BM61002: Corporate Finance

Portfolio Theory and CAPM

Abhijeet Chandra, PhD

Vinod Gupta School of Management

Indian Institute of Technology Kharagpur

January 27, 2015

Ecient Frontier

Figure: Ecient Frontier

CAPM: Assumptions

Everyone in the market agrees about E (ri )'s, i 's, and ij 's.

Everybody does the same mean-variance optimization, holding

same ecient frontier portfolio(s).

The market is ecient.

There exists only one risk-free asset.

No market frictions: no taxes, no transaction costs.

Unlimited shorting is allowed.

What does it mean?

Every investor puts their money into two types of assets:

The risk-free asset that pays an interest rate rf , andA single portfolio of risky assets, let's call it the tangent portfolio.

All investors hold the risky assets in same proportions.

That is, they hold the same risky portfolio and the same

tangent portfolio.

Thus the tangent portfolio, T, is the market portfolio, M.

CAPM

The (marginal) return-to-risk ratio of risky asset i in aportfolio p is:

Marginal_returnMarginal_risk =

rirf(ip/p)

Since T = M, the return-to-risk ratio of all risky assets mustbe the same:

rA1rf(A1M/M )

= rA2rf(A2M/M ) = ... =rAirf

(AiM/M )= rMrfM

If we re-write this as follows:

r irf(iM/M )

= rMrfMleading to:

r i rf = iM2M (rM rf ) = iM(rM rf )or:

r i = rf + iM(rM rf ) [CAPM]

CAPM: Features

r i = rf + iM(rM rf ) [CAPM] iM is the measure of the systematic risk (non-diversiablepart of risk) of the asset i .

rM rf is the market risk premium, that is, the premium perunit of systematic risk.

The risk premium on asset i equals the amount of itssystematic risk multiplied with the premium per unit of the

risk.

Security Market Line

The relation between an asset's premium and its market beta is

called the Security Market Line (SML).

Figure: Security Market Line

Investor's dream shattered: Given an asset's beta/expected return,

we can determine its expected return/beta.

Decomposing Sources of Variations in Returns

r i rf = i + iM(rM rf ) + iwhere

E (i ) = 0Covar[rM , i ] = 0

Three components:

Alpha (i ): CAPM assumes alpha () to be zero for all assets.

Beta (iM): connotes an asset's systematic risk; assets withhigher betas are more sensitive to the market.

Sigma (i ): measures its non-systematic risk; uncorrelatedwith systematic risk.

CAPM: Issues

A number of unrealistic underlying assumptions;

Investors had identical preferences, had the same information,

and hold the same portfolio (the market)

Dicult to measure (universal) market return; the market

portfolio (comprising every single available asset) is

unobservable.

Eugene Fama is quoted as saying:

"...beta as the sole variable explaining returns on stocks is

dead." Eric N. Berg, New York Times, 18 Feb., 1992.

The relation between average return and beta is completely

at. Michael Peltz, Institutional Investor, June 1992

Asset returns can not be merely a function of single factor,

such as beta.

Flattening Beta-return Relationship

1

The relation between beta and actual average return has been much weaker

since the mid-1960s.

Risk-return relationship has been appended in recent times. FT, 2014.

1

Source: Beta and return, F. Black, Journal of Portfolio Management, v20, 1993

Flat Beta in Indian Market

2

2

Source: Betting against beta in the Indian market, Agarwalla et al., IIMA WP 2014-07-01

Other Factors Aecting Returns

3

Stocks with small market cap have outperformed large-cap stocks.

Stocks with low ratios of market-to-book value have outperformed

stocks with high ratios.

3

Source: The cross section of expected stock returns, Fama and French, Jo Finance, v47, 1992

Companies possessing similar characteristics may, in a given

month, show returns that are dierent from the other companies.

The pattern of diering shows up as the factor relation."

4

4

Source: Extra market components of covariance in security markets, Rogenberg et al., JFQA, 1974

Alternative Pricing Models

Conditional CAPM: Do away with static risk factor, and use all

available information

5

.

E [R ti |t1] E [r tf |t1] = t1iM (E [R tM |t1] E [r tf |t1])where

t1iM = Cov [Rti ,R

tM |t1]

Var [RtM|t1] t1 represents all available information.BAPM: Uses SDF as pricing kernel (Shefrin, 2008).

Investors with hetergeneous beliefs

where one investor predicts (price) continuation and the other investor

(price) predicts reversal.

Models heterogeneity through the discounted probability.

The discounted probability is a relative wealth-weighted convex

combination of the individual investors' probabilities.

Family of CAPM: Intertemporal CAPM, Consumption-based CAPM, International

CAPM, Production-based CAPM, and so on...

5

The conditional CAPM and the cross-section..., Jagannathan and Wang, Jo Finance, 1996

Multiple Risk Factors: An example

Only two sources of risk for a company, Synfosis:

Technological risk: Upgradation in software used (RS),

Macroeconomic risk: Change in exchange rate (RE )

RM = RS + RECAPM: rSyn rf = Syn(rM rf )

Based on CAPM, the beta of Synfosis will be:

Syn = Cov(RSyn,RM )Var(RM ) =Cov(RSyn,RS )+Cov(RSyn,RE )

Var(RM )

Both covariances should be priced same (as per CAPM).

Possible??

What if there are n unique/unrelated risk factors?

Fama-French 3-factor model:

Beta (), Size (SMB), Book-to-price ratio (HML)

rSyn = rf + Syn(rM rf ) + bsSMB + bvHML +

Arbitrage Pricing Theory (APT)

Multi-factor model of asset pricing.

Based on law of one price and no arbitrage.

Makes much more sense as we don't assume:

There is only one risk factor;

Everyone is optimizing same mean-variance frontier;

Statistical model (unlike an equilibrium CAPM)

Yet, we assume:

Finite expected return and risk associated with each asset.

No market frictions: taxes, costs, etc.

There exists no arbitrage opportunity.

Someone out there forms well-diversied portfolio(s) for

relative assessment.

APT...

The market-risk model can be extended to multiple risk factors:

r i rf = bi11 + bi22 + ...+ bijjwhere,

bij is factor loadings for each of the risk factos. j is a set of risk premia.

The expected return of an asset i is a linear function of theasset's sensitivities (measured in terms of factor loadings, bij)to the j risk factors.

Risk in APT = f (Risk factor, Factor loadings, Factor riskpremium, Arbitrage)

APT: Example

6

Suppose that we know that IBM is going to pay a liquidating

dividend in exactly one year, and this is the only payment that

it will make.

However, that dividend that IBM pays is uncertain its size

depends on how well the economy is doing.

If the economy is in an expansion, then IBM will pay a

dividend of $140.

However, if the economy is in a recession, then its dividend will

be only $100.

If the two states are equally likely, the expected cash ow from

IBM is E (CF IBM1

) = $120

Note that, consistent with our denition of uncertainty, we

know exactly what will happen to IBM in each scenario, but

we don't know which scenario will occur.

6

D. Papanikolaou, Kellogg-NWU.

APT Exapmple ...

IBM

Boom Payo (Pr=0.5) 140

Bust Payo (Pr=0.5) 100

E(CF1

) 120

Price at t0

100

Discount Rate 20%

Assuming that the price of IBM is $100, we see that investors in

this economy are applying a discount rate of 20% to IBM's

expected cash-ows.

The way they come up with the price of IBM is to take the expected

cash ow at time 1 from IBM, E (CF IBM1

) = $120, and discount thiscash-ow back to the present at the "appropriate" discount rate,

which is apparently 20%.

Equivalently, we can say that the expected return of IBM is 20%.

APT Example...

Now let's consider a second security, DELL which, like IBM, will pay

a liquidating dividend in one year, and which has only two possible

cash ows, depending on whether the economy booms or goes into

a recession over the next year.

IBM DELL

Boom Payo (Pr=0.5) 140 160

Bust Payo (Pr=0.5) 100 80

E(CF1

) 120 120

Price at t0

100 ?

Discount Rate 20% ?

Now let's consider how investors will "price DELL.

Note the IBM and DELL have the same expected cash-ow, so one

might guess that a reasonable price for DELL might also be $100.

APT Example...

However, even though the expected cash-ows for DELL are the

same, we see that the pattern of cash ows across the two states

are probably worse for DELL:

DELL's payo is lower in the bust/recession state, which is when we

are more likely to need the money. It only does better when things

are good.

The recession is when the rest of our portfolio is more likely to do poorly,

and when we are more likely to lose our job, and our consulting income is

likely to be lower. We need the cash more in a recession.

In the boom/expansion, our portfolio will probably do better; we're more

likely to have a good job; we'll probably have more consulting income.

This means that if DELL were the same price as IBM, we would buy

IBM.

APT Example...

Therefore, to induce investors to buy all of the outstanding DELL

shares, it will have to be the case that DELL's price is lower than

$100.

Also, note that the three are statements are equivalent:

The price investors will pay for DELL will be less than $100, even though

DELL's expected cash ows are the same as IBM's.

The discount rate investors will apply to DELL's cash ows will be higher

than the 20% applied to IBM's cash ows.

The expected return investors will require from DELL will be higher than

the IBM's expected return of 20%.

Let's assume that investors are only willing to buy up all of DELL's

shares if the price of DELL is $90, or, equivalently, that the discount

rate that they will apply to DELL is 33.33%:

E (RDELL) = E(CFDELL1

)PDELLPDELL

= 1209090

= 3090

= 0.3333

APT Example...

IBM DELL

Boom Payo (Pr=0.5) 140 160

Bust Payo (Pr=0.5) 100 80

E(CF1

) 120 120

Price at t0

$100 $90

Discount Rate 20% 33.33%

The way to think about expected returns is that this is something

that investors determine.

After looking at the pattern of cash-ows from any investment, and

deciding whether they like or dislike this pattern, investors

determine what rate they will discount these cash ows at (to

determine the price.)

This rate is what we then call the "expected return".

Since investors accurately calculate the expected cash-ows, the

average return they realize on the investment will be the discount

rate.

APT: Implications (1)

Now let's see how all of this relates to the APT equations:

Calculating the business cycle factor fBC in the two states:

First, assume that we use the NBER (National Bureau of

Economic Research) business cycle indicator to construct our

factor.

The indicator is one (at the end of the next year) if the economy is

in an expansion, and zero if the economy is in a recession.

However, remember that for the factor, we need the

unexpected component of the business cycle.

Assuming there is a 50%/50% chance that we will be in an

expansion/recession, the expected value of the indicator is 0.5.

This means that the business-cycle factor has a value of

0.5 = 1 0.5 if the economy booms, and 0.5 = 0 0.5 if theeconomy goes bust.

APT: Implications (2)

Calculating the factor loadings for the two securities:

The way of doing this is to run a time-series regression of the

returns of IBM on the factor. Let's do this rst for IBM:

rIBM,t = E(rIBM) + bIBM,BC fBC ,t + eIBM,twhere E (rIBM) is the intercept and bIBM,BC is the slopecoecient.

Here, we have only two points, so we can t a line exactly:

0.40 = E (rIBM) + bIBM,BC 0.5 (Boom)

0.00 = E (rIBM) + bIBM,BC 0.5 (Bust)

Solving these gives E (rIBM) = 0.20 (which we already knew)and bIBM,BC = 0.4.

Similar calculations for DELL gives E (rDELL) = 0.3333 andbDELL,BC = 0.8889

APT: Implications (3)

Interpreting the factor loadings:

The factor loadings bIBM,BC and bDELL,BC tell us how muchrisk IBM and DELL have.

Risk means that the security moves up or down when the

factor (the business cycle) moves up and down.

Based on the way that we have characterized uncertainty, the

expected return and the factor loading bi,BC of each securitytell us everything that we need to know to calculate all of the

payo, per dollar, in every state of nature.

This will be true for every well-diversied portfolio, in

every factor model.

One way of thinking about this is that the expected

return tells us what the reward is, and the b's tell us whatthe risk of the security is.

APT Implications (4)

Calculating the factor risk premia ('s):The Factor Model tells us nothing about why investors are

discounting the cash ows from the dierent securities at

dierent rates.

To determine how investors view the risks associated with each

of the risks in the economy, we have to evaluate the APT

Pricing Equation.

Now that we have the factor loadings, we can determine the

factor risk-premia (or 's), by regressing expected returns Erion factor loadings bik .

Since there is only one factor and two stocks:

E(rIBM) = 0 + BC bIBM,BC E(rDELL) = 0 + BC bDELL,BCto which the solutions are 0

= 0.0909 and BC = 0.2727.

APT Implications (5)

Interpreting the risk premia ('s):DELL is considered to be a riskier security than IBM, in the

sense that investors discount DELL's cashows at a higher rate

than IBM.

This is reected in DELL's higher loading on the BC factor.

The BC is a measure of how much more investors discount astock as a result of having one extra unit of risk relating to the

BC factor.

0

tells you how high a return investors require if a security

has no risk.

APT: Summary

Using the set of securities that we used to calculated the 's for thepricing equations, we can always construct a portfolio p with any setof factor loadings (bp,k).

The pricing equation then tells us what the expected return (or

discount rate) for the portfolio of securities is.

Alternatively, this equation tells us, if we nd a (well-diversied)

portfolio with certain factor loadings, what discount-rate that

portfolio must have for there to be no arbitrage.

Finally, if we add a third security to the mix and calculate its factor

loadings, the APT will tell us what its expected return must be in

order to avoid arbitrage opportunities.

Essentially what we doing is pricing securities relative to other

securities, in much the same way we are pricing derivatives relative

to the underlying security.

APT Application: Finding Arbitrage

Arbitrage arises when the price of risk diers across securities.

1. If investors are pricing risk inconsistently across securities, than,

assuming these securities are well diversied, arbitrage will be

possible.

2. To see this, let's extend the example to include a risk-free asset

which we can buy or sell at a rate of 5%.

That is the rate for borrowing or lending is 5%.

3. Since 0

= 0.0909, we know that we can combine DELL and IBM insuch a way that we can create a synthetic risk-free asset with a

return of 9.09%.

4. So we borrow money at 5% (by selling the risk-free asset) and lend

money at 9.09%

APT Application: Buidling the arbitrage portfolio

To determine how much of IBM and DELL we buy/sell, we

solve the equation for the weights on IBM and DELL in a

portfolio which is risk-free:

wIBM bIBM,BC + (1 wIBM) bDELL,BC = bp,BC = 0or

wIBM = bDELL,BCbDELL,BCbIBM,BC = 1.8182This means that, to create an arbitrage portfolio, we can

invest $1.8182 in IBM, short $0.8182 worth of DELL, and

short $1 worth of the risk-free asset.

This portfolio requires zero initial investment and has a

positive payo in all states.

Building Arbitrage Portfolio...

To verify that this works, lets look at the payo to the

IBM/DELL risk-free portfolio in the two states:

Payo (boom) =

wIBM (140/100) + (1 wIBM) (160/90) = 1.0909Payo (bust) =

wIBM (100/100) + (1 wIBM) (80/90) = 1.0909This means that the payo from the zero-investment portfolio

is $0.0409 = 1.0909 - 1.05 in both the boom and bust states.

So this portfolio (in this simple economy) is risk free.

Of course, we can scale this up as much as we like. For $1

million investment in the long and short portfolios, we would

get a risk-free payo of $40,900 (assuming prices did not move

with our trades).

APT Application: Interpreting the arbitrage

What is going on here is that investors are pricing risk inconsistently

across securities.

With any pair of securities, we can calculate 0

and BC , however,each pair of securities will give you a dierent set of 's:

This means that, to nd the arbitrage, we could have:

1. Calculated the 's using any pair of securities, and then2. Calculated the expected return (or discount rate) for the third security,

3. Bought the high return and sold the low return.

APT Application

Here, we have used a simple example to understand how

investors price risky securities.

The idea behind the APT is that investors require dierent

rates of return from dierent securities, depending on the

riskiness of the securities.

If, however, risk is priced inconsistently across securities, then

there will be arbitrage opportunities.

Arbitrageurs will take advantage of these arbitrage

opportunities until prices are pushed back into line, risk is

priced consistently across securities, and arbitrages disappear.

What we have learnt so far...

Risk, Return, and Portfolio Theory

Risk-return relationship

Ecient frontier

Introducing risk-free asset

Mean-variance set

SML and CML

Capital Asset Pricing Model (CAPM)

Alternative asset pricing models

Arbitrage Pricing Theory