Return and Risk for
Capital Market Securities
Rate of Return Concepts
• Dollar return
• Number of $ received over a period
(one year, say)
• Sum of cash distributed plus capital
gain (loss)
• Percentage return
• Dollar return/(beginning-of-period
value)
• % cash distribution + % capital gain
• Real vs. Nominal return
• Real % return, r, related to nominal %
return, R, by: 1+R = (1+r)(1+h)
Standard Deviation as a Measure of Risk
1 T
2
Var ( R) 
(
R

R
)

t
T  1 t 1
SD( R)    Var ( R)
• Variance and standard deviation give equal weight to
observations above & below mean
• These make sense as risk measures if the return
distribution is symmetrical
Investor Portfolio Choices:
Risk and Return
• How does holding securities in
portfolios affect the risk-return
combinations available to investors?
• When and why does it pay to diversify
across securities?
Return Distributions
• Expected Return and Standard Deviation
for stocks and portfolios with S possible
scenarios
S
E ( R)   Rs prob( s)
s 1
S
var( R)   ( R)   Rs  E ( R)  prob( s)
2
2
s 1
stdev( R)   ( R)   ( R)
2
Portfolio Return and Variance
S
N
s 1
i 1
E ( R p )   R ps prob ( s )   X i E ( Ri )
   R ps  E ( R p )  prob( s )
S
2
p
2
s 1
• What’s the relationship between portfolio variance
and variances of individual securities?
Covariance
S
cov( RA , RB )   RAs  E ( RA ) RBs  E ( RB )  prob ( s )
s 1
• Covariance measures the extent to
which two securities’ returns tend to
vary together
Correlation
 AB 
cov( RA , RB )
 A B
• Correlation is a “standardized” measure of
covariance
• AB varies between -1 (perfect negative
correlation and +1 (perfect positive
correlation)
Portfolio Variance and Correlation (2 Securities)
  X   X   2 X A X B cov( RA , RB )
2
p
2
A
2
A
2
B
2
B
  X   (1  X A )   2 X A (1  X A )  AB A B
2
p
2
A
p  
2
A
2
p
2
2
B
When Does Diversification Pay?
a) Combine two securities with the
lowest possible return correlation
b) Combine large numbers of identical
securities whose returns are less than
perfectly correlated
a) Diversifying with 2 Securities
• With perfect positive correlation, combining
securities does not improve the risk-return
possibilities (opportunity set)
• The lower the correlation, the more the
opportunity set improves
• With perfect negative correlation, we can
eliminate risk altogether
b) Diversifying Across Many,
Identical Securities
p 

(n  1)
2


n
n
2
n     p  
2
• Diversification can eliminate “unsystematic”
risk
• Systematic risk stems from the common
thread running through all securities’ returns
and cannot be diversified away
Diversification and the Reward for Risk
• Total Risk = Systematic Risk + Unsystematic
Risk
• Well diversified portfolios should contain
almost entirely systematic risk
• Investors shouldn’t expect a reward for
bearing unsystematic risk, since that can be
eliminated (fairly cheaply) through
diversification
Measuring Systematic Risk
• Suppose the return
on Security i at time
t takes the form:
Rit   i   i RMt   t
Expected Return
“Surprise”
Return
Measuring Systematic Risk Using Regression
• Suppose we regress returns on Security i against the
returns on the market portfolio
• The regression error term represents it, the
“surprise” return component
• The slope coefficient, it, represents the extent to
which i moves with the market
• R2 = % total risk that is systematic
(1-R2 = % unsystematic)
Properties of Beta
• From the properties of linear regression, we
can say about beta:
cov( Ri , RM )  iM  ( Ri ) ( RM )
i 

2
 ( RM )
 2 ( RM )
  iM
 ( Ri )
 M  1
 ( RM )
Beta Estimates (Table 11.5)
(updated from finance.yahoo.com)
Company
Beta
McGraw-Hill
0.89
MMM
0.82
McDonald’s
1.12
Bed, Bath & Beyond
1.46
Home Depot
1.01
Dell
1.13
eBay
3.90
Computer Associates (CA)
1.86
Portfolio Beta
 p   X i i
i
• Portfolio beta is the weighted average of the
individual security betas
• Since M = 1, the average beta of all
securities is equal to 1
The Security Market Line
• All securities should plot along the same
security market line
• If they didn’t, investors would shun one
security in favor of another
E ( RA )  R f
A

E ( RB )  R f
B
Capital Asset Pricing Model (CAPM)
• Since the market portfolio should also plot
along the security market line, for any
security i:
E ( Ri )  R f
i

E ( RM )  R f
M

 E ( RM )  R f
E ( Ri )  R f   i E ( RM )  R f

Interpreting CAPM
E ( Ri )  R f   i E ( RM )  R f 
Pure time
value
Reward for bearing
systematic risk