FIN 312 Exam 2
Derive the CAPM Formula
RRi= rf+RPi = rf+ (pr)price of risk x amount of risk i =rf +pr x covari,m =rf + [E(rm)-rf/𝜎m^2] x covi,m = rf+ [covi,m/ 𝜎m^2] x E(rm)-rf RRi= rf + βi[E(rm) - rf]
The slope of the CML
(𝐸(𝑟𝑚)− 𝑟𝑓)/𝜎𝑚
Regarding the Utility function (E(r)-0.005Aσ^2) if A>0 what is their risk attitude
Risk Averse
What is another way to write E(rc) with Y= σc /σp
= rf + [ E(rp)-rf /σp ] x σc
Minimum Variance Portfolio
A portfolio of individually risky assets that, when taken together, result in the lowest possible risk level for the rate of expected return.
Regarding the Utility function (E(r)-0.005Aσ^2) if A<0 what is their risk attitude
Risk Loving
Regarding the Utility function (E(r)-0.005Aσ^2) if A=0 what is their risk attitude
Risk Neutral
What is considered efficient on the efficient frontier
All portfolios to the right(including) of the minimum variance portfolio
Find the risk of the portfolio with the following data. E(rp) = 10%, σp = 20%, rf = 4% If y = 70% and 1 - y = 30%
E(r)= rf + y[E(rp)-rf 4% + .7x6% = 8.2% σc= yσp =.7x20= 14%
You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 30%. The T-bill rate is 6%.Your client chooses to invest 65% of a portfolio in your fund and 35% in an essentially risk-free money market fund. What are the expected return and standard deviation of the rate of return on his portfolio?
E(r)=WbillxE(rtbill) Windex xE(risk) σ^2=y^2σ^2 .65(18)+ .35(6)= 13.80 σ^2=.65^2(30^2)= 380.25 σ= 19.5
Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 29% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 4%. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights as shown below. Wbill/ W index 0.0/ 1.0 0.2/ 0.8 0.4/ 0.6 0.6/0.4 0.8/0.2 1.0/0.0
E(r)=WbillxE(rtbill) Windex xE(rsp) .2(4)+.8(12)= 10.4 .4(4)+.6(12)= 8.80 .6(4)+.4(12)=7.20 .8(4)+.2(12)=5.60 1x4 = 4 σ2= Y^2σ^2 .8^2(29^2) =538 σ= 23.2 .6^2(29^2)=302.76 σ= 17.4 .4^2(29^2)=134.56 σ=11.6 .2^2(29^2)= 33.64 σ=5.8
Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 25% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 4%.Calculate the utility levels of each portfolio for an investor with A = 2. Wbill/Windex 0.0/ 1.0 .2/ .8 .4/ .6 .6/ .4 .8/ .2 1/ 0
E(r)=WbillxE(rtbill) Windex xE(rsp) σ2= Y^2σ^2 U= E(r)-0.005Aσ^2 =.2(4)+.8(12)=10.4 σ = 20 U= 10.4-0.05(2)(20^2)= 6.4 =.4(4)+.6(12)=8.8 σ= 15 U=8.8-0.005(2)(15^2)=6.55 .6(4)+.4(12)=7.2 σ=10 U=7.2-0.005(2)(10^2)=6.2 .8(4)+.2(12)=5.6 σ=5 U=5.6-0.005(2)(5^2)= 5.35
The same example above. If an investor likes to yield an 8.5% return for her complete portfolio, then solve for Y, 1-y and σ
E(rc) = rf + y[E(rp) - rf] E(rc) 8.5= 4+y[10-4] 8.5=4+6y 6y=4.5 Y=75% 1-y=25% σ= 75X.20= 15%
Formula for Return of a complete portfolio
E(rc) = yE(rp) + (1 - y) rf E(rc) = rf + y[E(rp) - rf] rf= risk free investment Y=inv weight in a risky portfolio 1-Y= inv weight in a risk free asset, f rc= return on a complete portfolio that consist of the risk free asset and the risky portfolio
You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%.Your risky portfolio includes the following investments in the given proportions: Stock A=25% Stock B=32% Stock C=43% Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%. What is the proportion y? What are your client's investment proportions in your three stocks and the T-bill fund?
E(rc)= rf + y[E(rp)-rf] 16=8 + Y(10) Y=80 1-Y=20 Tbill= 20% A= .8x25 = 20 B= .8x32= 25.6 C= .8X43= 34.40
Based on historical data for corporate bonds and large stocks: E(rB) = 6%; σB = 9% E(rS) = 10%; σS=20% If wB = ws = 0.50, then Expected return is
E(rp) = wBE(rB) + wSE(rS) = 0.5X6 +0.5X10%= 8%
For a portfolio that consists of two investments (Bond and Stock) what is Expected return and Variation formulas
E(rp) = wBE(rB) + wSE(rS) σ_p^2 = w_B^2 σ_B^2 + w_S^2 σ_S^2 + 2wBwSσBσSPBS
Slope(Reward to risk ratio) for CAL
E(rp)-rf / σp
What do more risk-averse investors prefer when E(rx) > E(ry) and σx = σy
E(rx) or the one with higher returns
True or false Regarding the Utility function (E(r)-0.005Aσ^2) if A increases the investor is NOT more risk averse
False
2 stocks: A and B Expected return for A (O) = 8% Expected return for B (U) = 17% βO = 0.8; βU = 2.0 rf = 4% and E(rm) = 10% Find Required Return for both and describe if we would buy it
RRa= 4+ 0.8[10-4] =8.8% RRb= 4+ 2[10-4] = 16% We only buy stocks that have Expected return > Required return so for stock A (8<8.8%) we would short it or sell it. For stock B (17>16%) we would buy it or hold it.
How do investors choose an optimal risky portfolio on the efficient frontier?
Introduce a risk-free investment and then generate a CAL by investing in the risk-free asset and a risky portfolio on the efficient frontier.
If Expected return of Two investment are equal but Investment B Stand Dev > Investment A Stand what would a risk averse person choose
Investment A
If Expected return of Two investment are equal but Investment B Stand Dev > Investment A Stand what would a risk Loving person choose
Investment B because there is a chance for higher return
You manage a risky portfolio with an expected rate of return of 22% and a standard deviation of 34%. The T-bill rate is 6%. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. Suppose that your risky portfolio includes the following investments in the given proportions: Stock A=31% Stock B= 36% Stock C= 33% What are the investment proportions of your client's overall portfolio, including the position in T-bills?
Investment Proportions= Wrisky x Stock weight Tbill= 30% A= .70X31= 21.7 B= .70X36= 25.2 C= .70X33 = 23.1
How does a risk-averse investor choose a point on the CAL?
Investor choose the tangent line of their utility curve that touches the CAL
Based on historical data for corporate bonds and large stocks: E(rB) = 6%; σB = 9% E(rS) = 10%; σS=20% If wB = ws = 0.50, then find standard deviation when P=1, 0, -1
P(1)=0.5^2x 9^2 + 0.5^2×20^2 + 2×0.5x0.5×9×20×PB,S = 120.25+ 90PBS σp= [120.25 + 90(1)]1/2 = 14.5% P(0)=σp= [120.25 +90(0)]1/2 = 11% P(-1)= σp= [120.25 + 90(-1)]1/2 = 5.5%
You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 31%. The T-bill rate is 4%. Your client's degree of risk aversion is A = 2.4, assuming a utility function U = E(r) − ½Aσ². What proportion, y, of the total investment should be invested in your fund? What are the expected value and standard deviation of the rate of return on your client's optimized portfolio?
Portion y = E(rport)-rf/0.01A(σ^2p) 18-4/0.01(2.4)(33^2)= 60.70 E(r)= .6070(18)+ .3929(4)= 12.498 σ=.6070(31)=18.817
You manage a risky portfolio with an expected rate of return of 19% and a standard deviation of 33%. The T-bill rate is 7%. Your client chooses to invest 80% of a portfolio in your fund and 20% in a T-bill money market fund. What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your client's?
R to R You = E(rp)-Rf/σp 19-7/33= .3636 R to R client E(R)= .8(19)+.2(7)=16.60 σ=Yσ = .8(33)=16.4 16.60-7/33= .3636
Among the infinite numbers of CAL that investors can create, which CAL to use for making asset allocation decision?
The CAL that is tangent to the efficient frontier, called CALO. - The tangency portfolio is called the optimal risky portfolio, O.
The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = 0.85; Corr(A, C) = 0.60; Corr(A, D) = 0.45. Each stock has an expected return of 8% and a standard deviation of 20%. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose:
The correct choice is Stock D. Intuitively, we note that since all stocks have the same expected rate of return and standard deviation, we choose the stock that will result in lowest risk. This is the stock that has the lowest correlation with Stock A.
Markowitz Efficient Frontier
The graph of the set of portfolios offering the maximum expected return for their level of risk (standard deviation of return).
If Expected return of Two investment are equal but Investment B Stand Dev > Investment A Stand what would a risk neutral person choose
They are indifferent
True or false Risk of a security ↑ (↓) -> Required return ↑ (↓)
True
True or false risk only matters if it cannot be diversified away
True
True or false when Pbs decreases σp declines
True (ρB,S ↓ -> σp ↓ -> diversification benefit)
A risky investment: E(r) = 10% and s = 20% A risk-free investment: rf = 4% •For an investor whose A's value is 2 (this is a less risk-averse investor compared to the first one), then (s)he will choose the risky investment over the risk-free investment, because:
U (risky investment) = 10 - 0.005 x2x20^2 = 6 U (risk-free investment) = 4 - 0.005x2 x0 = 4 Choose the one with higher return
A risky investment: E(r) = 10% and s = 20% A risk-free investment: rf = 4% For an investor whose A's value is 3, then (s)he is indifferent between choosing the risky investment or the risk-free investment, because
U (risky investment) = 10 - 0.005 x3x20^2 = 4 U (risk-free investment) = 4 - 0.005 x3x0^2 = 4
See data below and find U when an investors A Value is 4 so find U E(r) (%) 10, 15, 20, 25 σ(%) 20, 25.5, 30.0, 33.9
U= E(r)- 0.005Aσ^2 U(10)= 10-0.005(4)(20^2) = 2 U(15)= 15-0.005(4)(25.5)^2 = 1.995 or est 2 U(20)= 30-0.005(4)(20)^2 =2 U(25)= 25-0.005(4)(33.9)^2 =2.0158 or est 2
Regarding the Utility function U=(E(r)-0.005Aσ^2) if A Increases what happens to U?
Utility will decrease
Regarding the Utility function U=(E(r)-0.005Aσ^2) if σ Increases (Decrease) what happens to U?
Utility will decrease (If σ decreases U will Increase)
Regarding the Utility function U=(E(r)-0.005Aσ^2) if E(r) Increases (Decrease) what happens to U?
Utility will increase (If decrease U will decrease)
What is the Utility Function for Risk
Utility= Expected Return E(r)-0.005Aσ^2 A= Index of an investors risk attitude
Equation for finding Weight of least risky asset for Minimum variance portfolio
Wb(min var)= σS^2 - PbsσBσS/ σB^2 + σS^2 -2PbsσBσS
Using Historical Data of large cap stocks where Pbs=0.2 E(rb)=6% E(rs)=10% σB=9% σS=20% find WB, WS, E(rmin var) and σMin var
Wb= 20^2 - 0.2x9x20/ 9^2+ 20^2 - 2x0.2x9x20 = 89%= Y Ws=1-Y= 1-.89= 11% E(rmin var)= 0.89x6 + 0.11x10 =6.44% σ^2 Min= 0.89^2×9^2 + 0.11^2×20^2 + 2×0.89×0.11×0.2×9×20 = 76.0489)^1/2 = 8.72%
A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows The correlation between the fund returns is 0.10: Exp return /Stand Dev Stock = 19/31 Bond 14/23 What are the investment proportions in the minimum-variance portfolio of the two risky funds? What are the expected value and standard deviation of the minimum-variance portfolio rate of return?
Wb= 31^2-.10(31)(23)/23^2 +31^2-2(.10)(31)(23)= .6603 Ws= 1-Wb= .3397 E(r)= .6603(14) + .3397(19)= 15.69 σ= .6603^2 x (14^2) + .3397^2 x (19^2) - 2(.6603)(.3397)(.10)(23)(31)= 373.946^1/2= 19.33
A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 7%. The characteristics of the risky funds are as follows The correlation between the fund returns is 0.12. E(r)/Stand Dev Stock= 16/31 Bond 12/21 Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio.
Wb= [12-7]38^2 - [16-7]X.12(38)(21)/[12-7]38^2 + [16-7]21^2 - [(12-7) +(16-7)]X(.12)(21)(38)= .6456 Ws= 1-.6456= .3544 E(ro)= .6456x12 + .34544x16= 13.42 σ=.2022
What does it mean when an investor has over 100% in a risky asset making the risk free asset negative?
When Y>100% it means you borrowed money (margin purchase) and the interest rate you pay is your rf (4% in example above)
With the Derivative of Utility when σp Increases (Decreases) what happens to Y
Y will decrease (Increase)
With the Derivative of Utility when A Increases (Decreases) what happens to Y
Y will decrease (increase)
With the Derivative of Utility when E(rp) Increases (Decreases) what happens to Y
Y will increase (decrease)
the optimal complete portfolio Formula
Y*O= E(rp)-rf/ 0.01Aσ^2p
The example above: E(rp) = 10%, σp = 20%, and rf = 4%. For an investor whose A's value is 1, then the optimal investment weight in the risky portfolio p is? Also, find the return and standard deviation and rtr ratio
Y= .10-.04/0.01(1)(.20)^2 Y=150% Risky asset 1-Y = -50% risk free asset ERC=yE(rp)+(1-y)rf 1.5(10)+ -.50(4)= 13% σ= yσp = 1.5 X20 =30% RTR= 13-4/30=0.3
The example above: E(rp) = 10%, σp = 20%, and rf = 4%. For an investor whose A's value is 1.5, then the optimal investment weight in the risky portfolio p is? Also, find the return and standard deviation and rtr ratio
Y= .10-.04/0.01(1.5)(.20)^2 Y=100 Risky asset 1-Y = 0 risk free asset ERC=0X.4 + 1.0X10= 10% σ= yσp = 1X20 = 20% RTR= 10-4/20=0.3
What is the first derivative of Utility (How to find the point on CAL)
Y= E(rp)-rf/ 0.01Aσ^2p
The example above: E(rp) = 10%, σp = 20%, and rf = 4%. For an investor whose A's value is 4, then the optimal investment weight in the risky portfolio p is? Also, find the return and standard deviation and rtr ratio
Y= E(rp)-rf/ 0.01Aσ^2p Y= (.10-.04)/0.01(4)(.20)^2 Y= 37.5% risky asset 1-Y- 62.5% Risk free asset E(Rc)= (1-y)rf+yE(rp) = .625x4 + .375x10 = 6.25% σ= yσp = .375x20= 7.5% rtr= E(rc)-rf/ σc= 6.25-4/7.25 = 0.3
What is another way to find σc = yσp
Y=σc /σp
Beta of individual risky asset
covi,m/ 𝜎m^2
Capital Allocation Line (CAL)
plot of risk-return combinations available by varying portfolio allocation between a risk-free asset and a risky portfolio
Risk of a complete portfolio
σc = yσp σc = return standard deviation of a complete portfolio
What is the formula for standard deviation for optimal complete portfolio
σc= y*o x σo Y*o is weight of risky asset in the optimal risky portfolio O σO is the standard deviation of optimal portfolio
What do more risk-averse investors prefer when E(rx) = E(ry) and σx < σy
σx Or the one with smaller risk
Capital Market Line (CML)
𝐸(𝑟𝑝)=𝑟𝑓+[ (𝐸(𝑟𝑚)− 𝑟𝑓)/𝜎𝑚 ]σp