Chapter 15
International Portfolio Theory and
Diversification
1
International Portfolio Theory and
Diversification
• Total risk of a portfolio and its components – diversifiable
and non-diversifiable
• Demonstration how both the diversifiable and nondiversifiable risks of an investor’s portfolio may be reduced
through international diversification
• Foreign exchange risk and international investments
• Optimal domestic portfolio and the optimal international
portfolio
• Recent history of equity market performance globally
• Market integration over time
• Extension of international portfolio theory to the
estimation of a company’s cost of equity using the
international CAPM
2
International Diversification & Risk
• Portfolio Risk Reduction
– The risk of a portfolio is measured by the ratio of the
variance of the portfolio’s return relative to the
variance of the market return
– This is defined as the beta of the portfolio
– As an investor increases the number of securities in
her portfolio, the portfolio’s risk declines rapidly at
first and then asymptotically approaches to the level
of systematic risk of the market
– A fully diversified portfolio would have a beta of 1.0
3
International Diversification & Risk
Variance of portfolio return
Variance of a typical stock’s return
100
80
Total Risk
60
=
Diversifiable Risk
(unsystematic)
+
Market Risk
(systematic)
p
Standard deviation of portfolio return
Variance of portfolio return


Standard deviation of a typical stock' s return  i
Variance of a typical stock' s return
40
Portfolio of
U.S. stocks
27%
20
1
Total
risk
Systematic
risk
10
20
30
40
50
Number of stocks in portfolio
By diversifying the portfolio, the variance of the portfolio’s return relative to the variance of a
typical stock is reduced to the level of systematic risk -- the risk of the market itself.
4
International Diversification & Risk
Variance of portfolio return
Variance of a typical stock’s return
100
80
60
p
Standard deviation of portfolio return
Variance of portfolio return


Standard deviation of a typical stock' s return  i
Variance of a typical stock' s return
40
Portfolio of
U.S. stocks
27%
20
11.7%
1
Portfolio of international stocks
10
20
30
40
50
Number of stocks in portfolio
By diversifying the portfolio, the variance of the portfolio’s return relative to the variance of a
typical stock is reduced to the level of systematic risk -- the risk of the market itself.
5
Foreign Exchange Risk
• The foreign exchange risks of a portfolio,
whether it be a securities portfolio or the
general portfolio of activities of the MNE, are
reduced through diversification
• Internationally diversified portfolios are the
same in principle because the investor is
attempting to combine assets which are less
than perfectly correlated, reducing the risk of
the portfolio
6
Foreign Exchange Risk
• An illustration with Japanese equity
• US investor takes $1,000,000 on 1/1/2002 and invests
in a stock traded on the Tokyo Stock Exchange (TSE)
• On 1/1/2002, the spot rate was S1= ¥130/$
• The investor purchases 6,500 shares valued at ¥20,000
for a total investment of ¥130,000,000
• At the end of the year, the investor sells the shares at a
price of ¥25,000 per share yielding ¥162,500,000
• On 1/1/2003, the spot rate was S2= ¥125/$
• The investor receives a 30% return on investment
($1,300,000 – $1,000,000) / $1,00,000 = 30%
7
Foreign Exchange Risk
• An illustration with Japanese equity
• The total return reflects not only the appreciation in stock
price but also the appreciation of the yen
• The formula for the total return from US perspective is



R $  1  r ¥/$ 1  r shares,¥  1
• r¥/$ = (S1 – S2) / S2 = (¥130 – ¥125) / ¥125 = 0.04 and
• rshares, ¥ = (¥25,000 – ¥20,000) / ¥20,000 = 0.25
R $  1  0.041  0.250  1  0.300
• If the investment is not for exactly one year then the return
can be annualized by:
1
t
AR$  ( 1  R $ )  1, where t is in year(s)
8
Domestic Portfolio
• Classic portfolio theory assumes that a typical investor is
risk-averse
– The typical investor wishes to maximize expected return per
unit of expected risk
• An investor may choose from an almost infinite choice of
securities
• This forms the domestic portfolio opportunity set
• The extreme left edge of this set is termed the efficient
frontier
– This represents the optimal portfolios of securities that possess
the minimum expected risk per unit of return
– The portfolio with the minimum risk among all those possible is
the minimum risk domestic portfolio
9
Domestic Portfolio
Expected Return
of Portfolio, Rp
Less Risk
Averse
More Risk
Averse
Domestic portfolio
opportunity set
Expected Risk
of Portfolio, σp
An investor may choose a portfolio of assets enclosed by the Domestic portfolio opportunity set. The optimal domestic portfolio is found at
DP, where the Security Market Line is tangent to the domestic portfolio opportunity set. The domestic portfolio with the minimum risk is
MRDP.
10
Domestic Portfolio
Capital Market
Line (Domestic)
Expected Return
of Portfolio, Rp
Optimal domestic
portfolio (DP)
DP
•
R DP
MRDP
Minimum risk (MRDP )
domestic portfolio
•
Domestic portfolio
opportunity set
Rf
 DP
Expected Risk
of Portfolio, σp
An investor may choose a portfolio of assets enclosed by the Domestic portfolio opportunity set. The optimal domestic portfolio is found at
DP, where the Security Market Line is tangent to the domestic portfolio opportunity set. The domestic portfolio with the minimum risk is
MRDP.
11
Domestic Portfolio
Capital Market
Line (Domestic)
Expected Return
of Portfolio, Rp
Optimal domestic
portfolio (DP)
DP
•
R DP
MRDP
Minimum risk (MRDP )
domestic portfolio
•
Domestic portfolio
opportunity set
Rf
 DP
Expected Risk
of Portfolio, σp
An investor may choose a portfolio of assets enclosed by the Domestic portfolio opportunity set. The optimal domestic portfolio is found at
DP, where the Security Market Line is tangent to the domestic portfolio opportunity set. The domestic portfolio with the minimum risk is
MRDP.
12
Domestic Portfolio
Capital Market
Line (Domestic)
Expected Return
of Portfolio, Rp
Optimal domestic
portfolio (DP)
DP
•
R DP
MRDP
Minimum risk (MRDP )
domestic portfolio
•
Domestic portfolio
opportunity set
Rf
 DP
Expected Risk
of Portfolio, σp
An investor may choose a portfolio of assets enclosed by the Domestic portfolio opportunity set. The optimal domestic portfolio is found at
DP, where the Security Market Line is tangent to the domestic portfolio opportunity set. The domestic portfolio with the minimum risk is
MRDP.
13
Internationalizing the Domestic
Portfolio
• If the investor is allowed to choose among an
internationally diversified set of securities, the
efficient frontier shifts upward and to the left
• This is called the internationally diversified
portfolio opportunity set
14
Internationalizing the Domestic
Portfolio
Capital Market
Line (Domestic)
Expected Return
of Portfolio, Rp
Optimal domestic
portfolio (DP)
DP
•
R DP
MRDP
Minimum risk (MRDP )
domestic portfolio
•
Domestic portfolio
opportunity set
Rf
 DP
Expected Risk
of Portfolio, σp
Internationally diversified
portfolio opportunity set
15
Internationalizing the Domestic
Portfolio
• This new opportunity set allows the investor a
new choice for portfolio optimization
• The optimal international portfolio (IP) allows
the investor to maximize return per unit of risk
more so than would be received with just a
domestic portfolio
16
Internationalizing the Domestic
Portfolio
Expected Return
of Portfolio, Rp
R IP
Optimal
international
portfolio
CML (International)
CML (Domestic)
IP
•
R DP
DP
•
Internationally diversified
portfolio opportunity set
Domestic portfolio
opportunity set
Rf
 IP
 DP
Expected Risk
of Portfolio, σp
17
Internationalizing the
Domestic Portfolio
Expected Return
of Portfolio, Rp
R IP
Optimal
international
portfolio
CML (International)
CML (Domestic)
IP
•
R DP
DP
•
Internationally diversified
portfolio opportunity set
Domestic portfolio
opportunity set
Rf
 IP
 DP
Expected Risk
of Portfolio, σp
Slide 18 18
Calculating Portfolio Risk and Return
• The two-asset model consists of two components
– The expected return of the portfolio
– The expected risk of the portfolio
• The expected return is calculated as
E(RP )  wA E(RA )  wB E(RB )
• Where:
–
–
–
–
A = first asset
B = second asset
w = weights (respectively)
E(R) = expected return of assets
19
Calculating Portfolio Risk and Return
• Example of two-asset model
E(RP )  wUS E(RUS )  wGER E(RGER )
• Where:
– E(RUS) = expected return on US security = 14%
– E(RGER) = expected return on German security = 18%
– wUS
= weight of US security
– wUS
= weight of German security
– E(RP) = expected return of portfolio
20
Calculating Portfolio Risk and Return
• The expected risk is calculated as
σ P  wA2 σ A2  wB2 σ B2  2wA wB σ Aσ B ρ AB
• Where:
– A = first asset
– B = second asset
– w = weights (respectively)
– σ = standard deviation of assets
– ρ = correlation coefficient of the two assets
21
Population Covariance and Correlation
• The formulation of population covariance and
correlation
– When we compute expected returns based on a
probability distribution we would have the
following equations. Note that Pi is referring to
probabilities with “n” different states.
n
E R A    Pi Ri
i 1
n
VAR A     Pi [ Ri  E ( Ri )] 2
2
A
 A  VARA   A2
i 1
n
COV A, B   Pi [ Ri  E ( R A )][ R j  E ( R B )]
i 1
j 1
 A, B 
COV A, B
 A B
COV A, B   A, B A B
22
Sample Covariance and Correlation
• The formulation of sample covariance and
correlation
– When we compute expected returns based on a
sample we use the following equations. Note that
there are “N” observations.
1
RA 
N
N
 Ri
COV A, B
i 1
1 N
2


VAR A   
R

R
 i A
N  1 i 1
2
A
1 N

[ Ri  R A ][ R j  R B ]

N  1 i 1
j 1
 A  VARA   A2
 A, B 
COV A, B
 A B
COV A, B   A, B A B
23
Example based on a sample
Month
Jan-03
Feb-03
Mar-03
Apr-03
May-03
Jun-03
Jul-03
Aug-03
Sep-03
Oct-03
Nov-03
Dec-03
Sum
N
Average
Average
A
-0.0645
-0.0340
0.0015
0.1336
0.0628
0.0280
0.0470
0.0010
0.0172
0.0073
0.0637
0.0187
0.2823
12
0.0235
B
Month
-0.0248
Jan-03
-0.0324
Feb-03
0.0501
Mar-03
0.1086
Apr-03
0.0743
May-03
0.0456
Jun-03
-0.0293
Jul-03
0.1080
Aug-03
-0.0060
Sep-03
0.1260
Oct-03
-0.0604
Nov-03
0.0553
Dec-03
0.4150 Sum
12
N-1
0.0346 Variance
Std
Excel's COVAR
Excel's CORREL
Variance
A
0.007748
0.003309
0.000485
0.012117
0.001543
0.000020
0.000551
0.000507
0.000040
0.000263
0.001614
0.000023
0.028221
11
0.002566
0.050651
0.000850
0.28828
B
0.003526
0.004487
0.000241
0.005478
0.001577
0.000121
0.004081
0.005390
0.001647
0.008357
0.009022
0.000429
0.044357
11
0.004032
0.063502
Adjusted
0.000927
Covariance
Month
A&B
Jan-03
0.00523
Feb-03
0.00385
Mar-03
-0.00034
Apr-03
0.00815
May-03
0.00156
Jun-03
0.00005
Jul-03
-0.00150
Aug-03
-0.00165
Sep-03
0.00026
Oct-03
-0.00148
Nov-03
-0.00382
Dec-03
-0.00010
Sum
0.01020
N-1
11
Covariance 0.000927
Correlation
0.28828
0.000850 × 12 / 11
24
Degree of Correlation
A
y = 0.3614x + 0.0261
A vs. B
A and B Returns over time
R2 = 0.0831
B
0.1500
0.1000
0.1000
0.0500
0.0500
B
Return
0.1500
-0.1000
Dec-03
Nov-03
Oct-03
Sep-03
Aug-03
Jul-03
Jun-03
May-03
Apr-03
Feb-03
Mar-03
-0.0500
Jan-03
0.0000
-0.1000
0.0000
-0.0500
0.0000
0.0500
0.1000
0.1500
-0.0500
-0.1000
Month
A
Note: If you have only one independent variable in a regression then:
CorrelationA, B   A, B  R 2  0.0831  0.2883 *
* Sign of the correlation coefficient is the same as the sign of slope coefficient.
25
Degree of Correlation
R² = 0.2038
Singapore vs. Japan: 01/94 - 06/09
Denmark vs. Mexico: 01/94 - 06/09
y = 0.5689x + 0.0035
15.00%
30.00%
10.00%
25.00%
5.00%
20.00%
y = 0.1313x + 0.0072
R² = 0.0419
-20.00%
0.00%
-10.00%
0.00%
-5.00%
-10.00%
15.00%
10.00%
20.00%
30.00%
Denmark
Singapore
-30.00%
10.00%
5.00%
-15.00%
-60.00%
-40.00%
-20.00%
0.00%
0.00%
-20.00%
-5.00%
-25.00%
-10.00%
-30.00%
Japan
20.00%
40.00%
-15.00%
Mexico
26
Calculating Portfolio Risk and Return
• Example of two-asset model
2
2
2
2
σ P  wUS
σUS
 wGER
σGER
 2wUS wGERσUS σGER ρUS/GER
• Where:
–
–
–
–
–
–
–
0.151 
US = US security
GER = German security
wUS = weight of US security: 40%
wGER = weight of German security: 60%
σUS = standard deviation of US security: 15%
σGER = standard deviation of GER security: 20%
ρ = correlation coefficient of the two assets: 0.34
2
2
2
2
(0.40) (0.15)  (0.60) (0.20)  2(0.40)(0.60)(0.15)(0.20)(0.34)
27
Calculating Portfolio Risk and Return
Expected Portfolio
Return (%)
The example portfolio.
•
18
17
16
•
15
•
Initial portfolio
(40% US & 60% GER)
Minimum risk combination
(70% US & 30% GER)
•
14
Maximum
return &
maximum risk
(100% GER)
Domestic only portfolio
(100% US)
13
12
0
11
12
13
14
15
16
17
18
19
20
Expected
Portfolio
Risk (σ )
28
Calculating Portfolio Risk and Return
• The multiple asset model for portfolio return
N
E(rP )  Σ wi E(ri )
i 1
• The multiple asset model for portfolio risk
N
N-1
N
σ P  Σ w σ  Σ Σ wi w j σ i σ j ρij
i 1
2 2
i i
i 1 j i 1
29
Calculating Portfolio Risk and Return
Efficient Portfolios (All returns are measured in $)
Historical Returns and Exchange Rates: January 1994 - June 2009
2.0%
1.5%
1.0%
Return
Denmark
Australia
SP500
0.5%
Mexico
Singapore
New Zealand
0.0%
Japan
-0.5%
2%
3%
4%
5%
6%
7%
8%
9%
10%
Std
Data Sources: Market indexes from Yahoo.com and Exchange Rates from FRED.
30
National Equity Market Performance
January 1994 – June 2009
Mexico
Denmark
Japan
Singapore
Australia
New Zealand
SP500
Mean
0.92%
0.84%
-0.02%
0.34%
0.55%
0.32%
0.47%
Std
9.73%
6.24%
6.22%
7.83%
5.34%
5.04%
4.49%
Beta
1.240
0.641
0.715
1.074
0.782
0.333
1.000
Sharpe @ 4%
0.0605
0.0810
-0.0564
0.0008
0.0398
-0.0027
0.0299
Treynor @ 4%
0.0047
0.0079
-0.0049
0.0001
0.0027
-0.0004
0.0013
Mexico
Denmark
Japan
Singapore
Australia
New Zealand
SP500
Mexico
1.00
0.20
0.35
0.52
0.56
0.13
0.57
Denmark
0.20
1.00
0.33
0.35
0.46
0.47
0.46
Correlations Matrix
Japan
Singapore
0.35
0.52
0.33
0.35
1.00
0.45
0.45
1.00
0.58
0.60
0.27
0.34
0.51
0.61
Australia New Zealand
0.56
0.13
0.46
0.47
0.58
0.27
0.60
0.34
1.00
0.51
0.51
1.00
0.65
0.30
SP500
0.57
0.46
0.51
0.61
0.65
0.30
1.00
Data Sources: Market indexes from Yahoo.com and Exchange Rates from FRED.
31
Sharpe and Treynor Performance
Measures
• Investors should not examine returns in
isolation but rather the amount of return per
unit risk
• To consider both risk and return for portfolio
performance there are two main measures
– The Sharpe measure
– The Treynor measure
32
Sharpe and Treynor Performance
Measures
• The Sharpe measure calculates the average
return over and above the risk-free rate per
unit of portfolio risk
Sharpe measure 
Ri  Rf
i
• Where:
– Ri = average portfolio return
– Rf = risk-free rate of return
– σ = risk of the portfolio
33
Sharpe and Treynor Performance
Measures
• The Treynor measure is similar to Sharpe’s measure
except that it measures return over the portfolio’s beta
• The measures are similar depending on the
diversification of the portfolio
– If the portfolio is poorly diversified, the Treynor measure
will show a high ranking and vice versa for the Sharpe
measure
Treynor measure 
Ri  Rf
i
• Where:
– Ri = average portfolio return
– Rf = risk-free rate of return
– β = beta of the portfolio
34
Sharpe and Treynor Performance
Measures
• Example:
– Hong Kong average return was 1.5% per month
– Assume risk free rate of 5%
– Standard deviation is 9.61% and Beta is 1.09
0.05
0.015 
12  0.113
Sharpe measure 
0.0961
0.05
0.015 
12  0.0100
Treynor measure 
1.09
35
Sharpe and Treynor Performance
Measures
• For each unit of risk the Hong Kong market
rewarded an investor with a monthly excess
return of 0.113%
• The Treynor measure for Hong Kong was the
second highest among the global markets and
the Sharpe measure was eighth
• This indicates that the Hong Kong market
portfolio was not very well diversified from
the world market perspective
36
The International CAPM
• Recall that CAPM is
k e  k rf   (k m  k f )
• The difference for the international CAPM is that the
beta calculation would be relevant for the global equity
market for analysis instead of the domestic market
j
 i   jm
m
• Where:
– β = beta of the security
– ρ = correlation coefficient of the market and the security
– σ = standard deviation of return
37
The International CAPM
38