlation coefficient ranges between –1.0 and +1.0.


~ ~
ρiM   Cov Ri , RM /σi  σ M
•
Where i = standard deviation of the return on asset i.
Security Market Line versus Capital
Market Line

 E R

 β
CML  E R p   R f  E RM   R f / M   p
SML  E Ri   R f
M
 Rf
i
* SML explains the expected return for all assets.
* CML explains the expected return for efficient portfolios.
Risk of a Portfolio
Variance of a two-asset portfolio is calculated as:
weighted variance of the expected return for
each asset in the portfolio
+
twice the weighted covariance of the expected
return on the first asset with the expected
return on the second
Example—Risk of a Portfolio
Weighting
0.3
0.7
Asset A
Asset B
Std Deviation
0.26
0.13
The covariance of the expected returns between A and B is
0.017.
Variance  0.3  0.26  0.7  0.13  2  0.3  0.7  0.017 
2
2
 0.006084  0.008281  0.00714
 0.0215
Std dev  0.1466
Problems with CAPM
• Difficulties in estimating beta
- thin trading
- non-constant beta
• Using CAPM
- adding explanatory variables
- measure of market return