Fundamental
of Financial
Management
Chapter 08
Risk and Rate of
Return
By:S.Zakir Abbas Zaidi
1
Key Concepts and Skills
• Know how to calculate expected returns
• Understand the impact of diversification
• Understand the systematic risk principle
• Understand the security market line
• Understand the risk-return trade-off
2
Simple Returns
• The return from holding an investment over some
period – say, a year – is simply any cash payments
received due to ownership, plus the change in market
price, divided by the beginning price.1 You might, for
example, buy for $106 a security that and one year from
worth $107 one year later. The return would be ($107 +
$106)/$100 = 1%.
3
Expected Returns
• Expected returns are based on the probabilities of possible outcomes
• In this context, “expected” means “average” if the process is
repeated many times
• The “expected” return does not even have to be a possible return
n
E ( R)   pi Ri
i 1
4
Example: Expected Returns
• Suppose you have predicted the following returns for stocks C and T in
three possible states of nature. What are the expected returns?
•
•
•
•
State
Probability
Boom
0.3
Normal
0.5
Recession
???
C
0.15
0.10
0.02
T
0.25
0.20
0.01
• E(RC) = .3(.15) + .5(.10) + .2(.02) = .099 OR 9.9%
• E(RT) = .3(.25) + .5(.20) + .2(.01) = .177
OR 17.7%
5
Variance and Standard Deviation
• Variance and standard deviation still measure the volatility of returns
• Using unequal probabilities for the entire range of possibilities
• Weighted average of squared deviations
n
σ 2   pi ( Ri  E ( R )) 2
i 1
6
Example: Variance and Standard Deviation
• Consider the previous example. What are the variance and
standard deviation for each stock?
• Stock C
 2 = .3(.15-.099)2 + .5(.1-.099)2 + .2(.02-.099)2
= .002029
 = .045
• Stock T
 2 = .3(.25-.177)2 + .5(.2-.177)2 + .2(.01-.177)2
= .007441
 = .0863
7
Use of Standard Deviation Information
• So far we have been working with a discrete (noncontinuous) probability distribution, one where a
random variable, like return, can take on only certain values within an interval. In such cases we do
not have to calculate the standard deviation in order to determine the probability of specific
outcomes. To determine the probability of the actual return in our example being less than zero, we
look at the shaded section of Table 5.1 and see that the probability is 0.05 + 0.10 = 15%.
• The procedure is slightly more complex when we deal with a continuous distribution.
• Suppose that our return distribution had been approximately normal with an expected return equal
to 9 percent and a standard deviation of 8.38 percent. Let’s say that we wish to find the probability
that the actual future return will be less than zero. We first determine how many standard
deviations 0 percent is from the mean (9 percent). To do this we take the difference between these
two values, which happens to be −9 percent, and divide it by the standard deviation. In this case
the result is −0.09/0.0838 = −1.07 standard deviations. (The negative sign reminds us that we are
looking to the left of the mean.) In general, we can make use of the formula
8
Use of Standard Deviation Information
• So far we have been working with a discrete (noncontinuous) probability distribution, one where a
random variable, like return, can take on only certain values within an interval. In such cases we do
not have to calculate the standard deviation in order to determine the probability of specific
outcomes. To determine the probability of the actual return in our example being less than zero, we
look at the shaded section of Table 5.1 and see that the probability is 0.05 + 0.10 = 15%.
• The procedure is slightly more complex when we deal with a continuous distribution.
• Suppose that our return distribution had been approximately normal with an expected return equal
to 9 percent and a standard deviation of 8.38 percent. Let’s say that we wish to find the probability
that the actual future return will be less than zero. We first determine how many standard
deviations 0 percent is from the mean (9 percent). To do this we take the difference between these
two values, which happens to be −9 percent, and divide it by the standard deviation. In this case
the result is −0.09/0.0838 = −1.07 standard deviations. (The negative sign reminds us that we are
looking to the left of the mean.) In general, we can make use of the formula
9
Another Example
• Consider the following information:
•
•
•
•
•
State
Boom
Normal
Slowdown
Recession
Probability
.25
.50
.15
.10
Ret. on ABC, Inc.
.15
.08
.04
-.03
• What is the expected return?
• What is the variance?
• What is the standard deviation?
10
Coefficient of Variation
• Coefficient of Variation StockC = 0.045/.099 = 0.45
• Coefficient of Variation StockT = 0.0863/0.177= 0.49
• Thus, StockT is more riskier than StockC on the basis of this criterion.
11
The Scale Problem: an Example
Prob
10%
15%
50%
15%
10%
E(R)
Variance
Std. Dev.
C.V.
Potential Returns
ABC
XYZ
-12%
-24%
-5%
-10%
2%
4%
9%
18%
16%
32%
2.0%
4.0%
0.00539
0.02156
7.34%
14.68%
3.6708
3.6708
Is XYZ really twice
as risky as ABC?
No!
Measuring Stand-Alone Risk
• Standard deviation measures the stand-alone risk of an investment.
• The larger the standard deviation, the higher the probability that
returns will be far below the expected return.
• Coefficient of variation is an alternative measure of stand-alone risk.
Portfolios
• A portfolio is a collection of assets
• An asset’s risk and return are important to how the stock affects the
risk and return of the portfolio
• The risk-return trade-off for a portfolio is measured by the portfolio
expected return and standard deviation, just as with individual assets
14
The Expected Return of a Portfolio
Example: Portfolio Weights
• Suppose you have $15,000 to invest and you have purchased
securities in the following amounts. What are your portfolio weights
in each security?
•
•
•
•
$2,000 of DCLK
$3,000 of KO
$4,000 of INTC
$6,000 of KEI
•DCLK: 2/15 = .133
•KO: 3/15 = .2
•INTC: 4/15 = .267
•KEI: 6/15 = .4
16
Portfolio Expected Returns
• The expected return of a portfolio is the weighted
average of the expected returns of the respective
assets 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 as we did with
individual securities
17
Example: Expected Portfolio Returns
• Consider the portfolio weights computed previously. If the individual
stocks have the following expected returns, what is the expected
return for the portfolio?
• DCLK: 19.65%
• KO: 8.96%
• INTC: 9.67%
• KEI: 8.13%
• E(RP) = .133(19.65) + .2(8.96) + .267(9.67) + .4(8.13) = 10.24%
18
Determining Covariance and Correlation
• To find the risk of a portfolio, one must
know the degree to which the stocks’ returns move together.
The Portfolio Standard Deviation
• The portfolio standard deviation can be thought of as a weighted
average of the individual standard deviations plus terms that
account for the co-movement of returns
• For a two-security portfolio:
 P  w   w   2r1,2 1 2 w1w2
2
1
2
1
2
2
2
2
An Example: Perfect Pos. Correlation
State of Economy Probability
Recession
25%
Moderate Growth
50%
Boom
25%
Expected Return
Standard Deviation
Correlation
P 
Potential Returns
ABC
XYZ
50/50 Portfolio
2%
2%
2%
8%
8%
8%
14%
14%
14%
8%
8%
8%
4.24%
4.24%
4.24%
1.00
.52  0.0424 .52  0.0424  21.00 0.0424 0.0424 0.5 0.5  0.0424
2
2
An Example: Perfect Neg. Correlation
State of Economy Probability
Recession
25%
Moderate Growth
50%
Boom
25%
Expected Return
Standard Deviation
Correlation
P 
Potential Returns
ABC
XYZ
50/50 Portfolio
2%
14%
8%
8%
8%
8%
14%
2%
8%
8%
8%
8%
4.24%
4.24%
0.00%
-1.00
.52  0.0424 .52  0.0424  2  1.00 0.0424 0.0424 0.5 0.5  0.00
2
2
An Example: Zero Correlation
State of Economy Probability
Recession
25%
Moderate Growth
50%
Boom
25%
Expected Return
Standard Deviation
Correlation
Potential Returns
ABC
XYZ
50/50 Portfolio
2%
2%
2%
8%
2%
5%
14%
2%
8%
8%
2%
5%
4.24%
0.00%
2.12%
0.00
 P  .52  0.0424 .52  0.0424  2 0 0.0424 0.0424 0.5 0.5  0.0212
2
2
Determining Covariance
and Correlation (cont'd)
• Covariance
• The expected product of the deviations of two returns from their means
• Covariance between Returns Ri and Rj
Cov(Ri ,R j )  E[(Ri  E[ Ri ]) (R j  E[ R j ])]
• Estimate of the Covariance from Historical Data
1
Cov(Ri ,R j ) 
(Ri ,t  Ri ) (R j ,t  R j )

t
1returns tend to move together.
If the covariance is positive,TN
thetwo
•
• If the covariance is negative, the two returns tend to move in opposite
directions.
Determining Covariance
and Correlation (cont'd)
• Correlation
• A measure of the common risk shared by stocks that does not depend on their
volatility
Corr (Ri ,R j ) 
Cov(Ri ,R j )
SD(Ri ) SD(R j )
• The correlation between two stocks will always be between –1 and +1.
Diversification & Correlation
• The extent to which adding stocks to a portfolio
reduces its risk depends on the degree of
correlation among the stocks: The smaller the
correlation coefficients, the lower the risk in a large
portfolio. If we could find a set of stocks whose
correlations were zero or negative, all risk could be
eliminated. However, in the real world, the
correlations among the individual stocks are
generally positive but less than -1.0, so some but
not all risk can be eliminated.
26
Variance, Correlation and Beta from Historical Data
Year X
M
M^2
(X-Avg) (M - Avg) (X -Avg).(M -Avg)
1 14% 12%
0.020 0.014
6%
7%
0.41%
2 19% 10%
0.036 0.010
11%
5%
0.53%
3 -16% -12%
0.026 0.014
-24%
-17%
4.13%
4
1%
0.001 0.000
-5%
-4%
0.21%
5 20% 15%
0.040 0.023
12%
10%
1.18%
5% 0.1222 0.0614
0%
0%
6.45%
3%
8%
SD
X^2
0.15 0.109
COV
1.61%
CORR
0.98
Beta
1.35
The Concept of Beta
• A stock’s risk consists of two components, market risk and
diversifiable risk
• Diversifiable risk can be eliminated by diversification
• Beta Coefficient, b – is a metric that shows the extent to which a
given stock’s returns move up and down with the stock market. Beta
thus measures market risk.
28
Calculation of beta
29
Unsystematic risk Vs systematic Risk
Diversifiable Risk
• That part of a security’s risk associated with random events; it can be
eliminated by proper diversification. This risk is also known as
company specific, or unsystematic, risk.
Market Risk
• The risk that remains in a portfolio after diversification has
eliminated all company-specific risk. This risk is also known as
nondiversifiable or systematic or beta risk.
Beta measures market Risk
THE RELATIONSHIP BETWEEN RISK AND
RATES OF RETURN
Security Market Line
(SML) Equation
Betas: Relative Volatility of Stocks H, A, and L
The Security Market Line (SML)
Shift in the SML Caused by an Increase in Expected Inflation
Shift in the SML Caused by Increased Risk Aversion
Beta of a Portfolio
• The beta of a portfolio is a weighted average of its individual
securities’ betas:
• For example, if an investor holds a $100,000 portfolio consisting of
$33,333.33 invested in each of three stocks, and if each of the stocks
has a beta of 0.7, then the portfolio’s beta will be
Exercise
Quiz # 04
• Define the following terms using graphs or equations to illustrate your
answers whenever feasible:
Quiz # 04