BUS 422 Review HW 3
If an investment provides a 1.25% return quarterly, its effective annual rate is
(1.0125)4 - 1 = 5.09%.
If a portfolio had a return of 18%, the risk-free asset return was 5%, and the standard deviation of the portfolio's excess returns was 34%, the risk premium would be
18 - 5 = 13%.
Variance and Standard deviation
A measure of dispersion around the mean Var(R) = sigma ^2 Std dev = Var(R) ^ 1/2
Annual percentage rate vs Effective annual rate
APR= periodic interest rate x number of period in a year EAR= [( 1 + (APR/n))^n ] - 1 APR= [(1 + EAR)^(1/n) -1] x n
Capital allocation line (CAL)
Capital allocation line (CAL): Plot of risk-return combinations between a risk-free asset and a risky portfolio
Capital market line (CML)
Capital market line (CML): CAL that uses the market portfolio as the risky asset
What is the standard deviation of a random variable q with the following probability distribution: (Do not round intermediate calculations. Enter your answer in numbers not in percentage. Round your answer to 4 decimal places.) Value of q Probability 0 0.28 1 0.23 2 0.49
E(q) = (0 × 0.28) + (1 × 0.23) + (2 × 0.49) = 1.21 σq = [0.28 × (0 - 1.21)2 + 0.23 × (1 - 1.21)2 + 0.49 × (2 - 1.21)2]1/2 = 0.8520
Suppose your expectations regarding the stock price are as follows: Market Probability Ending Price HPR (including dividends) Boom 0.22 $140 52.5% Normal growth 0.30 110 18.5 Recession 0.48 80 -17.5 compute the mean and standard deviation of the HPR on stocks.
E(r) = [0.22 × 52.5%] + [0.30 × 18.5%] + [0.48 × (-17.5%)] = 0.0870 or 8.70 σ2 = [0.22 × (52.5 - 8.70)2] + [0.30 × (18.5 - 8.70)2] + [0.48 × (-17.5 - 8.70)2] = 780.360 σ = 0.2793 or 27.93 The mean is unchanged, but the standard deviation has increased, as the probabilities of the high and low returns have increased.
Risk Premium
Expected return in excess of the risk-free rate. E(R) + Rf
Expected Return
Mean value of the distribution of HPR E(R)
N in EAR/ APR problem
N is the number of periods of length m in a year (n=year/m) 1 quarter: n=4 1 month: n=12 2 months : n=6 8 days: n=365/8=45.625
Which of the following statement(s) is(are) true? Inflation has no effect on the nominal rate of interest. The realized nominal rate of interest is always greater than the real rate of interest. Certificates of deposit offer a guaranteed real rate of interest. None of the options is true.
None of the options is true. Expected inflation rates are a determinant of nominal interest rates. The realized nominal rate of interest would be negative if the difference between actual and anticipated inflation rates exceeded the real rate. The realized nominal rate of interest would be less than the real rate if the unexpected inflation were greater than the real rate of interest. Certificates of deposit contain a real rate based on an estimate of inflation that is not guaranteed.
Passive strategy
Passive strategy: Investment policy that avoids security analysis Benefits of passive strategy are that (1) active strategy is costly, and (2) passive strategy has a free rider benefit.
Real rate of return (r)
Percentage change in buying power, ie, return adjusted for inflation 1 + R = ( 1+ r) * (1 + i) Because future inflation is uncertain, real rate of return is always risky even if the nominal rate of return can be risk-free.
Nominal rate of return (R)
Percentage change in dollars, ie, return unadjusted for inflation
Asset Allocation
Portfolio weights: Wp + Wf = 1 E(Rc) -Rf= Wp * [(E(Rp) - Rf] sigma c = Wp * sigma p
Holding-period return (HPR)
Rate of growth over an investment period, ie, dollars earned per dollar invested. HPR= Capital gains yield + Dividend yield HPR= [(Ending Price - Beginning Price) / Beginning Price] +[Cash Dividend / Beginning Price]
Inflation (i)
Rate of increase in price level
Risk-free rate
Rate of return that can be earned with certainty Rf
Excess Rate
Realized return in excess of T-Bill return E(Rp) - Rf
Risk Aversion
Reluctance to accept risk E(Rp) - Rf = A *sigma ^2 *p A = risk aversion factor, higher risk, requires higher expected return
Reward-to-variability ratio
Reward-to-variability ratio: Ratio of risk premium to standard deviation, i.e. slope of CAL, aka Sharpe Ratio Slope = E(Rp) - Rf / sigma p
Geometric average
Single per-period return that gives the same cumulative performance as actual terms
Arithmetic average
Sum of periodic returns divided by number of periods
Assume that you manage a risky portfolio with an expected rate of return of 22% and a standard deviation of 35%. The T-bill rate is 6%. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio's standard deviation will not exceed 19%. a. What is the investment proportion, y? b. What is the expected rate of return on the complete portfolio?
a. σC = y × 35% If your client prefers a standard deviation of at most 19%, then: y = 19/35 = 0.5429 = 54.29% invested in the risky portfolio. b. E(rc) = (1 - y) × T-bill Rate + (y) × risky rate 14.69% = (1 - 0.5429) × 0.06 + 0.5429 × .22
You are bearish on Telecom and decide to sell short 170 shares at the current market price of $70 per share. a. How much in cash or securities must you put into your brokerage account if the broker's initial margin requirement is 50% of the value of the short position? Cash or securities to be put into brokerage account b. How high can the price of the stock go before you get a margin call if the maintenance margin is 30% of the value of the short position? Margin call will be made at price
a. Initial margin is 50% of $11,900, or $5,950. b. Total assets are $17,850 ($11,900 from the sale of the stock and $5,950 put up for margin). Liabilities are 170P. Therefore, equity is ($17,850 - 170P). A margin call will be issued when: ($17,850 − 170P) / 170P = 0.30 when P = $80.77 or higher
The risk premium for common stocks
cannot be zero, for investors would be unwilling to invest in common stocks and must always be positive, in theory.
When comparing investments with different horizons, the ____________ provides the more accurate comparison.
effective annual rate The effective annual rate provides the more accurate comparison of investments with different horizons because it expresses the returns in a common period.
The capital allocation line can be described as the
investment opportunity set formed with a risky asset and a risk-free asset. The CAL has an intercept equal to the risk-free rate. It is a straight line through the point representing the risk-free asset and the risky portfolio, in expected-return/standard deviation space.
During a period of severe inflation, a bond offered a nominal HPR of 85% per year. The inflation rate was 76% per year. a. What was the real HPR on the bond over the year? b. Find the approximation rr ≈ rn - i
rr = (1 + m)/ (1+i) (0.85 - 0.76) / 1.76 = 5.11% rr ~ m - i 85% - 76% = 9% Clearly, the approximation gives a real HPR that is too high.
ou have been given this probability distribution for the holding-period return for KMP stock: State of Economy Probability HPR Boom .3 18% Normal Growth .5 12% Recession .2 -5% What is the expected standard deviation for KMP stock?
s = [.30 (18 - 10.4)2 + .50 (12 - 10.4)2 + .20 (-5 - 10.4)2]1/2 = 8.13%.
Annual percentage rates (APRs) are computed using
simple interest.
The holding-period return (HPR) on a share of stock is equal to
the capital gain yield during the period, plus the dividend yield.
