EXAMPLE: PORTFOLIO RISK & RETURN
Suppose:
Georgia Thermo
Pacific Electron
Expected return ( r )
Variance ( )
Standard deviation ( )
2
Note:
15%
784
28
21%
1764
42
Expected return = average of possible returns
Variance = average of possible squared deviations from expected return
Standard deviation = square root of variance
1
PORTFOLIO RISK
Portfolio variance = sum of boxes
X 
2
2
1
1
X X   
1

X X   
1



12
2
X X 
1
1
12
2
2
2
X X 
1
1
12
2
X 
2
12
2
2
2
2
12
 covariance
 correlatio
of returns
n of returns
2
PORTFOLIO RISK: EXAMPLE
Georgia Pacific
Georgia
Pacific
Thermo
Electron
.62 *282
=282
.6*.4*.4*28*42
=113
Thermo Electron
.6*.4*.4*28*42
=113
.42 *422
=282
Variance = 282 +282 + (2 X 113) = 790
Std dev. = (790)1/2 = 28.1%
Note: assumes correlation is .4
3
Return and Risk for Portfolios
N
Expected
return of a portfolio
E(
R
p
X
)
i 1
i
E ( R i)
N
X
subject to
i 1
Portfolio

variance
i
2
P
1
N

X 
i 1
2
2
i
i

N
N
i 1
j 1
 X X 
i
j
ij
where
 Proportion
X
i
E(
R
i
)  Expectedre
N  Number


2
i
ij
of funds invested
turn on security
of securities
 Variance
 Covariance
in security
i
in portfolio
of return security
of returns
i
i
of security
i and j  Corr(
R R ) 
A
B
A
B
4
SUPPOSE EXPECTED RETURNS ARE AS FOLLOWS:
USA
CANADA
BELGIUM
FRANCE
GERMANY
ITALY
NETHERLANDS
SWITZERLAND
JAPAN
UK
13.7 Percent
18.3
12.5
14.4
12.5
19.7
17.4
13.0
11.3
17.2
5
EXAMPLE:INTERNATIONAL PORTFOLIO SELECTION
1 VARIABILITY OF DIFFERENT MARKETS 1980-85
USA
CANADA
BELGIUM
FRANCE
GERMANY
ITALY
NETHERLANDS
SWITZERLAND
JAPAN
UK
14.9 Percent
20.8
16.3
18.4
13.5
* 28.4
18.4
11.8
11.4
17.0
6
CORRELATIONS BETWEEN RETURNS ON
DIFFERENT MARKETS
USA
CAN
BEL
FRA
GER
ITA
NET
SWI
JAP
UK
USA
1.00
0.78
0.11
0.27
0.37
0.12
0.28
0.50
0.34
0.46
CAN
BEL
FRA
GER
ITA
NET
SWI
JAP
UK
1.00
0.09
0.19
0.33
0.29
0.35
0.56
0.32
0.51
1.00
0.41
0.30
0.08
0.38
0.36
0.30
0.34
1.00
0.28
0.19
0.31
0.38
0.23
0.31
1.00
0.04
0.52
0.59
0.36
0.47
1.00
0.43
0.20
0.21
0.40
1.00
0.49
0.45
0.45
1.00
0.37
0.58
1.00
0.55
1.00
7
EFFICIENT PORTFOLIOS
PERCENT INVESTED IN EACH COUNTRY
Port
1
2
3
4
5
6
7
8
9
10
11
12
13
Exp Std
ret
dev
19.7 28.4
19.0 19.9
18.7 18.2
18.0 15.2
17.4 13.9
17.1 13.4
16.8 13.1
16.6 12.8
16.2 12.3
13.6
9.5
13.4
9.4
13.4
9.3
12.6
9.1
USA
2.0
5.0
13.0
17.0
16.0
12.0
CAN
50.0
47.0
30.0
24.0
23.0
22.0
20.0
17.0
2.0
-
BEL
5.0
7.0
8.0
9.0
11.0
11.0
11.0
11.0
FRA
13.0
14.0
14.0
14.0
13.0
7.0
7.0
6.0
4.0
GER ITA
100.0
50.0
39.0
16.0
10.0
9.0
4.0
9.0
5.0
9.0
7.0 10.0
13.0 10.0
14.0 10.0
14.0 10.0
13.0 3.0
NET
14.0
28.0
26.0
23.0
21.0
20.0
17.0
2.0
0.0
-
SWI JAP UK
26.0
27.0
25.0
23.0
22.0
3.0
19.0
18.0 21.0
19.0 23.0
19.0 23.0
20.0 37.0
8
Security Market Line (SML)
R isk an d retu rn relatio n sh ip fo r in d iv id u al secu rity
U n d er C A P M , in v esto r is co n cern ed w ith th e risk o f th e
2
m ark et p o rtfo lio ,  M
R isk o f in d iv id u al secu rity sh o u ld b e m easu red b y its
co n trib u tio n to th e to tal risk o f th e m ark et p o rtfo lio ; th erefo re,
th e relev an t risk m easu re o f in d iv id u al secu rity is  iM ; n o t  i
Expected return
on asset
E q u atio n o f S M L
E ( R i) 
Set


i
E ( R i) 
r
f



iM
 E(
 
2
RM)


 M 


M
r
f

RM
M
iM
r
f

2
RF
 E ( R
i
M
)
r
f

0
1
9
U n d ersta n d in g b eta , 
 T w o m ajo r co m p o n en ts o f risk :
I.
C o m p an y -u n iq u e risk
II.
S y stem atic risk
 S y stem atic risk is a sto ck 's resp o n siv en ess to m o v em en ts in
th e "g en eral m ark et," w h ich can n o t b e elim in ated th ro u g h
d iv ersificatio n .
 S y stem atic risk is m easu red b y th e slo p e o f th e "
ch aracteristic lin e," as sh o w n b elo w :
Stock
Returns
10
5
5
10
15
Market
Returns
 T h e slo p e o f th e ch aracteristic lin e is called "b eta" in
fin an ce p arlen ce.
 T h e d isp ersio n ab o ut th e ch aracteristic lin e (reg ressio n lin e)
rep resen ts th e co m p an y u n iq u e risk .
 W h at w ill h ap p en to th e sy stem atic risk an d th e co m p an y u n iq u e risk as w e ad d o th er sto ck s to o u r p o rtfo lio ?
10
Capital Market Line (CML)
• Equilibrium relationship between E(Rp) and σp for efficient
portfolios
• Linear efficient set of CAPM by combining Market portfolio
with risk free (rf) borrowing and lending
• CML only permits to well-diversified portfolios; portfolios not
employing M, the market portfolio, will plot below the CML
• Equation of CML:
E(Rp)=rf + [(E(RM)-Rf )/ σM] σ(Rp)
• Slope of CML: price of risk
{E(RM) – Rf }/ σM
• Price of time: Rf
11
Capital Asset Pricing Model
(CAPM)
Developed by Sharpe, Treynor, Lintner and Mossin
•
• An equilibrium theory of how to price and measure risk of
portfolios as well as individual security
• Concerning decomposition of risk into two components:
systematic (market, non-diversifiable) and unsystematic
(unique, diversifiable)
• Stating that required return on any investment is the risk free
return plus a risk premium measured by its systematic risk
E(ri)=rf+[E(rm)-rf]β
where β = covariance risk of security i
E(rm)-rf = market risk premium
12
Feasible Set of Risky Portfolios
Expected portfolio
Return Kp
B
C
A
D
E
Feasible, or
Attainable, Set
Risk, σp
13
Optimal Portfolio Selection
Expected portfolio
Return Kp
B
C
A
D
E
Optimal
Portfolio
Investor B
Optimal
Portfolio
Investor A
Risk, σp
14
Efficient Frontier with Risk-Free Asset
Expected portfolio
Return Kp
new efficient
portfolio
Z
B
KM
C
M
A
D
kRF
Y=mx+b
Ki=Krf+σi/σm(Km-Krf)
b = intercept
m = slope
= Km-Krf/σm
E
rm
Risk, σp
15