Transcript CH10.ppt

Chapter 10
Portfolio Theory
Investments
Figure 3 : Histogram US real return (Feb 1915 – April 2004)
120
100
Frequency
80
60
40
20
0
-0.15
-0.11
-0.07
-0.03
0.01
0.05
0.09
0.13
Correlations and Covariance
3
 Correlation is the degree of association between
two variables
 Covariance is the product moment of two random
variables about their means
 Correlation and covariance are related and generally
measure the same phenomenon
Correlations and Covariance (cont’d)
4
COV ( A, B)   AB  E ( A  A)( B  B ) 
 AB 
COV ( A, B)
 A B
Correlations and Covariance (cont’d)
5
COV ( A, B)   AB  E ( A  A)( B  B ) 
 AB 
COV ( A, B)
 A B
Portfolio Standard deviation
Figure 1 : Random selection of stocks
22%
Specific/Diversifiable /
idiosyncratic risk
10.8%
??%
9.8%
Market / Systematic/
non-diversifiable risk
0
1
2 ... 10
20
35
Number of shares in portfolio
© K. Cuthbertson and D. Nitzsche
Table 2 : Individual stock returns
Returns
State Interest Growth
Prob.
Stock 1
Stock 2
1
High
Low
0.25
-5
45
2
High
High
0.25
5
35
3
Low
Low
0.25
10
10
4
Low
High
0.25
25
-5
© K. Cuthbertson and D. Nitzsche
Table 3 : Summary statistics
Stocks
Stock 1
Stock 2
Mean, ERi
8.75%
21.25%
Std. dev, i
10.83%
19.80%
Correlation
-0.9549
Covariance
-204.688
© K. Cuthbertson and D. Nitzsche
Table 4 : Risky portfolio
Alternativ
e risky
portf.
Share of
Portfolio
Stock 2,
w2
0
ERp
p
A
Stock 1,
w1
1
8.75%
10.83%
G
0.75
0.25
11.88%
3.70%
P
0.5
0.5
15%
5%
Z
0
1
21.25%
19.80%
© K. Cuthbertson and D. Nitzsche
Figure 2 : Efficient frontier
25
(0, 1)
Expected return (%)
20
Z
(0.5, 0.5)
P
G
(0.75, 0.25)
15
10
A
(1, 0 )
5
0
0
5
10
15
Standard deviation
© K. Cuthbertson and D. Nitzsche
20
25
Table 3 : Capital allocation line
50
45
Expected Return (%)
40
L
35
0.25 lending +
0.75 in risky portfolio
30
P
25
-0.5 borrowing
from bank
1.5 in risky portfolio
D
20
No borrowing/
no lending
15
10
5
F
0
0
All lending in bank deposit
At risk-free rate
5
10
15
Standard Deviation
© K. Cuthbertson and D. Nitzsche
20
25
30
Figure 4 : Efficient frontier and CML
45
40
35
Expected Return (%)
Q
30
25
Z
CML
20
CAL
15
X
P
10
5
F
A
G
0
0
5
10
15
Standard Deviation
20
25
Figure 5 : CML and market portfolio
Expected
Return
SRp = (ERp-r)/p
CML
L
ERL
P
ERp
ERD
r
wi - optimal proportions at P
(market portfolio)
ERp - r
D
p
D
© K. Cuthbertson and D. Nitzsche
M’s L less risk averse
than M’s D
p
L
Standard deviation
Figure 6 : Efficient frontier and correlation
Expected return (%)
25
20
 = -1
15
 = +0.5
 = +1
10
 = -0.5
=0
5
0
0
10
20
Std. dev.
© K. Cuthbertson and D. Nitzsche
30
Figure 7 : Efficient frontier, many stocks
ERp, % p.m.
=1.1%
=1.0%
x
B
x
x
x
xx x x x x
x x x x
x
x P1
x
xx x
x P2
x
x x
x
x
A
x C
p , % p.m.
© K. Cuthbertson and D. Nitzsche