Investments - Chapter 10

Assume that both portfolios A and B are well diversified, that E(rA) = 12%, and E(rB) = 9%. If the economy has only one factor, and βA = 1.2, whereas βB = .8, what must be the risk-free rate?

(12 - rf) / 1.2 = (9 - rf) / .8 Rf = 3%

Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial production, IP, and the inflation rate, IR. IP is expected to be 3%, and IR 5%. A stock with a beta of 1 on IP and .5 on IR currently is expected to provide a rate of return of 12%. If industrial production actually grows by 5%, while the inflation rate turns out to be 8%, what is your revised estimate of the expected rate of return on the stock?

1. 12% + 1(5 - 3) + .5(8 - 5) 12 + 2 + 3 = 17% revised expected return

Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%. Portfolio Expected Return Beta X 16% 1.00 Y 12 0.25 In this situation you would conclude that portfolios X and Y:

1. 16% = 8% + 1 * (Rm - 8) Rm = 16% 12% = 8% + .25 * (Rm - 8) Rm = 24% The answer is B. offer an arbitrage opportunity

Jeffrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In comparing CAPM and APT, the consultant makes the following arguments: A. Both the CAPM and APT require a mean-variance efficient market portfolio. B. Neither the CAPM nor the APT assumes normally distributed security returns. C. The CAPM assumes that one specific factor explains security returns but APT does not. State whether each of the consultant's arguments is correct or incorrect. Indicate, for each incorrect argument, why the argument is incorrect.

1. False, only CAPM relies on mean-variance efficient market portfolios Correct Correct

3 propositions of APT

1. Security returns can be described by a factor model (single, multiple etc.) 2. There are sufficient securities to diversify away idiosyncratic risk (ep) 3. Well functioning security markets do not allow for the persistence of arbitrage opportunities

APT and CAPM Compared

APT applies to well diversified portfolios but due to various sizes, may have residual risk. Therefore, 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 mean variance assumption APT can be extended to multifactor models as it is anchored in observable portfolios. CAPM is not testable. APT may be insufficient to enforce when residual risk is high or not subject to pure arbitrage

MFM example: electric utility vs. airline

An electric utility would be less sensitive to changes in GDP. But more sensitive to changes in interest rates. An airline would be more sensitive to GDP changes and less sensitive to interest rate changes.

Arbitrage definition

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 quickly disappear.

Arbitrage Pricing Theory (APT)

Builds on ideas from single-index models like CAPM, APT allows us to further decompose risk (not just firm specific and market) and create "multi factor models" for estimates of risk and returns. APT is a combination of these models with no arbitrage conditions (no alpha opportunity) that lead to simple relationships between expected returns and risk.

An investor takes as large a position as possible when an equilibrium price relationship is violated. This is an example of:

C. Arbitrage activity.

According to the theory of arbitrage:

C. Positive alpha investment opportunities will quickly disappear.

Examples of single index models

CAPM is a single index model, everything centers around the relationship of a stock or portfolio relative to the market.

Returns on a single security come from two sources

Common macro-economic factor (e.g. GDP growth/shrink or interest rate rise/fall) and firm specific events (ei)

In contrast to the capital asset pricing model, arbitrage pricing theory:

D. Does not require the restrictive assumptions concerning the market portfolio.

The general arbitrage pricing theory (APT) differs from the single-factor capital asset pricing model (CAPM) because the APT:

D. Recognizes multiple systematic risk factors.

The feature of the general version of the arbitrage pricing theory (APT) that offers the greatest potential advantage over the simple CAPM is the:

D. Use of several factors instead of a single market index to explain the risk-return relationship.

A zero-investment portfolio with a positive alpha could arise if:

D. a risk-free arbitrage opportunity exists

What should be the Expected Rate of Return given the following? Northeast Airlines GDP Beta of 1.2 Interest rate Beta of -0.3 GDP risk premium for exposure of 1 unit of risk is 6% Interest Rate risk premium for exposure of 1 unit of risk is -7% Risk-free rate = 4%

E(r) = rf + (BiGDP*RiskPremiumGDP) + (BiIR*RiskPremiumIR) E(r) = 4% + (1.2*6%) + (-0.3*-7%) = 13.3% Where 1.2*6% = 7.2% risk premium for exposure to GDP Where -0.3*-7% = 2.1% risk premium for exposure to interest rate risk.

What does F&F help explain?

Fama and French conducted studies to test their model and found that when size and value factors are combined with the beta factor, they could then explain as much as 95% of the return in a diversified stock portfolio.

If the APT is to be a useful theory, the number of systematic factors in the economy must be small. Why?

From a statistical regression perspective, which is typically used to calculate beta, a model that can explain 99% of variations with 5 variables rather than 1,000 variables is a lot easier to explain to others. This is why the number of systematic factors in the economy must only be 5 or 6 because any more factors would make the regression not be explainable.

Where is the shortfall in the single index model thinking?

It assumes that stocks all have the same relative sensitivity to the risk factors that make up the economy. Lumping all systemic sources of risk into one variable such as return to the market may not be valid.

Simply, what is APT?

It is an approach that holds that expected return of an individual stock can be modeled as a linear function of various economic factors. It is viewed as an alternative to CAPM. CAPM uses market expected return while APT uses E(ri) and risk premium of a number of macro-economic factors. Identifying a security's theoretical price could lead to an arbitrage opportunity against market prices.

Multi Factor Models in practice

MFMs use more than one factor in addition to market return: GDP, interest rates, expected inflation, etc. Then, estimate a beta for each factor by using multiple regression.

The APT itself does not provide guidance concerning the factors that one might expect to determine risk premiums. How should researchers decide which factors to investigate? Why, for example, is industrial production a reasonable factor to test for a risk premium?

The APT factors must correlate with sources of uncertainty that are of concern to many investors. Researchers should investigate factors that correlate with uncertainty in areas of the economy and investment opportunities. GDP, the inflation rate, and interest rates are among the factors that can be expected to determine risk premiums. In particular, industrial production is a good indicator of changes in the business cycle. Thus, IP is a candidate for a factor that is highly correlated with uncertainties.

APT equation again

rP = E (rP) + bPF + eP F = some factor For a well-diversified portfolio: eP approaches zero Nonsystematic risk cancels out

F&F Five Factor Model

The new model adds: Profitability - companies reporting higher future earnings have higher returns in the stock market. Capital Investment - companies directing profit towards major growth projects are likely to experience losses in the stock market.

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 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 M = Market index

Suppose we have a stock, GE for example, that has a Beta sensitivity of 1.2 to "macro factors". Expectations are that the GDP will increase 4% but new information suggests that it will increase only 3% -- a negative 1% "shock".

Then GE stock should be impacted by 1.2 x -1% or -1.2% relative to the market. This macro surprise together with the firm specific surprise (e) will determine the stocks total-return-departure from the stock's return from the original expected value.

F&F Three Factor Model Debate

There is a lot of debate about whether the outperformance tendency is due to market efficiency or market inefficiency. In support of market efficiency, excess returns are due to: Excess risk of small cap and value stocks - higher WACC and business risk In support of inefficiency: Participants incorrectly pricing these securities

Example Multifactor Model Equation

Two Factor Equation: ri= E(ri ) + BiGDP GDP + BiIR IR + ei ri = Return for security i BiGDP= Factor sensitivity for GDP BiIR = Factor sensitivity for Interest Rate ei = Firm specific events Note: Both factors have zero expectations and only represent changes in the variable that have not already been anticipated.

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 Interesting and often used

Multi Factor models

models of security returns that respond to several systematic factors Allows us to model and quantify other sources of risk that can affect rates of return

Single Factor Model Equation

ri = E(ri) + BF + ei ri = return on security E(ri) = expected return of said security B = Factor sensitivity, beta, or loading F = Surprise in Macro factor (could be positive negative or zero) ei = Firm specific events Assuming no macro surprises, the return of a security will equal its previous expected return plus firm specific events only. But macro surprises can be also positive or negative.