finance week 9

IPCOR421 FinanceAlex Kane 1

CLASS NOTES

WEEK IX

READING ASSIGNMENT

BMA 8


Portfolio characteristics

• To make things easy, the following is excel-based• Universe of N securities• Expected excess returns (risk premium = E(r)–rf )

denoted by: R1, R2, . . . , RN• Covariance matrix (NxN) denoted by C (from Excel)• Given pf weights: x1, x2, . . . , xN• Pf risk premium: Rp = sumproduct (x, R)

• Pf SD: Border C by x to compute p• Pf Sharpe ratio:

Rp / p


The optimal risky pf• From the capital allocation decision we know that an

optimal pf will maximize the Sharpe ratio• We get the Solver to:

Target cell: Sp = Rp / p

to be maximized

By changing x

Subject to: Sum(x)=1• Solver returns the weights (x), the portfolio risk premium,

standard deviation and Sharpe ratio• This is the optimal portfolio


Note on data used in portfolio analysis

• We estimate the covariance matrix (C ) from past data. While we do so by using Excel’s ‘covariance’, professionals use improved techniques to extrapolate changes of covariance values over time

• In real work, we NEVER use past data to estimate risk premiums. These estimates must come from forecasts based on analysis. When we use averages from past data, it is for exercise purposes only !!


The efficient frontier of risky assets• The optimization program we used can be extended to

find the optimal portfolios for various required risk premiums -- we will not do this in this course. However, you need to understand the results

• The efficient frontier contains the best risk-return tradeoffs given the available universe of stocks

• All individual stocks and inefficient Pfs lie within the efficient frontier

• Other things equal, the lower the correlations among stocks, the better the frontier -- less risk for any level of mean

• This work earned a Nobel prize to H. Markowitz


Output of the optimization program

Pf Mean

Pf SD

Individual stocks

efficient frontier of risky assets

Global minimum variance Pf


The capital allocation line (CAL) and the efficient frontier of risky assets

• The optimal risky pf that we identified is at the tangency point of the capital allocation line with the efficient frontier of risky assets. This is why we call this pf ‘the tangency pf’ denoted by T

• This means that indeed, the capital allocation line will provide any investor with the the highest reward-to-variability ratio regardless of where they wish to be on the capital allocation line (CAL)


Choice of risky Pf with a risk-free asset

Pf mean excess return

Pf SDGA

T

G, A and T are efficient frontier Pfs. F is the risk-free asset. Combinations of F and G are inferior to F and A. Combinations of F and T are best. T is the tangency point of lines from F to the efficient frontier. Notice that at T, xF=0, we invest 100% in T (xT=1).

F

combinations of F with T

A

G


The probability distribution of excess returns

• It turns out that rates of return on portfolios are well approximated by a normal distribution

• Approximating returns by a normal distribution simplifies pf analysis

• The most important implication is that the distribution is symmetric (see next slide with mean of 13.1% and SD of 20.2% = historical stats) and hence the SD is a good measure of risk


QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.


The normal distribution• Location parameters: mean=median=mode

• The entire distribution, that is, the probability of any point, can be generated with knowledge of just the mean and variance

• In this case, the 5% quantile Q(0.05) known as VaR=Value at Risk, adds no new information

• With a symmetric distribution, variance represents risk since deviations on the down side are same as upside


Distribution of excess returns• The normal distribution is prominent in nature. For

example, as error function: A machine designed to fill 1 gallon of liquid at a time into a container, will have deviations (actual fill minus 1 gallon) that are distributed normally

• We find that a normal distribution is a reasonable approximation for stocks, and more so for Pfs

• A critical advantage of assuming normality comes from the following property: Sum of normal returns (Pfs of securities) are also normally distributed ! Very few distributions have this property


Equilibrium models in economics

• The standard model of equilibrium in economics is a preference model which works as follows

1. Set up the demands of individuals from preference (utility) functions and aggregate it. Equate aggregate demand to supply, and solve for the relevant variables

2. Analyze the solution to derive the properties of equilibrium


Capital-market assumptions• The supply side is made up by the risk-free rate and the

parameters (mean, covariance matrix) of all available securities -- all assets are traded (leaves out human capital, private business and public assets)

• We assume that all individuals agree on these parameters (strong assumption called ‘homogeneous expectations’)

• Individual demands are derived from the M-V efficiency program (Markowitz) with each individual choosing a position on the capital allocation line, based on risk aversion

• Equilibrium requires that aggregate demand for each security i equals supply of this i-th security


Additional assumptions• To solve (simple version) of the model we need to

make a few other assumptions– No taxes and no transaction costs (equalizes net returns

from a stock to all investors)– No individual has market power -- would result in

complicating strategic behavior– Individuals optimize with respect to next period and

ignore all future periods (in jargon, they are myopic)

• More complex versions of the model relax various assumptions. The form of the solution turns out to be the same


The Capital Asset Pricing Model (CAPM)• The model solution turns out surprisingly simple

• Its properties derive from one result:– The tangency Pf (which serves all investors) is the

market portfolio (M). That is, T=M by the CAPM– The risk premium (RM) is related to two variables:

(1) Average risk aversion in the economy (A) and the Variance of M, M^2

– RM = A* M^2– The tangency line to M is now the investment

opportunity set relevant to all investors, called the capital market line (CML) -- the CAL in equilibrium


Choice of risky Pf with a risk-free asset

Pf mean excess return

Pf SD

M

M is the tangency Pf. No security can lie above the CML (can be tested empirically). The weight of each stock in M equals the market value of the corporation divided by the total market value of all corporations

F

The Capital Market Line CML (CAL in equilibrium)


The useful result from the efficiency of M

• The efficiency of M results in the excess returns (actual, not expected) equation:

Ri = i*RM+i• “beta” is the slope coefficient in the regression of

the excess return on security i on the market Pf excess returns

• The intercept in these regressions is always zero, that is: i = 0 for all i

• To the extent that i ≠ 0 we say the security is mispriced


Unique (diversifiable) risk

• The major reason for diversification is reducing risk -- the proverbial eggs in many baskets

• The mean-beta equation makes it easy to decompose the variance of a security to two parts:

i^2= i^2*M^2 + i^2

• Since the i-s are uncorrelated, the unique risk of a well diversified Pf, P, will have negligible variance

• A pf of securities has two components: the systematic (p*RM) and unique (p). The variance of the latter falls to zero with diversification


Market (systematic, non-diversifiable) risk

• The risk of the systematic (non-diversifiable) component of security is i^2*M^2, is a square of the product of two variables: beta (i) and the market SD (M). The beta of a pf is the weighted average of the betas of the securities (p=xi*i)

• The market SD appears in any security’s market risk. Therefore, beta provides a complete measure of market risk of a security. For this reason, the systematic risk component of a security is also called “beta risk”


Implication to Corp Finance

• For well diversified Pfs, unique risk (p=xi*i; p^2= xi^2*i^2) is negligible, hence the entire risk is systematic

• The only risk that counts for an individual asset is measured by beta

• The beta of a project determines its required rate of return (or expected excess return)

• This simple, linear mean-beta relationship -- called the security market line (SML) is used in corporate finance


The security market line (SML)

Beta

beta=1.0

Mean excess return (risk premium)

SML

M

Good projects: positive alpha and NPV

Bad projects: negative alpha and NPV


Mispriced securities

• The CAPM says that all alphas must be zero

• If an analyst claims a security is underpriced (the average investor doesn’t know its true value), it is equivalent to claiming this security has a positive alpha

• Similarly, an overpriced security has a negative alpha

• The job of Pf managers is to invest in positive alpha stocks and sell negative alpha stocks


Joint normality and factor structure• When you assume a joint normal distribution of n

security returns, you actually assume what is called a single, linear factor structure.

• Joint normality says there is one factor (a RV) that drives all securities. Denote this factor by RM

• With a factor structure each of the n security excess returns, Ri, is linearly driven by RM according to the equation: Ri = i+i*RM+i , where E(i)=0, and the i-s are uncorrelated: (i,j)=0 for i≠j

• The uniqueness of each Ri comes from: i, i, (i)


The realism of the model• A multivariate joint normal distribution is one where a

number of factors (k) drive securities• In this model,

Ri = i+( i1*RM1 + … + ik*RMk ) + i• In this way, we allow for more macro (or industry) factors

to drive securities• Surely, this model with a modest k is very realistic• Yet the multi-factor model is conceptually identical to the

single-factor model. You learn one you understand them all

• Finally, as you will see, we can make the single factor model quite realistic. Proof: its wide use in industry and research


The common sense of the factor structure• Assuming a common factor (RM) and a unique

individual “disturbance,” i, in the equationRi = i+i*RM+i, so that:

E(Ri) = i+i*E(RM), makes a lot of sense

• We view RM (with mean=E(RM) and SD=M) driven by macro forces that affect all securities

• Each security has unique: (1) sensitivity (i) to the macro factor (RM), (2) addition, i, to the factor-driven mean, i*E(RM), (3) additional (unique) risk expressed by the zero-mean, i


Identifying the macro factor• It is important to identify the factor that drives securities,

since we need to know E(RM) and M• One can think of a number of candidates for the factor, e.g.,

GDP growth• But quite a few other variables represent important macro

forces, e.g., inflation, interest rates, default risk on corporate bonds, etc.

• One quite satisfying assumption is that the resultant of all macro forces is expressed in the return on a broad stock market index (portfolio), e.g., the S&P 500. This type of approach is used most often in industry and research

• Hence, RM stands for ‘return on the market Pf’


Practicality of the factor structure

• The factor equations allow for easy estimation of the coefficients for each security from a regression of Ri on RM

• In addition to estimates of E(RM) and M, “all” we need is to run n regressions to get the n sets: i, i, (i)

• In a regression equation y=a+bx+u, we have: y^2=b^2*x^2 + u^2.

Hence, for security i,i^2= i^2*M^2 + i^2


Another simple aspect of factor structure• Suppose we construct a portfolio of the n securities

with weights xi, i=1,…,n

• What are the characteristic of the portfolio? Simple!

• The parameters P, P, are weighted averages: P = xii ; P = xii

and its systematic risk is: P^2*M^2

• The unique risk of the Pf is: P^2=xi^2*i^2 We know that for large Pfs, P^2 will go to zero

• The total Pf risk is: P^2= P^2*M^2 + P^2, With most risk coming from the systematic factor


Well diversified Pfs and beta• Suppose a well diversified Pf A (with negligible unique

risk) has a beta of 1.2. Its total risk (SD) is therefore 1.2*M

• We like this portfolio, but we want less risk, say beta of 0.8. We can easily construct a Pf (B) from PfA with the desired beta

• This is achieved by investing a proportion xF in the risk-free asset (F, with beta of zero), and xA in A

• Recall that a Pf beta is the average of the betas of its components

• The resultant beta is: B=0*xF + A*xA=0.8• Therefore, xA=B/A=0.67 and since xF+xA=1, xF=0.33


Arbitrage

• Suppose two securities provide identical future CFs under any circumstances. This means both have same expected returns and SD

• If these securities are not equally priced, an investor can perform arbitrage: sell short the expensive (overpriced) security and use the proceeds to invest in the cheap (underpriced) security

• When exercised, such arbitrage opportunity will yield a sure profit from zero investment


Portfolios and arbitrage

• An arbitrage opportunity requires that there will be no risk involved. It would be rare to find such opportunity with individual securities, even when they are clearly mispriced

• But when a well diversified Pf is mispriced relative to another, both portfolios will have negligible unique risk and we can equalize the betas (and risk) by using the risk-free asset. Thus, mispriced portfolios provide a true arbitrage opportunity


Arbitrage Pricing Theory (APT)

• APT (developed by Stephen Ross, 1976) poses the question: “What can we say about security returns if all we know about the capital market is that there are no arbitrage opportunities?”

• Because arbitrage opportunities require that there be no risk, the theory deals with well diversified Pfs

• The theory predicts the risk return relationship between Pf risk premium (mean excess return) and SD=beta*M (unique risk is negligible)


The assumptions needed for the APT• These are same as before -- absent the assumption that

individual investors optimize their pfs– Homogenous expectations– No taxes and no transaction costs (equalizes

individual net returns from a stock)– No individual has market power -- would result in

complicating strategic behavior– Individuals demand for the next period ignores all

future periods (in jargon, the demand is myopic)– Ignore labor income


CAPM and APT• The CAPM and APT trade one assumption:

• CAPM– assumes individual investors optimize (M-V

efficiency)– it does not concern itself with arbitrage opportunities

• APT– assumes no arbitrage opportunities– ignores preferences and optimization by individual

investors


What is superior about the APT?• Preference models (CAPM) require that all

individuals (investors) behave in a rational way (M-V optimization)

• But most investors know little about M-V optimization. So we appeal to the idea that, on average, they behave as if they do

• The APT requires much less: if there are arbitrage opportunities, it is feasible for a few smart investors to raise huge amounts of capital to invest in them. These trades will move prices until the arbitrage opportunity disappears


What is superior about the CAPM?

• The CAPM is a theory that predicts the rate of return for any and all assets and portfolios

• The APT applies only to well diversified (large n) Pfs

• It is true that if the APT applies to all Pfs, it will approximately apply to individual assets. But this approximation can be off the mark by quite a bit

• Conclusion Both models are important• What does the APT say about risk premiums?


CAPM and APT arrive at same results

• Although the CAPM and APT are very different in the major assumption they use, they arrive at the same result

• Risk premiums of securities must be proportional to their betas

• Both theories validate the CML (mean-variance efficiency) and the SML (discount rates for projects)


Empirical validity of asset pricing theories

• The first thing to know is that the econometric difficulties of testing the models are daunting– The huge variances make it difficult to get significance levels

– We observe returns, we can never observe expectations, variances and covariances directly; we estimate them from time series

– But we know that, at least to some degree, these parameters constantly change, hence will be measured with large errors

– Regression analysis is severely biased with errors in variables

• Result: it is very likely to reject the hypothesis when it’s true (low power of tests), simply because of lacking stats

• These assets theories are indeed rejected when we use standard significant levels


A note of hypothesis testing

• A Bayesian approach says this– We enter an experiment with a prior belief– The experiment generates a conclusion from a sample, a

statistical statement– Our posterior (what we believe post-experiment) is a

weighted average of the prior and the experiment result– The weights are taken from the strength of the prior and

the experiment. With a strong prior and a low power experiment with different results, we may end up with a posterior that is still closer to the prior than to the experiment result


What about the inconclusive evidence on asset pricing theories?

• The compelling logic of the models seems to dominate

• In industry and government applications, as well as in court cases, practices imply the CAPM and APT are valid


Multifactor models• The inconclusive results drives us to constantly

look for improvements• One logical improvement is that if we add a

couple of factors we may get better results• So far, one currently popular three-factor model

adds two factors: firm size and the ratio of book value to market value

• This model seems to fit the data better• But, it is clearly not the last word. Better test

methods keep closing the gap between the single factor model and others


How does a three-factor model work?• It is identical in essence to the single factor

model. The equation is as in slide 25

• Ri =i+( i1*RM1 + i2*RM2 + i3*RM3 ) +i

• Instead of single variable regressions we run regressions with 3 independent variables that produce 3 betas for each stock, as well as alpha and residual variance (= i)

• We add 2 dimensions to the CML and SML

• Otherwise, there is no difference