P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com
BODIE, CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND RETURN DESCRIBE THE INPUTS, INCLUDING FACTOR BETAS, TO A MULTI FACTOR MODEL.... 3 CALCULATE THE EXPECTED RETURN [AND VARIANCE] OF AN ASSET USING A SINGLE-FACTOR AND A MULTI-FACTOR MODEL.... 5 INTERPRET THE LAW OF ONE PRICE AND ASSESS WHETHER AN ARBITRAGE SITUATION EXISTS USING A MULTI-FACTOR MODEL.... 7 2
Bodie, Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return Describe the inputs, including factor betas, to a multi factor model. Calculate the expected return of an asset using a single-factor and a multi-factor model. Interpret the Law of One Price and assess whether an arbitrage situation exists using a multi-factor model. Describe properties of well-diversified portfolios and explain the impact of diversification on the residual risk of a portfolio. Explain how to construct a portfolio to hedge exposure to multiple factors. Compare the Arbitrage Pricing Theory (APT), the CAPM, and the Fama-French threefactor model, and evaluate the underlying assumptions of each. Describe the inputs, including factor betas, to a multi factor model. The single-factor model parses asset returns into systematic versus firm-specific risks. The single-factor model is given by: r E( r ) F e i i i i In this model, E(r i) is the expected return on stock (i), (F) is the deviation of the common factor from its expected value β(i) is the sensitivity of the firm (i) to the common factor (F), and e(i) is the firm-specific disturbance Note the following about the single-factor model: If the macro factor has a value of zero (0) in any particular period (i.e., no macro surprises), the return on the security will equal its previously expected value, E(r i), plus the effect of firm-specific events only. The nonsystematic components of returns, the e(i)s, are assumed to be uncorrelated among themselves and uncorrelated with the factor (F) The advantage of this model is its decomposition of returns into systematic and firm-specific components; but its disadvantage is that it confines systematic risk to only a single factor. 3
A multi-factor model should be better In reality, the systematic (or macro) factor in the single-factor model arises from several sources; e.g., business cycle uncertainty, interest rates, inflation. Its market return reflects both macro factors and the average sensitivity of firms to those factors. When we estimate a singleindex regression, therefore, we implicitly impose an (incorrect) assumption that each stock has the same relative sensitivity to each risk factor. If stocks actually differ in their betas relative to the various macroeconomic factors, then lumping all systematic sources of risk into one variable such as the return on the market index will ignore the nuances that better explain individualstock returns. Models that allow for several factors multifactor models in theory provide better descriptions of security returns. Apart from their use in building models of equilibrium security pricing, multifactor models are useful in risk management applications. These models give us a simple way to measure our exposure to various macroeconomic risks, and construct portfolios to hedge those risks. Generalizing to two factors Suppose the two most important macroeconomic sources of risk are: Business cycle uncertainty, as measured by unanticipated growth in GDP; and Unexpected changes in interest rates, denoted by IR. The return on any stock will respond both to sources of macro risk and to its own firm-specific influences. The two-factor model describing the rate of return on stock (i) is given by: r E( r ) GDP IR e i i igdp iir i Please note the following: The two macro factors on the right-hand side of the equation comprise the systematic factors in the economy. As in the single-factor model, both of these macro factors have zero expectation: they represent changes in these variables that have not already been anticipated. The coefficients of each factor measure the sensitivity of share returns to that factor. This is why the coefficients are sometimes called factor sensitivities, factor loadings, or, equivalently, factor betas. An increase in interest rates is bad news for most firms, so we would expect interest rate betas generally to be negative. As before, e(i) reflects firm-specific influences. 4
Generalizing to multiple factors The two-factor model is given by r E( r ) F F e i i i1 1 i 2 2 i This model can easily be generalized to multiple (several) factors but still in a linear model: r E( r ) F F... F e i i i1 1 i 2 2 in N i As before: E(r i) is the expected return on stock (i), F(n) is the deviation of the common factor from its expected value β(in) is the sensitivity of the firm (i) to the common factor F(n), and e(i) is the firm-specific disturbance Calculate the expected return [and variance] of an asset using a single-factor and a multi-factor model. Expected return of factor model The multifactor model is no more than a description of the factors that affect security returns. There is no theory in the equation. The obvious question left unanswered by a multi-factor model is, where does E(r) comes from? In other words, what determines a security s expected rate of return? This is where we need a theoretical model of equilibrium security returns. We can generalize from the familiar single-factor model: the security market line (SML) of the capital asset pricing model (CAPM). The CAPM asserts that securities will be priced to give investors an expected return comprised of two components: The risk-free rate, r(f), which is compensation for the time value of money, and A risk premium, RP, determined by multiplying a benchmark risk premium (i.e., the risk premium offered by the market portfolio) times the relative measure of risk, beta: E r r r r f M f How does this single-factor view generalize once we recognize the presence of multiple sources of systematic risk? Not surprisingly, a multifactor index model gives rise to a multifactor security market line in which the risk premium is determined by the exposure to each systematic risk factor, and by a risk premium associated with each of those factors. 5
For example, in a two-factor economy we might conclude that the expected rate of return on a security would be the sum of: 1. The risk-free rate of return. 2. The sensitivity to GDP risk (GDP beta) times the risk premium for bearing GDP risk. 3. The sensitivity to interest rate risk (i.e., the interest rate beta) times the risk premium for bearing interest rate risk. This assertion can be expresses by a two-factor security market line: f GDP GDP IR IR E r r RP RP Where β(gdp) denotes the sensitivity of the security return to unexpected changes in GDP growth, and RP(GDP) is the risk premium associated with one unit of GDP exposure; i.e., the exposure corresponding to a GDP beta of 1.0. The general K-factor model can be given by: E r r F K f k k k 1 Bodie s Concept Check 10.1: Suppose that the macro factor, F, is taken to be news about the state of the business cycle, measured by the unexpected percentage change in gross domestic product (GDP), and that the consensus is that GDP will increase by 4% this year. Suppose also that a stock s β value is 1.20. Further, suppose you currently expect the stock to earn a 10% rate of return. Then some macroeconomic news suggests that GDP growth will come in at 5% instead of 4%. How will you revise your estimate of the stock s expected rate of return? Answer: The GDP beta is 1.2 and GDP growth is 1% better than previously expected. So you will increase your forecast for the stock return by 1.2 * 1.0% = 1.2%. The revised forecast is for an 11.2% return. Bodie s Example 10.2: Suppose we estimate this two-factor model for Northeast Airlines: R = 0.133 + 1.2(GDP) 0.3(IR) + e This tells us that the expected excess rate of return for Northeast is 13.3%, but that for every percentage point increase in GDP beyond current expectations, the return on Northeast shares increases on average by 1.2%, while for every unanticipated percentage point that interest rates increases, Northeast s shares fall on average by.3%. 6
Variance of factor model If the expected return on of a portfolio is given by: r E r F e P P P P Then, under assumptions, the portfolio variance s parses into a systematic (the beta term) and non-systematic variances: 2 2 2 2 P P F P Why is this? e This is due to the variance properties. The return function is similar to Y = A + B*X + Z, where (A) and (B) are constants; and (X) and (Z) are the random variables Assuming independence between (X) and (Z), which drops out the covariance term, the variance (Y) = variance (BX) + variance (Z) = B^2*variance(X) + variance(z) A key assumption here is that the nonsystematic component, (e) or (Z) is uncorrelated with the common factor, (F) or (X). Interpret the Law of One Price and assess whether an arbitrage situation exists using a multi-factor model. An arbitrage is a riskless profit An arbitrage opportunity arises when an investor can earn riskless profits without making a net investment. A trivial example of an arbitrage opportunity would arise if shares of a stock sold for different prices on two different exchanges. For example, suppose IBM sold for $95 on the NYSE but only $93 on NASDAQ. Then you could buy the shares on NASDAQ and simultaneously sell them on the NYSE, clearing a riskless profit of $2 per share without tying up any of your own capital. The Law of One Price states that if two assets are equivalent in all economically relevant respects, then they should have the same market price. The Law of One Price is (should be) enforced by arbitrageurs: if they observe a violation of the law, they will engage in arbitrage activity simultaneously buying the asset where it is cheap and selling where it is expensive. In the process, they will bid up the price where it is low and force it down where it is high until the arbitrage opportunity is eliminated. The idea that market prices will move to rule out arbitrage opportunities is perhaps the most fundamental concept in capital market theory. Violation of this restriction would indicate the grossest form of market irrationality. 7
The critical property of a risk-free arbitrage portfolio is that any investor, regardless of risk aversion or wealth, will want to take an infinite position in it. Because those large positions will quickly force prices up or down until the opportunity vanishes, security prices should satisfy a no-arbitrage condition; i.e., a condition that rules out the existence of arbitrage opportunities. Arbitrage versus risk-return dominance There is an important difference between arbitrage and risk return dominance arguments in support of equilibrium price relationships. A dominance argument holds that when an equilibrium price relationship is violated, many investors will make limited portfolio changes, depending on their degree of risk aversion. Aggregation of these limited portfolio changes is required to create a large volume of buying and selling, which in turn restores equilibrium prices. By contrast, when arbitrage opportunities exist, each investor wants to take as large a position as possible; hence it will not take many investors to bring about the price pressures necessary to restore equilibrium. Therefore, implications for prices derived from no-arbitrage arguments are stronger than implications derived from a risk return dominance argument. The CAPM is an example of a dominance argument, implying that all investors hold mean-variance efficient portfolios. If a security is mispriced, then investors will tilt their portfolios toward the underpriced and away from the overpriced securities. Pressure on equilibrium prices results from many investors shifting their portfolios, each by a relatively small dollar amount. The assumption that a large number of investors are mean-variance sensitive is critical. In contrast, the implication of a no-arbitrage condition is that a few investors who identify an arbitrage opportunity will mobilize large dollar amounts and quickly restore equilibrium. Practitioners often use the terms arbitrage and arbitrageurs more loosely than our strict definition. Arbitrageur often refers to a professional searching for mispriced securities in specific areas such as merger-target stocks, rather than to one who seeks strict (risk-free) arbitrage opportunities. Such activity is sometimes called risk arbitrage to distinguish it from pure arbitrage. 8