Risk and Rates of Return



Stand-alone risk
Portfolio risk
Risk & return: CAPM / SML
5-1
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%.
5-2
What is investment risk?

Two types of investment risk




Stand-alone risk
Portfolio risk
Stand-alone risk: The risk an investor
would face if he or she held only one
asset.
Portfolio risk: The riskiness of assets
held in portfolios.
5-3
Why is the T-bill return independent of
the economy? Do T-bills promise a
completely risk-free return?



T-bills will return the promised 8%, regardless of
the economy.
No, T-bills do not provide a 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 risk-free in the default sense of the
word.
5-4
Expected Rate of return

Company
IBM
The rate of return expected to be realized
from an investment.
Expected Rate of Return
-22%
-2
20
35
50
Probability
10%
20
40
20
10
5-5
Return: Calculating the expected
return for each alternative
^
k  expected
^
k
rate of return
n

k i Pi
i 1
^
k IBM  (-22%) (0.1)  (-2%) (0.2)
 (20%) (0.4)  (35%) (0.2)
 (50%) (0.1)  17.4%
5-6
Summary of expected returns
for all alternatives
IBM
Market
USR
T-bill
Shell
Exp return
17.4%
15.0%
13.8%
8.0%
1.7%
IBM has the highest expected return, and appears
to be the best investment alternative, but is it
really? Have we failed to account for risk?
5-7
Risk: Calculating the standard
deviation for each alternative
  Standard deviation
 
 
Variance
n



2
2
ˆ
( k i  k ) Pi
i 1
5-8
Standard deviation calculation
n
 

^
(k i  k ) Pi
2
i 1
 (-22.0 - 17.4) (0.1)  (-2.0 - 17.4) (0.2)

2
2
   (20.0 - 17.4) (0.4)  (35.0 - 17.4) (0.2)
2


(50.0
17.4)
(0.1)

2
 IBM
2
 IBM  20.04%
 T - bills  0.0%





1
2
 Shell  13.4%
 USR  13.8%
M
 15.3%
5-9
Comments on standard
deviation as a measure of risk



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.
Difficult to compare standard
deviations, because return has not
been accounted for.
5-10
Comparing risk and return
Security
Expected
return
8.0%
Risk, σ
IBM
17.4%
20.04%
Shell
1.7%
13.4%
USR
13.8%
13.8%
Market
15.0%
15.3%
T-bills
0.0%
5-11
Coefficient of Variation (CV)
A standardized measure of dispersion about the
expected value, that shows the risk per unit of
return.
 Very useful in comparing the risk of assets that
have different expected returns.

CV 
Std dev
Mean


^
k
5-12
Risk rankings,
by coefficient of variation
T-bill
IBM
Shell
USR
Market


CV
0.000
1.152
7.882
1.000
1.020
Shell has the highest degree of risk per unit of
return.
IBM, despite having the highest standard
deviation of returns, has a relatively average CV.
5-13
Investor attitude towards risk


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.
5-14
Portfolio construction:
Risk and return
Assume a two-stock portfolio is created with
$50,000 invested in both IBM and Shell.

Expected return of a portfolio is a
weighted average of each of the
component assets of the portfolio.
5-15
Calculating portfolio expected return
^
kp 
n

^
wi ki
i 1
^
k p  0.5 (17.4%)  0.5 (1.7%)  9.6%
5-16
Calculating portfolio standard
deviation
Forecasted return
Year IBM Shell
2004
2005
2006
2007
2008
8%
10
12
14
16
16%
14
12
10
8
Portfolio Return
Calculation
(.50*8%) + (.50*16%)
(.50*10%) + (.50*14%)
(.50*12%) + (.50*12%)
(.50*14%) + (.50*10%)
(.50*16%) + (.50*8%)
Expected
Portfolio
Return
12%
12%
12%
12%
12%
5-17
Calculating portfolio standard
deviation (cont.)

Expected value of portfolio return, 2004-2008
12% + 12% + 12% + 12% + 12%
KP =
5
= 12%
5-18
Calculating portfolio standard
deviation (cont.)
n
P 

2
(k i  k ) /n - 1
i 1
 P  (12% - 12%)
2  (12% - 12%) 2  (12% - 12%) 2  (12% - 12%) 2  (12% - 12%) 2 /( 5  1)
 0%
5-19
Alternative Formula for Calculating
portfolio standard deviation
p 
2
W1  1
2  W2 2  2 2  2W1W2  1 2 r 12
W1  Proportion
of Asset 1
W2  Proportion
of Asset 2
 1  Standard Deviation of Asset 1
 1  Standard Deviation of Asset 2
r 12  Correlatio n Coefficien t between th e return of assets 1 and 2
5-20
Returns distribution for two perfectly
negatively correlated stocks (ρ = -1.0)
Stock W
Stock M
Portfolio WM
25
25
25
15
15
15
0
0
0
-10
-10
-10
5-21
Returns distribution for two perfectly
positively correlated stocks (ρ = 1.0)
Stock M’
Stock M
Portfolio MM’
25
25
25
15
15
15
0
0
0
-10
-10
-10
5-22
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
5-23
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. (e.g. War,
Inflation, High Interest Rates)
Firm-specific risk – portion of a security’s
stand-alone risk that can be eliminated through
proper diversification.
5-24
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.
CAPM : Ke= Rf + β(Rm – Rf)
Rf = Risk free rate of return
Rm = Market Return
β = Beta Coefficient
Ke = Required Return
5-25
Beta


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.
5-26
Comments on beta






If beta = 1.0, the security is just as risky as the
average stock.
If beta > 1.0, the security is riskier than average.
If beta < 1.0, the security is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5.
The beta coefficient for the market = 1
Betas May be positive or negative. But, positive is
the norm.
5-27
The Security Market Line (SML):
Calculating required rates of return
SML: ki = kRF + (kM – kRF) βi



Assume kRF = 8%, kM = 15% and βi =1.3
The market (or equity) risk premium is
RPM = kM – kRF = 15% – 8% = 7%.
ki = 8.0% + (15.0% - 8.0%)(1.30)
= 8.0% + (7.0%)(1.30)
= 8.0% + 9.1%
= 17.10%
5-28
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.
5-29
An example:
Equally-weighted two-stock portfolio


Create a portfolio with 50% invested in
HT and 50% invested in Collections.
The beta of a portfolio is the weighted
average of each of the stock’s betas.
βP = w1 β1 + w2 β2
βP = 0.5 (1.30) + 0.5 (-0.87)
βP = 0.215
5-30
Factor that shifts the SML

What if investors raise inflation expectations
by 3%, what would happen to the SML?
ki (%)
D I = 3%
SML2
SML1
18
15
11
8
Risk, βi
0
0.5
1.0
1.5
5-31
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?
ki (%)
D RPM = 3%
SML2
SML1
18
15
11
8
Risk, βi
0
0.5
1.0
1.5
5-32