Portfolio Management

Answers to the recommended questions Portfolio Management 1. Investors hold diversified portfolios in order to reduce the variance of the portfolio, which is considered a measure of risk of the portfolio. A diversified portfolio should accomplish this because the returns for the alternative assets should not be correlated so the variance of the total portfolio will be reduced. 2. The covariance is equal to E [(Ri - E(Ri)(Rj - E(Rj)] and indicates the absolute amount of comovement between two series. If they constantly move in the same direction, it will be a large positive value and vice versa. Covariance is important in portfolio theory because the variance of a portfolio is a combination of individual variances and the covariances among all assets in the portfolio. It is also shown that in a portfolio with a large number of securities the variance of the portfolio becomes the average of all the covariances. 3. Similar assets like common stock or stock for companies in the same industry (e.g., auto industry) will have high positive covariances because the sales and profits for the firms are being influenced by common factors since their customers and suppliers are the same. Because their profits and risk factors move together you should expect the stock returns to likewise move together and have high covariance. The returns from different assets will not have as much covariance because the returns will not be as correlated. This is even more so for investments in different countries where the returns and risk factors are very unique. 4. The covariance between the returns of asset i and j is affected by the variability of these two returns. Therefore, it is difficult to interpret the covariance figures without taking into account the variability of each return series. In contrast, the correlation coefficient is obtained by standardizing the covariance for the individual variability of the two return series, that is: rij = covi,j/(Sdi SDj). Thus, the correlation coefficient can only vary in the range of -1 to +1. A value of +1 would indicate a perfect linear positive relationship between Ri and Rj. 5. The efficient frontier has a curvilinear shape because if the set of possible portfolios of assets is not perfectly correlated the set of relations will not be a straight line, but is curved depending on the correlation. The lower the correlation the more curved. 7. The necessary information for the program would be: 1) the expected rate of return 2) the expected variance of return 3) the expected covariance of return with every other feasible stock under consideration. 9. The optimal portfolio for a given investor is the point of tangency between his set of utility curves and the efficient frontier. This will most likely be a diversified portfolio because almost all the portfolios on the frontier are diversified except for the two end points--the minimum variance portfolio and the maximum return portfolio. These two could be significant. 10. The utility curves for an individual specify the trade-offs he/she is willing to make between expected return and risk. These utility curves are used in conjunction with the efficient frontier to determine which particular efficient portfolio is the best for a particular investor. Two investors will not choose the same portfolio from the efficient set unless their utility curves are identical. 1 13. C. Expected return = (.15)(.50)+(.10)(.40)+(.06)(.10) = 12.1% 14. A. Adding an investment that has a correlation of –1.0 will achieve maximum risk diversification. The Market for Foreign Exchange 1. Using Exhibit 4.4, calculate a cross-rate matrix for the euro, Swiss franc, Japanese yen, and the British pound. Use the most current American term quotes to calculate the cross-rates so that the triangular matrix resulting is similar to the portion above the diagonal in Exhibit 4.6. Solution: The cross-rate formula we want to use is: S(j/k) = S($/k)/S($/j). The triangular matrix will contain 4 x (4 + 1)/2 = 10 elements. ¥ Euro 115.78 Japan (100) SF £ $ 1.4676 .6393 .9764 1.2675 .5522 .8433 .4356 .6653 Switzerland U.K 1.5272 6. Using Exhibit 4.4, calculate the one-, three-, and six-month forward premium or discount for the British pound in American terms using the most current quotations. For simplicity, assume each month has 30 days. Solution: The formula we want to use is: fN,£v$ = [(FN($/£) - S($/£))/S($/£] x 360/N f1,£v$ = [(1.5242 - 1.5272)/1.5272] x 360/30 = -.0236 f3,£v$ = [(1.5188 - 1.5272)/1.5272] x 360/90 = -.0220 f6,£v$ = [(1.5104 - 1.5272)/1.5272] x 360/180 = -.0220 8. Assume you are a trader with Deutsche Bank. From the quote screen on your computer terminal, you notice that Dresdner Bank is quoting €1.0242/$1.00 and Credit Suisse is offering SF1.5030/$1.00. You learn that UBS is making a direct market between the Swiss franc and the euro, with a current €/SF quote of .6750. Show how you can make a triangular arbitrage profit by trading at these prices. (Ignore bid-ask spreads for this problem.) Assume you have $5,000,000 with which to conduct the arbitrage. What happens if you initially sell dollars for Swiss francs? What €/SF price will eliminate triangular arbitrage? Solution: To make a triangular arbitrage profit the Deutsche Bank trader would sell $5,000,000 to Dresdner Bank at €1.0242/$1.00. This trade would yield €5,121,000= $5,000,000 x 1.0242. The Deutsche Bank trader would then sell the euros for Swiss francs to Union Bank of Switzerland at a price of 2 €0.6750/SF1.00, yielding SF7,586,667 = €5,121,000/.6750. The Deutsche Bank trader will resell the Swiss francs to Credit Suisse for $5,047,683 = SF7,586,667/1.5030, yielding a triangular arbitrage profit of $47,683. If the Deutsche Bank trader initially sold $5,000,000 for Swiss francs, instead of euros, the trade would yield SF7,515,000 = $5,000,000 x 1.5030. The Swiss francs would in turn be traded for euros to UBS for €5,072,625= SF7,515,000 x .6750. The euros would be resold to Dresdner Bank for $4,952,768 = €5,072,625/1.0242, or a loss of $47,232. Thus, it is necessary to conduct the triangular arbitrage in the correct order. The S(€/SF) cross exchange rate should be 1.0242/1.5030 = .6814. This is an equilibrium rate at which a triangular arbitrage profit will not exist. (The student can determine this for himself.) A profit results from the triangular arbitrage when dollars are first sold for euros because Swiss francs are purchased for euros at too low a rate in comparison to the equilibrium cross-rate, i.e., Swiss francs are purchased for only €0.6750/SF1.00 instead of the no-arbitrage rate of €0.6814/SF1.00. Similarly, when dollars are first sold for Swiss francs, an arbitrage loss results because Swiss francs are sold for euros at too low a rate, resulting in too few euros. That is, each Swiss franc is sold for €0.6750/SF1.00 instead of the higher noarbitrage rate of €0.6814/SF1.00. 9. The current spot exchange rate is $1.55/£ and the three-month forward rate is $1.50/£. Based on your analysis of the exchange rate, you are pretty confident that the spot exchange rate will be $1.52/£ in three months. Assume that you would like to buy or sell £1,000,000. a. What actions do you need to take to speculate in the forward market? What is the expected dollar profit from speculation? b. What would be your speculative profit in dollar terms if the spot exchange rate actually turns out to be $1.46/£. Solution: a. If you believe the spot exchange rate will be $1.52/£ in three months, you should buy £1,000,000 forward for $1.50/£. Your expected profit will be: $20,000 = £1,000,000 x ($1.52 -$1.50). b. If the spot exchange rate actually turns out to be $1.46/£ in three months, your loss from the long position will be: -$40,000 = £1,000,000 x ($1.46 -$1.50). 3 International Portfolio Investment 1. What factors are responsible for the recent surge in international portfolio investment (IPI)? Answer: The recent surge in international portfolio investments reflects the globalization of financial markets. Specifically, many countries have liberalized and deregulated their capital and foreign exchange markets in recent years. In addition, commercial and investment banks have facilitated international investments by introducing such products as American Depository Receipts (ADRs) and country funds. Also, recent advancements in computer and telecommunication technologies led to a major reduction in transaction and information costs associated with international investments. In addition, investors might have become more aware of the potential gains from international investments. 2. Security returns are found to be less correlated across countries than within a country. Why can this be? Answer: Security returns are less correlated probably because countries are different from each other in terms of industry structure, resource endowments, macroeconomic policies, and have non-synchronous business cycles. Securities from a same country are subject to the same business cycle and macroeconomic policies, thus causing high correlations among their returns. 5. Explain how exchange rate fluctuations affect the return from a foreign market measured in dollar terms. Discuss the empirical evidence on the effect of exchange rate uncertainty on the risk of foreign investment. Answer: It is useful to refer to Equations 11.4 and 11.5 of the text. Exchange rate fluctuations mostly contribute to the risk of foreign investment through its own volatility as well as its covariance with the local market returns. The covariance tends to be positive in most of the cases, implying that exchange rate changes tend to add to exchange risk, rather than offset it. Exchange risk is found to be much more significant in bond investments than in stock investments. 6. Would exchange rate changes always increase the risk of foreign investment? Discuss the condition under which exchange rate changes may actually reduce the risk of foreign investment. Answer: Exchange rate changes need not always increase the risk of foreign investment. When the covariance between exchange rate changes and the local market returns is sufficiently negative to offset the positive variance of exchange rate changes, exchange rate volatility can actually reduce the risk of foreign investment. 10. Why do investors invest the lion’s share of their funds in domestic securities? Answer: Investors invest heavily in their domestic securities because there are significant barriers to investing overseas. The barriers may include excessive transaction costs, information costs for foreign 4 securities, legal and institutional restrictions, extra taxes, exchange risk and political risk associated with overseas investments, etc. 11. What are the advantages of investing via international mutual funds? Answer: The advantages of investing via international mutual funds include: (1) save transaction/information costs, (2) circumvent legal/institutional barriers, and (3) benefit from the expertise of professional fund managers. Problems: 1. Suppose you are a euro-based investor who just sold Microsoft shares that you had bought six months ago. You had invested 10,000 euros to buy Microsoft shares for $120 per share; the exchange rate was $1.15 per euro. You sold the stock for $135 per share and converted the dollar proceeds into euro at the exchange rate of $1.06 per euro. First, determine the profit from this investment in euro terms. Second, compute the rate of return on your investment in euro terms. How much of the return is due to the exchange rate movement? Solution: It is useful first to compute the rate of return in euro terms: C r ≅ r$ + e 1   1 −    135 − 120   1.06 1.15  = + 1   120     1 . 15   = 0.125 + 0.085 = 0.210 This indicates that this euro-based investor benefited from an appreciation of dollar against the euro, as well as from an appreciation of the dollar value of Microsoft shares. The profit in euro terms is about C2,100, and the rate of return is about 21% in euro terms, of which 8.5% is due to the exchange rate movement. 4. Japan Life Insurance Company invested $10,000,000 in pure-discount U.S. bonds in May 1995 when the exchange rate was 80 yen per dollar. The company liquidated the investment one year later for $10,650,000. The exchange rate turned out to be 110 yen per dollar at the time of liquidation. What rate of return did Japan Life realize on this investment in yen terms? Solution: Japan Life Insurance Company spent ¥800,000,000 to buy $10,000,000 that was invested in U.S. 5 bonds. The liquidation value of this investment is ¥1,171,500,000, which is obtained from multiplying $10,650,000 by ¥110/$. The rate of return in terms of yen is: [(¥1,171,500,000 - ¥800,000,000)/ ¥800,000,000]x100 = 46.44%. 6