CFA Review 3CFA Review 3

Modern Portfolio Theory, Modern Portfolio Theory, Asset Pricing and Portfolio Asset Pricing and Portfolio

EvaluationEvaluation

policy statementpolicy statement

TOTAL RETURN= INCOME YIELD + CAPITAL GAIN YIELD TOTAL RETURN= INCOME YIELD + CAPITAL GAIN YIELD Objectives Objectives – Think in terms of risk and return to find the – Think in terms of risk and return to find the

“best” weights—i.e.,“best” weights—i.e., Capital preservation (high income, low capital Capital preservation (high income, low capital

gain)gain) Low to moderate risk Low to moderate risk Balanced return (Balanced capital gains and Balanced return (Balanced capital gains and

income reinvestment)income reinvestment)moderate to high riskmoderate to high risk Pure Capital appreciation (high capital gains, low to Pure Capital appreciation (high capital gains, low to

no income)no income)High riskHigh risk ConstraintsConstraints - liquidity, time horizon, tax factors, legal - liquidity, time horizon, tax factors, legal

and regulatory constraints, and unique needs and and regulatory constraints, and unique needs and preferencespreferences

ManagementManagement - Define an allowable allocation ranges - Define an allowable allocation ranges based on policy weightsbased on policy weights

SelectionSelection - Define guideline to pick securities to - Define guideline to pick securities to purchase for the portfolio (optional)purchase for the portfolio (optional)

Measuring Returns Measuring Returns r=FV/PV-1 or r=ln(FV/PV)r=FV/PV-1 or r=ln(FV/PV) When you have many returns:When you have many returns:

Arithmetic mean (R=1/n x Arithmetic mean (R=1/n x rrii)--2 weaknesses, 1) extreme )--2 weaknesses, 1) extreme values effect, 2)# of observation need to be knownvalues effect, 2)# of observation need to be known

Historical (R=1/n x Historical (R=1/n x rrtt)--)-- Cross-sectional (R=1/n x Cross-sectional (R=1/n x rrii) or Probability weighted (R=) or Probability weighted (R=ppiirrii))

Geometric mean(R=[Geometric mean(R=[rrii)])]1/n 1/n -1)-1) Historical (R=[Historical (R=[rrtt)])]1/n 1/n -1)-1) Cross-sectional (R=[Cross-sectional (R=[rrii)])]1/n 1/n -1)-1)

Example 1: 3 returns (10%, 12%, 14%)Example 1: 3 returns (10%, 12%, 14%) AM =(10%+12%+14%)/3=12%AM =(10%+12%+14%)/3=12% GM =[(1+10%)x(1+12%)x(1+14%)]GM =[(1+10%)x(1+12%)x(1+14%)]1/31/3-1=11.988%-1=11.988%

Example 2:Example 2:0.20 0.10 0.02000.30 0.11 0.03300.30 0.12 0.03600.20 0.13 0.0260

E(Rpor i) = 0.1150

Expected Portfolio

Return (Pi X Ri)

Expected Security

Return (Ri)

Probability (Pi)

Measuring RiskMeasuring RiskStandard deviation of returns with equally weighted deviations:Standard deviation of returns with equally weighted deviations:

Standard deviation of returns with unequally weighted deviations:Standard deviation of returns with unequally weighted deviations:

Standard deviation of unequally weighted standard deviations:Standard deviation of unequally weighted standard deviations:

n

1ii

2ii P)]E(R-[R Deviation Standard

2n

1i average)(r

1n

1DeviationStandard

jiijijij

2i

i

n

1i

n

1iijj

n

1ii

2i

2i

n

1i

n

1iijjij

n

1ii

2i

2i

σσcorCov wherej, and i assetsfor return of rates ebetween th covariance theCov

iasset for return of rates of variancetheσ

portfolio in the valueof proportion by the determined are weights

whereportfolio, in the assets individual theof weightstheW

:where

Covwwσwσor Corσσwwσwσ

Example 3Example 3::

)E( )E(R Asset ii

A .10 .07

B .20 .1

Case WA WB E(Ri) E(port) E(port) E(port) E(port) E(port)

Cor(A,B)--> 1 0.5 0 -0.5 -1f 0.00 1.00 0.20 0.1000 0.1000 0.1000 0.1000 0.1000g 0.20 0.80 0.18 0.0940 0.0878 0.0812 0.0740 0.0660h 0.40 0.60 0.16 0.0880 0.0779 0.0662 0.0520 0.0320i 0.50 0.50 0.15 0.0850 0.0740 0.0610 0.0444 0.0150j 0.60 0.40 0.14 0.0820 0.0710 0.0580 0.0410 0.0020k 0.80 0.20 0.12 0.0760 0.0682 0.0595 0.0492 0.0360l 1.00 0.00 0.10 0.0700 0.0700 0.0700 0.0700 0.0700

Portfolio Risk-Return Plots for Portfolio Risk-Return Plots for Different WeightsDifferent Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

Rij = -1.00

Rij = +0.50

f

gh

ij

kA

B

With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk

Rij = -0.50

The Capital Market Line (CML)The Capital Market Line (CML)

Imagine two portfolio: (1) a Imagine two portfolio: (1) a COMPLETELY DIVERSIFIEDCOMPLETELY DIVERSIFIED risky portfolio with an expected return of Rm and a risky portfolio with an expected return of Rm and a standard deviation of standard deviation of σm and (2) a riskless portfolio of t-σm and (2) a riskless portfolio of t-bills with an expected of Rf and a standard deviation bills with an expected of Rf and a standard deviation close to zero.close to zero.

You allocate Wrf in the riskless portfolio and (1-Wrf) in You allocate Wrf in the riskless portfolio and (1-Wrf) in the risky (best of the best portfolio)the risky (best of the best portfolio)

The standard deviation and expected return of this The standard deviation and expected return of this portfolio shall be:portfolio shall be:

σp=(1-Wrf) x σm or Wrf=1- σp/σm, thenσp=(1-Wrf) x σm or Wrf=1- σp/σm, thenRp=Wrf x Rf + (1-Wrf) x Rp replace Wrf by 1- σp/σmRp=Wrf x Rf + (1-Wrf) x Rp replace Wrf by 1- σp/σm

RP= Rf RP= Rf ++ (Rm –Rf) /σm (Rm –Rf) /σm x σp x σp Capital Capital Market Line Market Line

(CML)(CML)

Rp= intercept Rp= intercept + slope + slope x σpx σp

CML and leverageCML and leverageHow is the concept of leverage included in the CML? Wrf>0How is the concept of leverage included in the CML? Wrf>0 less risk; Wrf<0 less risk; Wrf<0

more riskmore risk Notice that the decision to invest has been separated from the decision to Notice that the decision to invest has been separated from the decision to

select specific investments (separation theorem). In a CML world, you do not select specific investments (separation theorem). In a CML world, you do not worry about “what to invest in since all investors will buy the same portfolio worry about “what to invest in since all investors will buy the same portfolio (M)—i.e., the investment selection process has been simplified from stock (M)—i.e., the investment selection process has been simplified from stock analysis and picking to efficient portfolio construction through diversification!analysis and picking to efficient portfolio construction through diversification!

Wrf>0 Wrf<0

Wrf=0

Where, Wrf=1-(P/M)

Utility Value of an InvestmentUtility Value of an InvestmentThere is a trade off between risk and return and “intelligent” investors like more returns and less risk. An investor is indifferent of the trade-off between risk and return at each indifference curve. However investors like I2 better than I1. Combining utility curves and CML, the optimal portfolio for each investor is the highest indifference that is tangent to the CML.

Risk aversion refers to the fact that investors prefer less risk than more risk. Utility is often described in a quadratic format:

Up=Rp-0.5 x Rz x p2

Where

Up is an investor’s utility for holding portfolio p

Rp is the expected return of portfolio p

Rz is the risk aversion value

p2 is the variance of portfolio p

More preferred direction for indifference curves

Portfolio Return St. Dev. Variance UtilityA 16.00% 18.00% 3.24% 9.52%B 12.80% 14.00% 1.96% 8.88%C 9.60% 11.00% 1.21% 7.18%

Example 4: Joe L’Outre has a risk aversion value of 4, which of portfolios A, B, and C is best for him?

Risk and DiversificationRisk and DiversificationReturnReturn == expectedexpected ++ unexpectedunexpected

Risk (return)= Risk (return)= 0 0 + market risk + business risk+ market risk + business risk

Risk (return)= Risk (return)= 0 + Systematic Risk + Non-systematic 0 + Systematic Risk + Non-systematic riskrisk

RRii=R=RFF + R + RRPRP, then…, then… Security risk premium = Security risk premium = (R(Rii- R- RFF), ), Market risk premium = (Market risk premium = (RRmm- R- RFF))

If security risk premium=If security risk premium=β x market risk premiumβ x market risk premium Then, (RThen, (Rii- R- RFF) =) =β x (Rβ x (Rmm- R- RFF) That is, ) That is, RRii = R = RFF +β x (R +β x (Rmm - R - RFF))

CAPM AssumptionsCAPM Assumptions RRii = R = RFF +β x (R +β x (Rmm - R - RFF) where ) where = cov(i,m)/ = cov(i,m)/mm

22

8 underlying assumptions (learn by heart):8 underlying assumptions (learn by heart): All investors use the (Markowitz) mean-varianc All investors use the (Markowitz) mean-varianc

eframework to select securities (that is, investors eframework to select securities (that is, investors have quadratic utility functions and security returns have quadratic utility functions and security returns are normally distributed)are normally distributed)

Investors can borrow and lend any amount of money Investors can borrow and lend any amount of money at the risk-free rateat the risk-free rate

Investors have homogenous expectations (that is, if Investors have homogenous expectations (that is, if they look at a stock, they all see the same they look at a stock, they all see the same risk/return distribution)risk/return distribution)

All investors have a 1-period time horizonAll investors have a 1-period time horizon All investments are indefinitely divisible (that is, you All investments are indefinitely divisible (that is, you

can buy/sell fraction of shares of a stock or portfoliocan buy/sell fraction of shares of a stock or portfolio No taxes and no transaction costsNo taxes and no transaction costs No inflation and interest rates do not changeNo inflation and interest rates do not change Capital markets are in equilibriumCapital markets are in equilibrium

CAPM ImplicationsCAPM Implications 4 implications4 implications

Because all investors have the same Because all investors have the same expectations all use mean-variance analysis, expectations all use mean-variance analysis, they identify the same optimal portfolio and they identify the same optimal portfolio and combine it with a risk free asset to create combine it with a risk free asset to create there own portfolio.there own portfolio.

Because all investors hold the same risky Because all investors hold the same risky portfolio, the weight on each asset must equal portfolio, the weight on each asset must equal to the proportion of its market value to the to the proportion of its market value to the total value of the portfolio.total value of the portfolio.

Since all investors hold the market with some Since all investors hold the market with some proportion of the risk free asset, the market proportion of the risk free asset, the market portfolio must be the point of tangency portfolio must be the point of tangency between the CML and the efficient frontier.between the CML and the efficient frontier.

The SML describes the relationship between The SML describes the relationship between the expected return and risk of all assets – the expected return and risk of all assets – individual securities and portfolios.individual securities and portfolios.

Example 5Example 5

RFR = 6% RFR = 6%

RRMM = 12% = 12%

Stock Beta

A 0.70B 1.00C 1.15D 1.40E -0.30

RFR)-(RRFR)E(R Mi i

E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%

E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%

E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%

E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%

E(RE) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%

Comparison of Required Rate of Comparison of Required Rate of Return to Estimated Rate of ReturnReturn to Estimated Rate of Return

Stock (Pi) Expected Price (Pt+1) (Dt+1) of Return (Percent)

A 25 27 0.50 10.0 %B 40 42 0.50 6.2C 33 39 1.00 21.2D 64 65 1.10 3.3E 50 54 0.00 8.0

Current Price Expected Dividend Expected Future Rate

Stock Beta E(Ri) Estimated Return Minus E(Ri) Evaluation

A 0.70 10.2% 10.0 -0.2 Properly ValuedB 1.00 12.0% 6.2 -5.8 OvervaluedC 1.15 12.9% 21.2 8.3 UndervaluedD 1.40 14.4% 3.3 -11.1 OvervaluedE -0.30 4.2% 8.0 3.8 Undervalued

Required Return Estimated Return

Plot of Estimated ReturnsPlot of Estimated Returnson SML Graphon SML Graph

)E(R i

Beta0.1

mRSML

0 .20 .40 .60 .80 1.20 1.40 1.60 1.80-.40 -.20

.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02

AB

C

D

E

Example 6Example 6 You gather the following information about two You gather the following information about two

stocks A and B, the SP500 and the treasury bill:stocks A and B, the SP500 and the treasury bill:

StateState Prob.Prob. E(Ra)E(Ra) E(Rb)E(Rb) E(SP500)E(SP500) RtbillRtbill

BadBad 25%25% 20%20% -20%-20% 0%0% 2%2%

AverageAverage 40%40% 10%10% 20%20% 5%5% 2%2%

GoodGood 35%35% -5%-5% 40%40% 10%10% 2%2%

CovariancCovariancee

AA BB SP500SP500 TbillTbill

AA 0.0096190.009619

BB --0.021330.02133 0.05310.0531

SP500SP500 --0.00373740.0037374

0.008650.00865 0.00147490.0014749

TbillTbill 00 00 00 00

What would be the allocation to A and B if you chose the minimum What would be the allocation to A and B if you chose the minimum risk portfolio?risk portfolio?

If Wa=W then Wb=1-W and the variance of the portfolio isIf Wa=W then Wb=1-W and the variance of the portfolio is

%4.29

%6.70

,

%6.7002133.020531.0009619.0

02133.00531.0

),(2

),(

),(4),(22220dW

d

is that ,0dW

d means variance

),(2),(22

),()1(2)1(

),(2

22

2

2222p

2p

22222222

22222

22222

B

A

BA

B

BBA

BBBAp

BAp

BABBAAp

W

W

Thus

BACOV

BACOVW

BAWCOVBACOVWW

Minimum

BACOVWBAWCOVWWW

BACOVWWWW

BACOVWWWW

2. Which stock would you consider as an addition to a portfolio made of the 2. Which stock would you consider as an addition to a portfolio made of the SP500? Which stock would you consider for stand-alone portfolio?SP500? Which stock would you consider for stand-alone portfolio?

2.A Stock to consider as an addition to a portfolio made of the SP5002.A Stock to consider as an addition to a portfolio made of the SP500Get the alpha of each stock—i.e., first get the theoretical (CAPM) return, then Get the alpha of each stock—i.e., first get the theoretical (CAPM) return, then

subtract it to the expected return. subtract it to the expected return. To get CAPM return:To get CAPM return:Ra=Rf+BETA(A) x (Rm-Rf)Ra=Rf+BETA(A) x (Rm-Rf)Rb=Rf+BETA(B) x (Rm-Rf)Rb=Rf+BETA(B) x (Rm-Rf)Rf is the treasury bill return=2%Rf is the treasury bill return=2%Rm is the sp500 return=5.5% (it is the weighted average return for sp500)Rm is the sp500 return=5.5% (it is the weighted average return for sp500)M=(0.0014749)M=(0.0014749)1/21/2=3.84%=3.84%BETA(A)=COV(A,M)/VAR(M)= -0.0037374/ 0.0014749=-2.53BETA(A)=COV(A,M)/VAR(M)= -0.0037374/ 0.0014749=-2.53BETA(B)= COV(B,M)/VAR(M)= 0.00865/ 0.0014749=5.85BETA(B)= COV(B,M)/VAR(M)= 0.00865/ 0.0014749=5.85ThenThenRa=Rf+BETA(A) x (Rm-Rf)=2%-2.53*3.5%=-6.86%Ra=Rf+BETA(A) x (Rm-Rf)=2%-2.53*3.5%=-6.86%Rb=Rf+BETA(B) x (Rm-Rf)=2%+5.85*3.5%=22.48%Rb=Rf+BETA(B) x (Rm-Rf)=2%+5.85*3.5%=22.48%ALPHA(A)=7.25%-(-6.86%)=ALPHA(A)=7.25%-(-6.86%)=14.11%14.11%UndervaluedUndervaluedALPHA(B)=17%-22.48%= ALPHA(B)=17%-22.48%= -5.48%-5.48%OvervaluedOvervaluedThen you would A to a well-diversified portfolio like AThen you would A to a well-diversified portfolio like A2.B Stock to consider for stand-alone portfolio2.B Stock to consider for stand-alone portfolio get the Coefficient of get the Coefficient of

VariationVariationCalculate the Coefficient of Variation:Calculate the Coefficient of Variation: CV(A)=9.8%/7.25%=CV(A)=9.8%/7.25%=1.351.35 CV(B)=23.04%/17%=CV(B)=23.04%/17%=1.351.35 There are basically equivalent in terms of reward to risk in a stand-alone portfolioThere are basically equivalent in terms of reward to risk in a stand-alone portfolio

3. How much (proportions) would you invest in A and B in order to get a 3. How much (proportions) would you invest in A and B in order to get a portfolio as risky as the market?portfolio as risky as the market?

The market has a beta of 1; Solve a system of two equations:The market has a beta of 1; Solve a system of two equations:Wa x BETA(A)+Wb x BETA(B)=1Wa x BETA(A)+Wb x BETA(B)=1Wa+Wb=100%Wa+Wb=100%

Then, Wb=[1-BETA(A)]/[BETA(B)-BETA(A)]Then, Wb=[1-BETA(A)]/[BETA(B)-BETA(A)]BETA(A)=COV(A,M)/VAR(M)= -0.0037374/ 0.0014749=-2.53BETA(A)=COV(A,M)/VAR(M)= -0.0037374/ 0.0014749=-2.53BETA(B)= COV(B,M)/VAR(M)= 0.00865/ 0.0014749=5.85BETA(B)= COV(B,M)/VAR(M)= 0.00865/ 0.0014749=5.85

Wb=Wb=42%42%So, Wa=So, Wa=58%58%

4.You have created your AB portfolio, then you decide to sell A and invest the 4.You have created your AB portfolio, then you decide to sell A and invest the proceed in T-bills. What the new portfolio Expected return, standard proceed in T-bills. What the new portfolio Expected return, standard deviation and beta?deviation and beta?

Wb=42%; Wa=58%Wb=42%; Wa=58% sell A, buy TBills sell A, buy TBills Wrf=58% Wrf=58%BETA(new portfolio)=Wb x BETA(B) +Wrf x BETA(Rf) and of course BETA(new portfolio)=Wb x BETA(B) +Wrf x BETA(Rf) and of course

BETA(Rf)=0; BETA(Rf)=0; Rf = 0Rf = 0 BETA(new portfolio)=.42 x 5.85=BETA(new portfolio)=.42 x 5.85=2.462.46 E(new portfolio)=.42 x 17% + .58 x 2%=E(new portfolio)=.42 x 17% + .58 x 2%=8.3%8.3% (new portfolio)=.42 x 23.04%=(new portfolio)=.42 x 23.04%=9.68%9.68% (from the portfolio risk (from the portfolio risk

equation with 2 assets, it simplifies a lot as equation with 2 assets, it simplifies a lot as Rf = 0)Rf = 0)

Efficient marketsEfficient markets An efficient capital market is one in which the current An efficient capital market is one in which the current

price fully reflects all the information currently price fully reflects all the information currently available for the security, including risk. available for the security, including risk.

An informationally efficient capital market is one in An informationally efficient capital market is one in which security prices adjust rapidly and completely to which security prices adjust rapidly and completely to new information, which enter the market in a random new information, which enter the market in a random and unpredictable manner and therefore causes stock and unpredictable manner and therefore causes stock prices to change in a random and unpredictable prices to change in a random and unpredictable fashion.fashion.

Market efficiency is based on the following set of Market efficiency is based on the following set of assumptions:assumptions:

A large number of profit maximizing participants are analyzing A large number of profit maximizing participants are analyzing and valuing securities independent of each other.and valuing securities independent of each other.

New information comes to the market on a random fashion, New information comes to the market on a random fashion, and news announcements are independent of each other in and news announcements are independent of each other in regard to timingregard to timing

Investors adjust price estimates rapidly to reflect their Investors adjust price estimates rapidly to reflect their interpretation of the new information received. Market interpretation of the new information received. Market efficiency does not assume that participants adjust prices efficiency does not assume that participants adjust prices correctly, just that price adjustments are unbiased.correctly, just that price adjustments are unbiased.

Expected returns explicitly include risk in the price of the Expected returns explicitly include risk in the price of the security.security.

Efficient market hypothesisEfficient market hypothesis Fama divided the efficient market hypothesis Fama divided the efficient market hypothesis

into 3 categories:into 3 categories: Weak-form efficient markets: current prices fully Weak-form efficient markets: current prices fully

reflect reflect all currently available security market all currently available security market informationinformation. Thus, past prices and volume . Thus, past prices and volume information has no predictive powerinformation has no predictive power an investor an investor cannot achieve excess return with technical analysis.cannot achieve excess return with technical analysis.

Semi-strong efficient markets: current prices rapidly Semi-strong efficient markets: current prices rapidly adjust to the arrival of new public information and adjust to the arrival of new public information and fully reflect fully reflect all publicly available informationall publicly available information. That . That is, all security market and non-market publicly is, all security market and non-market publicly available informationavailable information an investor cannot achieve an investor cannot achieve excess return with fundamental analysis.excess return with fundamental analysis.

Strong form efficient markets: prices fully reflect Strong form efficient markets: prices fully reflect all all information for public and private sources. information for public and private sources. That That is, all market, non-market public, and private (inside) is, all market, non-market public, and private (inside) information.information. There is no such thing as abnormal There is no such thing as abnormal returns.returns.

Technical Vs. Fundamental analysisTechnical Vs. Fundamental analysis Fundamentalist believes that he can predict Fundamentalist believes that he can predict

futures price patterns by analyzing earnings futures price patterns by analyzing earnings and other publicly available information (make and other publicly available information (make economic –risk/return--expectations on demand economic –risk/return--expectations on demand and supply shifts). Fundamentalists believe that and supply shifts). Fundamentalists believe that price adjustment to new info is rapid.price adjustment to new info is rapid.

The technical analyst believes he can predict The technical analyst believes he can predict prices based on historical prices and volume, prices based on historical prices and volume, e.g., price adjustment to new information is e.g., price adjustment to new information is slow. A technical analyst believes that prices slow. A technical analyst believes that prices are determined by supply and demand, which are determined by supply and demand, which are driven by both rational and irrational are driven by both rational and irrational behaviors.behaviors.

Technical AnalysisTechnical Analysis PRO: Quick, easy, no knowledge of accounting, incorporate PRO: Quick, easy, no knowledge of accounting, incorporate

psychological with econ reasons behind price changes, and it tells psychological with econ reasons behind price changes, and it tells when to buy.when to buy.

CON: Challenged by ENH, success strategy cannot be repeated, too CON: Challenged by ENH, success strategy cannot be repeated, too subjective, no underlying theory, rules change over time, self-fulfilling subjective, no underlying theory, rules change over time, self-fulfilling prophecy leading to prices that say only temporarily.prophecy leading to prices that say only temporarily. Dow theory: price moves in trends (major, intermediate, and short-Dow theory: price moves in trends (major, intermediate, and short-

term)term) Trading ratio (AKA TRIN)=(adv.issues/decl. issues)/ (adv.vol/decl. Trading ratio (AKA TRIN)=(adv.issues/decl. issues)/ (adv.vol/decl.

vol) –If>1, market overbought; If<1, market oversoldvol) –If>1, market overbought; If<1, market oversold Support and resistance levels: normal rang of price fluctuation – Support and resistance levels: normal rang of price fluctuation –

support price is cheap, resistance price is expensive.support price is cheap, resistance price is expensive. Moving average: if stock prices move by trends, then MA lines show Moving average: if stock prices move by trends, then MA lines show

these price trends—i.e, if 80% of stocks are above the market 200 these price trends—i.e, if 80% of stocks are above the market 200 MA, the market is considered overvalued. if 20% of stocks are MA, the market is considered overvalued. if 20% of stocks are above the market 200 MA, the market is considered undervaluedabove the market 200 MA, the market is considered undervalued

Relative Strength= stock price/market price. If increasing Relative Strength= stock price/market price. If increasing (decreasing) overtime, stock is overperforming (underperforming). (decreasing) overtime, stock is overperforming (underperforming). Help differentiate between stock-specific or market (macro) Help differentiate between stock-specific or market (macro) movements.movements.

Tests of EMHTests of EMH Event studies:Event studies:

Stock splits: no short-term or long term increase in Stock splits: no short-term or long term increase in abnormal returns as a result of a splits. Supports abnormal returns as a result of a splits. Supports EMH.EMH.

IPOs: If investment bankers underprice issues, then IPOs: If investment bankers underprice issues, then abnormal return should occur. Tests show that IPOs abnormal return should occur. Tests show that IPOs are underpriced by 15% (on average), but prices are underpriced by 15% (on average), but prices adjust within a day. Supports EMH.adjust within a day. Supports EMH.

Exchange listing: Does the choice of the exchange Exchange listing: Does the choice of the exchange increase liquidity and reputation (therefore increase liquidity and reputation (therefore abnormaly increase long-run value of stock prices)? abnormaly increase long-run value of stock prices)? No. Supports EMH. However, there are short term No. Supports EMH. However, there are short term profit opportunities around the listing date, which do profit opportunities around the listing date, which do not support EMH.not support EMH.

Accounting changes: Markets react quickly to Accounting changes: Markets react quickly to accounting changes. Supports EMH.accounting changes. Supports EMH.

Challenges to EMHChallenges to EMH Magnitude issues, selection bias, lucky event and Magnitude issues, selection bias, lucky event and

predictability testspredictability tests Magnitude issues: large investors (institutional) can influence Magnitude issues: large investors (institutional) can influence

prices – EMH still stands but the degree of efficiency still varies.prices – EMH still stands but the degree of efficiency still varies. Selection bias: if an investment strategy works, it will not be Selection bias: if an investment strategy works, it will not be

released to the public. That is, all public strategies have already released to the public. That is, all public strategies have already been tested and proved to fail: EMH cannot be proven using been tested and proved to fail: EMH cannot be proven using these strategies.these strategies.

Lucky Event: some investors simply beat the market by getting Lucky Event: some investors simply beat the market by getting lucky. On average, there are as many lucky as unlucky lucky. On average, there are as many lucky as unlucky investors.investors.

Predictability: Some variables are shown to have predictable Predictability: Some variables are shown to have predictable properties for stock returns (dividend yield, earnings yield, and properties for stock returns (dividend yield, earnings yield, and bond yield spreads). However, it can be argued that these bond yield spreads). However, it can be argued that these variables are rather linked to risk than market inefficiency. As variables are rather linked to risk than market inefficiency. As for dividend yields and earning yields, lower stock prices will for dividend yields and earning yields, lower stock prices will increase both dividend and earning yield and imply a higher increase both dividend and earning yield and imply a higher risk premium and thus higher expected returns.risk premium and thus higher expected returns.

Market AnomaliesMarket Anomalies Earning surprises to predict returns and identify stocks Earning surprises to predict returns and identify stocks

that could earn abnormal returns.that could earn abnormal returns. Calendar studies: January anomaly (profit by buying in Calendar studies: January anomaly (profit by buying in

late December an selling early January); weekend effect late December an selling early January); weekend effect (returns for week days are on average positive, and (returns for week days are on average positive, and negative from Friday close to Monday open)negative from Friday close to Monday open)

PE (low PE stocks have returned more than high PE PE (low PE stocks have returned more than high PE stocks)stocks)

Size: small firm have returned more than large firmsSize: small firm have returned more than large firms Neglected firms: firms with unusually low number of Neglected firms: firms with unusually low number of

analysts covering them have higher returns – the excess analysts covering them have higher returns – the excess return is likely attributed to the lack of institutional return is likely attributed to the lack of institutional interest and it affects firms of all sizes.interest and it affects firms of all sizes.

PB – the greater the book to price ratio (smaller PB), the PB – the greater the book to price ratio (smaller PB), the higher the returnshigher the returns

Many of these anomalies are at the origin of extended Many of these anomalies are at the origin of extended asset pricing model (extended from CAPM) such as the 3-asset pricing model (extended from CAPM) such as the 3-factor model of Fama and French which includes in factor model of Fama and French which includes in addition to the market premium, a size and value addition to the market premium, a size and value premiums. premiums.

Behavioral Interpretations and EMHBehavioral Interpretations and EMH Overconfidence (when one believes they can Overconfidence (when one believes they can

better interpret information than the average better interpret information than the average investor)investor)

Conservatism (underreaction to news events)Conservatism (underreaction to news events) Fear of Regret (hold on too long on an Fear of Regret (hold on too long on an

investment)investment) Reference Points (rather than looking at Reference Points (rather than looking at

outcomes, profits are estimated as compared to a outcomes, profits are estimated as compared to a reference point)reference point)

Effects of Past events (take more risk after Effects of Past events (take more risk after making money, and less risk after losing money)making money, and less risk after losing money)

Mental Accounting (profit target as a function Mental Accounting (profit target as a function Representativeness (purchase past winners, good Representativeness (purchase past winners, good

companies are good stocks)companies are good stocks)

Other issues affecting EMHOther issues affecting EMH Mutual Fund performance –no significant Mutual Fund performance –no significant

difference of performance between mutual funds difference of performance between mutual funds and indexes. Though, few portfolio managers and indexes. Though, few portfolio managers were able to consistently beat the index. In were able to consistently beat the index. In general terms, mutual funds tend to under-general terms, mutual funds tend to under-perform indices. Performance of funds is also perform indices. Performance of funds is also subjected to survivorship bias.subjected to survivorship bias.

Time Varying Volatility: Volatility responds to Time Varying Volatility: Volatility responds to news events and future volatility is forecastable news events and future volatility is forecastable using GARCH models.using GARCH models.

Equity Premium Puzzle: historically it is around Equity Premium Puzzle: historically it is around 5% if we use the history of the stock market. 5% if we use the history of the stock market. However, it has varied significantly – from 1872 However, it has varied significantly – from 1872 to 1949, historical and predicted were similar; to 1949, historical and predicted were similar; from 1950 to 1999, realized returns were higher from 1950 to 1999, realized returns were higher than predicted.than predicted.

Portfolio PerformancePortfolio Performance Measuring returnsMeasuring returns

Holding period 1Holding period 1 Beginning price: 100Beginning price: 100 Dividends paid : 2Dividends paid : 2 Ending price: 120Ending price: 120

Holding period 2 (buy 1 more share at the beginning)Holding period 2 (buy 1 more share at the beginning) Beginning price: 240 (2 shares)Beginning price: 240 (2 shares) Dividends paid : 4 ($2 per share)Dividends paid : 4 ($2 per share) Ending price: 260 (2 shares)Ending price: 260 (2 shares)

Time weighted returns (method of preference in the Time weighted returns (method of preference in the industry) industry)

HPR1=(122/100-1)=22% and HPR2=(264/240-1)=10%HPR1=(122/100-1)=22% and HPR2=(264/240-1)=10% Time weighted return = [(1+22%)x(1+10%)]Time weighted return = [(1+22%)x(1+10%)]1/21/2-1=15.84%-1=15.84%

Dollar-weighted returnsDollar-weighted returns Solve r: 100+120/(1+r)=2/(1+r)+264/(1+r)Solve r: 100+120/(1+r)=2/(1+r)+264/(1+r)1/21/2

In calc: cfo=-100; cf1=-118; cf2=264 comp IRR=13.86% In calc: cfo=-100; cf1=-118; cf2=264 comp IRR=13.86%

Composite Portfolio Performance Composite Portfolio Performance MeasuresMeasures

Treynor MeasureTreynor Measure SML ; T=(Rp-Rf)/ SML ; T=(Rp-Rf)/ββ

Sharpe MeasureSharpe Measure CML; S=(Rp-Rf)/ CML; S=(Rp-Rf)/σσ

Jensen MeasureJensen Measure SML; SML; J=J=αα=(R=(Rpp –Rf) – –Rf) – ββ (R (Rmm – Rf) – Rf)

Information ratioInformation ratioSML:SML:

jIR

)RfRm()RfRp(

Treynor versus Sharpe MeasuresTreynor versus Sharpe Measures Beta vs. Standard DeviationBeta vs. Standard Deviation

Treynor –> uses SML, thus focus on BetaTreynor –> uses SML, thus focus on Beta assumes that assumes that portfolio is well diversified.portfolio is well diversified.

Sharpe-> uses the CML, thus focus on standard Sharpe-> uses the CML, thus focus on standard deviationdeviationassumes that portfolio is assumes that portfolio is notnot well diversified. well diversified.

Ranking differences from different diversification levels. (SML vs. Ranking differences from different diversification levels. (SML vs. CML)CML)RR22 will tell you! will tell you!

Benchmark choice may affect the RBenchmark choice may affect the R22

The Jensen MeasureThe Jensen Measure Requires use of different RFR, RRequires use of different RFR, Rmm, and R, and Rjj, for each period. , for each period.

Assumes portfolio is well diversified and only considers systematic risk.Assumes portfolio is well diversified and only considers systematic risk. Provides inferences about abnormal gain/loss Provides inferences about abnormal gain/loss

Regression of (RRegression of (Rjj- RFR) and (R- RFR) and (Rmm - RFR). - RFR). RR22 can be useful as a measure of diversification. can be useful as a measure of diversification.

The Information Ratio Performance The Information Ratio Performance MeasureMeasure

Appraisal ratioAppraisal ratio measures average return in excess of benchmark measures average return in excess of benchmark

portfolio divided by the standard deviation of this portfolio divided by the standard deviation of this excess returnexcess return

jIR

)RfRm()RfRp(

MM22

CMLRfRm

RfR

RfRmRfM

pm

p

mp

)(

)(2

CML uses p/m and M2 uses m/p ; thus M2 is to be compared to the market return:

• If M2>Rm, Portfolio above CML it outperformed the market on a risk adjusted basis

• If M2<Rm, Portfolio above CML it underperformed the market on a risk adjusted basis

Potential Bias of One-Parameter Potential Bias of One-Parameter MeasuresMeasures

positive relationship between the composite positive relationship between the composite performance measures and the risk involvedperformance measures and the risk involved

alpha can be biased downward for those alpha can be biased downward for those portfolios designed to limit downside riskportfolios designed to limit downside risk

Need to break down performance:Need to break down performance: Performance attribution analysisPerformance attribution analysis Market timing abilityMarket timing ability

Decomposing overall performance into Decomposing overall performance into componentscomponents

Components are related to specific elements of Components are related to specific elements of performance:performance: AssetAsset Allocation Allocation Industry/SectorIndustry/Sector Allocation Allocation Security ChoiceSecurity Choice Selection Selection

Thus, Thus,

Contribution for asset and sector/industry allocation Contribution for asset and sector/industry allocation

+ Contribution for security selection + Contribution for security selection

= Total Contribution from asset class= Total Contribution from asset class

Performance Attribution AnalysisPerformance Attribution Analysis

Example : PAAExample : PAA

  Benchmark Manager A Manager B

  Weight Return Weight Return Weight Return

Stock 0.6 -5% 0.5 -4% 0.3 -5%

Bonds 0.3 -3.5 0.2 -2.5 0.4 -3.5

Cash 0.1 0.3 0.3 0.3 0.3 0.3

• Calculate the overall return of each portfolio and comment on whether these managers have under- or over-performed the benchmark fund.• Using attribution analysis, calculate (1) the asset allocation and (2) the sector allocation/stock selection (combined) effects. Combine your findings with those of (a.) and discuss each manager’s skills.

Example 7: Another example of PAAExample 7: Another example of PAA

1. Excess return1. Excess return

R(Benchmark)= weighted average returnR(Benchmark)= weighted average return

=60% x (–5%) +30% x (–3.5%) + 10% x =60% x (–5%) +30% x (–3.5%) + 10% x 0.3%0.3%

= - 4.02%= - 4.02%

R(a)= -2.41%R(a)= -2.41%

R(b)= -2.81%R(b)= -2.81%

SoSo

Excess return (a)= -2.41%-(-4.02%)= 1.61%Excess return (a)= -2.41%-(-4.02%)= 1.61%

Excess return (b)= -2.81%-(-4.02%)= 1.21%Excess return (b)= -2.81%-(-4.02%)= 1.21%

Example : PAAExample : PAA Allocation effect:Allocation effect:

AA W aW a WbenchWbench R benchR bench EffectEffect

StockStock 5050 6060 -5%-5% 0.5%0.5%

BondBond 2020 3030 -3.5%-3.5% 0.35%0.35%

cashcash 3030 1010 0.3%0.3% 0.06%0.06%

TotalTotal 0.91%0.91%

BB W bW b W W benchbench R benchR bench EffectEffect

StockStock 3030 6060 -5%-5% 1.5%1.5%

BondBond 4040 3030 -3.5%-3.5% -0.35%-0.35%

cashcash 3030 1010 0.3%0.3% 0.06%0.06%

TotalTotal 1.21%1.21%

Example : PAAExample : PAA

Selection Effect: Excess return –Allocation effectSelection Effect: Excess return –Allocation effect A: 1.61%-0.91%=0.7%A: 1.61%-0.91%=0.7% B: 1.21%-1.21% =0%B: 1.21%-1.21% =0% A is good at allocating and selectingA is good at allocating and selecting B is specialized in allocating among asset B is specialized in allocating among asset

classesclasses

Measuring Market Timing SkillsMeasuring Market Timing Skills Tactical asset allocation Tactical asset allocation

(TAA)(TAA) Attribution analysis is Attribution analysis is

inappropriateinappropriate indexes make selection indexes make selection

effect not relevanteffect not relevant multiple changes to asset multiple changes to asset

class weightings during an class weightings during an investment periodinvestment period

Regression-based Regression-based measurementmeasurement

t2

tBtgmintitBtPtpt RFRRRFRRRFRR

:approach Quadratic