1. ARBITRAGE PRICING THEORY
NMIMS/PTMBA FIN. DIV A & B
ARBITRAGE PRICING THEORY
Mahesh Parikh
August-09

2. Arbitrage Pricing Theory
Why & What is Need?
CAPM establishes relationship between risk and expected return for securities. (Where investors base their decision on comparision of return distribution of alternative portfolios.)
Equilibrium predictions of CAPM asserts that investors form portfolio consisting of riskless assets and market porttfolio; combination vary from investor to investor depending upon individual preference for expected risk and return.
Equilibrium expected return should be linear function of return on risk free assets (rf) plus market portfolio risk premium [E(rm) – rf] which is proportional to the beta of a security.

3. Arbitrage Pricing Theory
What was observed?
Small cap stocks outperformed large cap stocks on risk adjusted basis (finding by Mr. Banz)
e.g.
Tata Sponge Jindal Saw SBT LIC Housing
Equity
P/E
Market Price
( )
2. Low P/E stocks outperformed large P/E stocks; after adjustment for risk (finding by Mr. Banz)
P/E Market Price
TATA Steel
BPCL
SBI
Grasim

4. Value Stock Growth Stock
Value Stocks’ – Tata Chem, HLL (i.e. stocks with higher book to market value) generated larger returns than ‘Growth Stocks’ – JP Associates, GMR, RPL (stocks with low book-to-market value) on risk adjusted basis. (finding by Mr. Fama & French)
Value Stock Growth Stock
Low Earnings per share growth High Earnings per share growth
Low P/E Ratio High P/E Ratio
Low price to book-value ratio High price to book-value ratio
High Div. yield Low Div. yield
Betas tend to be less than one Beta tends to be >1
Out of Favour Popular
TATA Chem/HLL/Shipping Corp JP/GMR/RPL

5. Arbitrage Pricing Theory
What does this mean?
Are markets not efficient for long period of time?
Markets efficient but CAPM (a single factor model) does not capture risk adequately.
Look for Alternative Risk Return Model beyond CAPM
Hence
ARBITRAGE PRICING THEORY
(developed by Mr. Stephen Ross)

6. Arbitrage Pricing Model
Arbitrage Pricing Theory
Arbitrage Pricing Model
Say, there are 2 securities having same risk but Different Expected Return.
Investors to arbitrage / eliminate these differences.
Trading to buy Low Priced and sell High Priced to continue till both are at same expected return
Hence Ross developed a Model explaining
Asset’s Risk
Expected Return (required return) in terms of sensitivity to each of basic economic factors.

7. 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

8. Arbitrage Pricing Theory - APT
Three major assumptions:
Capital markets are perfectly competitive and no transaction cost. (This assumption makes possible the arbitraging of “ mispriced” securities, thus forcing an equilibrium price)
2. Investors always prefer more wealth to less wealth
3. Various factors give rise to returns on securities and the relation between the security return and that these factor is LINEAR. The stochastic process generating asset returns can be expressed as a linear function of a set of K risk factors or indexes 15

9. ARBITRAGE PRICING THEORY - APT
CAPM’s STRONG Assumption are not required
Single Period Investment horizon.
No Taxes.
Investors can freely borrow or lend at rf.
Investors select portfolio based on the expected mean and variance of return.

10. Arbitrage Pricing Theory
The return on any asset i, ri can be expressed as a linear Function of set of M factors/indexes
Ri = 10 + i1 I1 + i2 I2 + i3 I3 + ……. + iM IM + Ei
Ri = Return on asset I during a particular time period I = 1, 2 …n
10 = Expected Return on asset ‘i’, if all risk factors have zero charges (zero returns)
ij = Sensitivity of asset i’s return to the jth index i.e. Reaction in asset i’s returns to movement in a common factor. (Various systematic risk like interest rate risk etc.)
Ij = Value of Jth index j = 1,2, 3 …. M i.e. a common factor with zero mean that influences the returns of all assets i.e. Deviation of a systematic risk factor j from its expected value.
Ei = Random error term for asset i, with mean of zero and variance ϭ² E i.e. a unique effect on asset i’s return that by assumption is completely diversifiable in large portfolios and has a mean of zero
N = Nos. of assets
Note : 1] APT does not specify the risk factors I1, I2, …. Ik that have bearing on asset
return ] Error Term can be reduced to zero through appropriate diversification.

11. LINEAR RELATIONSHIP Arbitrage Pricing Theory
1] Depicts similar to Single Index Model except that under APT there are several indexes/ factors that may have influence on particular security’s index (Factor)
2] These indexes represent systematic influence on security’s return
e.g. Growth in GP
Changes in Inflation, Interest Rate Risk, Forex Rate Risk etc.
ONLY SYSTEMATIC FACTORS ARE IMPORTANT IN PRICING OF SECURITIES.
3] ij element in equation is analogus to i in Single Index Model. They represent the impact that a particular factor has on a particular security’s return e.g. a ij value of 3.0 means that for every 1% change in factor j, security i’s return is expected to change 3% or thrice the change in the value of Ij.

12. Arbitrage Pricing Theory (APT)
* ij determines how each asset reacts to the common factor
* Each asset may be affected by (though effects are not identical)
- Growth in GP
- Changes in expected inflation
- Changes in the market risk premium
- Changes in oil prices etc.
* Systematic risks (i.e. Non-diversifiable) factors are important in the pricing of securities.

13. Arbitrage Pricing Theory
APT
APT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away
The expected return on any well-diversified portfolio i (Ei) can be expressed as:

14. APT 0 = Expected Return on asset with zero systematic risk
Arbitrage Pricing Theory
APT
E(ri) = 0 + 1 i1 + 2 i2 + 3 i3 + …. + k ik
Where
E(ri) = Expected Return on asset I i.e. on well diversified portfolio
 = Expected Return on asset with zero systematic risk
 = The risk premium related to each of the common factor e.g. the risk premium
related to interest rate risk.
ij = the pricing relationship between the risk premium and asset i i.e. how responsive
asset i is to this common factor j i.e. sensitivity or beta coefficient for security i
that is associated with index j

15. TWO FACTOR MODEL
Assume, There are two factors which generate returns on assets “I”. APT Model becomes
E (ri) = λ0 + λ1bi1 + λ2bi2

16. ONE FACTOR MODEL Arbitrage Pricing Theory
Assume, there is only one factor which generates returns on asset i, APT Model boils down to
E(ri) = io + ij1
io = Risk Free Return or Zero Beta Security
Slope of arbitrage price line is  and intercept is io. The arbitrage price line shows the equilibrium relation between an asset’s systematic risk and expected return.

17. Risk Premium Factor %  is Response to Portfolio Response to Portfolio
Rate of Inflation 1 1 = bx by
Growth in GNP 2 = bx by
Zero Systematic 3 3 = 0.03
Risk Asset
E1 = 0 + 1bi1 + 2bi2
= (0.01) bi1 + (0.02) bi2
Ex = (0.01) (0.50) + (0.02) (1.50)
= = 6.5%
Ey = (0.01) bi1 + (0.02) bi2
= (0.01) (2) + (0.02) (1.75)
= = 8.5%

18. Example of Two Stocks and a Two-Factor Model
Arbitrage Pricing Theory
Example of Two Stocks and a Two-Factor Model
= changes in the rate of inflation. The risk premium related to this factor is 1 percent for every 1 percent change in the rate
= percent growth in real GNP. The average risk premium related to this factor is 2 percent for every 1 percent change in the rate
= the rate of return on a zero-systematic-risk asset (zero beta: boj=0) is 3 percent

19. Example of Two Stocks and a Two-Factor Model
Arbitrage Pricing Theory
Example of Two Stocks and a Two-Factor Model
= the response of portfolio X to changes in the rate of inflation is 0.50
= the response of portfolio Y to changes in the rate of inflation is 2.00
= the response of portfolio X to changes in the growth rate of real GNP is 1.50
= the response of portfolio Y to changes in the growth rate of real GNP is 1.75

20. Example of Two portfolios and a Two-Factor Model
Arbitrage Pricing Theory
Example of Two portfolios and a Two-Factor Model
= (0.01)bi (0.02)bi2
Ex = (0.01)(0.50) + (0.02)(1.50)
= = 6.5%
Ey = (0.01)(2.00) + (0.02)(1.75)
= = 8.5%

21. Arbitrage Mechanism Arbitrage Pricing Theory For same risks
Asset ‘U’ has higher return than Asset ‘O’
Asset ‘U’ is underpriced and assets ‘O’ is overpriced.
Sell asset ‘O” or go short on ‘O’
Buy asset ‘U’ or go long on ‘U’
Investor makes Riskless profit
Impact
Demand on asset ‘U’ goes up and supply of ‘O’ also goes up
Price of ‘U’ increases and price of ‘O’ decreases
Thus, Arbitrage goes on till prices are traded at same level.

22. ARBITRAGE Status as on 07-08-09 Arbitrage Pricing Theory Cash
Derivative
Lot Size
What is Needed
RIL
1995 (buy) Rs
2000 (sale) Rs
300
Funds
BHEL
2180 (buy) Rs
2200 (sale) Rs
EDUCOMP
3905(sale) Rs
3891 (buy) Rs 75
Shares
DLF
368 (buy) Rs
372 (sale) Rs
1600
L & T
1465 (buy) Rs
1470 (sale) Rs
400
SBI
1742 (buy) Rs
1750 (sale) Rs
264

23. Arbitrage Pricing Theory (APT)
Examples of multiple factors expected to have an impact on all assets:
Inflation
Growth in GNP
Major political upheavals
Changes in interest rates
And many more….
Contrast with CAPM insistence that only beta is relevant

24. Arbitrage Pricing Theory
APT and CAPM Compared
APT applies to well diversified portfolios and not necessarily to individual stocks
With APT it is possible for some individual stocks to be mispriced in equilibrium - not lie on the SML
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
CAPM
APT
Nature of relation
Linear
Number of risk factors 1 k
Factor risk premium
[E (RM) – Rf] λj
Factor risk sensitivity i
bij
Zero-beta return Rf λ0

25. IMPLICATIONS Arbitrage Pricing Theory
While the CAPM is still probably the best available estimate of risk for most corporate investment decision, managers must recognise that their stock prices may fluctuate more than what the standard theory suggests.
The market is usually smarter than the individual. Hence managers should weight the evidence of the market over the evidence of experts.
Markets function well when participants pursue diverse decision rules and their errors are independent. Markets, however, can become very fragile when participants display herd-like behaviour, imitating one another.
It may be futile to identify the cause of a crash or boom because in a non-linear system small things can cause large scale changes.
The discounted cash flow model provides an excellent framework for valuation. Indeed, it is the best model for figuring out the expectations embedded in stock prices.

26. Arbitrage Pricing Theory
APT – SUMMARY
Compared to the CAPM, the APT is much more general. It assumes that the return on any stock is linearly related to a set of systematic factors.
Ri = ai + bi1 I1 + bi2 I2 + ….. + bij Ij + ei
The equilibrium relationship according to the APT is
E(Ri) = 0 + i1 1 + bi2 2 + …. bij I
Some studies suggest that, in comparison to the CAPM, the APT explains stock returns better; other studies hardly find any difference between the two.

27. EXAMPLES Arbitrage Pricing Theory
1] Three portfolio with risk return characteristics as under –
Portfolio E(ri) Risk (i) A B U
Portfolio U offers returns disproportionate to risk Portfolios A & B returns are commensurate with the risk
Opportunity to profit
Riskless arbitrage conversion. How?
Construct Portfolio ‘F’ with same risk level as ‘U’
This can be done by investing equal amounts in Portfolio A & B
 of Portfolio F is X beta of A X beta of B
= 0.50 X X 1.20
= = 1.00
Return on Portfolio F is = X X 13.4
= 12%
This return is in line with the Portfolio’s risk.

28. EXAMPLES Arbitrage Pricing Theory
2] How arbitrage Portfolio can be constructed to benefit from differential return given by portfolio F & U having same level of risk.
Ans. Strategy is short sell ‘F’
Go Long on ‘U’
Portfolio Investment(Rs.) Return (Rs. ) Risk (i)
F ,
U ,
Arbitrage Portfolio
Arbitrage Portfolio ‘O’
Short sell F  Realise Rs. 10,000 i.e. –ve value of investment
Invest in U  Invest Rs. 10,000
Expected Return on F & 12% and 15%
Net Return on Arbitrage Rs. 300
Since eta of F & U are equal but opposite in value, Beta of arbitrage portfolio is zero.
Arbitrage Portfolio offers profit without any Net Investment or Risk.

29. EXAMPLES (two factor Arbitraging Process)
Arbitrage Pricing Theory
EXAMPLES (two factor Arbitraging Process)
Returns generated by two indices or factors as per following pricing relation
E(ri) = i i2
Ans
E(ri) Risk (i1) Risk (i2) A B O
A & B are correctly priced. Return ‘O’ is not commensurate with risk.
Scope of arbitrate
Construct Portfolio ‘F’ (same risk as ‘O’)
E(rf) = (0.50 X 15.4) + (0.50 X 16.6) = 16
Bf1 = (0.50 x 0.80) + (0.5 X 1.20) = 1.0
Bf2 = (0.50 X 1.20) + (0.5 X 0.80) = 1.0
Portfolio F has the same systematic risk as portfolio ‘O’
Arbitrage portfolio can be created by selling overprice i.e. ‘O’ and going long on ‘F’.

30. Example Three portfolios as under:- Portfolio E(ri) Bi1 Risk Bi2 Risk
A & B are correctly priced.
Return on O is not commensurate with underlying risk measured by Bil & Biz
We can construct a Portfolio F which also has same risk characteristic as portfolio O.
This portfolio f can be constructed by combining portfolios A & B in equal proportion.
i.e. E(rf) = (0.50)(15.4) + (0.50)(16.6)= = 16
βF1 = (0.50)(0.80) + (0.50)(1.20) = = 1
βF2 = (0.50)(1.20) + (0.50)(0.80) = = 1
Arbitrage Pricing Theory

31. Portfolio Investment Return Risk Bi1 Risk Bi2 O -10,000 -1400 -1 -1
Arbitrage Pricing Theory
Portfolio F has same systematic risk as portfolio O differs on higher return (i.e. 16% & not 14%). This offers an arbitrage opportunity.
Portfolio Investment Return Risk Bi Risk Bi2
O ,
F ,
________ ________ ____ ______

32. Examples
1. Consider the following data relating to beta and expected return of two portfolios:-
Portfolio Return % Beta A B
Given one factor arbitrage pricing model is E(ri) = βi1 show the arbitrage opportunity, if any, can be exploited.
Ans.: E (rA) = βi1 = 6 + 8(1) = 14
E (rB) = βi1 = 6 + 8(1) = 14
The return from B is than 14% expected. Hence, we should go long on B & short sell A suppose we have Rs. 1,00,000/-
Portfolio Investment Return Risk (Beta)
A ,00, ,
B ,00, ,
+2,000