BKM Chapter 10 - APT and multi-factor Models Flashcards
Multi-factor model
An extention of the single factor model Ri = E(Ri) + ßiF + ei
We can write the multifactor model as:
Ri = E(Ri) + ßi,GDP *GDP + ßi,IR *IR + ei
The betas in the equation above, referred to as factor loadings or factor betas when used in a multifactor model, reflect the degree of sensitivity of the excess return to the factors.
Multifactor SML
Difference between multifactor model and multifactor SML.
Recall that the Security Market Line (SML) is an equation for the the expected return, of the following form: E(ri) = rf + ßi [E(rm) - rf] ⇒ E(Ri) = ßiRM
For obvious reasons, it would be relatively trivial to extend this to include multiple sources of risk premiums that result from sensitivity to multiple risk factors.
An important point to keep in mind is that the multifactor model and the multifactor SML are NOT the same thing. The former is a statement about what causes actual returns to deviate from their expected values; the latter is a statement, based on some underlying theory, of what the expected returns are. CAPM provided us with one theory for the expected returns and risk premiums. The APT will provide us with another theory.
The Fundamental APT Formula
The APT equation for the expected return on a risky portfolio of assets looks almost identical to a multifactor CAPM:
E(Ri) = ß1RP1 + ß2RP2 + …+ ßNRPN
The only differences between that formula and the CAPM formula for the expected return is that there may be more than one risk factor and none of the risk factors need to be the market index return.
Tow fundamental assumptions of APT
Assumption 1 — Returns are generated by a factor model. This means that actual returns for any portfolio consist of its expected return, plus some return due to sensitivity to unexpected shocks that are common across all portfolios, plus an idiosyncratic risk term that is uncorrelated with the shocks.
Assumption 2 — All assets can be bought or sold in any quantities at the market price, so in equilibrium arbitrage profits cannot be earned.
Law of One Price
Under the Law of One Price, two assets that are identical in all economically relevant aspects should have the same price. If not, an arbitrage opportunity exists. Investors would be able to simultaneously buy and sell identical assets at two different prices and earn a net profit equal to the difference as a risk free profit. This activity would put upward pressure on the low-priced version and downward pressure on the high-priced version, until their prices were indeed equal.
Examples of violation of law of one price
- “Siamese Twin Companies”
Two companies merged into one firm. The two original company, which continued to trade seperately, agreed to split profit from the join company on a 60/40 basis. One would expect the company received 60% of the profit should sell for 60/40=1.5 times the price of the other company. But the values of the two firms has departed considerably from this ratio for extended periods of time.
Explanation: ??
- Equity Carve-Outs
The case of 3Com decided to spin off its Palm division. Each 3com shareholder would receive 1.5 shares of Palm in the spinoff.
Once Palm share started trading, the share price of 3com should have been at least 1.5 times that of Palm. Instead, Palm shares at IPO actually sold for more than 3com shares.
Explanation: inability to sell Palm short
- Closed-End Funds
Closed-end funds often sell for substantial discount or premium from net asset value. One would expect the value of the funds to = the value of the shares it holds.
Explanation: The fund incurs expense that ultimatly are paid for by investor
Outline of Key Ideas behind APT
Porfolio return for a well-diversified portfolio is: Rp = E(Rp) + ßiF
Notice what would happen if we had two well diversified portfolios, A and B, with different expected excess returns but the same betas (that is, the same degree of systematic risk). Assume E(RA) = 10% and E(RB) = 9% but both had ßA=ßB=1.
Intuitively we can see that this probably couldn’t last for long. Since well diversified portfolios are free of any non-factor risk, these two portfolios have equal risk but unequal returns. Whatever the realized value for F turns out to be, Portfolio A would always have a higher actual return than Portfolio B. This would entice investors to want to own A and not B, pushing up the price of A and pushing down its expected return (and the reverse for B).
Consider this again but with respect to a transaction designed to exploit this arbitrage opportunity. Suppose we invested $1 million in A and sold (short) $1 million of B. The actual returns in dollars for each of these portfolios and for the aggregate portfolio would be as shown in the attached picture.
Risk Premiums: relationship between risk premium relative to their betas.
E(RA)/ßA = E(RB)/ßB
This is the crucial result! It shows that since we would be able to combine two well diversified portfolios together and take advantage of arbitrage opportunities, in equilibrium all well diversified portfolios must have the same risk premiums relative to their betas.
Relationship to CAPM
Consider a reference portfolio such as the market portfolio and assume that this market portfolio represents the systematic risk factor in a single factor model.
In this case, ßM=1 and therefore the previous relationship (which showed that excess returns relative to betas have to be equal across all well-diversified portfolios, can be written with the market portfolio parameters and then rearranged:
E(R<sub>A</sub>)/ß<sub>A</sub> = E(R<sub>M</sub>)/1
⇒ E(RA) = ßA * E(RM)
⇒ E(rA) = rf + ßA[E(rm)-rf] ⇒Same retult as CAPM without restrictive and unrealistic assumptions.
Factor Portfolios
Since we have so many assets to choose from, think about constructing factor portfolios that are well diversified portfolios that have factor sensitivities (betas), of 1.0 to only one factor and factor sensitivities of zero to all other factors.
Given the expected returns for these two factor portfolios and the risk free rate, we can easily determine the risk premiums for each of the factor portfolios. Then, given the factor sensitivities of any particular portfolio, we can easily determine that the total risk premium for this portfolio must equal the sum of its factor sensitivities times the associated factor portfolio risk premiums.
The Chen, Roll and Ross model and the Fama-French 3-Factor model are both presented in the BKM textbook as examples of multi-factor equilibrium return models.
Contrast the way in which the factors were formed in the two models.
The CRR Model:
- developed by identifying factors that appeared to be plausible sources of risk for which investors would demand risk premiums (e.g. decline in GDP)
- after identified factors, CRR then collected historical data and estimated how well differences in expected return across portfolios can be explained.
The Fama-French Three Factor Model:
It was developed to help understand why small stocks (measured in terms of the total market value of the firm’s equity) tended to earn higher returns than large stocks and why firms with high ratios of book value to market value also tend to earn higher returns than firms with low book to market ratios.
They started with a universe of risky assets and formed three portfolios (market portfolio, SMB portfolio and HML portfolio).
E(ri)-rf = bi[E(rM) - rf] + siE(SMB) + hiE(HML)
The FF model is assuming that the SMB and HML time series serve as proxies for some other unidentified risk factors. High book-to-market firms are likely to be in financial distress and small firms are more sensitive to business conditions, so it seems plausible that these indices are capturing risk factors that exist in the macroeconomy.
The different implications regarding the number of investors who are required to behave according to our model’s assumptions in order to ensure that equilibrium prices achieve those in our models between CAPM & APT.
Dominance arguments result in each investor making small changes to their holdings, depending on their degree of risk aversion. To influence the prices of mispriced assets, we would need a large number of investors to be trying to optimize the mean and variance of their portfolios so that the aggregate pressure on prices was sufficient to bring about equilibrium.
In contrast, arbitrage arguments require only one investor willing and able to identify the arbitrage opportunity and then mobilize large dollar amounts (recall that in APT we assumed unlimited ability to buy or sell any asset) in order to restore equilibrium.
Since the arbitrage argument relies on changes by fewer investors, the arbitrage argument is stronger.
