Numerical Flashcards
You invest $100 in a risky asset with an expected rate of return of 10% and a standard deviation of 18% and a T-bill with a rate of return of 4%. The slope of the Capital Allocation Line formed with the risky asset and the risk-free asset is equal to _______
ERP / StdDev
For the Capital Allocation Line the risk free rate (T-bill at 4%).
To calculate the slope of the CAL, the numerator is:
E(r) of the risky asset (10%) minus the risk free rate (4%).
The denominator is the standard deviation of the risky asset (18%).
The resulting slope calculation is (10% - 4%) / 18% = .33
You invest $100 in a T-bill that pays 4% and a risky portfolio, P, with an expected return of 10% and a standard deviation of 18%. If you want to form a portfolio with a standard deviation of 13.5%, what percentages of your money must be invested in P and the T-bill, respectively?
The standard deviation of a portfolio comprising of y% in risky portfolio P and the remaining in the risk free asset:
New StdDev = StDev*(y)
- 5% = 18%*(y) Solving: y = 13.5% / 18% = 75%.
* 75% of your money must be invested in the risky portfolio, and 25% in the T-bill.*
Suppose you buy 100 shares of Anushka.com at the beginning of year 1 for $42.
The stock price at the end of the year 1 is $46, the price is $54 at the end of year 2, the price is $62 at the end of year 3, and the price is $59 at the end of year 4.
If Anushka.com pays no dividends, what is the geometric mean return (annual) during the four-year period?
Calculate the return for each period (end - beg)/beg:
Year 1: (46-42)/42 = 9.52% +1 = 1.0952
Year 2: (54-46)/46 = 17.39% +1 = 1.1739
Year 3: (62-54)/54 = 14.81% +1 = 1.1481
Year 4: (59-62)/62 = -4.84% +1 = .9516
To calculate the Geometric Mean: Calculate the product of all of the returns –
1.0952*1.1739*1.1481*.9516 = 1.4047
Raise this result to the ¼ power, as there are four observations. Then subtract 1. This gives a result of 8.87%
Assume that all five portfolios below lie on the investment opportunity set of Two-risky assets (stock and bond), and one among them is the optimal portfolio.
PORTFOLOIO - Expected Return (%) Standard Deviation (%)
Portfolio A - 11.0 14.20
Portfolio B 9.5 11.70
Portfolio C 8.5 11.56
Portfolio D 10.0 12.25
Portfolio E 10.5 12.10
If the risk free rate is 5%, identify the optimal portfolio
For each portfolio, calculate the Sharpe ratio.
The portfolio with the highest Sharpe ratio is the optimal portfolio. Sharpe Ratio = (Expected Portfolio return – risk free rate) / Standard Deviation of the Portfolio
Portfolio E has the highest Sharpe Ratio: (10.5% - 5%) / 12.1% = .4545
Consider a portfolio that offers an expected return of 10% and a standard deviation of 16%. T-bills offer a risk-free 7% rate of return. Describe the entire range of A for which the risky portfolio is still preferred to bills. You may assume the common form of utility function: U = E(R) – (0.5*A*σ^2). Investors are risk averse.
The utility of the risk free rate is 7%.
Therefore, the utility acceptable with the risky portfolio needs to exceed 7%. To find the A value, we set U=7%.
Then, the equation is as follows:
rf = Er - (.5* A * StDv^2))
.07 = .1 – (.5*A*(.16^2))
–>
Solve for A –> A = .03 / .0128 = 2.34
–>
A value of A > 2.34 would cause the Utility of risky portfolio to be lower than 7%.
Therefore, the value of A < 2.34.
Consider two risky assets (A & B) that are perfectly negatively correlated. A has an expected rate of return of 10% and a standard deviation of 18%. B has an expected rate of return of 8% and a standard deviation of 24%. The weights of A and B in the minimum variance portfolio are _____ and _____, respectively.
The minimum variance that is possible when two risky assets are perfectly negatively correlated is zero. The standard deviation of the portfolio in this scenario is:
Standard Deviation of A* w1 - Standard Deviation of B * (1-w1) = 0
Substituting -
18*X - 24*(1-X) = 0
–>
18*X - 24 + 24X = 0
–>
42X = 24
–>
X=.57
The weight of A is .57. The weight of B is .43.
