CHAPTER 5
Risk and Rates of Return
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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
Risk



The chance that some unfavorable
event will occur.
In case of investment it is the chance
that return will fall.
Risk is higher in stock investment than
debt investment.
5-3
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.
5-4
Probability Distribution

A listing of all possible outcomes, and
the probability (chance of occurrence
out of 1) of each occurrence.
5-5
Probability distributions

Can be shown graphically.
5-6
Expected Rate of return


The rate of return expected to be realized from an
investment.
The weighted average of probability distribution of
possible results.
Company
IBM
Rate of Return (Ki)
-22%
-2
20
35
50
Probability(Pi)
10%
20
40
20
10
5-7
Return: Calculating the expected
return for each alternative
^
k  expected rate of return
^
n
k   k i Pi
i 1
^
k IBM  (-22%) (0.1)  (-2%) (0.2)
 (20%) (0.4)  (35%) (0.2)
 (50%) (0.1)  17.4%
5-8
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?
5-9
Standard Deviation




The tighter the probability distribution the lower the
risk, since the range of possible returns become
smaller.
Standard deviation is a statistical measure of the
variability of a set of observation.
Using sigma we can see how much the return can
deviate away from the expected or weighted
average. This is the measure of stand alone risk.
The smaller the standard deviation the tighter the
deviating range of returns are and thus the lower the
risk.
5-10
Standard Deviation

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
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
Pi = Probability of outcome
ri = Rate of return
r^ = Expected rate of return
N = number of observations
∑ = Summation
√ = Square root
5-11
Risk: Calculating the standard
deviation for each alternative
  Standard deviation
  Variance  2

n
 (k  k̂ ) P
i1
2
i
i
5-12
Standard deviation calculation

n
 (k
i 1
^
i
 k ) 2 Pi
(-22.0 - 17.4) (0.1)  (-2.0 - 17.4) (0.2) 


  (20.0 - 17.4) 2 (0.4)  (35.0 - 17.4) 2 (0.2) 
 (50.0 - 17.4) 2 (0.1)



2
 IBM
 IBM  20.04%
 T -bills  0.0%
2
1
2
 Shell  13.4%
 USR  13.8%
 M  15.3%
5-13
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.
The larger σi is, thus higher the risk.
Difficult to compare standard deviations,
because return has not been accounted
for.
5-14
Comparing risk and return:
Coefficient of Variation
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
Lets compare two stocks
Stock A:



Stock B:

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Expected Return = 8%
Standard Deviation = 15%
Expected Return = 15%
Standard Deviation = 29%
Which stock to invest in?
5-15
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.

Std dev 
CV 
 ^
Mean
k
5-16
Coefficient of Variation

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Coefficient of Variation is the ratio of
stand alone risk and expected return.
CV = σ/r^
Stock A (CV) = 15/8 = 1.875
Stock B (CV) = 29/15 = 1.933
So it is better to invest in Stock A since
it has lower risk per unit of return
compared to Stock B
5-17
Investor attitude towards risk


Risk aversion – It is assumed that a
rational investor is risk-averse. Risk-averse
investors dislike risk and will not purchase
risky asset unless compensated with
higher rate of return.
Risk premium – The extra amount of
return expected from a riskier asset
compared to a less risky asset is called risk
premium, which serves as compensation
for investors to hold riskier securities.
5-18
Portfolio Risk & Return



In cases of stocks or shares, investors rarely
invest in only one. They invest their money
in a group of shares to create a portfolio.
Risk and return of an individual stock should
be analyzed in terms of how the security
affects the risk and return of the portfolio in
which it is held.
So instead of stand alone risks and expected
returns of each of the stocks, what becomes
important is the entire portfolio’s risk and
5-19
return.
Portfolio construction:
Risk and return
Assume a two-stock portfolio is created with
$50,000 invested in IBM and $50,000 in Shell.

Expected return of a portfolio is a
weighted average of each of the
component assets of the portfolio.
5-20
Calculating portfolio Expected Return
The weighted average of the expected
returns on the individual assets held in the
portfolio

^
kp 
n

^
wi ki
i 1
^
k p  0.5 (17.4%)  0.5 (1.7%)  9.6%
5-21
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 (ki)
12%
12%
12%
12%
12%
5-22
Calculating portfolio standard
deviation (cont.)

Expected value of portfolio return, 20042008
12% + 12% + 12% + 12% + 12%
KP =
5
= 12%
5-23
Calculating portfolio standard
deviation (cont.)
n
P 

P 
(12% -12%) 2  (12% -12%) 2  (12% -12%) 2  (12% -12%)2  (12%-12%)2 / (5  1)
i 1
(k i  kp) 2 /n-1
 0%
5-24
Alternative Formula for Calculating
portfolio standard deviation
 p  W12  12  W22  2 2  2W1W2 1 2r12
W1  Proportion of Asset 1
W2  Proportion of Asset 2
 1  Standard Deviation of Asset 1
 1  Standard Deviation of Asset 2
r12  Correlation Coefficient between the return of assets 1 and 2
5-25
Portfolio Risk


Unlike expected portfolio return, portfolio risk is
not the weighted average of individual stock’s
standard deviation (stand alone risk).
Portfolio risk depends on correlation between the
stocks i.e. the tendency of two variables to move
together.

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This gives an idea of the degree of
diversification in the portfolio.
Correlation coefficient (ρ) is a standardized
measure, in between -1 and 1, for the degree of
relationship between two variables.
5-26
Diversification
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Securities held in portfolio reduces the overall
risk an investor is exposed to.
This is due to the effect of Diversification.
One stock can earn extra return to cover up
the loss of another stock’s negative return.
Since it is assumed that rational investors are
risk averse thus most investments are done in
portfolio of stocks
5-27
Graphical Presentation of
Diversification
5-28
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
5-29
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
5-30
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
5-31
Breaking down sources of risk
Stand-alone risk = Market risk + Diversifiable risk


Market risk – Market risk is the risk associated with the
entire market and cannot be diversified away.
 It is the risk that remains after diversification and
thus known as nondiversifiable, or systematic or
beta risk. Caused by political and macroeconomic
factors.(e.g. War, Inflation, High Interest Rates)
Diversifiable risk–is that portion of a security’s standalone risk associated with random events, or news.
 It can be completely eliminated by proper
diversification.
 Also known as company specific risk, unsystematic
risk
5-32
5-33
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.
5-34
Relevant Risk & Beta Coefficient

Relevant risk the portion of an
individual stock’s risk that contributes to
the market risk of it’s portfolio.


It is the extent to which a given stock’s
returns move up and down with the
market, measured by Beta Coefficient (ϐ).
It is considered that all other risks are
diversified away in a portfolio except
the relevant risks of the individual
5-35
stocks.
Comments on beta
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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.
5-36
Finding Beta Coefficient
Covariance of Returns between th e stock and the market
Variance of the Market Returns
Cov(R s R M )
Beta 
Var(R M )
Beta 



Covariance is a measure of the degree to which
returns on two risky assets move in tandem. A
positive covariance means that asset returns move
together. A negative covariance means returns move
inversely. Similar to Correlation Coefficient.
Variance is the square of standard deviation
Beta of Market = 1.0
5-37
Portfolio Beta

Beta of a portfolio of securities is the
weighted average of the individual
securities’ beta.
5-38
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
5-39
Capital Asset Pricing Model
(CAPM)

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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.
Required rate of return on a stock = Risk-free rate of return
+ Risk premium of that stock.
ri = rRF + (rM – rRF)bi
CAPM : Ke= Rf + β(Rm – Rf)
Rf = Risk free rate of return
Rm = Market Return
β = Beta Coefficient
5-40
Ke = Required Return
Returns and Premiums
5-41
Security Market Line (SML)

A graph of CAPM equation. It shows the relationship
between risk as measured by beta and required
rates of return on individual securities.
5-42
The Security Market Line (SML):
Calculating required rates of return
SML: ki = kRF + (kM – kRF) βi

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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%
5-43