CHAPTER 8
Risk and Rates of Return
Outline



Stand-alone return and risk
Return
Expected return
Stand-alone risk
Portfolio return and risk
Portfolio return
Portfolio risk
Link Risk & return: CAPM / SML
Beta
CAPM and computing
SML
5-1
I-1: Return: What is my
reward of investing?
5-2
Investment returns
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%.
The rate of return on an investment can be
calculated as follows:

Return =
(Amount received – Amount invested)
________________________
Amount invested
5-3
Rates of Return: stocks
P D
P
HPR 
P
1
0
1
0
HPR = Holding Period Return
P1 = Ending price
P0 = Beginning price
D1 = Dividend during period one
Define return?
Your gain per dollar investment
5-4
Rates of Return: Example
Ending Price =
Beginning Price =
Dividend =
24
20
1
HPR = ( 24 - 20 + 1 )/ ( 20) = 25%
5-5
I-2: Expected return: describe
the uncertainty
5-6
Calculating expected return


Two scenarios and the concept of
expected return
Extending to more than two scenarios
5-7
Investment alternatives
Economy
Prob.
T-Bill
HT
Coll
USR
MP
Recession
0.1
5.5%
-27.0%
27.0%
6.0%
-17.0%
weak
0.2
5.5%
-7.0%
13.0%
-14.0%
-3.0%
normal
0.4
5.5%
15.0%
0.0%
3.0%
10.0%
strong
0.2
5.5%
30.0%
-11.0%
41.0%
25.0%
Boom
0.1
5.5%
45.0%
-21.0%
26.0%
38.0%
5-8
Calculating the expected return
^
r  expected rate of return
^
N
r   ri Pi
i 1
^
r HT  (-27%) (0.1)  (-7%) (0.2)
 (15%) (0.4)  (30%) (0.2)
 (45%) (0.1)  12.4%
5-9
Summary of expected returns
HT
Market
USR
T-bill
Coll.
Expected return
12.4%
10.5%
9.8%
5.5%
1.0%
HT has the highest expected return, and appears
to be the best investment alternative, but is it
really? Have we failed to account for risk?
5-10
I-3. Stand-alone risk
5-11
Calculating standard deviation
  Standard deviation
  Variance  2
σ 
N
2
ˆ
(r

r
)
Pi
 i
i 1
5-12
Standard deviation for each investment

N

i 1
^
(ri  r )2 Pi
(5.5 - 5.5) (0.1)  (5.5 - 5.5) (0.2) 


2
2
  (5.5 - 5.5) (0.4)  (5.5 - 5.5) (0.2) 


2

(5.5
5.5)
(0.1)


2
 T  bills
 T  bills  0.0%
 HT  20.0%
2
1
2
 Coll  13.2%
 USR  18.8%
 M  15.2%
5-13
Comparing standard deviations
Prob.
T - bill
USR
HT
0
5.5 9.8
12.4
Rate of Return (%)
5-14
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.
Larger σi is associated with a wider
probability distribution of returns.
5-15
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 a risk free
asset, which serves as compensation for
investors to hold riskier securities.
5-16
Comparing risk and return
Security
Risk, σ
T-bills
HT
Expected
return, ^
r
5.5%
12.4%
Coll*
USR*
Market
1.0%
9.8%
10.5%
13.2%
18.8%
15.2%
0.0%
20.0%
* Seem out of place.
5-17
Selected Realized Returns,
1926 – 2001
Small-company stocks
Large-company stocks
L-T corporate bonds
Average
Return
17.3%
12.7
6.1
Standard
Deviation
33.2%
20.2
8.6
Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation
Edition) 2002 Yearbook (Chicago: Ibbotson Associates, 2002), 28.
5-18
Coefficient of Variation (CV)
A standardized measure of dispersion about
the expected value, that shows the risk per
unit of return.
Standard deviation 
CV 

Expectedreturn
rˆ
5-19
Risk rankings,
by coefficient of variation
T-bill
HT
Coll.
USR
Market


CV
0.0
1.6
13.2
1.9
1.4
Collections has the highest degree of risk per unit
of return.
HT, despite having the highest standard deviation
of returns, has a relatively average CV.
5-20
II: Risk and return in a
portfolio
5-21
Portfolio construction:
Risk and return
Assume a two-stock portfolio is created with
$50,000 invested in both HT and Collections.


Expected return of a portfolio is a
weighted average of each of the
component assets of the portfolio.
Standard deviation is a little more tricky
and requires that a new probability
distribution for the portfolio returns be
devised.
5-22
II-1. Portfolio return
5-23
Calculating portfolio expected return
Economy
Prob.
HT
Coll
Recession
0.1
-27.0% 27.0%
weak
0.2
-7.0%
13.0%
normal
0.4
15.0%
0.0%
strong
0.2
30.0% -11.0%
Boom
0.1
45.0% -21.0%
Port.
7.5%
^
r p  0.10 (0.0%)  0.20 (3.0%)  0.40 (7.5%)
 0.20 (9.5%)  0.10 (12.0%)  6.7%
5-24
Calculating portfolio expected return
Economy
Prob.
HT
Coll
Port.
Recession
0.1
-27.0% 27.0%
0.0%
weak
0.2
-7.0%
13.0%
3.0%
normal
0.4
15.0%
0.0%
7.5%
strong
0.2
30.0% -11.0%
Boom
0.1
45.0% -21.0% 12.0%
9.5%
^
r p  0.10 (0.0%)  0.20 (3.0%)  0.40 (7.5%)
 0.20 (9.5%)  0.10 (12.0%)  6.7%
5-25
An alternative method for determining
portfolio expected return
^
r p is a weighted average :
^
N
^
r p   wi r i
i 1
^
r p  0.5 (12.4%)  0.5 (1.0%)  6.7%
5-26
II-2. Portfolio risk and beta
5-27
Calculating portfolio standard
deviation and CV
 0.10 (0.0 - 6.7) 

2 
 0.20 (3.0 - 6.7) 
 p   0.40 (7.5 - 6.7)2 
 0.20 (9.5 - 6.7)2 


2
 0.10 (12.0 - 6.7) 
2
1
2
 3.4%
3.4%
CVp 
 0.51
6.7%
5-28
Comments on portfolio risk
measures




σp = 3.4% is much lower than the σi of
either stock (σHT = 20.0%; σColl. = 13.2%).
σp = 3.4% is lower than the weighted
average of HT and Coll.’s σ (16.6%).
Therefore, the portfolio provides the
average return of component stocks, but
lower than the average risk.
Why? Negative correlation between stocks.
5-29
Returns distribution for two perfectly
negatively correlated stocks (ρ = -1.0)
Stock W
Stock M
Portfolio WM
40
40
40
15
15
15
0
0
0
-10
-10
-10
5-30
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
5-31
Creating a portfolio:
Beginning with one stock and adding
randomly selected stocks to portfolio



σp decreases as stocks 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%.
5-32
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
5-33
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.
5-34
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.
Portfolio beta is a weighted average of
its individual securities’ beta
5-35
Calculating betas


Run a regression of past returns of a
security against past returns on the
market.
The slope of the regression line is
defined as the beta coefficient for the
security.
5-36
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.
5-37
III: CAPM
5-38
What risk do we care?


Stand alone?
Risk that can not be diversified?
5-39
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.
5-40
Capital Asset Pricing Model
(CAPM)



Model linking risk and required returns. CAPM
suggests that a stock’s required return equals
the risk-free return plus a risk premium that
reflects the stock’s risk after diversification.
ri = rRF + (rM – rRF) bi
Risk premium RP: additional return to take
additional risk
The market (or equity) risk premium is (rM – rRF)
5-41
Calculating required rates of return





rHT
rM
rUSR
rT-bill
rColl
=
=
=
=
=
=
5.5%
5.5%
5.5%
5.5%
5.5%
5.5%
+
+
+
+
+
+
(5.0%)(1.32)
6.6%
(5.0%)(1.00)
(5.0%)(0.88)
(5.0%)(0.00)
(5.0%)(-0.87)
=
=
=
=
=
12.10%
10.50%
9.90%
5.50%
1.15%
5-42
Applying CAPM




Portfolio beta: Beta of a portfolio is a weighted
average of its individual securities’ betas.
Computing other variables: risk free rate, market
return, market risk premium
Computing the difference of return between two
stocks.
Computing price in the future when current price is
given
5-43
CAPM in a graph: the
Security Market Line
SML: ri = 5.5% + (5.0%) bi
ri (%)
SML
.
..
HT
rM = 10.5
rRF = 5.5
-1
.
Coll.
. T-bills
0
USR
1
2
Risk, bi
5-44
Applying CAPM in real world(optional)



Total Risk vs. Beta. An experiment
The difference between commonly
referred risk and beta (Are these high
beta stocks really high beta)
High risk( total risk), low beta stock can
hedge your portfolio (reduce portfolio
risk)
5-45
Problems with CAPM (optional)




Measurement error of beta
Empirical relationship between beta and
return is weak
Size and Book-to-market factors
Momentum
5-46
Optional: diversification in real world





Stock Index ETF
Style: Value vs. Growth
Style: Small vs. Big
Performance, Risk, Expense(0.1% is low,
0.5% is about average)
Examples:




Vanguard Small Cap Value ETF VBR
Small growth: VBK
Large value: VTV
Large growth: VUG
5-47
diversification in real world

Foreign ETF:RBL

Pros:



More diversification
Low PE ratio
cons




Higher risk
Higher expense: 0.6% vs. 0.1%
Higher spread
Poor prior performance
5-48