fin 300 math
risk free rate symbol
rf
expected return =
rf + β × [E(rM) − rf]
expected return
risk free rate + beta (market risk premium)
expected return
riskfree rate + beta (market premium -risk free rate)
b. Could the equilibrium rƒ be greater than rate of return?
should be the same
Asset A has an expected return of 14% and a standard deviation of 20%. The risk-free rate is 10%. What is the reward-to-variability ratio?
(0.14 − 0.10)/0.20 = 0.20
holding period return
(Sold share+dividend-purchase price)/purchase price
A portfolio is composed of two stocks, A and B. Stock A has a standard deviation of return of 22%, while stock B has a standard deviation of return of 28%. Stock A comprises 60% of the portfolio, while stock B comprises 40% of the portfolio. If the variance of return on the portfolio is 0.046, the correlation coefficient between the returns on A and B is __________.
0.046 = (0.62)(0.222) + (0.42)(0.282) + 2(0.6)(0.4)(0.22)(0.28) ρ; ρ = 0.542\
What would be the expected rate of return for each company, according to the capital asset pricing model (CAPM)? (Round your answers to 2 decimal places.)
1 discount everything 5 E(r) = rf + β × [E(rM) − rf] = 0.046 + β × 0.056 E(r$1 Discount Store) = 0.046 + 1.5 × 0.056 = 13.00 = 13.00% E(rEverything $5) = 0.046 + 1.0 × 0.056 = 0.1000 = 10.20%
what is market risk
17.3% = 11% + 0.9 × (MRP)
You put up $40 at the beginning of the year for an investment. The value of the investment grows 5% and you earn a dividend of $4.50.
5%+(4.5/40)=16.3% Investment grows +(earn dividend/beginning investment)
Reward to volatility ratio
= Portfolio risk premium/Standard deviation of portfolio excess return
sharpe ratios for different portfolios risk free : expected return 5 std 0 Market expected return 10.6 std 23 A expected return 8.6 std 12
SA = (8.6 − 5)/12 = 0.30 SM = (10.6 − 5)/23 = 0.24
rate of return
E(rC) = rf + y × [E(rP) − rf]
Rate of return
Expected cash flow - value of portfolio/ value of portfolio.
HPR
Investment grows +(earn dividend/beginning investment)
covariance
Standard deviation a * standard deviation B * correlation coefficient
when there is a negative correlation
Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rƒ. The characteristics of two of the stocks are as follows: StockExpected ReturnStandard DeviationA9%35%B14%65%Correlation = −1 9%*65%+35%*14% USE THE OPPOSITES
ou manage an equity fund with an expected risk premium of 11% and a standard deviation of 24%. The rate on Treasury bills is 6.2%. Your client chooses to invest $80,000 of her portfolio in your equity fund and $20,000 in a T-bill money market fund. What are the expected return and standard deviation of your client's portfolio? expected return stanrad deviation
expected return = 11+6.2=17.2 (0.8*17.2%)+(0.2*6.2%)=15% standard deviation 0.8*24% =19.2 %
reward-to-variability ratio?
expected return - risk free rate/standard deviation
how to find alpha
expected return found- the expected return given -e
treasury bond you want higher or lower
higher bond coupon
call option higher or lower
lower call option
Expected return
Get expected return for portfolio Treasury bills + risk premium = % Do the money for portfolio and divide of the money 0.#*that risk premium + 0.#*that risk premium=
A stock has a correlation with the market of 0.57. The standard deviation of the market is 23%, and the standard deviation of the stock is 31%. What is the stock's beta?
(0.57)(0.31)(0.23)/0.23^2 = 0.77
Total compound return over
(1+/- depend on loss or profit ) X (1+/- depend on loss or profit ) -1
An investment earns 40% the first year, earns 45% the second year, and loses 42% the third year. The total compound return over the 3 years was __________
(1.40) × (1.45) × (1 − 0.42) − 1 = 17.74% Total compound return over three years (1+/- depend on loss or profit ) X (1+/- depend on loss or profit ) -1
A portfolio with a 30% standard deviation generated a return of 15% last year when T-bills were paying 6.0%. This portfolio had a Sharpe ratio o
(15% − 6.0%)/30% = 0.30
how to find mean with probability and hpr
(mulitply * HPR)
A portfolio with a 30% standard deviation generated a return of 15% last year when T-bills were paying 6.0%. This portfolio had a Sharpe ratio o
(15% − 6.0%)/30% = 0.30 return-t bills/ std
geometric mean
[(1+0.#)+(1+#)^1/2]-1
put option higher or lower
lower put option.
investment proportion y
standard devation/ standard deviation
Arithmetic mean
→ average
expected return symbol
E(rP)
equation for expected return
E(rP) = rf + β[E(rM) − rf]
What is the expected rate of return for a stock that has a beta of 1 if the expected return on the market is 12%?
keeps the same at 12
You invest $1,600 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 17% and a standard deviation of 20% and a Treasury bill with a rate of return of 8%. __________ of your complete portfolio should be invested in the risky portfolio if you want your complete portfolio to have a standard deviation of 11%.
σC = y × σp 11% = y × 20% y = 11%/20% = 55% The standard deviation of risky portfolio = y * standard deviation of given thing
prob (0.4,0.4,0.2) hpr (41, 15, -19) Use above equations to compute the mean and standard deviation of the HPR on stock mean standard deviation
Mean [0.4*41]+[0.4*15]+[0.2*-19]=18.6% Variance 0.4 * [0.41 - 0.186]2 + 0.4 * [ 0.15 - 0.186]2 + 0.2 * [ -0.19 - 0.186]2=0.049 To get standard deviation → square root of variance
A stock quote indicates a stock price of $76 and a dividend yield of 4%. The latest quarterly dividend received by stock investors must have been ______ per share.
(76*0.04)/4=0.76
return for the period?
(future year - previous year)/ previous year → find that for the three → then find the geo mean
expected return
(probability * HPR)
expected cash flow
(probability× $) + (probability × $) =
value portfolio
(probability× $) + (probability × $) = expected cash flow (Value of portfolio)+(1+(risk premium+T-bills))= expected cash flow
Sharpe Ratio
(return - risk free rate) / standard deviation
The arithmetic average of -27%, 47%, and 52% is __________.
(−27% + 47% + 52%)/3 = 24.00%
A portfolio earned a rate of return equal to 10.5% last year with a standard deviation of 15.0%. Treasury Bills returned 1.7%. excess return rate sharpe ratio
10.5-1.7= 0.105-1.7/0.15
beta
Beta = correlation * standard deviation of stock* standard deviation of the market/standard deviation of market ^ 2
The standard deviation of return on investment A is 28%, while the standard deviation of return on investment B is 23%. If the correlation coefficient between the returns on A and B is −0.248, the covariance of returns on A and B is __________.
Covariance = −0.248(0.28)(0.23) = −0.0160
Market Risk Premium (RPm)
Rm-Rrf
A stock has an expected return of 5%. What is its beta? Assume the risk-free rate is 6% and the expected rate of return on the market is 16%. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)
Using the SML: 5% = 6% + β × (16% − 6%) ⇒⇒ β = −1/10 = −0.10
The geometric average of −20%, 45%, and 50% is __________.
[(1 + −0.20) × (1 + 0.45) × (1 + 0.50)]^1/3 ]− 1 = 20.28%
variance
probability *(HPR - Mean) ^2 + add them
the correlation coefficient between returns
covariance/(standard deviation*standard deviation)
rate of return
t bill + risky (expected rate - t bill)
Standard deviation
the square root of the variance
Assume that you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 36%. The T-bill rate is 4.5% A client prefers to invest in your portfolio a proportion (y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio's standard deviation will not exceed 25%. investment proportion y b. What is the expected rate of return on the overall portfolio?
y = (0.25/0.36) = 0.69440 = 69.44% standard devation/ standard deviation He should invest, at most, 69.44% in the risky fund. rate of return E(rC) = rf + y × [E(rP) − rf] = 0.04 + 0.69440 × 0.13 = 0.13181 or 13.18%
You have the following rates of return for a risky portfolio for several recent years. Assume that the stock pays no dividends. beginning prices $85 $90 $86 $89 What is the geometric average return for the period?
yr 1 ($90 − $85)/$85 = 5.88% yr 2 ($86 − $90)/$90 = -4.44% yr 3 ($89 − $86)/$86 = 3.49% [(1.06)(1 + −0.0444)(1.0349)]1/3 − 1 = 1.54%
The standard deviation of return on investment A is 17%, while the standard deviation of return on investment B is 12%. If the covariance of returns on A and B is 0.010, the correlation coefficient between the returns on A and B is __________.
Correlation = 0.010/[0.17(0.12)] = 0.490
What must be the beta of a portfolio with E(rP) = 16.70%, if rf = 5% and E(rM) = 14%? (Round your answer to 2 decimal places.)
E(rP) = rf + β[E(rM) − rf] Given rf = 5% and E(rM) = 14%, we can calculate β: 16.70% = 5% + β(14% − 5%) ⇒⇒ β = 1.30
You manage an equity fund with an expected risk premium of 13.8% and a standard deviation of 52%. The rate on Treasury bills is 3.6%. Your client chooses to invest $120,000 of her portfolio in your equity fund and $30,000 in a T-bill money market fund. What is the reward-to-volatility (Sharpe) ratio for the equity fund? (Round your answer to 4 decimal places.)
Reward to volatility ratio = Portfolio risk premium/Standard deviation of portfolio excess return = 13.8%/52% = 0.2654
how to find excess return
a. The excess return equals rate minus the risk-free rate (Treasury Bills):
