Finance

1. Consider a 10-year zero coupon bond with a face amount of $1,000 and an annually compounded yield of 3%? a. What is its price?
   b. What will the price of the bond be in 1 year if the yield rises to 4%? Drops to 2%?
2. Consider a 3-year, risk-free bond with a coupon rate of 6% (annual coupons) and a face amount of $1,000. a. What is price of this bond if the YTM is 5%? 6%? 7%?
   b. What is the YTM if the price is $1,100 (in percent)?
3. The yields on 1-year, 2-year and 3-year, risk-free, zero-coupon bonds are 2%, 2.5% and 3%, respectively. What is the value of a 3-year, risk-free bond with a coupon rate of 4% (annual coupons) and a face amount of $1,000?
4. Assume that over the next year you believe there are only 2 possible states of the world, each with probability 50%. The table below gives the returns on the US and Brazilian equity markets in these two scenarios. a. What are the expected returns on the US and Brazil (in percent)?
   b. What are the volatilities (standard deviations) of the returns on the US and Brazil (in percent)?
   c. What is the covariance between the returns on the US and Brazil?
   d. What is the correlation between the returns on the US and Brazil?

Prob. US return Braz. return
50% -10% -15%
50% 20% 35%

1. 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% 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. (The spreadsheet will plot the investment opportunity set given these calculations.)
   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? (The spreadsheet will plot the CAL for this portfolio given these calculations.) (Trial and error is a viable strategy, but you can also use the Solver tool in Excel. The pre-module materials take you through how to install and use Solver.)
   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 a portfolio that itself is invested 40% in the US and 60% in Brazil.
   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 asset and the 40/60 portfolio from d(ii)? What is the standard deviation of this portfolio?

The correlation between the returns is 0.3, and the (annual) risk-free (T-bill) rate is 2%.

Sample Solution