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**International Portfolio Theory and Diversification**

Chapter 15 International Portfolio Theory and Diversification

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**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

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**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

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**International Diversification & Risk**

20 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.

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**International Diversification & Risk**

20 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.

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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

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**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%

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**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:

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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

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**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.

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**• 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.

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**• 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.

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**• 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.

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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.

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**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.

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**Example based on a sample**

× 12 / 11

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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.

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Degree of Correlation

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**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

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**Calculating Portfolio Risk and Return**

11 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)

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**Calculating Portfolio Risk and Return**

The multiple asset model for portfolio return The multiple asset model for portfolio risk

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**Calculating Portfolio Risk and Return**

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**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.

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**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

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**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

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**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

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**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

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**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

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**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

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**The International CAPM**

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