Portfolio analysis
Management 4430.02
Class notes by A.P. Palasvirta
2
Geometric average return
 Annualized return if you know the purchase price
and the sale price of an asset
Rt

4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
 PT 


P

0 
1
T
1
7/17/2016
3
Forecasting risk & return
 Expected return

E
n

Ri  Pri
i 1
 Standard deviation
SD
 n

i  1

R
i
 E xyz

2

 Pri 

0.5
 Covariance
Covx, y

 R
n
i, x
i 1
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.

 E x  Ri, y  E y
  Pr
i
7/17/2016
4
Historical risk & return
 Mean return

R
1
T
T

Rt
t 1
 Standard deviation
 Covariance

SD
Covx, y
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.


1


 T 1

 R
2
T
t
R
t 1






0.5
T
1
R t , x  Rx  R t , y  Ry

T  1 t 1

7/17/2016
5
Contrasting methods
 Forecasting
 Forward looking
 Must determine possible outcomes
 Probabilities of those outcomes
 Completely speculative
 Historical
 Backward looking
 Using time series data
 Completely definitive
 Statistics derived mean exactly the same thing
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
6
Covariance/variance matrix
Covariance Matrix
1
2
3
4
5
M
1
0.1240
0.3521
0.1910
2
0.0676
0.0792
3
0.0650
0.0642
0.0628
4
0.0396
0.0341
0.0309
0.0343
5
0.0104
0.0120
0.0097
0.0117
0.0071
Standard Deviation for Assets
0.2814 0.2506 0.1852 0.0843
Expected Return for Assets
0.1750 0.1590 0.1520 0.0920
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
M
0.0283
0.0242
0.0216
0.0196
0.0083
0.0135
0.1162
0.1220
7/17/2016
7
Correlation
 Correlation normalizes covariance
 a, b

Cov a, b
SD a  SD b
 value of +1 means perfect positive correlation
 value of 0 means independence between data
series
 value of -1 means perfect negative correlation
1 
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.

 1
7/17/2016
8
Diversification
 Correlation is a measure of how two
assets react together to economic
market conditions
• High positive correlation
 Two assets are affected similarly by economic
events
• High negative correlation
 Two assets are affected in completely opposite
ways by economic events
• Independence -zero correlation
 Two assets are act independent of each other
relative to economic events
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
9
Portfolio Statistics
Expected return to the portfolio of multiple assets
E Port

n

i 1
xi  E
n
i
s. t .

i 1
xi  1
Standard deviation to a portfolio of multiple assets
SD port

 n
 
i  1
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
n

j1

x i  x j  COVi , j 

0.5
7/17/2016
10
Two-asset Portfolio
Ep
Preferred Portfolio
Efficient Set
Opportunity Set
Risk preferences
SDP
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
11
Implications of Correlation
Statistics
 The lower the pair-wise correlation of two assets,
the greater the diversification benefit of adding
those assets to your portfolio
 Adding assets which have low pair-wise
correlation with each other to your portfolio
reduces overall portfolio risk
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
12
Affects of Correlation
Ep
High
Correlation
Low
Correlation
SDP
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
13
Three-asset Portfolio
Ep
SDP
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
14
Calculating Portfolio values three asset portfolio
EP
3
x

i1
i
 Ei

x1  E1  x2  E 2  x3  E 3
3 3

SDP      xi  x j  Covi , j 
i  1 j  1

0.5

 x1  x1  Cov1, 1  x2  x2  Cov2 , 2  x3  x3  Cov3, 3 

 2  x1  x2  Cov1, 2  2  x1  x3  Cov1, 3  2  x2  x3  Cov2 , 3
x  SD  x  SD  x  SD  2  x  x  Cov
2
1
2
1
2
2
2
2
2
3
2
3
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
1
2
1, 2



0.5

 2  x1  x3  Cov1, 3  2  x2  x3  Cov2 , 3

0.5
7/17/2016
15
Multiple-asset Portfolio
Ep
SDP
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
16
Systematic Risk
SDp
SDM
N
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
17
Systematic Risk
 As you diversify your portfolio by
adding assets, your portfolio standard
deviation decreases
 Optimal approx 20 assets = 21.68%
 20*100*$20 = $40,000 buying individual stocks
 Buy funds which are already diversified
 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
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
18
Optimal Portfolio
U1
More
risk
averse
U2
Capital
Market
Line
EM
Less risk
averse
SDM
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
19
Capital market line
 Any efficient portfolio will be found on the capital
market line
 
E RP
 Rf 
 
SD  R 
E RM  R f
 
 SD R P
M
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
20
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
 Risk-free rate of return
 Beta
 Risk premium
Ri


R f  i  RM  R
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
f

7/17/2016
21
The Risk-free Asset
The risk-free asset does not exist
except as a theoretical concept
assets used t-bills or t-bonds
 Low default risk - government backing
 For calculation of the company β, companies
often use the t-bond rate that has a similar term
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
22
Beta
 β measures only the systematic risk of an assets
 β is also a covariance that is normalized by
something, the variance of the market
 XYZ 
Cova, market
Covm, m

Cova, m
Varm

Cova, m
SDm2
 β measures the risk of holding that firms stock in a
fully diversified portfolio
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
23
Risk premium
 Return to the market portfolio less the return to
the risk-free rate of return
 Return to the market portfolio
 Uses the 90 day return to the a broad based stock
index
 Toronto stock exchange index
 Dow index
 S&P 500 index
 T-bond rate of appropriate term
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
24
Statistics
Correlation Matrix
1
1
2
3
4
5
Beta
1.0000
0.6821
0.7366
0.6072
0.3505
2.0963
1.0000
0.9103
0.6543
0.5060
1.7926
1.0000
0.6658
0.4594
1.6000
1.0000
0.7690
1.7926
1.0000
0.0083
2
3
4
5
M
1.0000
Required Rate of Return on Assets
0.2009
0.1791
0.1652
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
0.1791
0.0506
0.1220
7/17/2016
25
Security Market Line
RM
β=1
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
26
Security Market Line Change in rate of inflation
RM
RM
Inflation adjustment
β=1
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016
27
Security Market Line Change in Risk Premium
RM
RM
Slope change
reflecting increased
Systematic Risk
β=1
4430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.
7/17/2016