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

2. 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

3. 9.1 MULTIFACTOR MODELS

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 9.2 ARBITRAGE PRICING THEORY

14. 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

15. 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

16. 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

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

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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 9.3 A MULTIFACTOR APT

29. 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

30. 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

31. 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%

32. 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

33. 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

34. 9.4 WHERE TO LOOK FOR FACTORS34

35. 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

36. 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

37. 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

38. 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

39. 9.5 THE MULTIFACOTOR CAPM AND THE APT39

40. 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)

41. 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

42. 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