1 Portfolio Theory Capital Market Theory Capital Asset Pricing Model “ Do not put all your eggs in...

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Portfolio Theory Capital Market Theory

Capital Asset Pricing Model

“Do not put all your eggs in one basket

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Expected Returns

Expected return: average return on a risky asset expected in the future.

where:Pi = probability of each state of the economyRi = expected return under each state

and

Risk premium = E ( R ) - Rf = Expected return – Risk free rate

n

i ii=1

E(R) = P *R

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Standard Deviation

Standard deviation is calculated as:

where:Pi = probability of each state of the economyRi = expected return under each stateE(R) = expected return for the security

n2

i ii=1

= P *(R ( ))E R

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Example; Stock A gives average return 4% on the normal situation. During the crisis,Stock A plumped and generated negative return -2%. However,Stock A expected to generate return 10% on bullish economy. Given the possibility of crisis, normal, and bullish to be 30%,50%, and 20% respectively. What is the expected return and standard deviation (total risk) of stock A?

E (R) = 0.30 x ( -2%) + 0.50 x (4%) + 0.20 x (10%)E (R) = 3.4%

SDA = [0.3 x(-2 - 3.4)2 + 0.5 x(4 - 3.4)2 + 0.2x(10 – 3.4)2 ]0.5

SDA = 4.2%

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Calculation of Expected Return

State of the Economy

Probability of the State Occurring

Expected Return under this State

Expected Return Pi x Ri

Recession 10.0% -60.0% -6.00%Downturn 20.0% -20.0% -4.00%Normal 40.0% 20.0% 8.00%Upturn 20.0% 60.0% 12.00%Boom 10.0% 100.0% 10.00%

E(R) = 20.0%

n

i ii=1

E(R) = P *R

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Calculation of Standard Deviation

Probability of the State Occurring

Return under this State Ri - E(R) (Ri - E(R))2 (Ri - E(R))2Pi

10.0% -60.0% -80.0% 6400 64020.0% -20.0% -40.0% 1600 32040.0% 20.0% 0.0% 0 020.0% 60.0% 40.0% 1600 32010.0% 100.0% 80.0% 6400 640

variance = 1,920standard deviation = 43.8%

n2

i ii=1

= P *(R ( ))E R

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Portfolio Portfolio: Group of assets such as stocks and

bonds held by an investor.

Efficient portfolio : portfolio that maximize the expected return given a level of risk.

Optimal portfolio : the most preferred efficient portfolio that an investor selects.

Risk averse investor : with the same expected return but two different risks, he will prefer the lower risk.

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Portfolio Portfolio weights: Percentage of a portfolio's

total value invested in a particular asset.

Portfolio expected returns: the weighted average combination of the expected returns of the assets in the portfolio.

Portfolio Variance:it is the combination of the weighted average of the individual security's variance and securities’ covariance factor.

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Portfolio Expected Returns

Two ways to calculate portfolio expected return:

Xj = weight of each stock in the portfolioE(Rj) = expected return of each stock in the portfolion = total number of different stocks in the portfolio

E(Rps) = expected return of portfolio under state sPs = Probability of state sm = total number of all states in the future

1 1

( ) * ( ) * ( )n m

P j j s psj s

E R X E R P E R

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Portfolio Risk Portfolio variance:

cov (Ri, Rj) = p1[ri1 - E(Rj) ][rj1 – E(Rj)] +

p2[ri2 - E(Rj) ][rj2 – E(Rj)] +

…. +

pN[riN - E(Rj) ][rjN – E(Rj)]

Where ; rin and rjn = the nth possible rate of return for asset i and j respectively

pn = the probability of attaining the rate of return n for assets i and j

N = the number of possible outcomes for the rate of return

var(Rp) = Wi2 var(Ri) + Wj

2 var (Rj) + 2WiWj cov(Ri,Rj)2 2 2 2 2 2P A A B B A B A B ABx x x x

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Covariance and Correlation Relationship between covariance and correlation:

Correlation represent only the direction between two assets. It values between -1 to +1

Covariance represent both direction and magnitude between two assets.

corr (Ri, Rj) = cov (Ri, Rj)

SD(Ri) SD(Rj)

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Calculating covariance and correlation :

cov (Ri, Rj) = 0.50(15-11)( 8- 8) +

0.30(10-11)(11-8) +

0.13( 5 -11)( 6- 8) +

0.05( 0 -11)( 0- 8) +

0.02(-5-11)(-4- 8)

= 8.90

corr (Ri, Rj) = 8.9 / [(4.91)(3.05)]

= 0.60

nReturn for

stock i (%)

Return for

stock j (%)Prob.

1 15 8 0.502 10 11 0.303 5 6 0.134 0 0 0.055 -5 -4 0.02

Stock i Stock j

Expected

return (%)11.05 8.00

Variance 24.15 9.30SD (%) 4.91 3.05

Covariance and Correlation

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nReturn for

stock i (%)

Return for

stock j (%)Prob. Stock i Stock j Portfolio

1 15 8 0.50 Weight 0.50 0.50 1.002 10 11 0.30 Return (%) 11.05 8.003 5 6 0.13 Variance 24.15 9.304 0 0 0.05 SD (%) 4.91 3.055 -5 -4 0.02 Cov (i,j) 8.9

Corr (i,j) 0.6

Example: Portfolio Return and Risk

= 0.50 x 11.05% + 0.50 x 8%

9.53

= (0.502)( 4.912) + (0.502)(3.052) + 2(0.50)(0.50)(8.90)

12.81

Diversification!

3.58

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Feasible and Efficient Portfolios

Stock C E(RC) 10%SD(RC) 30%Stock D E(RD) 25%SD(RD) 60%

Profolio WC WD E(RP)SD(RP)1 1.00 0.00 10.0%30.0%2 0.75 0.25 13.8%3.9%3 0.50 0.50 17.5%6.8%4 0.25 0.75 21.2%17.4%5 0.00 1.00 25.0%60.0%

Corr (RC,RD) -0.50

Figure 3.2 ; Feasible and efficient sets of portfolios for stocks C and D

E(Rp)

SD(Rp)

10%

30%

20%

10% 20% 40%30% 60%50%

1

3

2

5

4

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Feasible and Efficient PortfoliosFigure 3.2 ; Feasible and efficient sets of portfolios for stocks C and D

E(Rp)

SD(Rp)

10%

30%

20%

10% 20% 40%30% 60%50%

1

3

2

5

4

6

1 is not include in Markowitz efficient set Because it was dominated by 2,3, and 4

Space on the left side of 2,3,4, and 5 are not attainable from combinations of C and D.

Space on the right side of 2,3,4, and 5 are not include in Markowitz efficient set. i.e. 6 has same return as 2 but 6 gives higher risk. In another word, 6 has same risk as 4 but 6 gives lower return

Markowitz efficient frontier

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E(r)E(r)The minimum-variance The minimum-variance frontier of risky assetsfrontier of risky assets

EfficientEfficientfrontierfrontier

GlobalGlobalminimumminimumvariancevarianceportfolioportfolio MinimumMinimum

variancevariancefrontierfrontier

IndividualIndividualassetsassets

St. Dev.

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Markowitz Efficient Frontier

Efficient Frontier:

Set of portfolios with the maximum return for a given risk level.

Data needed to find Efficient Frontier:– Expected returns on all assets– Standard deviations on all assets– Correlation coefficients between every pair of

assets

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Feasible and Efficient Portfolios

E(Rp)

SD(Rp)

10%

30%

20%

10% 20% 40%30% 60%50%

Moving from left to right on the frontier, although risk increases, so does the expect return. Which point is the best portfolio to hold?

The answer is optimal portfolio depend on the investor’s preference or utility as the trade-off between risk and return.

U1

U2

U3

= Optimal portfolio

U1,U2,U3 = Indifference curves with U1 < U2 < U3

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Correlation Coefficient Correlation: The tendency of the returns on two

assets to move together. Corr(RA, RB) or A,B

-1.0 +1.0 Perfect positive correlation: +1.0 gives no risk

reduction Perfect negative correlation: -1.0 gives

complete risk reduction Correlation coefficient between -1.0 and +1.0

gives some, but not all, risk reduction The smaller the correlation, the greater the risk

reduction potential

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Portfolio Diversification

Portfolio Diversification

0%

10%

20%

30%

40%

50%

60%

0 10 20 30 40 50 60 70 80 90 100

No. of Stocks in Portfolio

An

nu

al S

tan

dar

d D

evi

atio

n

Diversification: Spreading an investment across a number of assets will eliminate some, but not all, of the risk.

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Returns Distributions for Two Perfectly Positively Correlated Stocks ( = +1.0)

Stock A

0 0

Stock B

0

Portfolio AB

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Returns Distribution for Two Perfectly Negatively Correlated Stocks ( = -1.0)

0 0 0

Stock A Stock B Portfolio AB

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= = 00

E(r)E(r)

= 1= 1 = -1= -1

= -1= -1

= .3= .3

13%13%

8%8%

12%12% 20%20% St. DevSt. Dev

Risk & Return with 2 Assets

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Problem 3Use the following information to calculate the expected

return and standard deviation of a portfolio that is 40% invested in Kuipers and 60% invested in SuCo.

Kuipers SuCoExpected return, E(R) 30% 28%Standard deviation , .65 .45Correlation .30

E(RP) = .4 x (.30) + .6 x (.28) = 28.8%

P = .402 x (.65)2 + .602 x (.45)2 + 2 (0.4)(0.6)(0.65 x .45 x .30

= (0.18262)1/2 = 42.73%

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CAPM : Capital Asset Pricing Model Assumptions ;

1.Two-Parameter Model : Investors rely on E(R) and Risk in making decision

2.Rational and Risk Averse : Investors are rational and risk averse.

3.One-Period Investment Horizon : Investors all invest for the same period of time

4.Homogeneous Expectations : Investors share all expectations about assets

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CAPM : Capital Asset Pricing Model Assumptions ;

5.Existence of a risk-free asset and unlimited borrowing and lending at the risk-free rate : Investors can borrow and lend any amount at the risk-free rate.

6.Capital markets are completely competitive and frictionless : The sufficiently large number of buyers&sellers. Also, no transaction costs.

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CML : Capital Market LineFrom;

Var(Rp) = (WRf)2 Var(Rf) + (Wm)2 Var(Rm) + 2WRfWmcov(Rf,Rm)

Since, risk-free asset has no variability, and therefore does not move at all with the return on the market portfolio

Var(Rp) = (Wm)2 Var(Rm)

Wm = SD(Rp) / SD(Rm)

From;

E(Rp) = WRfRf + Wm E(Rm) ; which WRf + Wm = 1

E(Rp) = (1-Wm)Rf + Wm E(Rm)

E(Rp) = Rf + Wm (E(Rm) - Rf )

Therefore, E(Rp) = Rf + [(E(Rm) - Rf )/ SD(Rm)] SD(Rp)

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CML : Capital Market LineE

xp

ec

ted

Re

turn

Standard Deviation

Rf

Efficient Frontier

MarketPortfolio

M

BorrowingPortfolio

LendingPortfolio

Capital market line:CML

Rp = Rf + [( Rm - Rf )/ SD(Rm)] SD(Rp)

Every combination of the risk-free asset and the Markowitz efficient portfolio M is shown on the capital market line (CML)

With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope

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CML : Capital Market Line Because investors have the same optimal risky

portfolio given the risk-free rate, all investors have the same CML. What if they have different levels of risk-free rate?

100% bonds

100% stocks

retu

rn

First Optimal Risky Portfolio

Second Optimal Risky Portfolio

CML 0 CML 1

0fr

1fr

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Announcements & News

Announcement = expected part + surprise

the part of the information that the market uses to form the expectation, E(R).

the news that influences the unanticipated return on the stock, U, real news which results in risk.

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Expected Return or Unexpected Return

Total Return = Expected Return + Unexpected Return

Total Return - Expected Return = Unexpected Return

return theofpart unexpected theis

return theofpart expected theis

where

U

R

URR

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Risk: The unanticipated part of the return, the portion resulting from surprises.

Types of risk:– Systematic or “market”

– Unsystematic or “unique” or “asset-specific”

Risk that influences a single company or a small group of companies. Also called unique or asset-specific risk.

Risk: Systematic & Unsystematic

Risk that influences a large number of assets. Also called market risk.

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Components of Risk

R - E(R) = Systematic portion + Unsystematic portion

R - E(R) = U = m +

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Components of RiskP

ort

folio

Ris

k -

Std

. De

v.

Number of Stocks in Portfolio

Market Risk

Company-specific risk

Total risk

Diversifiable riskUnsystematic risk

Non-diversifiable or Systematic risk

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Unsystematic Risk:– “Unsystematic risk is essentially eliminated by

diversification, so a portfolio with many assets has almost no unsystematic risk.”

– Diversifiable risk / unique risk / asset-specific risk

Systematic Risk:– Systematic risk affects all assets and can not be

diversified away (even in a larger portfolio).– Non diversifiable risk / market risk

Components of Risk

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Systematic Risk (Beta) Systematic risk can not be eliminated by

diversification. Since unsystematic risk can be eliminated at no cost,

there is no reward for bearing it. Systematic Risk Principle:

Measuring Systematic Risk: Beta or

The reward (expected return) for bearing risk depends only on the systematic risk of an investment.

Measure of the relative systematic risk of an asset. Assets with betas larger (smaller) than 1 have more (less) systematic risk than market average and will have greater (lower) expected returns.

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Beta Coefficients

Beta

Company Coefficient ( i)

Exxon 0.65

AT&T 0.90

IBM 0.95

Wal-Mart 1.10

General Motors 1.15

Microsoft 1.30

Harley-Davidson 1.65

America Online 2.40

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Portfolio Beta

With a large number of assets in a portfolio, multiply each asset's beta by its portfolio weight, and then sum the results to get the portfolio's beta:

n

P ii=1

= iX

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Portfolio Beta

Amount PortfolioStock Invested Weights Beta

(1) (2) (3) (4) (3) x (4)

Haskell Mfg. $ 6,000 50% 0.90 0.450Cleaver, Inc. 4,000 33% 1.10 0.367Rutherford Co. 2,000 17% 1.30 0.217

Portfolio $12,000 100% 1.034

n

P ii=1

= iX

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Portfolio Beta

High beta security is more sensitive to market movements

Low beta security is relatively insensitive to market movements.

Sensitivity depends on:– How closely is the security’s return

correlated with the market– How volatile the security is relative to the

market

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Portfolio Beta

Beta is computed as:

M

jMjj xR,RCorr

)(

)(2

,

M

Mii R

RRCov

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Portfolio Beta Beta is estimated as:

Sec

uri

ty R

etu

rns

Sec

uri

ty R

etu

rns

Return on Return on market %market %

RRii = = ii + + iiRRmm + + eeii

Slope = Slope = iiCharacte

ristic

Line

Characteris

tic Line

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Beta & Expected Return of a portfolio with one risky and one risk free asset

If you invest in WA in Asset A and the rest (1-WA) in Risk-Free Asset (with ß=0), then

ßP = WA ßA + (1 - WA) 0 = WA ßA

If you own 40% in A with an expected return of 18%, with the remainder in the Risk-Free Asset with a 8% return. The beta on Asset A is 1.4.

E(RP) = (0.40) (0.18) + (1 - 0.40) (0.08) = 12%

ßP = WA ßA + (1 - WA ) 0 = (0.40)(1.4) = 0.56

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Why do Betas Differ?

Using daily, weekly, monthly, quarterly, or annual returns.

Estimating betas over short periods (a few weeks) versus long periods (5-10 years).

Choice of the market index (S&P 500 index versus all risky assets).

Some sources adjust betas for statistical and fundamental reasons (such as Value Line).

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E(Ri) = Rf + [(E(Rm) - Rf )/ SD(Rm)] SD(Ri)

by assuming the unsystematic risk is zero and since,

,therefore,

E(Ri) = Rf + [cov (Ri, Rm) / var(Rm )] (E(Rm) - Rf )

SML shows that it is not the variance or standard deviation of an asset that affects its return. Actually, Covariance of the asset’s return with the market’s return affects its return.

From equation above, positive covariance implies higher expected return than the risk-free asset. However, with positive covariance, it increases the risk of an asset in a portfolio, that’s why investors will buy that asset only if they expect to earn a return higher than the risk-free asset.

SML : Security Market Line

M

jMjj xR,RCorr

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SML : Security Market Line

Assume you wish to hold a portfolio consisting of Asset A and a risk-less asset. Given the following information:

Asset A has a beta of 1.2 and an expected return of 18%. The risk-free rate is 7%.

Use Asset A weights of: 0%, 25%, 50%, 75%, and 100%.

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SML : Security Market Line

Proportion

Proportion Invested in Portfolio

Invested in Risk-Free Expected Portfolio

Asset A (%) Asset (%) Return (%) Beta

0 100 7.00 0.00

25 75 9.75 0.30

50 50 12.50 0.60

75 25 15.25 0.90

100 0 18.00 1.20

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Expected Returns & Betas for Asset A

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Beta

Ex

pec

ted

Ret

urn Security market

line (SML)

SML : Security Market Line

Graphical representation of the linear relationship between systematic risk and expected return in financial markets

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Reward-to-Risk Ratio

• Risk premium of X% per unit of systematic risk

• Based on CAPM assumptions, the reward-to-risk ratio is the same for all securities. Hence,

A

fA R)R(ESlope

fMj

fj R)R(ER)R(E

fMjfj R)R(ER)R(E

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SML : Security Market Line

E(RA)

Rf

E(RM)

ßM= 1.0 ßA

Slope of SML= [E(Rj) - Rf] / ßj or= [E(RM) - Rf]

Asset ExpectedReturn [E(Rj)]

Asset Beta ßj

SML

C

D

A

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Question;

Asset A has an expected return of 12% and a beta of 1.40. Asset B has an expected return of 8% and a beta of 0.80. Are these assets valued correctly relative to each other if the risk-free rate is 5%?

a. For A: (.12 - .05) / 1.40 = ________ b. For B: (.08 - .05) / 0.80 = ________

What would the risk-free rate have to be for these assets to be correctly valued?

(.12 - Rf) / 1.40 = (.08 - Rf) / 0.80 Rf = ________

0.0500.0375

0.0267

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CAPM:Capital Asset Pricing Model

This result gives us the CAPM:

The Capital Asset Pricing Model (CAPM) is an equilibrium model of the relationship between risk and return in a competitive capital market.

The CAPM shows the expected return for an asset depends on:

1.Pure time value of money: risk-free rate2.Reward for bearing systematic risk: market risk premium3.Amount of systematic risk: beta coefficient

fMjfj R)R(ER)R(E

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CAPM:Capital Asset Pricing Model

fMjfj R)R(ER)R(E

Expected return on a security

=Risk-

free rate+

Beta of the security

×Market risk

premium

• Assume j = 0, then the expected return is RF.• Assume j = 1, then Mi RR

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Risk & Return Summary Total risk: variance (standard deviation) of an asset’s

return

Total return: expected return + unexpected return

Total unexpected return= Systematic risk: + Unsystematic risk

Systematic risk: unanticipated events that affect almost all assets to some degree

Unsystematic risk: unanticipated events that affect single assets or small groups of assets

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Risk & Return Summary Effect of diversification: elimination of unsystematic

risk via the combination of assets into a portfolio

Systematic risk principle & beta: reward for bearing risk depends only on it’s level of systematic risk

Reward-to-risk ratio: ratio of an asset’s risk premium to it’s beta

Capital asset pricing model: expected return on an asset can be written as:

E(Rj) = Rf + [E(RM) - Rf] x ßj

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Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit.

Since no investment is required, an investor can create large positions to secure large levels of profit.

In efficient markets, profitable arbitrage opportunities will quickly disappear.

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APT: Arbitrage Pricing Theory Model

E(Ri) = Rf + bi,f1[E(Rf1) – Rf] + bi,f2[E(Rf2) – Rf] + ….

+ bi,fn[E(Rfn) – Rf]

Concept : The return on a security is linearly related to H “factors”. The APT does not specify what these factors are, but it is assumed that the relationship between security returns and the factors is linear.

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Question;You own a stock portfolio invested 30% in Stock Q, 20% in Stock R, 10% in Stock S, and 40% in Stock T. The betas for these for stocks are 1.2, 0.6, 1.5, and 0.8, respectively. What is the portfolio beta?

Solution:

ßP = .3(1.2) + .2(.6) + .1(1.5) + .4(.8) = .95

Question;

A stock has a beta of 1.2, the expected return on the market is 17%, and the risk-free rate is 8%. What must the expected return on this stock be?

Solution:

E[Rj] = .08 + (.17 - .08)(1.2) = .188 = 18.8%

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Question;

Stock Y has a beta of 1.59 and an expected return of 25%. Stock Z has a beta of .44 and an expected return of 12%. If the risk-free rate is 6% and the market risk premium is 11.3%, are these stocks correctly priced?

Solution:

E[rj] = .06 + .113ßj

E[rY] = .06 + .113(1.59) = .2397 < .25

so Y plots above the SML and is undervalued

E[rZ] = .06 + .113(.44) = .1097 < .12

so Z plots above the SML and is undervalued

Question;

what would the risk-fee rate have to be for the two stocks to be correctly priced?

Solution:

[.25 - Rf] / 1.59 = [.12 - Rf] / 0.44 , therefore, Rf = .0703

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Risks associated with investing in Financial AssetsPrice risk An asset’s value drops.

Default risk The issuer of an asset cannot meet its obligations.

Inflation risk The rate of inflation erodes the value of an asset.

Currency risk The rate of exchange erodes the value of a foreign-denominated asset.

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Risks associated with investing in Financial Assets

Reinvestment risk The cash flow received must be reinvested in a

similar vehicle that offers a lower return.

Liquidity risk An asset cannot easily be sold at a fair price.

Call risk The issuer of an asset exercises its right to pay off the amount borrowed.