1. Investments and Portfolio Management
WEEKS 24 & 28 1

2. Arbitrage Pricing Theory (APT)
Stephen Ross
APT 1976
Objectives :
Understand the reasons for the development of the APT
Calculate the expected return using the APT
Compare APT and CAPM models
Understand the empirical tests on the APT
Discuss multifactor models 2

3. Reasons for development of APT
CAPM is criticised because of
The many unrealistic assumptions
The difficulties in selecting a proxy for the market portfolio as a benchmark
An alternative pricing theory with fewer assumptions was developed: Arbitrage Pricing Theory (APT) 3

4. Weaknesses of CAPM Assumptions: Market anomalies 4
Investors are rational
Transaction costs don’t exist
Taxation and inflation don’t exist
Investors are homogenous (they believe the same outcomes will occur and use the same time period horizon)
There is one single risk factor/risk premium
Market anomalies 4

5. Arbitrage Pricing Theory
The expected return on a financial asset can be described by its relationship with several common risk factors
E(Ri)=λ0+ λ1bi1+ λ2bi2+…+ λnbin 5

6. Assumptions Three Major Assumptions:
Capital markets are perfectly competitive
Investors always prefer more wealth to less wealth with certainty
All unsystematic risk diversified away
In contrast to CAPM, APT doesn’t assume
There is only one systematic risk factor present
Stability of individual betas
No inflation or taxation 6 15

7. Arbitrage Pricing Theory
Unlike CAPM, which is a single-factor model, APT is a multifactor pricing model
BUT APT model does not specifically identify these risk factors
The multiple factors may include
Inflation (both changes in expected and unanticipated)
Growth in GNP
Major political upheavals
Unanticipated changes in interest rates
Unanticipated changes in default risk 7 15

8. The APT Model E(Ri)=λ0+ λ1bi1+ λ2bi2+…+ λnbin where:
λ0=the expected return on an asset with zero systematic risk
λn=the risk premium related to the nth common risk factor – e.g. inflation
bin=the pricing relationship between the risk premium and the asset; that is, how responsive asset i is to the nth common factor.
Factor betas 8 15

9. A Comparison with CAPM E(Ri)=Rf + βi[(Rm-Rf)]
In CAPM, the relationship is as follows:
E(Ri)=Rf + βi[(Rm-Rf)]
Comparing CAPM and APT
CAPM APT
Form of Equation Linear Linear / planar
Number of Risk Factors (≥ 1)
Factor Risk Premium Rm – Rf λn
Factor Risk Sensitivity βi bin
“Zero-Beta” Return Rf λ0
(risk-free return) 9 15

10. Using the APT Selecting Risk Factors
The primary challenge with using the APT in security valuation is identifying the risk factors
For this illustration, assume that there are two common factors
First risk factor: Unanticipated changes in the rate of inflation
Second risk factor: Unexpected changes in the growth rate of real GDP 10

11. Determining the Risk Premium
Assume the following have been identified:
The risk premium related to unanticipated changes in the rate of inflation is 2% for every 1% change in the rate
Let us define this as λ1=0.02
λ2: The risk premium related to changes in the growth rate of real GDP is 3% for every 1% change in the rate of growth
Let us define this as λ2=0.03 11

12. Determining the Risk Premium
Assume the following have been identified:
The rate of return on a zero-systematic risk asset (i.e. zero beta) is 4%
Let us define this as λ0= 0.04 12

13. Determining the Sensitivities for Asset X and Asset Y
Assume the following have been identified:
The response of asset x to changes in the inflation factor is 0.50, whereas the response of asset y to changes in the inflation factor is 2.00
Defined as bx1 = 0.50, by1 = 2.00
The response of asset x to changes in the GDP factor is 1.50, whereas the response of asset y to changes in the GDP factor is 1.75
Defined as bx2 = 1.50, by2 = 1.75
These are the factor betas for the individual assets 13

14. Estimating the Expected Return
APT Model
= (0.02)bi1 + (0.03)bi2
Asset X
E(Rx) = (0.02)(0.50) + (0.03)(1.50)
= = 9.5%
Asset Y
E(Ry) = (0.02)(2.00) + (0.03)(1.75)
= = 13.25% 14

15. Security Valuation with the APT:
Three stocks (A, B, C) and two common systematic risk factors have the following relationship
Assume λ0=0
E(RA)=(0.8) λ1 + (0.9) λ2
E(RB)=(-0.2) λ1 + (1.3) λ2
E(RC)=(1.8) λ1 + (0.5) λ2 15

16. Security Valuation with the APT:
If λ1=4% and λ2=5%, then it is easy to calculate the expected returns for the stocks:
E(RA)=(0.8) (0.04) + (0.9) (0.05) = 7.7%
E(RB)=(-0.2) (0.04) + (1.3) (0.05) = 5.7%
E(RC)=(1.8) (0.04)+ (0.5) (0.05) = 9.7% 16

17. Calculating Expected Prices ….One Year Later
Assume that all three stocks are currently priced at £35 and do not pay a dividend
Estimate their prices one year later as…..
E(PA)=£35(1.077)=£37.70
E(PB)=£35(1.057)=£37.00
E(PC)=£35(1.097)=£38.40 17

18. Example 18

19. currently overpriced securities with some new assets.
James Ltd, an investment management company, is considering rebalancing its asset portfolio and replacing some
currently overpriced securities with some new assets.
As an investment analyst, you have been presented with the following information regarding two shares, A and B, and two common risk factors, 1 and 2:
Security
bi1
bi2 A
4.1
2.8
18.4% B
3.5
1.4
22.5%
E(r)
The securities are expected to reach a price level of 42 pence and 57 pence for A and B respectively in one-years time.
Also, neither stock is expected to pay any dividend over the next year.
The current risk-free rate is rf = 6% 19

20. (a) Using the Arbitrage Pricing Theory model of Steven Ross, calculate the levels of the factor risk premia which are consistent with the above reported values for the two securities.
Using the APT model, we can structure the above scenario as a system of simultaneous equations. Therefore we have:
𝐸 𝑅 𝐴 = 𝜆 𝜆 2 =18.4%
𝐸 𝑅 𝐴 = 𝜆 𝜆 2 =22.5% 20

21. i 𝜆 𝜆 2 =18.4%
ii 𝜆 𝜆 2 =22.5% 21

22. 22

23. (b) What should be the price of each stock today in order to be consistent with the expected return levels reported in the above table? 23

24. Arbitrage
A trading strategy designed to generate guaranteed profits, with no capital commitment or risk to the trader. E.g. A simultaneous purchase and sale of the same security in different markets at different prices at the same time. 24

25. Arbitrage Opportunity
If one “knows” actual future prices for these stocks are likely to be different from those previously estimated, then these stocks are either undervalued or overvalued
This is a similar concept to that shown in CAPM (above SML = underpriced, below SML = overpriced)
Arbitrage trading will take place until arbitrage opportunity disappears (see example over next few slides) 25

26. Arbitrage Opportunity
Assume the actual prices of stocks are estimated to be:
A £37.20, B £37.80 and C £38.50 one year later:
Are each of the stocks over or under-valued at the current price of £35 per share?
E(PA)= £37.70 (overvalued)
E(PB)= £37.00 (undervalued)
E(PC)= £38.40 (slightly undervalued) 26

27. Arbitrage Opportunity
Short sell shares of A, to purchase shares of B and C.
E.g. short sell 2 shares of A at £70 to buy 1 share each of B and C (total £70) giving 0 net investment i.e. No capital commitment.
Net exposure to risk factors:
Investment proportions
Weighted exposure from..
Factor 1 bi1
Factor 2 bi2
Stock A
-2 x 0.8
-2 x 0.9
Stock B
1 x -0.2
1 x 1.3
Stock C
1 x 1.8
1 x 0.5
Net risk exposure
No risk 27

28. Arbitrage Opportunity
Going back to the prices that we estimated (on slide 20):
A £37.20, B £37.80 and C £38.50 one year later
The benefit obtained from the arbitrage opportunity would be:
2 x (£ £37.20) + (£ £35.00) + (£ £35.00) = £1.90
Therefore a profit has been made from a position of no capital commitment and no risk 28

29. Arbitrage Opportunity
Eventually, arbitrage trading will lead to new current prices:
E(PA)=£37.20 / =£34.54
E(PB)=£37.80 / 1.057=£35.76
E(PC)=£38.50 / 1.097=£35.10 29

30. Empirical Tests of the APT
Roll-Ross Study (1980)
The methodology used in the study is as follows
Used time-series data to estimate the expected returns and the factor coefficients
Use these estimates to test the pricing conclusion implied by the APT
Specifically testing whether the same factors influenced different securities and whether λ0 was the same
The authors concluded that the evidence generally supported the APT, but acknowledged that their tests were not conclusive
Investors should structure their portfolio to minimise the impact of the risk factors identified 30

31. Empirical Tests of the APT
The APT and Stock Market Anomalies
Small-firm Effect
Chen (1981): Supported the APT model over CAPM. Results have been questioned. Awaiting further studies!
January Anomaly
Gultekin and Gultekin: APT not better than CAPM
Burmeister and McElroy: Effect not captured by APT, but still better than CAPM 31

32. Problems of APT Main problems of APT:
As neither the number and type of risk factors are set, they must be identified in an ad-hoc manner:
Identifying the number of risk factors
Identifying what those risk factors are
Multifactor models, with an identified set of risk influences, have been developed to overcome this problem 32

33. Multifactor Models & Risk Estimation
Macroeconomic-Based Risk Factor Models: Risk factors are viewed as macroeconomic in nature (e.g. inflation, GDP)
Microeconomic-Based Risk Factor Models: Risk factors are viewed at a microeconomic level by focusing on relevant characteristics of the securities themselves (e.g. size of firm, financial ratios) 33

34. Macroeconomic-Based Risk Factor Models
Burmeister, Roll, and Ross (1994) analyzed the predictive ability of a model based on the following set of macroeconomic factors.
Confidence risk
Time horizon risk
Inflation risk
Business cycle risk
Market timing risk 34

35. Microeconomic-Based Risk Factor Models
Fama and French (1993)
Where:
ai is the expected return if all indices have a value of zero
ei is a random error term (or residual)
And risk factors are:
SMB (i.e. small minus big) is the return to a portfolio of small capitalization stocks less the return to a portfolio of large capitalization stocks
HML (i.e. high minus low) is the return to a portfolio of stocks with high ratios of book-to-market values less the return to a portfolio of low book-to-market value stocks it t i mt e
HML b
SMB
RFR R a + - = 3 2 1 ) ( 35

36. Microeconomic-Based Risk Factor Models
Carhart (1997), based on the Fama and French three factor model, developed a four-factor model by including a risk factor that accounts for the tendency for firms with positive past return to produce positive future return (momentum effect) 36

37. Summary
APT model has fewer assumptions than the CAPM and does not specifically require the designation of a market portfolio.
The APT says that expected security returns are related in a linear fashion to multiple common risk factors.
Unfortunately, the theory does not offer guidance as to how many factors exist or what their identifies might be.
Given the difficulty of identifying these on an ad-hoc nature, multifactor models of risk and return attempt to bridge the gap between the practice and theory by specifying a set of variables.
These models can be Macroeconomic or microeconomic in nature 37

38. Reading The slides for this week’s lecture have been based on:
Reilly, F. K. and Brown, K. C. (2012) Analysis of Investments and Management of Portfolios, 10th edition, Cengage Learning – Chapter 9
Other relevant reading could include:
Elton, E. J, Gruber, M.J et al. (2011) Investments and Portfolio Management, 8th edition, John Wiley & sons – Chapter 16
Jones, C. P., (2014) Investments, Principles and Concepts, John Wiley & sons – Chapter 9 38