Chapter 9 The Case for International Diversification Note:
In the sixth edition of Global Investments, the exchange rate quotation symbols differ from previous editions. We adopted the convention that the first currency is the quoted currency in terms of units of the second currency. For example, €:$ 1.4 indicates that one euro is priced at 1.4 dollars. In previous editions we used the reversed convention $/ € 1.4, meaning 1.4 dollars per euro. All problems in this test bank still use the old convention and have not been adapted to reflect the new quotation symbols used in the 6th edition.
Questions and Problems 1. The annualized performance, performance, in U.S. dollars, of the United States and EAFE stock indices are: ReturnUS
12%
ReturnEAFE 14.6%
US US
15.5%
EAFE
18.2%
Correlation 0.47 a. b.
What would be the return return and risk of a portfolio portfolio invested invested half in the U.S. market and half in in the EAFE index? What if the correlation increases increases to 0.6?
Solution
a.
Return portfolio 50% ReturnUS 50% ReturnEAFE 13.3%. portfolio
(0.5 US )2 (0.5 EAFE )2 2 0.47 0.5 0.5 US EAFE 2 09.166 .
portfolio
14.46%.
2
b.
The return of the portfolio remains remains unchanged at 13.3%. The risk is higher at: portfolio
(0.5 US )2 (0.5 EAFE )2 2 0.60 0.5 0.5 US EAFE 2 27.50
portfolio
15.08%.
2
2. The annualized performance, in U.S. dollars, of of the U.S. and European stock indices indices are: ReturnUS
10%
Returneurope 11%
US US
europe europe
Correlation 0.60
16% 18%
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a. b.
What would be the return and risk of a portfolio invested half in the U.S. market and half in the European index? What if the correlation increases to 0.8?
Solution
a.
Return portfolio 50% ReturnUS 50% Returneurope 10.5%. portfolio
(0.5 US )2 (0.5 europe )2 2 0.6 0.5 0.5 US europe 231.4
portfolio
15.21%.
2
b.
The return of the portfolio remains unchanged at 10.5%. The risk is higher at: portfolio
(0.5 US )2 (0.5 europe )2 2 0.8 0.5 0.5 US europe 260.2
portfolio
16.13%.
2
3. Here are the expected returns and risks of two assets:
a. b. c.
E( R1) 10%
1
16%
E( R2) 14%
2
16%
Assume a correlation of 0.5 and draw all the portfolios made up of the two assets in an Expected Return/Risk graph. Same question assuming successively a correlation of 1, 0, and 1. Looking at the four graphs, what do you conclude about the importance of correlation in risk-reduction?
Solution
Chapter 9
The Case for International Diversification
Conclusion: The lower the correlation the more risk reduction can be achieved without sacrificing return.
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4. Try to find some reasons why: a. b.
Stock and bond markets should be strongly correlated and, Stock and bond markets should be weakly correlated.
Solution
a.
Stock and bond markets should be strongly correlated for several reasons: — Both react negatively to interest rate movements. — Both are affected by the risk attitude of investors: when economic uncertainties or investors’ risk aversion increase, investors require a larger risk premium on assets and the price of bonds and stocks fall. — Foreign capital flows are influenced by exchange rate expectations. Foreigners may buy U.S. bonds and stocks when they believe the dollar is strong but sell their U.S. assets in fear of a dollar depreciation. This foreign attitude creates a positive correlation between the domestic bond and stock markets. b. Stock and bond markets should be weakly correlated for the following reason: — Stocks ar e “real” assets influenced mostly by real variables while bonds are “nominal” assets influenced by “monetary” variables. The two sets of variables are quite independent. 5. The French stock market has a sigma of 20%, when computed in euros. The U.S. stock market has a sigma of 16% in US$ and the €/US$ exchange rate has a sigma of 6%. a. b.
Assuming that the correlation between stock market and currency movements is zero, what is the sigma of the U.S. stock market when expressed in €. Using this number, calculate the sigma, in €, of a portfolio made up of 50% of French stocks and 50% of U.S. stocks (zero-correlation between the two markets).
Solution:
a.
b.
Total risk of U.S. stocks in €: 2 in €
in2 $ €2/$ 0 292
in €
17.1%.
Risk of a diversified portfolio 50 French/50 U.S., all measured in €, with zero correlation: 2
portfolio
(0.5 french)2 (0.5 US in €)2 2 0.5 0.5 0 french US in € 173
13.15%.
6. The Japanese stock market has a sigma of 18%, when computed in yen. The U.S. stock market has a sigma of 17% in US$ and the US$/¥ exchange rate has a sigma of 6%. The correlation between the Japanese stock market and $/¥ currency movements is 0.1; in other words, the Japanese stock market tends to go up when the yen goes down. The correlation between the Japanese and U.S. stock markets is equal to 0.4, measured either in local currency of in dollars. a. b.
What is the sigma of the Japanese market when expressed in dollars? Using this number, calculate the sigma, in dollars, of a portfolio made up of 50% of Japanese stocks and 50% of U.S. stocks.
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Solution
a.
b.
Total risk of Japanese stocks in $: 2 in $
in2 $/2 2 in $/ 338.4
in €
18.4%.
ゥ
ゥ
Risk of a diversified portfolio 50 US/50 Japan, measured in $: 2
portfolio
(0.5 US)2 (0.5 japan)2 2 0.5 0.5 0.4 US japan 219.45
14.8%.
7. Assume that the domestic volatility (standard deviation in yen) of the Japanese stock market is 18%. The volatility of the yen against the U.S. dollar is 6%. a. b.
What would the dollar volatility of the Japanese stock market be for a U.S. investor if the correlation between the Japanese stock market returns and e xchange rate movements were zero? Suppose the dollar volatility of the Japanese stock market is 18.4%, what can you conclude about the correlation between the Japanese stock market movements and exchange rate movements?
Solution
a.
b.
If the correlation between stock market returns and exchange rate movements were equal to zero, the dollar volatility of the Japanese stock market would be: 2 in $
in2 $/2 2 in $/ (18)2 (6)2 (2)(0)(18)(6) 360
2 in $
18.97%.
ゥ
ゥ
Because the actual dollar volatility is 18.4%, we conclude that the correlation between Japanese stock market returns and exchange rate movements is negative. The actual correlation is 0.1. This can be explained by the idea that a weak currency is associated with rising stock prices; a depreciation of the yen is good for Japanese corporations.
8. Assume that the domestic volatility (standard deviation in yen) of the Japanese bond market is 8%. The volatility of the yen against the U.S. dollar is 6%. a. b.
What would the dollar volatility of the Japanese bond market be for a U.S. investor if the correlation between the Japanese stock market returns and e xchange rate movements were zero? Suppose the dollar volatility of the Japanese stock market is 11.35%, what can you conclude about the correlation between the Japanese bond market movements and exchange rate movements?
Solution
a.
b.
If the correlation between bond market returns and exchange rate movements were equal to zero, the dollar volatility of the Japanese bond market would be 2 in $
in2 $/2 2 in $/ (8)2 (6)2 (2)(0)(8)(6) 100
2 in $
10%.
ゥ
ゥ
Because the actual dollar volatility is 11.35%, we conclude that the correlation between Japanese bond market returns and exchange rate movements is positive. The actual correlation is 0.30.
This can be explained by the idea that a weak currency is associated with rising interest rates (and negative bond returns) to defend the currency.
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9. In 1994, the United States was experiencing a fairly strong economic recovery, ahead of other nations. Fears of an overheating economy led to sudden inflationary fears for the next few years. a. b. c.
Would you expect U.S. interest rates to rise or drop? Would you expect the dollar to depreciate or appreciate? Would you expect a foreign bond portfolio to be a good investment compared to a U.S. dollar portfolio under this scenario?
Solution
a. b. c.
Inflationary fears will cause interest rates to rise, even if the real interest rate remains constant. Economic growth could also lead to higher real interest rates. The dollar is likely to depreciate because of higher expected inflation. This could be offset by higher real interest rates, which could attract foreign capital flows. The rise of U.S. interest rates will cause the market price of U.S. bonds to fall. It is, therefore, appropriate to invest in foreign bonds rather than U.S. bonds. The likely depreciation of the dollar makes such an investment even more interesting (the foreign currency in which the bonds are denominated will appreciate against the dollar during the investment period and the investor will make a foreign exchange profit when reselling his bonds).
10. Suppose that you overheard the following statements at a conference for institutional investors: ( A German national ): “My money manager knows the German firms very well; why should I bother to invest in French and American shares? I am not familiar with their names or their operations, and I will have to pay much higher costs to buy t hem.” ( A French national ): “Why should I buy German and American shares? The foreign brokers will give preferential treatment to their domestic clients, and I am going to get a lousy deal in terms of prices and costs. Furthermore, I can’t read the financi al statements of these companies, as they are written in German or English, and with different accounting methods.” ( An American national ): “I can’t even pronounce the names of these foreign companies; how could I defend investing abroad in front of my board of trustees? By the way, what is the capital of Switzerland: Geneva or Zurich?” How would you try to convince these people to diversify their portfolios if you were the marketing representative of a big international money manager? Solution
I would use all the arguments and evidence presented in Chapter 4 of International Asset Pricing . Two specific arguments quoted are that:
Foreign is exotic and therefore that an investor feels safer by staying at home.
Foreign investment would be more costly in terms of transaction cost and price execution.
The first argument is very practical. Today, a finance professional should be able to deal with the world and invest globally if it makes financial sense. The second argument is getting outdated by market deregulation and competition among brokers. Local brokers will do their utmost to attract foreign clients. It is often claimed that competition among brokers leads to lower costs and better prices for new foreign clients than for captive domestic clients. In any case, the automation of all stock exchanges reduces the risk for any investor of being treated unfairly.
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11. Assume that the domestic and foreign assets have standard deviations of d 16% and f 19%, respectively, with a correlation of df 0.6. The risk-free rate is equal to 5% in both countries. a.
The expected returns of the domestic and foreign assets are both equal to 10%, E ( Rd ) E ( R f ) 10%. Calculate the Sharpe ratios for the domestic asset, the foreign asset, and an internationallydiversified portfolio equally invested in the domestic and foreign assets. What do you conclude? b. Assume now that the expected return on the foreign asset is higher than on the domestic asset, E ( Rd ) 10% but E ( R f ) 12%. Calculate the Sharpe ratio for an internationally diversified portfolio equally invested in the domestic and foreign assets, and compare your findings to those in Question (a). Solution
a.
The domestic asset has an expected return of 10% and a standard deviation of 16%. Its Sharpe ratio is equal to Sharpe ratio
E (R) Risk-free rate
10% 5% 16%
0.313.
10% 5%
0.263. 19% A portfolio equally invested in the domestic and foreign assets has a n expected return of 10% and a standard deviation p given by: The foreign asset has a Sharpe ratio of
p2
0.52 d2 f2 (2df d f )
p2
0.52 [256 361 (2 0.6 15 19)] 245.45.
Hence, the standard deviation p is given by portfolio is equal to: Sharpe ratio
245.45, or 15.67%. The Sharpe ratio of the
E ( R) Risk-free rate
10% 5% 15.67%
0.319.
The foreign asset has a lesser Sharpe ratio than the domestic asset because it has the same expected return but a larger standard deviation. However, the equally-weighted portfolio benefits from risk diversification and a lower standard deviation. Hence, its Sharpe ratio is better than the ratios of both the domestic and the foreign assets. b.
A portfolio equally invested in the domestic and foreign asset now has an expected return of 5% 11% (0.510% 0.512% 11%). Hence, the Sharpe ratio is equal to 11% 0.383. The 15.67% portfolio’s Sharpe ratio is now better than that of the domestic asset (0.313), both because of risk diversification benefits and because of the superior expected return of the foreign asset [new 5% Sharpe ratio of 12% 0.368]. 19%
12. You consider investing in an emerging market. Its stock market volatility (standard deviation of returns measured in U.S. dollars) is 25%. The volatility of the World index of developed markets is 15%. The correlation between the emerging market and the World index is 0.2. a. b.
What would be the volatility of a portfolio invested 95% in the World index and 5% in this emerging market? Compare the result found in the previous question with the volatility of the World index and give an intuitive explanation.
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Solution
a.
b.
2 portfolio
(0.95 world )2 (0.05 em .)2 2 0.2 0.95 0.05 world em . 211.75
portfolio
14.55%.
The portfolio has a smaller volatility than the World index. Although we add an asset with high volatility (25% compared to 15%), the net result is a drop in total volatility. This is because this new investment has a low correlation with the World index of developed markets. There will be periods when the world index will experience a negative return while the emerging market will experience a positive return (and even a large positive return given its high volatility).
13. You consider investing in some emerging country. Its recent economic growth rate is around 7%, well above the average growth rate of developed countries estimated at 2% by the OECD. Its annual inflation rate is around 10%, well above the average inflation rate of developed countries estimated at 2% by the OECD. The currency of the emerging country has been depreciating at an annual rate of around 8% against major currencies. While the volatility of the World stock index (standard deviation of dollar returns) is around 15%, the stock market of this emerging country has a volatility of 25%. The correlation of this emerging stock market with the World index is only 0.2. a. b. c.
Are the high inflation rate and weak currency sufficient reasons to avoid investing in this emerging country? Is the high volatility of the local market a sufficient reason to avoid investing in this emerging country? Suggest why you would consider investing in this emerging country.
Solution
a.
The local currency depreciates because of the high local inflation rate (see Chapter 2). According to purchasing power parity, the currency depreciation is expected to be equal to the inflation differential between the emerging country and developed countries. But real economic growth is high. Inflation is not necessarily bad, if stock prices appreciate by the inflation rate plus a real return commensurate to the local growth rate. b. The local market volatility is high. But most of it will get diversified away in a globally diversified portfolio. The contribution of this emerging market to the global portfolio risk will be much smaller than indicated by its high volatility. c. This emerging market offers high growth potential, so the contribution to return could be high. Its low correlation also offers some risk-reduction potential. 14. You have collected some risk and return estimates for various market indexes. These indexes are the World stock market index of developed markets, the Morgan Stanley Capital International (MSCI) Europe, Australasia and Far East (EAFE) index, and the International Finance Corporation (IFC) Composite index of emerging markets. Here are some risk and return estimates for the future: Market
World EAFE Composite
Return
Risk
10% 12% 15%
16% 17% 25%
Chapter 9
The Case for International Diversification
All return and risk measures are calculated in U.S. dollars and are expressed in % per year. The correlation matrix is given below: World
EAFE
Composite
1.0 0.5 0.2
0.5 1.0 0.1
0.2 0.1 1.0
World EAFE Composite a.
b.
Calculate the return and risk of a portfolio invested in the following proportions: Portfolio
World
EAFE
Composite
A B C
50% 45% 40%
50% 45% 40%
0% 10% 20%
Try to derive some estimate of the efficient frontier obtained by using these three indexes (no short sales are allowed).
Solution
a.
Risk and return: Portfolio
Return%
Volatility%
A: 50%World 50%EAFE
11.00
14.29
B: 45%World 45%EAFE 10%Composite
11.40 11.80
13.52 13.24
C: 40%World 40%EAFE 20%Composite
b.
Efficient frontier:
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Mean Return
Volatility
% WORLD
% EAFE
% Composite
11.76 11.80 12.00 12.20 12.40 12.60 12.80 13.00 13.20 13.40 13.60 13.80 14.00 14.20 14.40 14.60 14.80 15.00
13.24 13.24 13.29 13.40 13.59 13.84 14.16 14.53 14.96 15.46 16.16 17.05 18.10 19.28 20.57 21.95 23.40 25.00
42 41 36 30 24 19 13 08 02 00 00 00 00 00 00 00 00 00
38 38 41 43 46 49 51 54 56 53 47 40 34 27 20 14 07 00
20 21 24 27 30 33 35 38 41 47 53 60 66 73 80 86 93 100
Note:
In this table the weights of the efficient portfolios have been rounded to the nearest percentage point. Mean return and volatility calculations use the exact weights. 15. The currencies of several emerging countries depreciate at a rapid pace. Does it imply that you should not invest in their stock markets? For example, the Polish zloty went from 15,767 to 21,444 zlotys per U.S. dollar in 1993. The Polish stock market went from 1,040 to 12,439 during the same period. Guess why the zloty depreciated. Solution
No. A rapid depreciation of a currency is generally caused by a rampant inflation. Stocks are claims on real assets and their prices tend to go up with inflation. The question is whether they go up faster or slower than the inflation rate? During 1993, the Polish inflation rate was around 40%, explaining the zloty’s depreciation. The Polish stock market appreciation was extremely high. The privatization program drew enormous interest in 1992 and 1993. The rise in stock prices was either based on expectations of very high future economic growth or was somewhat irrational. The 1994 performance of the stock market was very bad.
16. You consider investing in four very volatile emerging markets. These are small countries just opening up to foreign investment. You spread your money equally across them. After a year, the following observations are made on the performance of each market:
Country
A B C D
Return in Currency Local Currency Depreciation
400% 60% 0 – 100%
20% 10% 40% 80%
Comment
High inflation, high growth High inflation, low growth Foreigners got expropriated
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a.
Calculate the return, in dollars, on each market. The currency depreciation is equal to the drop in the dollar value of one unit of local currency. For example, if the peso moves from 1 dollar per peso to 0.8 dollar per peso, the depreciation of the peso is measured as 20%. b. What is the return on a portfolio equally invested in each market? Solution
a.
Let’s remember the relations linking the dollar and local -currency value of an asset: $
V
where
V S
$
V is the dollar value of asset, V is the local currency value of asset, S is the exchange rate ($/local currency). In rate of return form, we have:
1 R$ (1 R) (1 s ). Where R$, R, and s are the percentage variations in the above variables. Then:
b.
Country
Dollar Return
A B C D
300% 44% 40% 100%
The return on an equal-weighted portfolio is 51%.
17. Project : Take monthly values of the stock indexes of a selected group of developed and emerging stock markets over a period of ten years. a. b. c.
d.
Calculate the correlation of returns among them, in local currency. Break the period into two five-year subperiods and compare the calculated correlations over the two subperiods. Multiply the stock prices by the exchange rate and calculate the correlation of U.S. dollar returns. Do the figures change dramatically for developed markets? Do the figures change dramatically for emerging markets? Focus on emerging markets experiencing high inflation. Why is it important to perform the calculations using a single-base currency when looking at countries with high inflation from a foreign viewpoint?
