Portfolio theory and capm

Portfolio Theory and Capital Asset Pricing Model

Introduction

• Asset Pricing – how assets are priced?• Equilibrium concept• Portfolio Theory – ANY individual

investor’s optimal selection of portfolio (partial equilibrium)

• CAPM – equilibrium of ALL individual investors (and asset suppliers) (general equilibrium)

Our expectation

• Risky asset i:• Its price is such that:

E(return) = Risk-free rate of return + Risk premium specific to asset i

= Rf + (Market price of risk)x(quantity of risk of asset i)

CAPM tells us 1) what is the price of risk?2) what is the risk of asset i?

The Risk-Free Asset

• Only the government can issue default-free bonds– Guaranteed real rate only if the duration of

the bond is identical to the investor’s desire holding period

• T-bills viewed as the risk-free asset– Less sensitive to interest rate fluctuations

4

• It’s possible to split investment funds between safe and risky assets.

• Risk free asset: proxy; T-bills• Risky asset: stock (or a portfolio)

Portfolio Theory: Portfolios of One Risky Asset and a Risk-Free Asset

5

An example to motivateExpected Return Standard Deviation

Asset i 10.9% 4.45%

Asset j 5.4% 7.25%

E(return) = Risk-free rate of return + Risk premium specific to asset i= Rf + (Market price of risk)x(quantity of risk of asset i)

Question: According to the above equation, given that asset j has higher risk relative to asset i, why wouldn’t asset j has higher expected return as well?

Possible Answers: (1) the equation, as intuitive as it is, is completely wrong.(2) the equation is right. But market price of risk is different for different assets.(3) the equation is right. But quantity of risk of any risky asset is not equal to the standard deviation of its return.

CAPM’s AnswersE(return) = Risk-free rate of return + Risk premium specific to asset i


• The intuitive equation is right.• The equilibrium price of risk is the same

across all marketable assets• In the equation, the quantity of risk of any

asset, however, is only PART of the total risk (s.d) of the asset.

CAPM’s Answers• Specifically:

Total risk = systematic risk + unsystematic risk

CAPM says: (1)Unsystematic risk can be diversified away. Since there is

no free lunch, if there is something you bear but can be avoided by diversifying at NO cost, the market will not reward the holder of unsystematic risk at all.

(2)Systematic risk cannot be diversified away without cost. In other words, investors need to be compensated by a certain risk premium for bearing systematic risk.

CAPM resultsE(return) = Risk-free rate of return + Risk premium specific to asset i


Precisely:[1] Expected Return on asset i = E(Ri)[2] Equilibrium Risk-free rate of return = Rf[3] Quantity of risk of asset i = COV(Ri, RM)/Var(RM)[4] Market Price of risk = [E(RM)-Rf]

Thus, the equation known as the Capital Asset Pricing Model:

E(Ri) = Rf + [E(RM)-Rf] x [COV(Ri, RM)/Var(RM)]

Where [COV(Ri, RM)/Var(RM)] is also known as BETA of asset I

Or

E(Ri) = Rf + [E(RM)-Rf] x βi

Pictorial Result of CAPM

E(Ri)

E(RM)

Rf

SecurityMarketLine

β = [COV(Ri, RM)/Var(RM)]βΜ= 1.0

slope = [E(RM) - Rf] = Eqm. Price of risk

CAPM in Details: What is an equilibrium?

CONDITION 1: Individual investor’s equilibrium: Max U

• Assume: • [1] Market is frictionless

=> borrowing rate = lending rate => linear efficient set in the return-risk space

[2] Anyone can borrow or lend unlimited amount at risk-free rate• [3] All investors have homogenous beliefs

=> they perceive identical distribution of expected returns on ALL assets=> thus, they all perceive the SAME linear efficient set (we called the line: CAPITAL MARKET LINE=> the tangency point is the MARKET PORTFOLIO


CONDITION 1: Individual investor’s equilibrium: Max U

Rf

A

Market Portfolio

Q

B

Capital Market Line

σp

E(Rp)

E(RM)

σM


CONDITION 2: Demand = Supply for ALL risky assets

• Remember expected return is a function of price.• Market price of any asset is such that its expected return is just

enough to compensate its investors to rationally hold it.

CONDITION 3: Equilibrium weight of any risky assets

• The Market portfolio consists of all risky assets.• Market value of any asset i (Vi) = PixQi• Market portfolio has a value of ∑iVi• Market portfolio has N risky assets, each with a weight of wi

Such thatwi = Vi / ∑iVi for all i


CONDITION 4: Aggregate borrowing = Aggregate lending

• Risk-free rate is not exogenously given, but is determined by equating aggregate borrowing and aggregate lending.


Two-Fund Separation:

Given the assumptions of frictionless market, unlimited lending and borrowing, homogenous beliefs, and if the above 4 equilibrium conditions are satisfied, we then have the 2-fund separation.

TWO-FUND SEPARATION:Each investor will have a utility-maximizing portfolio that is a combination of the risk-free asset and a portfolio (or fund) of risky assets that is determined by the Capital market line tangent to the investor’s efficient set of risky assets

Analogy of Two-fund separationFisher Separation Theorem in a world of certainty

• The relationship depends on the correlation coefficient

• -1.0 < ρ < +1.0• The smaller the correlation, the greater the

risk reduction potential• If ρ = +1.0, no risk reduction is possible

Correlation Effects

The Sharpe Ratio

• Maximize the slope of the CAL for any possible portfolio, p

• The objective function is the slope:

( )P fP

P

E r rS

σ−

=

17

Figure: The Variance of an Equally Weighted Portfolio with Risk Coefficient βp in the Single-Factor Economy

18


Two-fund separation

Rf

A

Market Portfolio

Q

B

Capital Market Line

σp

E(Rp)

E(RM)

σM

The Role of Capital Market

Efficient set

U’’ U’

P

Endowment Point

E(rp)

σp

Borrow at the Risk-Free Rate and invest in stock.

Using 50% Leverage,rc = (-.5) (.07) + (1.5) (.15) = .19

σc = (1.5) (.22) = .33

Capital Allocation Line with Leverage

21

The Role of Capital Market

Efficient set

U’’’ U’’ U’

P

Endowment Point

E(rp)

σp

M

U-Max PointCapital Market Line

Rf

CAPM• 2 sets of Assumptions:[1] Perfect market:

• Frictionless, and perfect information• No imperfections like tax, regulations, restrictions to short

selling• All assets are publicly traded and perfectly divisible• Perfect competition – everyone is a price-taker

[2] Investors:• Same one-period horizon• Rational, and maximize expected utility over a mean-

variance space• Homogenous beliefs

Derivation of CAPM• Using equilibrium condition 3

wi = Vi / ∑iVi for all i

market value of individual assets (asset i)wi = ------------------------------------------------

market value of all assets (market portfolio)

• Consider the following portfolio:hold a% in asset iand (1-a%) in the market portfolio

Derivation of CAPM• The expected return and standard deviation of such a

portfolio can be written as:

E(Rp) = aE(Ri) + (1-a)E(Rm)σ(Rp) = [ a2σi

2 + (1-a)2σm2 + 2a (1-a) σim ] 1/2

• Since the market portfolio already contains asset i and, most importantly, the equilibrium value weight is wi

• therefore, the percent a in the above equations represent excess demands for a risky asset

• We know from equilibrium condition 2 that in equilibrium, Demand = Supply for all asset.

• Therefore, a = 0 has to be true in equilibrium.

Derivation of CAPME(Rp) = aE(Ri) + (1-a)E(Rm)

σ(Rp) = [ a2σi2 + (1-a)2σm

2 + 2a (1-a) σim ] 1/2

• Consider the change in the mean and standard deviation with respect to the percentage change in the portfolio invested in asset i

• Since a = 0 is an equilibrium for D = S, we must evaluate these partial derivatives at a = 0

)RE( - )RE( = a

)RE( mi

p

∂∂

] 4a - 2 + 2a + 2 - 2a [* ]a)-2a(1 + )a-(1 + a [21=

a )R(

imim2m

2m

2i

1/2-im

2m

22i

2p σσσσσσσσσ∂

∂

)RE( - )RE( = a

)RE( mi

p

∂∂

σσσσ

m

2mimp - =

a )R(

∂∂

(evaluated at a = 0)


Derivation of CAPM• the slope of the risk return trade-off evaluated at point M in

market equilibrium is

• but we know that the slope of the opportunity set at point M must also equal the slope of the capital market line. The slope of the capital market line is

• Therefore, setting the slope of the opportunity set equal to the slope of the capital market line

• rearranging,

σσσσ

m

2mim

mi

p

p

- )RE( - )RE( =

a)/R(a)/RE(

∂∂∂∂

σ m

fm R - )RE(

σσσσ m

fm

m2mim

mi R - )RE( = / ) - (

)RE(-)RE(

]R - )R[E( + R = )RE( fm2m

imfi

σσ


Derivation of CAPM• From previous page

• Rearranging

• Where

E(return) = Risk-free rate of return + Risk premium specific to asset iE(Ri) = Rf + (Market price of risk)x(quantity of risk of asset i)

CAPM Equation

)RVAR()R,RCOV( = =

m

mi2m

imi σ

σβ

β ifmfi ]R - )R[E( + R = )RE(

]R - )R[E( + R = )RE( fm2m

imfi

σσ

CML Equation

• Y = b + mX

This leads to the Security Market Line (SML)

( )

[ ]FMM

PF

PM

FMFP

RRERSDRSDR

RSDRSD

RRERRE

−+=

−+=

)()()(

gives grearrangin

)()(

)(

SML Equation

( ) ( )

( ) ( )

( )FMiF

FMM

MiF

MiM

FMFi

RRR

RR)R(VAR

RRCOVR

RRCOV)R(VAR

RRRRE

−β+=

−×+=

×−

+=

β is a measure of relative risk• β = 1 for the overall market. • β = 2 for a security with twice the systematic risk of

the overall market, • β = 0.5 for a security with one-half the systematic

risk of the market.

Pictorial Result of CAPM

E(Ri)

E(RM)

Rf

Security MarketLine

β = [COV(Ri, RM)/Var(RM)]βΜ= 1.0

slope = [E(RM) - Rf] = Eqm. Price of risk

Properties of CAPM• In equilibrium, every asset must be priced so that its risk-

adjusted required rate of return falls exactly on the security market line.

• Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk – a measure of how the asset co-varies withthe entire economy (cannot be diversified away)

e.g., interest rate, business cycle

Unsystematic Risk – idiosyncratic shocks specific to asset i,(can be diversified away)e.g., loss of key contract, death of CEO

• CAPM quantifies the systematic risk of any asset as its β

• Expected return of any risky asset depends linearly on itsexposure to the market (systematic) risk, measured by β.

• Assets with a higher β require a higher risk-adjusted rate ofreturn. In other words, in market equilibrium, investors are onlyrewarded for bearing the market risk.

CAPM and Portfolios

• How does adding a stock to an existing portfolio change the risk of the portfolio?– Standard Deviation as risk

• Correlation of new stock to every other stock– Beta

• Simple weighted average:

• Existing portfolio has a beta of 1.1 • New stock has a beta of 1.5. • The new portfolio would consist of 90% of the old portfolio

and 10% of the new stock • New portfolio’s beta would be 1.14 (=0.9×1.1 + 0.1×1.5)

∑=

×=n

iiiP w

1ββ

Estimating Beta

• Need– Risk free rate data– Market portfolio data

• S&P 500, DJIA, NASDAQ, etc.

– Stock return data• Interval

– Daily, monthly, annual, etc.• Length

– One year, five years, ten years, etc.

Market Index variations

Constant 0.005

Std Err of Y Est 0.006

R Squared 24.71%

No. of Observations 52

Degrees of Freedom 50

Beta estimate 0.995

Std Err of Coef. 0.246

t-statistic 4.05

Constant 0.004


R Squared 20.11%



Beta estimate 0.737


t-statistic 3.550

A. Google Beta Using S&P 500 Index

-12.00%

-8.00%

-4.00%

0.00%

4.00%

8.00%

12.00%

-6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00%

S&P 500 Index Returns

Goog

le Re

turn

sB. Google Beta Using Russell 2000

-12.00%

-8.00%

-4.00%

0.00%

4.00%

8.00%

12.00%

16.00%

20.00%

-8.00% -3.00% 2.00% 7.00%

Russell 2000 Returns

Goog

le Re

turn

s

A. GM Beta Estimation: Daily Data

-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

-25.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

S&P 500 Returns

GM

Ret

urns

Interval variations

Constant 0.0001


R Squared 31.37%



Beta estimate 1.047


t-statistic 66.26

Constant 0.019


R Squared 31.41%



Beta estimate 1.301


t-statistic 4.06

D. GM Beta Estimate: Yearly Returns

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

-30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

S&P 500 Returns

GM

Ret

urns

Problems using Beta

• Which market index?• Which time intervals?• Time length of data?• Non-stationary

– Beta estimates of a company change over time.– How useful is the beta you estimate now for thinking about

the future?

• Other factors seem to have a stronger empirical relationship between risk and return than beta– Not allowed in CAPM theory– Size and B/M

Use of CAPM

• For valuation of risky assets

• For estimating required rate of return of risky projects

EXHIBIT: Geometric Mean rates of Return and Standard Deviation for Sotheby's Indexes, S&P 500, Bond Market

Series, One-Year Bonds, and Inflation

Standard Deviation

1-Year Bond

CPI

Amer FumLBGC

Cont SilverEng Silver

Fr+Cont FurnUW Index

Eng Furn

Mod PaintCont Art

Imp Paint

Chinese Ceramic

S&P500Old Master

Cont Ceramic

19C Euro

Amer PaintFW Index

VW Index

Table: Risk Measures for Non-Normal Distributions

43

EXHIBIT: Alternative Investment Risk and Return Characteristics

FuturesArt and Antiques

Warrants and OptionsCoins and StampsCommercial Real Estate

Foreign Common StockUS Common Stocks

Real Estate (Personal Home)Foreign Corporate BondsUS Corporate Bonds

Foreign Government BondsUS Government Bonds

T-Bills

Arbitrage Pricing Theory (APT)

45

• Assume factor model such as

• And no arbitrage opportunities exist in equilibrium

• Then, we have

iiii eFr ++= 1βα

iirE βλλ 10)( +=

• APT applies to almost all individual securities• 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

APT and CAPM Compared

Multifactor APT

• Use of more than a single factor• Requires formation of factor portfolios• What factors?

– Factors that are important to performance of the general economy

– Fama-French Three Factor Model

Two-Factor Model

• The multifactor APT is similar to the one-factor case – But need to think in terms of a factor portfolio

1 1 2 2( )i i i i ir E r F F eβ β= + + +

Example of the Multifactor Approach

• Work of Chen, Roll, and Ross– Chose a set of factors based on the ability

of the factors to paint a broad picture of the macro-economy

– GDP factor, inflation factor, and interest rate factor

Another Example:Fama-French Three-Factor Model

• The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums

• Where:– SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in

excess of the return on a portfolio of large stocks– HML = High Minus Low, i.e., the return of a portfolio of stocks with a

high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio

it i iM Mt iSMB t iHML t itr R SMB HML eα β β β= + + + +

The Multifactor CAPM and the APT

• A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge

• The APT is largely silent on where to look for priced sources of risk

Using CAPM• Expected Return

– If the market is expected to increase 10% and the risk free rate is 5%, what is the expected return of assets with beta=1.5, 0.75, and -0.5?

• Beta = 1.5; E(R) = 5% + 1.5 × (10% - 5%) = 12.5%• Beta = 0.75; E(R) = 5% + 0.75 × (10% - 5%) = 8.75%• Beta = -0.5; E(R) = 5% + -0.5 × (10% - 5%) = 2.5%

• Finding Undervalued Stocks…(the SML)

Date post:	07-Aug-2015
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Portfolio theory and capm

Economy & Finance