FIN4514 Final Exam
Weak Form EMH tests: Findings
'50s and '60s: 1. no evidence of serial correlation. the price of a stock is just as likely to rise after a previous day's increase as after a previous day's decline 2. therefore, stock prices follow random walk
Sharpe Ratio
(rp-rf)/st devp Measures excess returns (or risk premium) per unit of total risk for a portfolio Use average standard deviation and average risk free rate
Technical Analysis
(weak form) -do not think the market is weak form efficient -believe investors are emotionally driven and predictable (can predict their predictability) -computers find patterns
APT and CAPM compared
-APT applies to well diversified portfolios and not necessarily to individual stocks -with APT it is possible for some individual stocks to be mispriced - not lie on the SML -APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio -APT can be extended to multifactor models
CAL vs CML
-CAL (capital allocation line) can be drawn from the risk-free rate to the portfolio. This is a line called a CAL. -the CML is a special case of a CAL -Sharpe ratio for any portfolio is the slope of its ex-post CAL
Jensen's Measure (Multiple Period Alpha)
-a positive alpha implies superior risk-adjusted performance -must use excess returns her or the alpha will not have the correct interpretation -p-value = level of reasonable doubt (you want it to be <5% -t stat > 2; evidence that alpha is different than zero
Efficient Market Hypothesis (EMH)
-all relevant information is reflected in the price -if you believe the market is efficient, there is no need to spend time to figure out which firms are over/underpriced (price says it all) -no need to worry about timing; market takes it into account -implications for business and corporate finance is affected by market efficiency
Risk and return for individual securities
-beta of security i measures how the return of i moves with the return of the market. In other words, it is a measure of the systematic risk -only systematic risk matters in determining the equilibrium expected return -unsystematic risk affects only a single security or a limited number of securities -systematic risk affects the entire market
M squared measure
-find a complete, or "adjusted" portfolio on the CAL plotted with the portfolio you are evaluating) that has the same standard deviation as the market portfolio (the adjusted portfolio is defined as P*) -compare the return of the adjusted portfolio to that of the market M2= rp* - rm
Assumptions of CAPM
-individual investors are price takers -single period investment horizon -investments are limited to traded financial assets -no taxes and transaction costs -information is costless and available to all investors -investors are rational mean-variance optimizers -there are homogeneous expectations
Evaluate the market timing and security selection abilities of four managers whose performances are plotted on the diagrams: (see graphs)
-intercept = measure of stock selection ability - if manager has positive excess returns even when the market is neutral (i.e. has 0 excess returns), then we conclude the manager made good stock picks; stock selection must be the source of positive returns -timing=curve -lines that become steeper as you move to the right along horizontal axis show good timing ability -steeper slope = manager maintained higher portfolio sensitivity to market swings (i.e. higher beta) in periods when market performed well -ability to choose market sensitive securities in anticipation of market upturns = good timing -declining slope as you move right means the portfolio is more sensitive to the market when the market did poorly and less sensitive when it did well = poor timing
A: E(r)=12%, B=1.2 F: E(r)=6%, B=0 E: E(r)=8%, B=0.6 how to arbitrage?
-make a portfolio D where its risk= risk of beta between extreme betas (port E) -choose A & F to put together (bc of extreme beta values) and they will meet E in the middle at .6 beta Bd= wa*Ba + wf*Bf 0.6= wa*1.2 + wf*0 wa=.5 wf=.5 E(rd) = .5(.12) + .5(.6) = 9% E(rarb) = 9%-8% = 1% you want to long D and short E (remember D contains A, you like A; you know you don't like E)
ramifications of predictability
-new information is unpredictable; if it could be predicted, then the prediction would be part of today's information -stock prices changes/movement is unpredictable and random
Trading on inside info
-not legal -SEC prosecutes offenders -rules protect the small investor
Fama-French Model
-positive alpha implies superior risk-adjusted performance SML>0 more small stocks than large HML>0 more high stocks than low
EMH Semi-Strong Form
-stock prices reflect ALL PUBLICLY AVAILABLE information about a firm -uses all info from firms -firm info can't predict the future -FUNDAMENTAL
EMH: Weak Form
-stock prices reflect all PAST market price and volume information -least amount of information reflected (historical price and volume) -can't make future predictions using old betas -TECHNICAL
For a well-diversified portfolio:
-systematic risk is captured by Bp, which is the weighted average of betas of the individual assets -unsystematic risks cancel each other out, therefore ep approaches zero, similar to CAPM
Weak Form EMH Tests: Method #2
-use historical price info to analyze "abnormal returns" over various time horizons -in general, this method involves investing in stocks that have performed in a certain manner in the past to see if these stocks will provide abnormal returns in the future AbReturns = Actual return - benchmark
Historical market risk premium (proxied by the S&P 500 index) is 8.2%, and the standard deviation of the market portfolio is 20.6%. Based on these statistics, what is the average coefficient of risk aversion? E(rm)-rf = Abar(st dev m)^2
0.082 = Abar *0.206^2 Abar = 1.932
difficult to determine if market is efficient
1. difficult to distinguish luck from skill 2. Selection bias (there may be hidden investors that do earn abnormal risk-adjusted returns) 3. difficult to measure risk-adjusted returns (is CAPM the "right" model to use? Did we use the "right" market proxy as out benchmark?)
If A1=2, A2=3, A3=4 What is A bar?
1/A = 1/3*(1/2 + 1/3 + 1/4) =1/3*(6+4+3/12) = 36 A=2.77
Assume that both portfolios A and B are well diversified, E(ra)=12% and E(rb)=9%. If the economy has only one factor, and Ba=1.2 and Bb=.8, what must be the risk free?
12%= rf + 1.2RP 9% = rf + .8RP rf=3% RP=7.5
The security market line depicts:
A security's expected return as a function of its systematic risk
Is this possible if CAPM is valid? Risk-Free: E(r): 10% St Dev: 0% Market: E(r): 18% St Dev: 24% Portfolio A: E(r): 16% St. Dev: 22%
A: 16-10/22 = 0.27 Mkt: 18-10/24 = 0.33 Possible because market out-weights portfolio A
What should be the total investment in the market portfolio? Let A1=1.5, A2=2, A3=3, E(rm)-rf= 9%, st dev(m)= 20%
A=1.5 .09/(1.5*2^2)= 1.5 y*=1.5 (1-y)= -0.5 A=2 .09/(2*2^2)= 1.125 y*=1.125 (1-y)= -0.125 A=3 .09/(3*2^2)= 0.75 y*= 0.75 (1-y)= .25 investor 3 is most averse to risk (they are lending to the market)
Active or Passive Management
Active: -security analysis -timing Passive: -accept EMH -buy and hold -index funds and ETFs -very low costs
Three-Factor Risk Measure
Alpha from Fama-French 3-factor model
Is this possible if CAPM is valid? Risk-Free: E(r): 10% Beta: 0 Market: E(r): 18% Beta: 1.0 Portfolio A: E(r): 16% Beta: 0.9
CAPM= 10 + 0.9(18-10) = 17.2 A is overpriced
Within the context of the CAPM, assume: E(r mkt) = 15% Rf = 8% E(r XYZ) = 17% Beta of XYZ = 1.25 Which of the following are correct? -XYZ is overpriced -XYZ is fairly priced -XYZ's alpha is -.25% -XYZ's alpha is .25%
E(r) = 8 +1.25(15-8) = 16.75 Actual exp return = 17% a = 17 - 16.75 = 0.25% XYZ's alpha is 0.25%
Interpretation of M2
M2= 1.59-1.00 = .59 If one were to invest in a portfolio composed of the risk-free asset and Windsor II fund, which offers the same risk as the market portfolio, one would have reached a return that is 0.59% higher each month M2 is the difference between the CAL and the CML when graphed
Market risk (B-CAPM) measures
Multiple-Period Alpha; Jensen's Measure; Treynor's Measure
Is this possible if CAPM is valid? Risk-Free: E(r): 10% St Dev: 0% Market: E(r): 18% St Dev: 24% Portfolio A: E(r): 20% St. Dev: 22%
Not possible. Portfolio dominates market portfolio.(A has lower standard deviation and higher E(r)
Weak Form EMH Tests: Method #1
Positive (+) Serial Correlation: -(+) returns follow (+) returns for a given stock or (-) returns follow (-) returns for a given stock -called "momentum" Negative (-) Serial Correlation: -(+) returns follow (-) returns for a given stock or (-) returns follow (+) returns for a given stock -called "reversals" if we find (+) or (-) serial correlation, this is evidence against the weak-form EMH as it implies that past prices can be used to predict future prices technical analysis looks for such patterns to exploit and earn abnormal returns
R squared
R^2= (Bi^2*stdevm^2)/st devi^2 -measures the degree of diversification -tells us the percentage of the movement of the portfolio that is explained by the market -can be obtained from regression output -"what percentage is described from market movement"
Fundamental Analysis
SEMI-STRONG -do not think the stock market is semi-strong form efficient -they use public info as relative analysis (can't use just this for future prices)
Port A: E(r): 12% B=1.2 Port F: E(r): 6% B=0 Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected return of 8%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?
Sa: 12-6/1.2 = 5 Se: 8-6/.6 = 3.33 yes, arbitrage opportunity exists create new portfolio G using A & F (extremes) .6= wa*1.2 + wf*0 wa=.5 wf=.5 E(rg) = .5*12 + .5*8 = 9% long G, short E E(rarb) = 9% - 8% = 1%
Is this possible if CAPM is valid? Risk-Free: E(r): 10% St Dev: 0% Market: E(r): 18% St Dev: 24% Portfolio A: E(r): 15% St. Dev: 12%
Sharpe Ratio: E(r) - rf/ st dev A: .16-.10/.12 = 0.5 Mkt: .18-.10/.12 = 0.33 impossible bc CAPM predicts the market is the most efficient portfolio. In this scenario, A is greater than the market
Total risk measures
Sharpe Ratio; M Squared Measure
X: E(r)=16%, B=1.0 Y: E(r)=12%, B=0.25 F: E(r)=8%, B=0 Is there an arbitrage opportunity?
Sx: 16-8/1= 8 Sy: 12-8/.25= 16 yes, sharpe ratio is different
How to test for market efficiency?
Weak -see if there are patterns in past prices Semi-Strong -see if new public info rapidly synthesized in market prices; can you profit from public info? Strong -see if private info can lead to profits
Windsor II average monthly returns = 1.39% Market Proxy: SPY avg monthly returns = 1%
Windsor II outperformed the index over the last 12 months
port R: ror= 11%, stdev=10%, beta=.5 s&p500: ror= 12%, stdev=12%, beta=1.0 a.) When plotting portfolio R relative to the SML, portfolio R lies: b.) When plotting portfolio R on the CML line, portfolio R lies:
a & b) you need know the risk free rate in order to determine both
A zero-investment portfolio with a positive alpha could arise if: -the expected return of the portfolio equals zero -the capital market line is tangent to the opportunity set -the Law of One Price remains unviolated -a risk-free arbitrage opportunity exists
a risk-free arbitrage opportunity exists
Suppose the rate of return on short-term government securities (perceived to be risk-free) is about 5%. Suppose also that the expected rate of return required by market for a portfolio with a beta of 1 is 12%. According to the CAPM: a.) What is the expected rate of return on the market portfolio? b.) What would be the expected rate of return on a stock with B=0? c.) Suppose you consider buying a share of stock at $40. The stock is expected to pay $3 dividends next year and you expect to sell it for $41. The stock risk has been evaluated at B= -.5. Is the stock over or underpriced?
a.) 12%; the market beta is 1 b.) 5%; the risk-free rate c.) E(r) = .05 + -.5(.12-.05) = 1.5% actual E(r) = 41+3/40 = 10% because the actually expected returns exceeds the fair return, the stock is underpriced
Consider the following multi-factor (APT) model of security returns for a particular stock: Inflation: B=1.2, RP=6% Ind.Prod: B=.5, RP=8 Oil Prices: B=.3, RP=3% a.) if T-bills offer 6% yield, find the expected rate of return on this stock if the market views the stock as fairly priced. b.) suppose the market expects the values for the tree macro factors given below, but the actual values turn out as given below. Calculate the revised expectations for the rate of return on the stock once the "surprises" become know. Inflation: Expected= 5%, Actual = 4% Ind.Prod: Expected= 3%, Actual = 6% Oil Prices: Expected= 2%, Actual = 0%
a.) E(r) = 6% + (1.2*6)+(.5*8)+(.3*3) = 18.1% b.) E(r) = 1.2*(4-5) + .5*(6-3) + (.3*(0-2) = =.3% E(r)= 18.1% - .3% = 17.8%
a.) A mutual fund with a beta of .8 has an expected return of 14%. If rf= 5%, and you expect the rate of return on the market portfolio to be 15%, should you invest in this fund? What is the funds alpha? b.) What passive portfolio comprised of market-index portfolio and money market account would have the same beta as the fund? Show that the difference between the expected rate of return on the passive portfolio and that of the fund equals the alpha from part (a)
a.) SML: 5+.8(10) = 13% a = 14% expected - 13% actual = 1% alpha you should invest because alpha is positive b.) the passive portfolio with the same beta as the fund should be invested 80% in the mkt-index and 20% in the money market account: E(rp)= (.8*15) + (.2*5) = 13% a = 14-13 = 1% alpha
Williamson: E(r)=22.1% St Dev= 16.8% Beta=1.2 Joyner: E(r)=24.2% St Dev=20.2% Beta=0.8 a.)calculate the sharpe ratio and treynor measure for both. b.) the investment committee notices that using the Sharpe ratio versus Treynor measure produces different performance rankings for williamson and Joyner. Explain why
a.) Sw= .221-.05/.168 = 1.02 Sj= .242-.05/.202= .95 Tw= 22.1-5/1.2= 14.5 Tj= 24.2-5/.8= 24 b.) Joyner had a higher Treynor ratio (24) and a lower sharpe ratio (.95) than Williamson (14.25,1.02), so Joyner must be less diversified than Williamson. Treynor's measure indicates that Joyner has a higher return per unit of systematic risk than Will, while the sharpe ratio indicated Joyner has a lower return per unit of total risk than Will
Two investment advisors are comparing performance. One averaged 19% rate of return and the other 16% rate of return. However, the beta of the first investor was 1.5, whereas that of the second investor was 1. a.) Can you tell which investor was a better selector of individual stocks (aside from the issue of general movements in the market)? b.) If the T-bill rate was 6% and the market return during the period was 14%, which investor would be considered the superior stock selector? c.) What if the T-bill rate was 3% and the market was 15%?
a.) To determine which investor was a better selector of individual stocks we look at abnormal return, which is the ex-post alpha; that is, the abnormal return is the difference between the actual return and that predicted by the SML. Without information about the parameters of this equation (risk-free rate and market rate of return) we cannot determine which investor was more accurate. b.) α1 = .19 - [.06 + 1.5 × (.14 - .06)] = .19 - .18 = 1% α2 = .16 - [.06 + 1 × (.14 - .06)] = .16 - .14 = 2% here, the second investor has the larger abnormal return and thus appears to be the superior stock selector. By making better predictions, the second investor appears to have tilted his portfolio toward underpriced stocks. c.) α1 = .19 - [.03 + 1.5 × (.15 - .03)] = .19 - .21 = -2% α2 = .16 - [.03+ 1 × (.15 - .03)] = .16 - .15 = 1% Here, not only does the second investor appear to be the superior stock investor, but the first investor's predictions appear valueless (or worse).
Annual return S&P = 14% Rf= 8% P: E(r)=17% st dev=20% Beta=1.1 Q: E(r)=24 st dev=18 beta=2.1 R: E(r)=11 st dev=10 beta=0.5 S: E(r)=16 st dev=14 beta=1.5 S&P: E(r)=14 st dev=12 beta=1.0 a.) what is the treynor performance for Porffolio P? b.) what is the sharpe ratio for portfolio Q?
a.) Treyp= 17-8/1.1 = 8.182 b.) Sq= .24-.08/.18 = .888
Analyst wants to evaluate portfolio X using Treynor's and Sharpe ratio. The table provides the average annual rate of return for X, the market portfolio, and T-bills for the past 8 years: X: E(r)= 10% st dev= 18% beta=0.6 S&P: E(r)=12 st dev=13 beta=1.00 T-bills E(r)= 6 a.) calculate treynor's measure for portfolio X and S&P 500. Briefly explain whether port X underperformed, equaled, or outperformed the S&P on a risk-adjusted basis using both Treynor and sharpe ratio. b.) on the basis of performance of port X relative to the S&P briefly explain the reason for the conflicting results when using Treynor measure versus sharpe ratio.
a.) Tx: 10-6/.6 = 6.67 T500= 12-6/1 = 6 Sx: .10-.06/.18 = .222 S500= .10-.06/.13 = .462 Port X outperforms the market based on Treynor's measure, but underperforms based on Sharpe ratio b.) the two measures use different measures of risk (T=beta S=st dev). Port X is less systematic than the market, as measured by its lower beta, but more total risk (volatility), as measured by its st dev.
Based on current dividend yields and expected capital gains, the expected rates of return on portfolios A and B are 12% and 16%. The beta of A is .7 while that of B is 1.4. The T-bill rate is currently 5%, whereas the expected rate of return of the S&P 500 is 13%. The standard deviation of portfolio A is 12% annually, that of B is 31%, and that of the S&P 500 is 18%. a.) If you currently hold a market-index portfolio, would you choose to add either of these portfolios to your holdings? Explain. b.) If instead you could invest only in T-bills and one of these portfolios, which would you choose?
a.) aa= 12% - [5% +0.7(13-5)] = 1.4% ab= 12% - [5% + 1.4(13-5)] = -0.2% you would want to take long position on A and short B b.) Sa= .12-.05/.12 = 0.583 Sb= .16-.05/.16 = 0.355 portfolio A is preferred bc it has the higher sharpe ratio
RF= 6% Mkt return = 15% Stock A: Index Regression Estimates = 1% + 1.2(rm-rf) R-square = 0.576 Res St Dev (e) = 10.3% St dev excess returns = 21.6% Stock B: Index Regression Estimates = 2% + 0.8(rm-rf) R-square = 0.436 Res St Dev (e) = 19.1% St dev excess returns = 24.9% a.) calculate the following: i.) alpha ii) information ratio iii) sharpe ratio iv) treynor's measure b.) which stock is the best under these circumstances: i.) this is the only risky asset to be held by investor ii) this stock will be mixed with the rest of the investor's portfolio, currently composed solely of holdings in the market index fund iii) this is one of many stocks that the invest is analyzing to form an actively managed stock portfolio
a.) i) A: 1% B: 2% ii.) A: 1%/10.3 = .09709 B: 2%/19.1 = .1047 iii.) solve index models Sa: 10.6/21.6= .491 Sb=8.4/24.9= .249 iv.) Ta= 10.6/1.2 Tb= 8.40.8 = 10.5 b.) i.) use sharpe ratio: stock A ii.) use info ratio to determine the contribution to the overall sharpe ratio; stock B iii.) use Treynor's measure since its one of the many stocks; stock B
Assume that security returns are generated by the single-index model (Ri=ai+Bi*Rm+ei) where Ri is the excess return for security i and Bm is the market's excess return. The rf=2%. A: Bi=0.8, E(r)=10%, stdev(e)=25% B: Bi=1.0, E(r)=12%, stdev(e)=10% C: Bi=1.2, E(r)=14%, stdev(e)=20% a.) If stdev market=20%, calculate the variance of returns of securities A, B, and C. b.) Now assume that there are an infinite number of assets with return characteristics identical to those of A, B, and C. If one forms a well-diversified portfolio with type A securities, what will be the mean and variance of the portfolio's excess returns? What about portfolios of type B and C stocks? c.) is there an arbitrage opportunity in this market? what is it?
a.) variance= B^2*stdev(mkt)^2+stdev(e)^2 A: (.8)^2*(.2)^2+.25^2 = .0881 B: 1^2*(.2)^2 + .10^2 = .0500 C: 1.2^2*(.2)^2 + .20^2 = .0976 b.) with infinite # of assets, well-diversified portfolios will have only systematic risk since non-systematic risk will approach zero with large n (ei=0) A: .8^2*.2^2 = .0256 B: 1^2*.2^2 = 0.0400 C: 1.2^2*.2^2 = 0.0576 mean will equal that of the individual stocks c.) there is no arbitrage opportunity bc the well-diversified portfolios all plot on the SML; bc they are fairly-priced, there is no arbitrage
SML relationship states that the expected risk premium on a security in a one-factor model must be directly proportional to the security's beta. Suppose this were not the case. For example, suppose that expected return rises more than proportionately with beta in the figure below. (upward curve with points A,C,B from right to left) a.) how could you construct an arbitrage portfolio? (hint: consider combinations of portfolios A and B, and compare the resultant portfolio C) b.) some researchers have examined the relationship between average returns on diversified portfolios and the beta and beta squared of those portfolios. What should they have discovered about the effect of B squared on portfolio return?
a.) you could create a new portfolio "D" comprised of A & B which will offer an expected return-beta tradeoff lying on a straight line between A & B (remember when the returns make a triangle, there is an arbitrage opportunity). therefore, you can choose weights so Bd=Bc but with expected return higher than C. Hence, combining D with short position in C will create an arbitrage profit with zero investment, zero beta, and positive rate of return b.) argument in part (a) concludes that the coefficient of B squared must be zero in order to prelude arbitrage opportunity
An investor takes as large a position as possible when an equilibrium price relationship is violated. This is an example of: -a dominance argument -the mean-variance efficient frontier -arbitrage activity -the capital asset pricing model
an arbitrage activity -investors will take on as large a position as possible only if the mis-pricing opportunity is an arbitrage -otherwise, considerations of risk and diversification will limit the position they attempt to take in the mis-priced security
Appraisal ratio (information ratio)
ap/st dev(ep) divides the alpha of the portfolio by the nonsystematic risk it measures the marginal benefits (alpha) per unit of cost (diversifiable non-systematic risk)
Grossman-Stiglitz Theorem
assumptions (two types of investors): -uninformed: liquidity or noise traders ex: shift investments from equity to bonds closer to retirement to reduce risk and steady your income Do NO research -informed: spend serious amounts of money to dig up info no one else has (spend resources to try to get ahead) Do research until marginal benefit = marginal cost
X: E(r)=16%, B=1.0 Y: E(r)=12%, B=0.25 F: E(r)=8%, B=0 How do you arbitrage?
create new portfolio D using x and rf asset (for extremes) to make beta D = beta Y .25=wx*1 + wy*0 wx=.25 wy=.75 E(rd)= .25(1) + .75(8) = 10% E(rarb) = 12-10= 2% long Y, short X (long better return, short lower return)
Alpha
difference between the actual expected rate of return and that dictated by the SML
In contrast to the CAPM, arbitrage theory: -requires that markets be in equilibrium -uses risk premiums based on micro variables -specifies the number and identifies specific factors that determine expected returns -does not require the restrictive assumptions concerning the market portfolio
does not require the restrictive assumptions concerning the market portfolio
A: E(r)=12%, B=1.2 F: E(r)=6%, B=0 E: E(r)=8%, B=0.6 Is there an arbitrage opportunity?
first, determine Sharpe ratio for each Sa: 12-6/1.2 = 5 Se: 8-6/0.6=3.33 A>E, so A is undervalued bc it offers greater sharpe ratio yes, there is an arbitrage opportunity bc the sharpe ratios are different CHECK: do betas match? if no, you have to adjust for risk. If yes, just do long/short
EMH: Strong Form
includes ALL INFORMATION (public and private) about a firm
Treynor's Ratio
measures excess return (or risk-adjusted return) per unit of systematic risk rp-rf/Bp Treynor's ratio for the market is the slope of the SML
Is this possible if CAPM is valid? Portfolio A: E(r): 20% Beta: 1.4 Portfolio B: E(r): 25% Beta: 1.2
not possible. A beta > B beta but B has higher expected return, so these two cannot exist in equilibrium
According to the theory of arbitrage: -high-beta stocks are consistently overpriced -low-beta stocks are consistently overpriced -positive alpha investment opportunities will quickly disappear -rational investors will pursue arbitrage opportunities consistent with their risk tolerances
positive alpha investment opportunities will disappear quickly
The general arbitrage pricing theory (APT) differs from the single-factor capital asset pricing model (CAPM) because the APT: -places more emphasis on market risk -minimizes the importance of diversification -recognizes multiple unsystematic risk factors -recognizes multiple systematic risk factors
recognizes multiple systematic risk factors
What is the expected return of a zero-beta security?
risk-free rate
According to CAPM, the expected return of a portfolio with a beta of 1.0 and an alpha of 0 is: -between rm and rf -the rf rate -B(rm-rf) -the expected return on the market, rm
rm
Is this possible if CAPM is valid? Portfolio A: E(r): 30% St Dev: 35% Portfolio B: E(r): 40% St Dev: 25%
st dev A>B this is possible as long as firm A's E(r)A<B is less than B's (standard deviation measures firm-specific risk)
The CAPM asserts that expected returns on individual securities are best explained by
systematic risk
Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%. X: E(r)=16% Beta= 1.00 Y: E(r)=12% Beta= .25 In this situation you could conclude that portfolios X and Y: -are in equilibrium -offer an arbitrage opportunity -are both underpriced -are both fairly priced
they offer an arbitrage opportunity since portfolio X has B=1 it's the market portfolio and E(rm)=16% using E(rm)=16% and rf=8%, the expected return for Y is not consistent
goal of an active manager
to earn positive abnormal risk-adjusted returns
goal of evaluator
try to identify superior or inferior risk-adjusted returns
X: E(r)=16%, B=1.25 Y: E(r)=14%, B=1.00 Z: E(r)=8%, B=0.75 Is there an arbitrage opportunity? How to evaluate?
use 2 extremes for marginal risk-return(sharpe) (change in risk/change in return) Sz-x: (16-8)/(1.25-.75) = 16 Sz-y: (14-8)/(1-.75) = 24 yes, arbitrage opportunity use highest and lowest portfolio (X and Z) 1.00=wx*1.25 + (1-wx)*.75 1.00=.5wx + .75 wx = .5 wz=.5 E(rd) = .5(16) + .5(8) = 12% long Y (better return), short D (lower return) E(rarb) = 14-12 = 2%
Is this possible if CAPM is valid? Risk-Free: E(r): 10% Beta: 0 Market: E(r): 18% Beta: 1.0 Portfolio A: E(r): 16% Beta: 1.5
use SML: E(r) = rf + B(mkt - rf) E(ra)= 10 + 1.5(18-10) = 22% A is consistent with CAPM bc E(ra)= 16%<22% so it is overpriced
The feature of APT that offers the greatest potential advantage over the simple CAPM is the: -identification of anticipated changes in production, inflation, and term structure of interest rates as key factors explaining the risk-return relationship -superior measurement of the risk-free rate of return over historical time periods -variability of coefficients of sensitivity to the APT factors for a given asset over time -use of several factors instead of a single market index to explain the risk-return relationship
use of several factors instead of a single market index to explain the risk-return relationship
