APT AND MULTIFACTOR MODELS OF RISK AND RETURN CHAPTER 9 APT AND MULTIFACTOR MODELS OF RISK AND RETURN

Arbitrage Exploitation of security mispricing, risk-free profits can be earned No arbitrage condition, equilibrium market prices are rational in that they rule out arbitrage opportunities 2

9.1 MULTIFACTOR MODELS

Single Factor Model Returns on a security come from two sources Common macro-economic factor Firm specific events Focus directly on the ultimate sources of risk, such as risk assessment when measuring one’s exposures to particular sources of uncertainty Factors models are tools that allow us to describe and quantify the different factors that affect the rate of return on a security

Single Factor Model ri = Return for security I = Factor sensitivity or factor loading or factor beta F = Surprise in macro-economic factor (F could be positive, negative or zero) ei = Firm specific events F and ei have zero expected value, uncorrelated

Single Factor Model Example Suppose F is taken to be news about the state of the business cycle, measured by the unexpected percentage change in GDP, the consensus is that GDP will increase by 4% this year. Suppose that a stock’s beta value is 1.2, if GDP increases by only 3%, then the value of F=? F=-1%, representing a 1% disappointment in actual growth versus expected growth, resulting in the stock’s return 1.2% lower than previously expected

Multifactor Models Macro factor summarized by the market return arises from a number of sources, a more explicit representation of systematic risk allowing for the possibility that different stocks exhibit different sensitivities to its various components Use more than one factor in addition to market return Examples include gross domestic product, expected inflation, interest rates etc. Estimate a beta or factor loading for each factor using multiple regression. Multifactor models, useful in risk management applications, to measure exposure to various macroeconomic risks, and to construct portfolios to hedge those risks

Multifactor Models Two factor models GDP, Unanticipated growth in GDP, zero expectation IR, Unanticipated decline in interest rate, zero expectation Multifactor model: Description of the factors that affect the security returns Factor betas

Multifactor Models Example One regulated electric-power utility (U), one airline (A), compare their betas on GDP and IR Beta on GDP: U low, A high, positive Beta on IR: U high, A low, negative When a good news suggesting the economy will expand, GDP and IR will both increase, is the news good or bad ? For U, dominant sensitivity is to rates, bad For A, dominant sensitivity is to GDP, good One-factor model cannot capture differential responses to varying sources of macroeconomic uncertainty 9

Multifactor Models Expected rate of return=13.3% 1% increase in GDP beyond current expectations, the stock’s return will increase by 1%*1.2

Multifactor Security Market Line Multifactor model, a description of the factors that affect security returns, what determines E(r) in multifactor model Expected return on a security (CAPM) Compensation for bearing the macroeconomic risk Compensation for time value of money

Multifactor Security Market Line Multifactor Security Market Line for multifactor index model, risk premium is determined by exposure to each systematic risk factor and its risk premium

9.2 ARBITRAGE PRICING THEORY

Arbitrage Pricing Theory Stephen Ross, 1976, APT, link expected returns to risk Three key propositions Security returns can be described by a factor model Sufficient securities to diversify away idiosyncratic risk Well-functioning security markets do not allow for the persistence of arbitrage opportunities

Arbitrage Pricing Theory Arbitrage - 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 will quickly disappear

Arbitrage Law of One Price Arbitrage activity If two assets are equivalent in all economically relevant respects, then they should have the same market price Arbitrage activity If two portfolios are mispriced, the investor could buy the low-priced portfolio and sell the high- priced portfolio Market price will move up to rule out arbitrage opportunities Security prices should satisfy a no-arbitrage condition

Well-diversified portfolios Well-diversified portfolio, the firm- specific risk negligible, only systematic risk remain n-stock portfolio

Well-diversified portfolios The portfolio variance If equally-weighted portfolio , the nonsystematic variance N lager, the nonsystematic variance approaches zero, the effect of diversification

Well-diversified portfolios This is true for other than equally weighted one Well-diversified portfolio is one that is diversified over a large enough number of securities with each weight small enough that the nonsystematic variance is negligible, eP approaches zero For a well-diversified portfolio

Betas and Expected Returns Only systematic risk should command a risk premium in market equilibrium Well-diversified portfolios with equal betas must have equal expected returns in market equilibrium, or arbitrage opportunities exist Expected return on all well-diversified portfolio must lie on the straight line from the risk-free asset

Betas and Expected Returns Only systematic risk should command a risk premium in market equilibrium Solid line: plot the return of A with beta=1 for various realization of the systematic factor (Rm) Expected rate=10%,completely determined by Rm Subject to nonsystematic risk

B: E(r)=8%. beta=1; A:E(r)=10%. beta=1 Arbitrage opportunity exist, so A and B can’t coexist Long in A, Short in B Factor risk cancels out across the long and short positions, zero net investment get risk-free profit infinitely large scale until return discrepancy disappears well-diversified portfolios with equal betas must have equal expected return in market equilibrium, or arbitrage opportunities exist

A: beta=1,E(r)=10%; What about different betas C: beta=0.5,E(r)=6%; D: 50% A and 50% risk-free (4%) asset, beta=0.5*1+0.5*0=0.5, E(r)=7% C and D have same beta (0.5) different expected return arbitrage opportunity

An arbitrage opportunity Expected Return % 10 Risk-free rate=4 Risk premium 0.5 beta 1 6 7 C D A A/C/D, well-diversified portfolio, D : 50% A and 50% risk-free asset, C and D have same beta (0.5), different expected return, arbitrage opportunity

M, market index portfolio, on the line and beta=1 no-arbitrage condition to obtain an expected return- beta relationship identical to that of CAPM

Market index, expected return=10%;Risk-free rate=4% EXAMPLE Market index, expected return=10%;Risk-free rate=4% Suppose any deviation from market index return can serve as the systematic factor E, beta=2/3, expected return=4%+2/3(10%-4%)=8% If E’s expected return=9%, arbitrage opportunity Construct a portfolio F with same beta as E, 2/3 in M, 1/3 in T-bill Long E, short F

One-Factor SML M, market index portfolio, as a well-diversified portfolio, no- arbitrage condition to obtain an expected return-beta relationship identical to that of CAPM three assumptions: a factor model, sufficient number of securities to form a well-diversified portfolios, absence of arbitrage opportunities APT does not require that the benchmark portfolio in SML be the true market portfolio

9.3 A MULTIFACTOR APT

Multifactor APT Use of more than a single factor Two-factor model Several factors driven by the business cycle that might affect stock returns Exposure to any of these factors will affect a stock’s risk and its expected return Two-factor model Each factor has zero expected value, surprise Factor 1, departure of GDP growth from expectations Factor 2, unanticipated change in IR e, zero expected ,firm-specific component of unexpected return

A MULTIFACTOR APT Requires formation of factor portfolios Well-diversified Beta of 1 for one factor Beta of 0 for any other Or Tracking portfolio: the return on such portfolio track the evolution of particular sources of macroeconomic risk, but are uncorrelated with other sources of risk Factor portfolios will serve as the benchmark portfolios for a multifactor SML

A MULTIFACTOR APT Example: Suppose two factor Portfolio 1, 2, Risk-free rate=4% Consider a well-diversified portfolio A ,with beta on the two factors Multifactor APT states that the overall risk premium on portfolio A must equal the sum of the risk premiums required as compensation for each source of systematic risk Total risk premium on the portfolio A: Total return on the portfolio A: 9%+4%=13%

A MULTIFACTOR APT Factor Portfolio 1 and 2, factor exposures of any portfolio P are given by its and Consider a portfolio Q formed by investing in factor portfolios with weights in portfolio 1 in portfolio 2 in T-bills Return of portfolio Q

A MULTIFACTOR APT Suppose return on A is 12% (not 13%), then arbitrage opportunity Form a portfolio Q from the factor portfolios with same betas as A, with weights: 0.5 in factor 1 portfolio 0.75 in factor 2 portfolio -0.25 in T-bill Invest $1 in Q, and sell % in A, net investment is 0, but with positive riskless profit Q has same exposure as A to the two sources of risk, their expected return also ought to be equal

9.4 WHERE TO LOOK FOR FACTORS 34

Multifactor APT Two principles when specify a reasonable list of factors Limit ourselves to systematic factors with considerable ability to explain security returns Choose factors that seem likely to be important risk factors, demand meaningful risk premiums to bear exposure to those sources of risk

Multifactor APT Chen, Roll, Ross 1986 Chose a set of factors based on the ability of the factors to paint a broad picture of the macro-economy IP: % change in industrial production EI: % change in expected inflation UI: % change in unexpected inflation CG: excess return of long-term corporate bonds over long- term government bonds GB: excess return of long-term government bonds over T-bill Multidimensional SCL, multiple regression, residual variance of the regression estimates the firm-specific risk

Multifactor APT Fama, French, three-factor model Use firm characteristics that seem on empirical grounds to proxy for exposure to systematic risk SMB: return of a portfolio of small stocks in excess of the return on a portfolio of large stocks HML: return of a portfolio of stocks with high book-to- market ratio in excess of the return on a portfolio of stocks with low ratio Market index is expected to capture systematic risk

Fama, French, three-factor model Long-standing observations that firm size and book- to-market ratio predict deviations of average stock returns from levels with the CAPM High ratios of book-to-market value are more likely to be in financial distress, small stocks may be more sensitive to changes in business conditions The variables may capture sensitivity to risk-factors in macroeconomy

9.5 THE MULTIFACOTOR CAPM AND THE APT 39

APT and CAPM Compared Many of the same functions: give a benchmark for rate of return. APT highlight the crucial distinction between factor risk and diversifiable risk APT assumption: rational equilibrium in capital markets precludes arbitrage opportunities (not necessarily to individual stocks) APT yields expected return-beta relationship using a well-diversified portfolio (not a market portfolio)

APT and CAPM Compared APT applies to well diversified portfolios and not necessarily to individual stocks APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio APT can be extended to multifactor models

The Multifactor CAPM and the APM A multi-index CAPM Derived from a multi-period consideration of a stream of consumption will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge, from a particular hedging motive The APT is largely silent on where to look for priced sources of risk