CAPM

Preliminaries Asset pricing Intuition Conclusion

Lecture 5: The Capital Asset Pricing ModelSAPM [Econ F412/FIN F313 ]

Ramana Sonti

BITS Pilani, Hyderabad Campus

Term II, 2014-15

1/27 Lecture 5: The Capital Asset Pricing Model Ramana Sonti


Agenda

1 PreliminariesIntroduction

2 Asset pricingThe market portfolioThe CAPM equationExampleThe security market line

3 IntuitionIn wordsIn pictures

4 ConclusionBeta as a regression coefficient



Introduction

Asset pricing models and the CAPM Needed: A model to relate risk to return this is the ambitious goal

of asset pricing useful to calculate the cost of capital useful as a key input in portfolio allocation useful in measuring portfolio performance

The CAPM is the earliest APM ... circa 1962 a theoretical APM ... derived wholly from first principles, and not

motivated by data an APM based on the concept of equilibrium ... meaning an economic

state where no one wants to do anything differently a very popular APM ... especially in the practitioner world does not seem to be confirmed by the data ... some claim it is not even

testable

Depends on a bunch of strong assumptions (see next page) Stringent assumptions are the price to pay for the models elegance Many of the assumptions can be relaxed



Introduction

Assumptions underlying the CAPM The heroic assortment of assumptions (which even finance

professors do not believe!) Investors are in perfect competition with each other. No one investor

has the ability to influence prices with her/his trade All investors have the same one-period horizon. Could be a year, a

month or a week All investors are Markowitz efficient investors, who care only about the

mean and variance of their portfolios All investors have homogeneous expectations i.e. have identical

probability distributions regarding asset returns Markets are frictionless, meaning

Investors can borrow or lend any amount at the risk-free rate Investments are infinitely divisible one can buy or sell any fraction of any

asset. No taxes or transactions costs There is no inflation or change in interest rates, or inflation is fully

anticipated.

Capital markets are in equilibrium all securities are properly pricedtaking into account their risk



The market portfolio

The market portfolio: Example

All investors have identical expectations they all graph the sameefficient frontier, and end up with the same MVE portfolio

Consider a highly simplified world with only 2 investors and 2 assets Assets: Infosys and Reliance stocks; Let MVE portfolio be 70% I and

30% R Investors: Mr. Squeamish and Ms. Bungee Jumper with Rs. 1 M

apiece S has an optimal complete portfolio of 40% T-bills and 60% MVE.

Then, BJ must have -40% T-bills and 140% MVEInvestor I R T-bills Total

S 420,000 180,000 400,000 1,000,000BJ 980,000 420,000 -400,000 1,000,000

Total 1,400,000 600,000 0 2,000,000

Note 1: Net supply of risk-free assets is zero Note 2: Fraction of I in market=1.4/2.0=70%; Fraction of R=30%

Lesson: Under CAPM assumptions, the market portfolio is MVE



The market portfolio

The market portfolio: Details

Definition: The market portfolio is a portfolio of all risky assets in theeconomy, with each asset in proportion to its market value

Weight of asset i in mkt. portfolio =Market value of asset i

Total market value of all risky assets

All risky assets includes stocks, bonds, currencies, derivatives, real estate, commodities,human capital etc.

What would happen if there is new information suggesting thatInfosys stock is underpriced? You would revise your expected return forecast upward for I. (With no change in the risk of

I) your revised MVE portfolio should contain a bit more than 70% of I However, at the same time, everybody else is doing the identical thing, arriving at the same

revised MVE portfolio Supply of I shares is fixed; the only thing that can happen is that the stock price of I

increases The price of I adjusts to a new level such that the new market portfolio is again MVE This is the logic of equilibrium



The CAPM equation

The pricing model: Part 1 Since we now know that the market portfolio is MVE, we can draw aCapital Market Line (CML)

Portfolio standard deviation

Port

folio

expe

cted

retu

rn

Market

CML

All efficient portfolios must lie along the CML, i.e, they have to satisfy

E(re) = rf +(E(rm) rf

m

)e



The CAPM equation

The pricing model: Part 2

What about inefficient assets, e.g, individual assets?

The CAPM: E(ri) = rf + i[E(rm) rf

],

where i =Cov(ri,rm)

2m

In plain English, the expected return on any asset over and abovethe risk free rate must be determined by the sensitivity (beta) of thatasset w.r.t. the market portfolio, and the risk premium on the marketportfolio



The CAPM equation

CAPM: Proof Heres an intuitive proof:

Recall the useful property of the MVE portfolio that says marginalreward-to-risk ratios of all risky assets must be the same at the MVE

In the CAPM context, substituting the market portfolio instead of theMVE, we can write:

E(ri) rfCov(ri, rm)

=E(rj) rfCov(rj, rm)

Since this is true for any two risky assets i and j, it must also be truefor the market portfolio itself, which is a valid risky asset

Substituting j = m, we have:E(ri) rfCov(ri, rm)

=E(rm) rfCov(rm, rm)

=E(rm) rfVar(rm)

which reduces to

E(ri) rf = Cov(ri ,rm)Var(rm)[E(rm) rf

] E(ri) = rf + i

[E(rm) rf

]9/27 Lecture 5: The Capital Asset Pricing Model Ramana Sonti


Example

CAPM: Example

Lets try and write out the CAPM for our 3-asset example from theprevious lecture

Recall that

= 0.100.200.15

=

0.0049 0.0007 00.0007 0.01 0.01080 0.0108 0.0144

If you remember, the MVE (or tangency) portfolio of these three

assets is wX = 0.2274,wY = 1.7793,wZ = 1.0067 Although the math is right, this cannot be a valid market portfolio(why?)



Example

CAPM: Example ... contd.

Lets tweak the numbers to

= 0.100.200.15

and = 0.0049 0.0032 0.00130.0032 0.01 0.00540.0013 0.0054 0.0144

The MVE (or tangency) portfolio of these three assets iswX = 0.0546,wY = 0.8424,wZ = 0.1030. It is easy to calculateE(rm) = 0.1894 and m = 0.0922

Calculate the betas of the individual assets:Asset Cov(ri, rm) Var(rm) i E(ri) Ratiorf 0 0.0085 0.0 0.05 -

Market 0.0085 0.0085 1.0 0.1894 16.3895X 0.0031 0.0085 0.36 0.10 16.3895Y 0.0092 0.0085 1.08 0.20 16.3895Z 0.0061 0.0085 0.72 0.15 16.3895

Obviously, beta and expected return go hand in hand



The security market line


The Security Market Line (SML) is a graph of the CAPM

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

Portfolio beta

Port

folio

expe

cted

retu

rn

Market

SML

X

Y

Z

In equilibrium, all assets should lie on the SML




CML versus SML

0 0.02 0.04 0.06 0.08 0.1 0.120

0.05

0.1

0.15

0.2

0.25


Port

folio

expe

cted

retu

rn

Market

CML

X

Y

Z

H0.055, 0.842,0.103L

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

Portfolio betaPo

rtfo

lioex

pect

edre

turn

Market

SML

X

Y

Z

All assets lie on the SML. Only the risk-free asset and the market lie on the CML Investments with same expected return could have different standard deviations, but

they must all have the same beta, and vice versa Entire lines of points in the CML diagram plot at the same point in the SML

diagram



In words

What the CAPM is saying (in English) Covariance (with the market portfolio), not variance, is the

appropriate measure of risk The marginal contribution of an asset to a portfolios risk is entirely due

to its covariance with the market High beta assets have high expected returns

High beta assets produce high returns when the market return is high,i.e., when the marginal value of an extra $1 is low. Low beta assetshave the opposite property

When you really need the money (in bad states of the world), high betaassets fare poorly. So, to induce investors to hold these assets, theyhave to offer high expected returns

Beta alone determines expected returns Once beta is taken into account, nothing else matters for expected

returns This is the implication that has been incompatible with historical data

Diversifiable risk is not priced, i.e., if an investor is dumb enough notto diversify, he will not be compensated Remember the formula: Return on Idiocy (ROI) =0



In pictures

Graphical intuition

Say we look at all combinations of Asset Z and the market portfolio In equilibrium, this combinations curve lies entirely inside the risky

asset efficient frontier, and is tangent to the CML There is nothing to be gained (in terms of Sharpe ratio) by adding (subtracting) Z to (from)

the market The same condition should be true for every risky asset This is a state of equilibrium, where the marginal reward-to-risk ratio is equal across all

assets

Now, suppose you have private information that E(rZ) = 0.20, not0.15 as everyone else in the market expects it to be (every othernumber being unchanged) The combinations curve lies partly outside the risky asset efficient frontier, and is no

longer tangent to the CML There is some Sharpe ratio to be gained by adding some Z to the market, represented by a

new (your own private) CAL This is a state of disequilibrium, where the marginal reward-to-risk ratio is not equal across

all assets Of course, this information will eventually find its way to the market, and everyone will

revise their portfolios etc., and the equilibrium condition will be restored



In pictures

Graph: Equilibrium

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.05

0.1

0.15

0.2

0.25


Port

folio

expe

cted

retu

rn

Mkt

Z

CML



In pictures

Graph: Disequilibrium

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.05

0.1

0.15

0.2

0.25


Port

folio

expe

cted

retu

rn

MktZ

CML

New CAL



In pictures

Disequilibrium and the SML Another way to see this is to plot the SML

Asset Z is off the SML, which cannot happen in the CAPM equilibrium

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

Portfolio beta

Port

folio

expe

cted

retu

rn

Market

SML

X

YZ



In pictures

Supply, Demand and CAPM equilibrium

Lets examine in detail an example. A simple set-up: The risk free rate is 5%. In addition, there are two

risky assets, Bottom Up (BU) and Top Down (TD). Details given below:

Asset BU TDNo. of shares 5M 4M

E(P1) 40 38D1 6.40 3.80 40% 20% 0.20 -

Investors are Sigma (S), an actively managed fund with a corpus of220M, and an Index fund (I) with a corpus of 130M

What should the current market prices of the stocks BU and TD bein equilibrium?



In pictures

Determining the price of BU

Lets start with a pair of prices (pBU, pTD) = (38, 39) These prices imply expected returns[(E(rBU),E(rTD)] = (0.2211, 0.0718)

Using these and the variances and covariances, we havewBU,MVE = 0.8964;wTD,MVE = 0.1036

The market portfolio is wBU,m = 385385+394 = 0.5491;wTD,m = 0.4509 Sigmas demand for BU is then 0.8964 220/38 = 5.19 M shares,

while the Index funds demand for BU is 0.5491 130/38 = 1.88 Mshares. Total demand is therefore=5.19 + 1.88 = 7.07 M shares

Substituting different values of pBU (keeping pTD = 39), we cangenerate a demand curve for BU shares

Finally, intersecting the demand curve and the supply curve, weobtain an equilibrium price for BU shares, which is pBU = 40.85



In pictures

BU price determination: Graph

35 36 37 38 39 40 410

2

4

6

8

10

BU Price

BUQu

antity

HMsh

ares

L

PHBUL=40.85



In pictures

TD price determination: Graph

35 36 37 38 39 400

2

4

6

8

10

TD Price

TDQu

antity

HMsh

ares

L

PHTDL=38.14



In pictures

Full Equilibrium For these prices to be a full equilibrium, (not just a partial one) we

need the prices of BU and TD to be consistent with each other Assuming pTD = 39, the equilibrium price of BU is 40.85 Now, using price pBU = 40.85, find the equilibrium price of TD. Is it

39? No, it is 38.90. In other words, (40.85, 39) is an inconsistent pair ofprices

Keep iterating the trial-and-error process until we end up with aconsistent pair of prices

It turns out that the full equilibrium pair of prices is (39.2228, 38.4714)(Verify this!)

At this equilibrium pair of prices Demand functions for S and I are derived, respectively, from the MVE

and market portfolios Market for BU shares clears, given price of TD Market for TD shares clears, given the price of BU

At the full equilibrium prices, what is the relationship between theMVE and the market portfolios? See graphs on following pages



In pictures

Full equilibrium

0 2 4 6 8 1035

36

37

38

39

40

TD Quantity HM sharesL

TDPr

ice

P*HTDL=38.47

35 36 37 38 39 4035

36

37

38

39

40

BU price

TDpr

ice

P*HBUL=39.22,P*HTDL=38.47

0

2

4

6

8

1035 36 37 38 39 40

BUQu

antity

HMsh

ares

L

BU Price

P*HBUL=39.22



In pictures

Convergence to full equilibrium

5 10 15 20 25 30 350.54

0.545

0.55

0.555

0.56

0.565

Iteration

BUw

eig

ht

As the market converges to full equilibrium, the MVE (red) and market (blue)portfolios converge to the same point

This process of convergence is called tatonnement (French for, roughly trial anderror)

Imagine this onerous process with thousands of stock and millions of investors!25/27 Lecture 5: The Capital Asset Pricing Model Ramana Sonti


Beta as a regression coefficient

Why beta is called beta

Lets start with the following regression equation:ri rf = i + i

(rm rf

)+ i

This is not the CAPM. It is simply a regression of asset is excess returns on the marketsexcess returns

In English: Excess returns of asset i depend on the markets excess returns. There is anidiosyncratic shock beyond the effect of the market

Taking expectations on both sides, we have:E (ri) rf = i + i

[E (rm) rf

]+ E(i)

From standard regression assumptions, we can write E(i) = 0,i.e, the idiosyncratic part ofreturn is unpredictable

If the CAPM is true, it must be true that i = 0,i.e., i must be interpreted as the systematicout-performance (or under-performance) of asset is return relative to the CAPM

Taking the variance of both sides, we have:2i =

2i

2m +

2i + 2iCov (i, rm)

Another standard regression assumption is that Cov (i, rm) = 0, i.e., i is idiosyncratic,precisely because it is uncorrelated with the markets return

Therefore, 2i = 2i 2m + 2i ,i.e., total risk is the sum of systematic risk and idiosyncratic risk;the latter is not priced according to the CAPM



Beta as a regression coefficient

Final thoughts

Remember that CAPM betas are linear, i.e.,p = w11 + w22 + + wnn

There have been several extensions of the CAPM, relaxing the basicassumptions. BKMM provides some details I will not coverextensions here

The CAPM does not seem to fit historical data well; however, westudy it so that we develop intuition before proceeding to morecomplicated models

Remember that regardless of the validity of the CAPM,mean-variance optimization remains a useful model for portfolioallocation


PreliminariesIntroduction

Asset pricingThe market portfolioThe CAPM equationExampleThe security market line

IntuitionIn wordsIn pictures

ConclusionBeta as a regression coefficient

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