1. Mary decides to set aside a small part of her wealth for investment in a port
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Question
1. Mary decides to set aside a small part of her wealth for investment in a portfolio that has greater risk than her previous investments because she anticipates that the overall market will generate attractive returns in the future. She assumes that she can borrow money at 3% (the risk-free rate) and achieve the same return on the S&P 500 as before: an expected return of 12% with a standard deviation of 22%.
A. Calculate Mary’s expected risk and return if she borrows 20%, 60% and 100% of her initial investment. Show your calculations.
B. What is the slope of he capital market line for part A? Show your calculations.
C. Assume that Mary can borrow at 6 percent instead the risk-free rate. What would be the expected return and risk if Ms. Moneypenny borrows 20%, 60% and 100% of her initial investment? Show your calculations.
D. How does the slope of the capital market line change from part A to part C? Explain why the slope of the capital market line changes. Show your calculations.
2. An investor is considering investing in a small-cap stock fund and a general bond fund. The correlation between the returns of the two funds is 0.10. The returns and standard deviations are given in the following table.
Expected Annual Return
Standard Deviation of Returns
Emerging Markets stock fund
28%
40%
Global Equities Fund
16%
21%
A. If the investor requires a portfolio return of 23 percent, what should the percentage invested in each fund be? Show your calculations.
B. What is the standard deviation of a portfolio constructed according to the weights computed in part A? Show your calculations.
Expected Annual Return
Standard Deviation of Returns
Emerging Markets stock fund
28%
40%
Global Equities Fund
16%
21%
Explanation / Answer
Risk-free rate = 3% which same as borrowing rate
Expected return on market = 12% Standard deviation = 22%
Answer (A)
Return on portfolio E(rp)= rf + (rm-rf)
= 0.03 + 0.22 (0.12 – 0.03) = 0.0498 or 4.98%
The return on a portfolio consisting 80% own amount and 20% borrowed amount
Return on portfolio rp= w1rf + (1-w1)*E(rm)
rp = 0.6 * 0.03 + (1-.06) * 0.12
= 0.006 + 0.096 = 0.102 or 10.2%
Risk of the portfolio SD = (1-w1) * Standard Deviation = (1-0.2) * 0.22 = 0.176 or 17.6%
Expected return to Mary = Portfolio return – weight of borrowing * interest on borrowing
= 0.102 – 0.2 * 0.03 = 0.102 – 0.006 = 0.096 or 9.6%
The return on a portfolio consisting 40% own amount and 60% borrowed amount
rp = 0.6 * 0.03 + (1-.06) * 0.12
= 0.018 + 0.048 = 0.066 or 6.60%
Risk of the portfolio = (1-w1) * Standard Deviation = (1-0.6) * 0.22 = 0.088 or 8.80%
Expected return to Mary = 0.066 – 0.6 * 0.03 = 0.066 – 0.018 = 0.048 or 4.80%
The return on a portfolio consisting 100% borrowed amount
rp = 1.0 * 0.03 + (1-1) * 0.12
= 0.03+0= 0.03 or 3%
Risk of the portfolio = (1-w1) * Standard Deviation = (1-1) * 0.22 = 0%
Expected return to Mary = 0.03 – 1.0 * 0.03 = 0%
Calculation of expected risk (standard deviation of portfolio)
Variance of returns = {(0.096 – 0.0498)^2 + (0.048 – 0.0498)^2 + (0. – 0.0498)^2}/3
= { 0.00213444 + 0.00000324 + 0.00248004}/3 = 0.004617724/3
= 0.00153924
Standard Deviation of Portfolio = Square Root of (0.00153924) = 0.0392 or 3.92%
Answer (B)
Slope of Capital Market Line =
Reward to Variability Ratio = (Portfolio Return – Risk-free return)/Portfolio Standard Deviation
= (0.0498 – 0.03)/0.0392 = 0.0198 / 0.0392 = 0.5051
Answer (C)
If Mary borrows at 6% instead of risk free rate of 3%,then the expected return to Mary is as follows
20% borrowed amount - Expected return = rp – 0.2 * 0.06 = 0.102 – 0.012 = 0.09 or 9%
60% borrowed amount - Expected return = rp – 0.6 * 0.06 = 0.066 – 0.036 = 0.03 or 3%
100% borrowed amount – Expected return = rp – 1.0 * 0.06 = 0.03 – 0.06 = - 0.03 or -3%
Answer (D)
Variance of Returns = { (0.09 – 0.0498)^2 + (0.03-0.0498)^2 + (-0.03-0.0498)^}/3
= {0.00161604 + 0.00039204 + 0.00636804}/3 = 0.00837612/3
= 0.00279204
Standared Deviation of Returns = Square root of (0.00279204) = 0.052839 or 5.28%
Slope of CML = Reward to variability Ratio = Portfolio Return – Riskfree return / SD of portfolio returns
= (0.0498 – 0.03) / 0.052839 = 0.0198 / 0.052839 = 0.3747
The slope changes indicates the decline in the rewards to variability ratio. This returns declined due to the change in borrowing rate which affected the returns available to investor net of borrowing cost.
Answer (2)(A)
Expected Return on Emerging Market Fund E(r) =28%
Expected Return on Global Equities Fund = 16%
Let w1 be the weight of Emerging Market Fund in a portfolio with an expected return of 23% consisting of the above two funds.
(1-w1) would be the weight of Global equities fund. Then
0.23 = w1 * 0.28 + (1-w1) *0.16 ==> 0.23 = 0.28 w1 + 0.16 – 0.16 w1 ===> 0.23 = 0.16 + 0.12 * w1
0.12* w1 = 0.23 – 0.16 ==> 0.12*w1 = 0.07 or w1 = 0.07/0.12 = 0.58333 or 58.33%
Proportion of Emerging Market Fund = 58.33%
Proportion of Global Equities Fund = 1- 0.5833 = 0.4167 or 41.67%
Answer (2)(B)
Variance of portfolio = w1^2 * SD1^2 + (1-w1)^2*SD^2 + 2*w1*(1-w1)*correlation coefficient
Variance of Portfolio= 0.5833^2 * 0.40^2 + 0.4167^2 + 0.21^2 + 2 * 0.5833 * 0.4167 * 0.10
= 0.34023889 * 0.16 + 0.17363889 * 0.0441 + 0.048612222
= 0.05443822224 + 0.007657475049 + 0.048612222
= 0.110707919289
Risk of Portfolio = Standard Deviation = Square Root of (0.110707919289) = 0.33273 or 33.27%
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