1. International Portfolio Theory and Diversification
Chapter 15
International Portfolio Theory and Diversification

2. International Portfolio Theory and Diversification
Total risk of a portfolio and its components – diversifiable and non-diversifiable
Demonstration how both the diversifiable and non-diversifiable 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

3. 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

4. International Diversification & Risk20 40 60 80
Number of stocks in portfolio 10 30 50 1
100
Total Risk = Diversifiable Risk Market Risk
(unsystematic) (systematic)
Variance of a typical stock’s return
Variance of portfolio return
Portfolio of
U.S. stocks
Total
risk
27%
Systematic
risk
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. International Diversification & Risk20 40 60 80
Number of stocks in portfolio 10 30 50 1
100
Variance of a typical stock’s return
Variance of portfolio return
Portfolio of
U.S. stocks
27%
Portfolio of international stocks
11.7%
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.

6. 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

7. 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%

8. 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¥/$ = (S1 – S2) / S2 = (¥130 – ¥125) / ¥125 = 0.04 and
rshares, ¥ = (¥25,000 – ¥20,000) / ¥20,000 = 0.25
If the investment is not for exactly one year then the return can be annualized by:

9. 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

10. Domestic Portfolio Less Risk Averse More Risk Averse Expected Return
of Portfolio, Rp
Expected Risk
of Portfolio, σp
Less Risk Averse
More Risk Averse
Domestic portfolio
opportunity set
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 R DP Rf  DP Expected Return of Portfolio, Rp DP
Expected Risk
of Portfolio, σp Rf
Capital Market
Line (Domestic) DP
Optimal domestic
portfolio (DP) •
 DP
R DP
Minimum risk (MRDP )
domestic portfolio
MRDP
Domestic portfolio
opportunity set
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 R DP Rf  DP Borrow at Rf and Invest in DP
Expected Return
of Portfolio, Rp
Expected Risk
of Portfolio, σp Rf
Capital Market
Line (Domestic) DP
Optimal domestic
portfolio (DP)
Sell DP and Lend at Rf •
 DP
R DP
Minimum risk (MRDP )
domestic portfolio
MRDP
Domestic portfolio
opportunity set
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. • Domestic Portfolio R DP Rf  DP Borrow at Rf and Invest in DP
Expected Return
of Portfolio, Rp
Expected Risk
of Portfolio, σp Rf
Capital Market
Line (Domestic) DP
Optimal domestic
portfolio (DP)
Sell DP and Lend at Rf •
 DP
R DP
Minimum risk (MRDP )
domestic portfolio
MRDP
Domestic portfolio
opportunity set
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.

14. 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

15. Internationalizing the Domestic Portfolio
Expected Return
of Portfolio, Rp
Expected Risk
of Portfolio, σp Rf
Capital Market
Line (Domestic) DP
Optimal domestic
portfolio (DP) •
 DP
R DP
Minimum risk (MRDP )
domestic portfolio
MRDP
Domestic portfolio
opportunity set
Internationally diversified portfolio opportunity set

16. 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

17. Internationalizing the Domestic Portfolio
Optimal international portfolio
Expected Return
of Portfolio, Rp
Expected Risk
of Portfolio, σp
CML (International) Rf
CML (Domestic)
Internationally diversified portfolio opportunity set
R IP •
 IP IP •
 DP
R DP
Domestic portfolio
opportunity set DP

18. Internationalizing the Domestic Portfolio
Optimal international portfolio
Expected Return
of Portfolio, Rp
Expected Risk
of Portfolio, σp
CML (International) Rf
CML (Domestic)
Internationally diversified portfolio opportunity set
R IP •
 IP IP •
 DP
R DP
Domestic portfolio
opportunity set DP
Slide 18

19. 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
Where:
A = first asset
B = second asset
w = weights (respectively)
E(R) = expected return of assets

20. Calculating Portfolio Risk and Return
Example of two-asset model
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

21. Calculating Portfolio Risk and Return
The expected risk is calculated as
Where:
A = first asset
B = second asset
w = weights (respectively)
σ = standard deviation of assets
ρ = correlation coefficient of the two assets

22. 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.

23. 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.

24. Example based on a sample
× 12 / 11

25. Degree of Correlation
Note: If you have only one independent variable in a regression then:
* Sign of the correlation coefficient is the same as the sign of slope coefficient.

26. Degree of Correlation

27. Calculating Portfolio Risk and Return
Example of two-asset model
Where:
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

28. Calculating Portfolio Risk and Return11 12 13 14 15 16 17 18 19 20
Expected
Portfolio
Risk (σ )
Expected Portfolio
Return (%)
The example portfolio. •
Maximum
return &
maximum risk
(100% GER) •
Initial portfolio
(40% US & 60% GER) •
Minimum risk combination
(70% US & 30% GER) •
Domestic only portfolio
(100% US)

29. Calculating Portfolio Risk and Return
The multiple asset model for portfolio return
The multiple asset model for portfolio risk

30. Calculating Portfolio Risk and Return

31. 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 4%
0.0605
0.0810
0.0008
0.0398
0.0299 4%
0.0047
0.0079
0.0001
0.0027
0.0013
Correlations Matrix
Mexico
Denmark
Japan
Singapore
Australia
New Zealand
SP500
1.00
0.20
0.35
0.52
0.56
0.13
0.57
0.33
0.46
0.47
0.45
0.58
0.27
0.51
0.60
0.34
0.61
0.65
0.30
Data Sources: Market indexes from Yahoo.com and Exchange Rates from FRED.

32. 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

33. Sharpe and Treynor Performance Measures
The Sharpe measure calculates the average return over and above the risk-free rate per unit of portfolio risk
Where:
Ri = average portfolio return
Rf = risk-free rate of return
σ = risk of the portfolio

34. 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
Where:
Ri = average portfolio return
Rf = risk-free rate of return
β = beta of the portfolio

35. 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

36. 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

37. The International CAPM
Recall that CAPM is
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
Where:
β = beta of the security
ρ = correlation coefficient of the market and the security
σ = standard deviation of return

38. The International CAPM