Chapter Twenty Two
Eiteman, Stonehill, & Moffett
April 15, 2020 Chapter 15 - Portfolio Investment 1

Relevant statistics for a return distributions for a single asset
 Expected return
 use the mean of historical returns to estimate
 Standard deviation
 use the standard deviation from historical returns to estimate
 Covariance
 use covariance with other return distributions to estimate correlation and asset betas
April 15, 2020 Chapter 15 - Portfolio Investment 2

Statistics.
r
R
 i
N 

1 r i
 prob i sd
R


 i
N 

1
( r i
 r )
2  prob i


0 .
5
Cov
R , ST
 i
N 

1
( r
R , i
 r
R
)

( r
ST , i
 r
ST
)
 prob i
April 15, 2020 Chapter 15 - Portfolio Investment 3

Correlation

R , ST

Cov
R , ST sd
R
 sd
ST

1
  
1
 Correlation normalizes covariance
 value of +1 means perfect positive correlation
 value of 0 means independence
 value of -1 means perfect negative correlation
April 15, 2020 Chapter 15 - Portfolio Investment 4

Relevant statistics for a portfolio of many assets -
E
P
 i
N 

1 x i
 r i s .
t i
N 

1 x i

1
SD
P



 i
N 

1 j
N 

1
 x i
 x j

Cov i , j
 


0 .
5
April 15, 2020 Chapter 15 - Portfolio Investment 5

Portfolio Risk
April 15, 2020
0.3
0.25
0.2
0.15
0.1
0.05
0
0 0.2
0.4
0.6 0.8
1
Chapter 15 - Portfolio Investment
Exp Return
Std Dev
6

Two-asset Portfolio
Preferred Portfolio
E p
Efficient Set
Risk preferences
Opportunity Set
Chapter 15 - Portfolio Investment
SD
P April 15, 2020 7

Implications of Correlation
Statistics
 The lower the pairwise correlation of two assets, the greater the diversification benefit of adding those assets to your portfolio
 Adding assets which have low pairwise correlation with each other to your portfolio reduces overall portfolio risk
April 15, 2020 Chapter 15 - Portfolio Investment 8

Affects of Correlation
E p
High positive
Correlation
April 15, 2020
Low positive
Correlation
Chapter 15 - Portfolio Investment
SD
P
9

Assets with high positive correlation r
April 15, 2020 Chapter 15 - Portfolio Investment
T
10

Assets with low positive correlation r
April 15, 2020 Chapter 15 - Portfolio Investment
T
11

Four-asset Portfolio
E p
Portfolios with assets A , B, C & D
Portfolios with assets C & D
Chapter 15 - Portfolio Investment
SD
P
Portfolios with assets A & B
12 April 15, 2020

Calculating Portfolio values four asset portfolio
E
P s .
t .
 x
1 x
1

 r
1 x
2

 x x
2
3
 r
2
 x
4
 x
3

1
 r
3
 x
4
 r
4
SD
P

[ x
1
2  sd
1
2  x
2
2  sd
2
2  x
3
2  sd
3
2  x
4
2  sd
4
2 
2 x
1 x
2
Cov
1 , 2

2 x
1 x
3
Cov
1 , 3

2 x
1 x
4
Cov
1 , 4

2 x
2 x
3
Cov
2 , 3

2 x
3 x
4
Cov
3 , 4
]
0 .
5
April 15, 2020 Chapter 15 - Portfolio Investment 13

Multiple-asset portfolios calculating portfolio values
E
P
 i
N 

1 x i
 r i s .
t i
N 

1 x i

1
SD
P



 i
N 

1 j
N 

1
 x i
 x j

Cov i , j
 


0 .
5
April 15, 2020 Chapter 15 - Portfolio Investment 14

Multiple-asset Portfolio
E p
April 15, 2020 Chapter 15 - Portfolio Investment
SD
P 15

Systematic Risk
SD p
SD
C
SD
W
April 15, 2020
International
Portfolio
Chapter 15 - Portfolio Investment
All Canadian equity portfolio
N
16

International Diversification
E p
World Equities
Efficient Frontier
Canadian Equities
Efficient Frontier
April 15, 2020 Chapter 15 - Portfolio Investment
SD
P 17

Systematic Risk
 As you diversify your portfolio by adding assets, your portfolio standard deviation decreases
 When you are fully diversified, your risk is the risk of the market portfolio
 By changing portfolio proportions you can modify risk to suit your preferences
April 15, 2020 Chapter 15 - Portfolio Investment 18

Optimal Portfolio
U
1
U
2
Less risk averse
Capital
Market
Line
E
M
More risk averse
April 15, 2020
SD
M
Chapter 15 - Portfolio Investment 19

Relevant statistic for a fully diversified portfolio - Beta
 Beta measures only the systematic risk of an assets
 Beta is also a covariance that is normalized by something, the variance of the market

R

Cov
R , m
Var m

0 .
0283
0 .
0243

1 .
165

ST

Cov
ST , M
Var
M
April 15, 2020


0 .
0095
 
0 .
391
0 .
0243
Chapter 15 - Portfolio Investment 20

The Risk-free Asset
 The risk-free asset does not exist except as a theoretical concept
 The least risky asset is the T-bill
 Low default risk - government backing
 Low interest rate risk - very short-term security
April 15, 2020 Chapter 15 - Portfolio Investment 21

Capital Asset Pricing Model
 Prices the risk of asset relative to its systematic risk
 gives the required rate of return relative to its systematic risk r i
 i
 r rf

Cov i , m
 
Var m i

( r
M
 r rf
)
April 15, 2020 Chapter 15 - Portfolio Investment 22

Security Market Line r
M
April 15, 2020
=1
Chapter 15 - Portfolio Investment 23

Security Market Line -
Change in rate of inflation r
M r
M
April 15, 2020
Inflation adjustment
=1
Chapter 15 - Portfolio Investment 24

Security Market Line -
Change in Risk Premium r
M r
M
April 15, 2020
Slope change reflecting increased
Systematic Risk
=1
Chapter 15 - Portfolio Investment 25

Country Markets
 Are the efficient
 perfect information
 few market reporting requirements
 price takers
 few buyers, few sellers
 no transactions costs
 capital controls
 foreign content rules
April 15, 2020 Chapter 15 - Portfolio Investment 26