Consider portfolios with positions in the US and Brazilian equity markets. The (annual) expected return and standard deviation of returns for the 2 markets are as follows:

US Brazil

E[r] 5% 10%

SD[r] 15% 25%

The correlation between the returns is 0.4, and the (annual) risk-free (T-bill) rate is 1%. (The Excel file attached "Finance for Executives - US-China" that contains the calculations for the 2-asset example may be helpful in answering this question.)

a.Calculate the expected returns (in percent), standard deviations (in percent), and Sharpe ratios of the portfolios for weights in the US ranging from 100% to 0% (in 10% increments), with the remainder invested in Brazil.

b.Find the weights in the US and Brazil for a portfolio with an expected return of 15%. What is the standard deviation of this portfolio (in percent)?

c.What are the approximate weights (to the nearest 1%) in the US and Brazil in the maximum Sharpe ratio portfolio, i.e., the tangency portfolio? What is the expected return (in percent), standard deviation (in percent), and Sharpe ratio of this portfolio?

(Trial and error is a viable strategy, but you can also use the Solver tool in Excel if you know how.)

d.Calculate the expected returns (in percent), standard deviations (in percent) and Sharpe ratios of the following portfolios:

(i)50% in the risk-free asset, 50% in the US

(ii)50% in the risk-free asset, 50% in the tangency portfolio from part c.

e.What are the portfolio weights in the risk-free asset, the US, and Brazil in the portfolio in part d(ii).

f.Find the weights (T-bill, US, Brazil) for a portfolio with the same expected return as Brazil (10%), using only a combination of the risk-free rate and the tangency portfolio in part c? What is the standard deviation of this portfolio?

US China rF E[r] 9.0% 13.0% 1% w(US) 0.02088 w(US) Min Var W(China) 2.0 -1.0 1.9 -0.9 1.8 -0.8 1.7 -0.7 1.6 -0.6 1.5 -0.5 1.4 -0.4 1.3 -0.3 1.2 -0.2 1.1 -0.1 1.0 0.0 0.9 0.1 0.8 0.2 0.7 0.3 0.6 0.4 0.5 0.5 0.4 0.6 0.3 0.7 0.2 0.8 0.1 0.9 0.0 1.0 -0.1 1.1 -0.2 1.2 -0.3 1.3 -0.4 1.4 -0.5 1.5 -0.6 1.6 -0.7 1.7 -0.8 1.8 -0.9 1.9 -1.0 2.0 W(China) 0.867 0.133 0.857 w(US) Max SR Vol 15.0% 30.0% 0.0% 0.756 0.756 w(Max SR) 0.0 3.0 W(China) 0.244 E[r] 1.00% 27.93% SR 0.533 0.400 Corr E[r] 0.20 Vol 5.00% 5.40% 5.80% 6.20% 6.60% 7.00% 7.40% 7.80% 8.20% 8.60% 9.00% 9.40% 9.80% 10.20% 10.60% 11.00% 11.40% 11.80% 12.20% 12.60% 13.00% 13.40% 13.80% 14.20% 14.60% 15.00% 15.40% 15.80% 16.20% 16.60% 17.00% 37.95% 35.12% 32.34% 29.62% 26.97% 24.42% 22.00% 19.77% 17.80% 16.17% 15.00% 14.40% 14.45% 15.13% 16.38% 18.06% 20.08% 22.34% 24.77% 27.34% 30.00% 32.73% 35.52% 38.35% 41.22% 44.12% 47.03% 49.97% 52.92% 55.89% 58.86% E[r] Vol 9.53% 14.35% SR 0.105 0.125 0.148 0.176 0.208 0.246 0.291 0.344 0.405 0.470 0.533 0.583 0.609 0.608 0.586 0.554 0.518 0.483 0.452 0.424 0.400 0.379 0.360 0.344 0.330 0.317 0.306 0.296 0.287 0.279 0.272 E[r] Vol SR 14.68% 0.612 0.00% 44.04% 0.90 Vol 58.48% 54.10% 49.71% 45.33% 40.96% 36.59% 32.23% 27.88% 23.55% 19.24% 15.00% 10.88% 7.10% 4.60% 5.53% 8.87% 12.87% 17.06% 21.34% 25.66% 30.00% 34.36% 38.72% 43.09% 47.47% 51.85% 56.24% 60.62% 65.01% 69.40% 73.79% 13.42% 12.50% 11.77% 11.28% 11.06% 11.12% 11.46% 12.06% 12.87% 13.86% 15.00% 16.25% 17.60% 19.01% 20.48% 21.99% 23.55% 25.13% 26.73% 28.36% 30.00% 31.66% 33.33% 35.00% 36.69% 38.39% 40.09% 41.80% 43.52% 45.23% 46.96% Expected R Expected R SR 0.595 9.98% Vol -0.90 Vol Expected R Expected R 14% 12% Tangency portfolio 10% Expected Return China 8% US 6% Expected R 4% 2% 0% 0% 5% 10% 15% 20% 25% 30% 35% Volatility 10.4% 10.2% Tangency portfolio 10.0% 9.8% Expected Return 9.6% 9.4% 9.2% US 9.0% 8.8% 12% 13% 14% Volatility 15% 16% w(US) 3 0 -3 E[r] 25% 1% -23% 14% Expected Return 20 13% Expected Return 15 China 12% 10 11% 5 10% 9% US rf 8% -5 15% 21% 27% 33% 12% 18% 24% 30% Volatility 14% 13% China 12% Expected Return 11% 10% 9% US 8% 12% 15% 18% 21% 24% 27% 30% 33% Volatility 0.20 14% 13% 12% 11% Expected Return 10% 9% 8% 0% -0.90 10% 0.90 20% Volatility 30% 0 40% Expected Return 11% 10% 9% 8% 0% 10% 20% 30% 40% Volatility 14% 14 12% 12 China 10% Expected Return 10 8 8% US 6% Expected Return 6 4% 4 2% 2 0% 0% 5% 10% 15% 20% 25% 30% 35% 0 Volatility Vol 45% 0% 45% 20% Expected Return 15% 10% 5% rf US 0% 0% 5% 10% 15% 20% 25% 30% -5% Volatility 14% 12% China 10% Expected Return 8% US 6% 4% 2% 0% 0% 5% 10% 15% 20% 25% 30% 35% Volatility Q5 US 5% 15% rf corr(US,Brazil) Brazil 10% 25% 1% 0.4 w(US) w(Brazil) 100% 0% 90% 10% 80% 20% 70% 30% 60% 40% 50% 50% 40% 60% 30% 70% 20% 80% 10% 90% 0% 100% E[r] E[r] SD[r] a SD[r] SR 14% 12% 10% 8% Expected Return IOS US Brazil CAL 6% 4% 2% 0% 0% 5% 10% 15% 20% 25% 30% Volatility b c d e f E[r] 15% w(US) w(Brazil) SD[r] CAL Maximum Sharpe ratio (tangency) portfolio w(US) w(Brazil) E[r] SD[r] w(rf) 1 0 -1 SR i w(rf) 50% w(US) 50% E[r] SD[r] SR ii w(rf) 50% w(TP) 50% E[r] SD[r] SR w(US) w(Brazil) SD[r] Portfolio weights in portfolio d(ii) w(rf) w(US) w(Brazil) E[r] 10% w(rf) w(TP) 0 1 2 E[r] 0.00% 0.00% 0.00% SD[r] 0.00% 0.00% 0.00%