chapter 15 international portfolio theory and diversification 1

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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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ST Portfolio ofU.S. stocks

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.

Systematicrisk

Totalrisk

Total Risk = Diversifiable Risk + Market Risk (unsystematic) (systematic)

20

40

60

80

Number of stocks in portfolio

10 20 30 40 501

100

27%

return sstock' typicala of Variance

return portfolio of Variance

return sstock' typicala ofdeviation Standard

return portfolio ofdeviation Standard

i

p

Varia

nce

of p

ortfo

lio re

turn

Varia

nce

of a

typi

cal s

tock

’s re

turn

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Portfolio of international stocks

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.

20

40

60

80

Number of stocks in portfolio

10 20 30 40 501

100

Portfolio ofU.S. stocks

Varia

nce

of p

ortfo

lio re

turn

Varia

nce

of a

typi

cal s

tock

’s re

turn

27%

11.7%

return sstock' typicala of Variance

return portfolio of Variance

return sstock' typicala ofdeviation Standard

return portfolio ofdeviation Standard

i

p

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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:

1r1r1R shares,¥¥/$$

300012501041 ..0.0R$

year(s) in is t where,)R(AR t$$ 111

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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 PortfolioExpected Returnof Portfolio, Rp

Expected Riskof Portfolio, σp

Domestic portfolioopportunity 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.

More Risk Averse

Less Risk Averse

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Domestic PortfolioExpected Returnof Portfolio, Rp




Rf

Capital MarketLine (Domestic)

•

DP

R DP

•Minimum risk (MRDP )domestic portfolio

MRDP

DP

Optimal domesticportfolio (DP)

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Domestic Portfolio

Expected Returnof Portfolio, Rp




Rf


•

DP

R DP


MRDP

DP


Sell DP and Lend at R f

Borrow at R f a

nd Invest in DP

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Rf


•

DP

R DP


MRDP

DP


Sell DP and Lend at R f

Borrow at R f a

nd Invest in DPDomestic Portfolio

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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 PortfolioExpected Returnof Portfolio, Rp


Domestic portfolioopportunity setRf


•

DP

R DP


MRDP

DP


Internationally diversified portfolio opportunity set

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• 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 PortfolioExpected Returnof Portfolio, Rp


Rf

CML (Domestic)

•

DP

R DP


DP


R IP •

IP

IP

Optimal international portfolio

CML (International)

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Slide 18



Rf

CML (Domestic)

•

DP

R DP


DP


R IP •

IP

IP

Optimal international portfolio

CML (International)

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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

)E(Rw)E(Rw)E(R BBAAP

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• 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

)E(Rw)E(Rw)E(R GERGERUSUSP

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• 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

ABBABABBAAP ρσσwwσwσwσ 22222

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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.

)]()][([

11

, BjAi

n

ji

iBA RERRERPCOV

BA

BABA

COV

,

,

i

n

iiA RPRE

1

2

1

2 )]([ ii

n

iiAA RERPVAR

BABABACOV ,,

2AAA VAR

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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.

N

ji

BjAiBA RRRRN

COV

11

, ]][[1

1

BA

BABA

COV

,

,

N

iiA R

NR

1

1

N

iAiAA RR

NVAR

1

22

1

1 2AAA VAR

BABABACOV ,,

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Example based on a sample

Month A B Month A B Month A&BJan-03 -0.0645 -0.0248 Jan-03 0.007748 0.003526 Jan-03 0.00523Feb-03 -0.0340 -0.0324 Feb-03 0.003309 0.004487 Feb-03 0.00385Mar-03 0.0015 0.0501 Mar-03 0.000485 0.000241 Mar-03 -0.00034Apr-03 0.1336 0.1086 Apr-03 0.012117 0.005478 Apr-03 0.00815May-03 0.0628 0.0743 May-03 0.001543 0.001577 May-03 0.00156Jun-03 0.0280 0.0456 Jun-03 0.000020 0.000121 Jun-03 0.00005Jul-03 0.0470 -0.0293 Jul-03 0.000551 0.004081 Jul-03 -0.00150

Aug-03 0.0010 0.1080 Aug-03 0.000507 0.005390 Aug-03 -0.00165Sep-03 0.0172 -0.0060 Sep-03 0.000040 0.001647 Sep-03 0.00026Oct-03 0.0073 0.1260 Oct-03 0.000263 0.008357 Oct-03 -0.00148Nov-03 0.0637 -0.0604 Nov-03 0.001614 0.009022 Nov-03 -0.00382Dec-03 0.0187 0.0553 Dec-03 0.000023 0.000429 Dec-03 -0.00010

Sum 0.2823 0.4150 Sum 0.028221 0.044357 Sum 0.01020N 12 12 N - 1 11 11 N - 1 11Average 0.0235 0.0346 Variance 0.002566 0.004032 Covariance 0.000927

Std 0.050651 0.063502 Correlation 0.28828

CovarianceVarianceAverage

AdjustedExcel's COVAR 0.000850 0.000927Excel's CORREL 0.28828

0.000850 × 12 / 11

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Degree of Correlation

nCorrelatioA vs. B y = 0.3614x + 0.0261

R2 = 0.0831

-0.1000

-0.0500

0.0000

0.0500

0.1000

0.1500

-0.1000 -0.0500 0.0000 0.0500 0.1000 0.1500

A

B

A and B Returns over time

-0.1000

-0.0500

0.0000

0.0500

0.1000

0.1500

Jan-

03

Feb

-03

Mar

-03

Apr

-03

May

-03

Jun-

03

Jul-

03

Aug

-03

Sep-

03

Oct

-03

Nov

-03

Dec

-03

Month

Ret

urn

A B

*2883.00831.02,, RnCorrelatio BABA

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

-30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00%

-30.00%

-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

f(x) = 0.568923647871492 x + 0.00349191444520973R² = 0.203787769818797

Singapore vs. Japan: 01/94 - 06/09

Japan

Sin

ga

po

re

-60.00% -40.00% -20.00% 0.00% 20.00% 40.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

f(x) = 0.131334264049143 x + 0.00717971025200196R² = 0.0418829945321518

Denmark vs. Mexico: 01/94 - 06/09

MexicoD

enm

ark

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• 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

US/GERGERUSGERUSGERGERUSUSP ρσσwwσwσwσ 22222

)34.0)(20.0)(15.0)(60.0)(40.0(22

)20.0(2

)60.0(2

)15.0(2

)40.0(151.0

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11 12 130 14 15 16 17 18 19 20

ExpectedPortfolioRisk (σ )

Expected PortfolioReturn (%)

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13

14

15

16

17

18 •Maximumreturn &maximum risk(100% GER)

• Minimum risk combination(70% US & 30% GER)

• Domestic only portfolio(100% US)

• Initial portfolio(40% US & 60% GER)

The example portfolio.

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• The multiple asset model for portfolio return

• The multiple asset model for portfolio risk

)E(rwΣ)E(r ii

N

iP

1

ijjiji

N

ij

N-

iii

N

iP ρσσwwΣΣσwΣσ

1

1

1

22

1

30


2% 3% 4% 5% 6% 7% 8% 9% 10%-0.2%

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

1.6%

1.8%

Efficient Portfolios (All returns are measured in $)Historical Returns and Exchange Rates: January 1994 - June 2009

Std

Ret

urn Denmark

Australia

New Zealand

Japan

Singapore

Mexico

Data Sources: Market indexes from Yahoo.com and Exchange Rates from FRED.

SP500

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National Equity Market PerformanceJanuary 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

Correlations Matrix

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

Japan 0.35 0.33 1.00 0.45 0.58 0.27 0.51

Singapore 0.52 0.35 0.45 1.00 0.60 0.34 0.61

Australia 0.56 0.46 0.58 0.60 1.00 0.51 0.65

New Zealand 0.13 0.47 0.27 0.34 0.51 1.00 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.

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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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• 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

i

fi RR measure Sharpe

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• 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

i

fi RR measureTreynor

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• 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

113.00.0961

12

05.0015.0

measure Sharpe

0100.01.09

12

05.0015.0

measureTreynor

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• 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

m

jjmi

)kk(kk fmrfe

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The International CAPM

chapter 15 international portfolio theory and diversification 1

Documents

risk of portfolio

securities portfolio

portfolio of assets

investors portfolio

expected return of portfolio

stocks variance of portfolio

risk averse slide

minimum expected risk