Chapter 7
Portfolio Mean And Variance
®1999 South-Western College Publishing
1
Weight Asset 1
Portfolio
of
Assets
Weight Asset 2
Weight Asset 3
Weights Sum to 100%
®1999 South-Western College Publishing
2
Expected Rate Of Return On
The Portfolio E(R)
• Calculate All Possible Return on the
Portfolio, and Then Calculate its E(R)
• Use Equation 7.2
W1 • E(R1) + W2 • E(R2) = E(Rp)
• Both Methods Provide the Same Results
®1999 South-Western College Publishing
3

The expected return of a portfolio is a
weighted average of the component
expected returns.
E R portfolio   x i E  Ri 
n
i 1
where xi = the proportion invested in security i
®1999 South-Western College Publishing
4
Measuring Portfolio Risk
• Variance
• Standard Deviation
• Degree of Dependency
– Positive
– Negative
• Lower risk
• Lower risk premium
®1999 South-Western College Publishing
5
What Does A Risk-Averse
Investor Require?
• A Risk Premium
– Requires a risk premium that decreases as
the degree of dependency decreases
– The required risk premium is a function
of the asset’s variance and its dependency
with other assets
®1999 South-Western College Publishing
6
What Is The Risk Premium?
• Required to Compensate Investors for Risk
• Higher the Variance or Standard Deviation, the
Higher the Required Risk Premium
• Degree of Dependency Affects the Risk Premium
– The more negative the degree of dependency
– The lower the risk of the portfolio
– The lower the required risk premium
®1999 South-Western College Publishing
7
Summary
• One Asset
– Variance is measure of risk
– Higher the variance, the higher the required risk
premium
• More Than One Asset
– Risk is a function of both
• The asset’s variance
• The degree of dependency
– Portfolio’s variance (Key factor)
• Larger the variance
• Higher the risk premium
• Larger the risk premium on each asset
®1999 South-Western College Publishing
8
Covariance
Expected value of the Product of
Deviations From the Mean
®1999 South-Western College Publishing
9
Covariance
• Measures the Degree of Dependency of 2 Assets
– Positive
• Rates of return moves together
– Zero
• Rates of return have independent movements
– Negative
• Rates of return move in opposite directions
®1999 South-Western College Publishing
10
Correlation Coefficient
• Correlation Coefficient of 2 Stocks
Cor(RA, RB)
A,B =
A B
• Strength of Dependency
– +1 perfectly positive
– 0 no correlation
– - 1 perfectly negative
• Correlations are Directly Comparable
no units or $
®1999 South-Western College Publishing
11
Portfolio Variance
• Direct Method (easy to calculate)
– Calculate rate of return
– Calculate variance
• Indirect Method (demonstrates relationship)
– Equation based on variance & covariance
– Sheds light on factors affecting risk
reduction
®1999 South-Western College Publishing
12

The total risk of a portfolio comes from the
variance of the components and from the
relationships among the components.
two-security
portfolio = riskA + riskB + interactive risk
n
risk
  x   x   2 xa xb  ab a b ,  xi  1
2
p
2
a
2
a
2
b
2
b
i 1
where   portfolio variance
x i  proportion of portfolio invested in stock i
 i  standard deviation of stock i
 ab  correlatio n coefficien t between a and b
2
p
13
Low Correlation
• Reduce Portfolio Fluctuation
• Achieved by Diversification
• Attractive to Risk-Averse Investors
®1999 South-Western College Publishing
14
Investment Alternatives
(Given in the problem)
Economy
Recession
Below avg.
Average
Above avg.
Boom
Prob.
0.1
0.2
0.4
0.2
0.1
1.0
©
1998South-Western
The Dryden Press
®1999
College Publishing
T-Bill
HT
Coll
8.0% -22.0% 28.0%
8.0
-2.0
14.7
8.0
20.0
0.0
8.0
35.0 -10.0
8.0
50.0 -20.0
USR
MP
10.0% -13.0%
-10.0
1.0
7.0
15.0
45.0
29.0
30.0
43.0
15
Portfolio Risk and Return
Assume a two-stock portfolio with
$50,000 in HT and $50,000 in
Collections.
^
Calculate kp and p.
©
1998South-Western
The Dryden Press
®1999
College Publishing
16
^
Portfolio Return, kp
^
kp is a weighted average:
n
^
^
kp = S wikw.
i=1
^
kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.
^
^
^
kp is between kHT and kCOLL.
©
1998South-Western
The Dryden Press
®1999
College Publishing
17
^
Portfolio Return, kp
Looking at this more closely:
^
kp = $50,000(17.4%) + $50,000(1.7%)
$100,000
= $8,700 + $850 =
$9,550
$100,000
$100,000
^
kp = 9.550%  9.6%.
®1999 South-Western College Publishing
18
Alternative Method
Estimated Return
Economy
Recession
Below avg.
Average
Above avg.
Boom
Prob.
HT
Coll.
Port.
0.10
0.20
0.40
0.20
0.10
-22.0%
-2.0
20.0
35.0
50.0
28.0%
14.7
0.0
-10.0
-20.0
3.0%
6.4
10.0
12.5
15.0
^
kp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40
+ (12.5%)0.20 + (15.0%)0.10 = 9.6%.
©
1998South-Western
The Dryden Press
®1999
College Publishing
19
p
1/ 2
  3.0 - 9.6  0 .10



   6 . 4 - 9 .6  2 0 .20 




2
=   10 . 0 - 9 .6  0 .40  = 3.3%.


2
  12 .5 - 9 .6  0 .20 


  15 . 0 - 9 .6  2 0 .10 


2
CVp = 3.3% = 0.34.
9.6%
©
1998South-Western
The Dryden Press
®1999
College Publishing
20
 p = 3.3% is lower than average for HT
and Coll of 16.7%.
• (in fact, p = 3.3% is much lower than that
of either stock (20% and 13.4%))
 \ Portfolio provides average k but much
lower than average risk.
• Reason: less than perfect correlation (and,
in this specific case, negative correlation).
©
1998South-Western
The Dryden Press
®1999
College Publishing
21