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MiF Art Durnev International Corporate Finance Spring 2012 CASE 3: HEDGING CURRENCY RISKS IN INTERNATIONAL PORTFOLIOS You are fresh out of the NES MiF program and have joined Troika Dialog’s Emerging Markets group. Your first assignment is to advise several partners on portfolio allocation strategies for their World Equity Fund. They have decided that the fund will allocate its capital across Western European, Asian, and North American countries. Initially, in each of these markets, the fund faces a constraint which makes short-selling the index prohibitively expensive. You begin by obtaining monthly market return and exchange rate data over the past 30 years. Combining this with information contained in forward rates and option prices, you obtain the estimates of annual percent expected returns, correlations, and risks for the $-valued returns and summarize them in an Excel file (portfolios_ex.xls). 1. The partners would like for the fund to be fully invested across all countries but would otherwise like to minimize the fund’s risk. Can you suggest a strategy for doing so? What are the risk, expected return, Sharpe ratio, and weights of such a minimum-risk all-equity portfolio? 2. Seeing the low returns offered by the minimum-risk portfolio, the partners are interested in learning about what additional returns they can get if they increase the fund’s risk level. How do the risk-return tradeoffs (efficiency frontier) offered by the above opportunities look? How do the weights change as you increase the portfolio risk? 3. Noticing the fund’s strange all-equity stance, you decide to convince the partners to permit the fund to deposit or borrow under the risk-free rate. a. How much additional expected return can your strategy offer them when the fund maintains the level of risk calculated in Question 1? How do we allocate the capital between the risk-free security and equities? What are the weights of the “Tangency” or “Efficient” portfolio (i.e. weights as a fraction of total equity weight)? b. Can you suggest an alternative way (other than maximizing the Sharpe Ratio) to find the “Tangency” portfolio? c. How do the risk-return tradeoff offered by the above opportunities look? 4. How does the efficiency frontier look if you allow to short-sell risky securities? Plot the frontier when you can invest in a risk-free asset and when you cannot. Compare them to the frontiers in questions 2 and 3. 5. Having convinced the partner’s to relax the all-equity constraint, you decided the next thing to do is to convince them to relax the countries’ focus. To demonstrate the merits of your plan, you consider the benefits of diversifying the fund, for the time being, into the RTS, a Russian index. The RTS’ expected return is estimated to be 24.57% with a standard deviation of 16.35%, and its correlations with the remaining markets are: 0.37, 0.39. 0.25, 0.25, 0.13, 0.07, 0.27, 0.32, 0.08, 0.17, 0.19, -0.21, -0.24, 0.33, and 0.17, respectively. a. How much additional return will your plan to obtain on all-equity fund if you maintain the same risk as in Question 1? b. What would be the risk, return, and weights of portfolio if investor has the preferences described by function U(R,) =R - 0.5. Assume that short-selling or investment in a risk-free security is not allowed. International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 Next, you set your sights on currency risk. The currency risk, you explain, can be easily hedged away using money market hedges – simply borrowing in the foreign currency and depositing in dollars to offset a position in the equity market. To demonstrate this, you set up a spreadsheet similar to the one found in “portfolios_ex.xls” You obtain decompositions of dollar return covariances into local currency return and exchange rate covariances. Also, you construct exchange rate expectations assuming uncovered interest parity holds. The fund has decided that with the uncertainty in financial markets which ensued following the U.S. presidential elections, the fund will retrench and allocate its capital solely across 4 countries: Korea, Hong Kong, Singapore, and Taiwan. Again, in each of these markets, the fund faces a constraint which makes short-selling the index prohibitively expensive. 6. How much return can you obtain on an all-Asian 100% equity portfolio if the partners permit you to take a position in each currency up to an amount which offsets the equity position (maintaining 18.468% risk )? Describe this portfolio’s positions. 7. At the presentation of your proposal, one of Troika’s exchange rate forecasters points out that they are forecasting a 1.5% appreciation of the Taiwanese dollar – not the 1.375% depreciation implied by the interest differential (that is, the 23.3% dollar-based Taiwanese return estimate actually consists of a 21.8% expected local return and a 1.5% anticipated appreciation). Assume that the fund maintains the level of risk in Question 6. With this new information, how do your recommendations change? Why? It is now one year later. Your fund has been an enormous success since your arrival. As a result, you find yourself with a large team of market and currency researchers and forecasters. After many weeks of work, they produce for you updated estimates of risks and correlations which remain similar to those in the Template. However, interest rates and your researcher’s expectations of market and currency returns have changed substantially. The following new returns and interest rates are anticipated (note that UIP is not expected to hold): Country Singapore Korea Taiwan Hong Kong Risk-free rate Market Exchange Rate Interest Rate 15.5 2.5 4.0 18.5 5.5 3.0 23.5 6.5 1.0 21.5 6.0 0.0 5.5 8. You have been informed that the returns of your fund will now be compared to the returns of other funds (i.e. other groups) during each month of the subsequent year. Your employers would like the average monthly ranking of your fund to be as high as possible. Your only constraints are that 1) capital is allocated across the 4 markets and risk-free security; 2) no markets are sold short; 3) leverage is no higher than 2 (i.e. the weights on the 4 markets add up to no more than 2); 3) no currency is sold short in greater magnitude than the long position in the country’s equity market. a. Can you suggest a set of portfolio weights that will accomplish this? b. Assuming that you are managing pension money, would it be a good investment strategy? International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 NOTE: As the case output, please, send me your Excel spreadsheet with all necessary formulas, graphs, and explanations. International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 Appendix A Optimal Portfolios A Three-Country Example Using Excel Refer to portfolios_ex.xls Step 1: Correlation coefficients and standard deviations appear in cells B5:E8. Create the Variance/Covariance Matrix V using the following formula: Cov(A,B)=Corr(A,B)*SD(A)*SD(B)). Place it in cells B18:D20. Step 2: Enter returns from Question 1. Then set up the vector of returns in excess of US risk-free rate, RRf: Enter this formula as a column vector in cells E11:E13. Step 3: Set up a vector of initial weights W for each country by entering 0.0 in cells B23:B25. Enter the total risky portfolio weight in B26 (“=SUM(B23:B25)”). Step 4: Set up a row vector of weights WT by entering the following formulas in cells C22 through E22: C22: “=23” D22: “=B24” E22: “=B25”. Step 5: Create the expected return cell by entering the following formula into B29: “=B14+MMULT(C22:E22,E11:E13)” This calculates Rf + WTR …the expected return given certain weights in each of the 3 risky markets and the reminder in the risk-free security. Step 6: Create the portfolio standard deviation cell by entering the following formula into cell B30: International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 “=MMULT(MMULT(C22:E22,B18:D20),B23:B25))^0.5” This calculates (WTVW)1/2. The expression “MMULT(C22:E22,B18:D20)” multiplies the row vector of weights (WT) in cells C22:E22 times the Variance-Covariance matrix in B18:D20. The rest multiplies this times the column vector of weights (W) in B23:B25 and takes it square root. Step 7: Run solver: -to solve for the minimum risk portfolio, enter B30 as the “Target Cell”. Set “Equal To:” to “Min”. Enter cells B23:B25 in the “By Changing Cells:” box. This says solver will attempt to minimize the risk level (B30) by altering the portfolio weights (B23:B25). To avoid investment in a risk-free security add B26=1 constraint. No short-sale constraint is B23:B25>=0. -to solve for the maximum return for a given level of risk, enter the desired level of risk in cell C30. Then enter B29 as the Target Cell. Set Equal To: to Max. Then add the constraint B30=C30. -to construct the efficient frontier, add an additional constraint that the risk is equal to some level in cell, B30=C30. By changing the number in C30 you will be getting a new risk-return combination. - to construct the optimal portfolio, that is the portfolio that maximizes the Sharpe Ratio = (R – Rf)/ enter =(B29-B14)/B30 in cell B31 and set the target cell =B31 in the Step 8: To allow borrowing at a risk-free rate, delete the B26=1 constraint. Step 9: To allow short-sales, delete the B23:B25>=0 constraint. International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 Appendix B Unhedged $ Portfolio Return Expected $ return for an unhedged portfolio with N $-denominated risky securities: R p (1 1 2 ... N )r f 1 R1 2 R2 ... N RN r f 1 R1 r f 2 R2 r f ... N RN r f , where Rp = $ portfolio return, wi = weight on security i, rf = risk-free rate and Ri = $expected return for security i. In matrix notations, R p rf T R e rf 1 2 ... N R1 rf R2 rf . ... RN rf The variance-covariance matrix is V p T V , where var( R1 ) cov( R R ) 2 1 V . cov( R N R1 ) International Corporate Finance cov( R1 R2 ) ... cov( R1 R N ) var( R2 ) ... cov( R2 R N ) . . . . . . cov( R N R1 ) ... var ( R N ) Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 Appendix C Hedged $ Portfolio Return Expected return of the hedged portfolio with N risky securities: R p r f 1 R1 %S1 r f 2 R2 %S 2 r f ... N RN %S N r f 1c r f r1 %S1 2c r f r2 %S 2 ...1c r f r2 %S 2 , where Ri = local-currency return on a security in country i Si = expected return on currency i, ri = interest rate in country i and ic = weight on currency i. Note that hedging currency i involves borrowing currency at rate ri, exchanging it to $, and depositing $ at rf. In matrix notations, R p rf T R e rf 1 2 ... N 1c 2 c ... N R1 %S1 rf R2 %S 2 rf ... RN %S N rf . r r % S f 1 1 r r %S 2 2 f ... rf rN %S N For practical purposes, it is more convenient to write the portfolio return as R p r f T R r f 1 2 ... N 1c 2 c ... N R1 %S1 r f R2 %S 2 r f ... R N %S N r f . % S r r 1 1 f %S r r 2 2 f ... %S N rN r f In this case, the optimization program will choose the currency weights as –w1c, -w2c, …, -wNc. International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 The variance-covariance matrix is: V p T VRR T VSS T VSR T VRS . where VRR, VSS, VSR, VRS are variance-covariance matrixes of returns, exchange rates and returns with exchange rates, respectively. They are: V RR cov( R1 R2 ) ... cov( R1 R N ) | 0 0...0 var( R1 ) cov( R R ) var( R ) ... cov( R2 R N ) | 0 0...0 2 1 2 . . . . . . cov( R N R1 ) cov( R N R1 ) ... var ( R N ) | 0 0...0 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0 ... 0 | 0 0...0 0 . . . | . 0 ... 0 | 0 0...0 0 To shorten notations we drop %notation. So Si stands for %Si. VSS cov( S1 S 2 ) ... cov( S1 S N ) | var( S1 ) cov( S1 S 2 ) ... cov( S1 S N ) var( S1 ) cov( S S ) var ( S ) ... cov( S 2 S N ) | cov( S 2 S1 ) var ( S 2 ) ... cov( S 2 S N ) 2 1 2 . . . | . . . cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) | cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) | cov( S1 S 2 ) ... cov( S1 S N ) | var( S1 ) cov( S1 S 2 ) ... cov( S1 S N ) var( S1 ) cov( S S ) var ( S ) ... cov( S 2 S N ) | cov( S 2 S1 ) var ( S 2 ) ... cov( S 2 S N ) 2 1 2 . . . |. . . cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) | cov( S N S1 ) cov( S N S 2 ) ... var ( S N ) International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 VSR VRS 0 cov( S1 R1 ) cov( S1 R2 ) ... cov( S1 R N ) | 0 0 ... cov( S R ) cov( S R ) ... cov( S R ) | 0 0 ... 0 2 1 2 2 2 N . . . .|. . cov( S N R1 ) cov( S N R2 ) ... cov( S N R N ) | 0 0 ... 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- | - - - - - - - - - - , 0 cov( S1 R1 ) cov( S1 R2 ) ... cov( S1 R N ) | 0 0 ... cov( S R ) cov( S R ) ... cov( S R ) | 0 0 ... 0 2 1 2 2 2 N . . . . . . cov( S N R1 ) cov( S N R2 ) ... cov( S N R N ) | 0 0 ... 0 cov( R1 S1 ) cov( R1 S 2 ) ... cov( R1 S N ) | cov( R1 S1 ) cov( R1 S 2 ) ... cov( R1 S N ) cov( R S ) cov( R S ) ... cov( R S ) | cov( R S ) cov( R S ) ... cov( R S ) 2 1 2 2 2 N 2 1 2 2 2 N . . . . . . cov( R N S1 ) cov( R N S 2 ) ... cov( R N S N ) | cov( R N S1 ) cov( R N S 2 ) ... cov( R N S N ) | 0 0 | 0 0 0 0 0 0 0 | 0 0 0 . . .| . . . 0 0 0 |0 0 0 International Corporate Finance Case3: Portfolios MiF Art Durnev International Corporate Finance Spring 2012 International Corporate Finance Case3: Portfolios