This ppt is direct make by rajesh katiyar from IBM csjmu kanpur under gidance of sruti mam...i am so glad upload in slideshare

1. BRANCH-FINANCE AND CONTROL 4TH
SEM.
C.S.J.M. UNIVERSITY KANPUR
INSTITUTE OF BUSINESS MANAGMENT
TERM PAPER OF PORTFOLIO MANAGEMENT
TOPIC-ARBITRAGE PRICING THEORY & FACTOR MODEL
UNDER GUIDANCE &
SUBMITTED TO-
ASS. PROF.-
MS. SHRUTI MISHRA
SUBMITTED BY-
GROUP NO-7
PRIYA SINGH
KUMUD SAGAR
PARUL SACHAN
RAJESHBABU KATIYAR
(BATCH=>2017-19)

2. ARBITRAGE PRICING THEORY
The Arbitrage Pricing Theory (APT) is a theory of asset pricing that holds that an
asset’s returns can be forecast using the linear relationship between the asset’s expected return
and a number of macroeconomic factors that affect the asset’s risk. This theory was created in
1976 by the economist, Stephen Ross. Arbitrage pricing theory offers analysts and investors a
multi-factor pricing model for securities based on the relationship between a financial asset’s
expected return and its risks.
The theory aims to pinpoint the fair market price of a security that may be temporarily
mispriced. The theory assumes that market action is less than always perfectly efficient, and
therefore occasionally results in assets being mispriced – either overvalued or undervalued – for
a brief period of time. However, market action should eventually correctthe situation, moving
price back to its fair market value. To an arbitrageur, temporarily mispriced securities represent
a short-term opportunity to profit virtually risk-free.
The APT is a more flexible and complex alternative to the Capital Asset Pricing Model
(CAPM). The theory provides investors and analysts with the opportunity to customize their
research. However, it is more difficult to apply, as it takes a considerable amount of time to
determine all the various risk factors that may influence the price of an asset.
The general idea behind APT is that two things can explain the expected return on a financial
asset: 1) macroeconomic/security-specific influences and 2) the asset's sensitivity to those
influences. This relationship takes the form of the linear regression formula above.
There are an infinite number of security-specific influences for any given security
including inflation, productionmeasures, investor confidence, exchange rates, market indices or
changes in interest rates. It is up to the analyst to decide which influences are relevant to the
asset being analyzed.
Once the analyst derives the asset's expected rate of return from the APT model, he or she can
determine what the "correct" price of the asset should be by plugging the rate into a
discounted cashflow model.
Note that APT can be applied to portfolios as well as individual securities. After all, a portfolio
can have exposures and sensitivities to certain kinds of risk factors as well.
Why it Matters:
The APT was a revolutionary model because it allows the user to adapt the model to the
security being analyzed. And as with other pricing models, it helps the user decide whether a
security is undervalued or overvalued and so he or she can profit from this information. APT is
also very useful for building portfolios becauseit allows managers to test whether their
portfolios are exposed to certain factors.

3. APT may be more customizable than CAPM, but it is also more difficult to apply because
determining which factors influence a stockor portfolio takes a considerable amount of
research. It can be virtually impossible to detect every influential factor much less determine
how sensitive the security is to a particular factor. But getting "closeenough" is often good
enough; in fact studies find that four or five factors will usually explain most of a security's
return: surprises in inflation, GNP, investor confidence and shifts in the yield curve.
Assumptions in the Arbitrage Pricing Theory
The Arbitrage Pricing Theory operates with a pricing model that factors in many sources of risk
and uncertainty. Unlike the Capital Asset Pricing Model (CAPM) which only takes into account
the single factor of the risk level of the overall market, the APT model looks at several
macroeconomic factors that, according to the theory, determine the risk and return of the
specific asset.
These factors provide risk premiums for investors to consider because the factors carry
the systematic riskthat cannot be eliminated by diversification of an investment portfolio.
The APT suggests that investors will diversify their portfolios, but that they will also choose
their own individual profile of risk and returns based on the premiums and sensitivity of the
macroeconomic risk factors. Risk-taking investors will exploit the differences in expected and
real return on the asset by using arbitrage.
Principal of Arbitrage
ARBITRAGE
Arbitrage is the practice of taking positive expected return from overvalued or undervalued
securities in the inefficient market without any incremental risk and zero additional investments.
Mechanics
In the APT context, arbitrage consists of trading in two assets – with at least one being
mispriced. The arbitrageur sells the asset which is relatively too expensive and uses the
proceeds to buy one which is relatively too cheap.
Under the APT, an asset is mispriced if its current price diverges from the price predicted by the
model. The asset price today should equal the sum of all future cashflows discounted at the
APT rate, where the expected return of the asset is a linear function of various factors, and
sensitivity to changes in each factor is represented by a factor-specific beta coefficient.
A correctly priced assethere may be in fact a synthetic asset - a portfolio consisting of other
correctly priced assets. This portfolio has the same exposure to each of the macroeconomic

4. factors as the mispriced asset. The arbitrageur creates the portfolio by identifying x correctly
priced assets (one per factor plus one) and then weighting the assets such that portfolio beta per
factor is the same as for the mispriced asset.
When the investor is long the asset and short the portfolio (or vice versa) he has created a
position which has a positive expected return (the difference between asset return and portfolio
return) and which has a net-zero exposure to any macroeconomic factor and is therefore risk
free (other than for firm specific risk). The arbitrageur is thus in a position to make a risk-free
profit:
Where today's price is too low:
The implication is that at the end of the
period the portfolio would have
appreciated at the rate implied by the
APT, whereas the mispriced asset would
have appreciated at more than this rate.
The arbitrageur could therefore:
Today:
1 short sell the portfolio
2 buy the mispriced asset with the
proceeds.
At the end of the period:
1 sell the mispriced asset
2 use the proceeds to buy back
the portfolio
3 pocket the difference.
Where today's price is too high:
The implication is that at the end of the
period the portfolio would have
appreciated at the rate implied by the
APT, whereas the mispriced asset would
have appreciated at less than this rate.
The arbitrageur could therefore:
Today:
1 short sell the mispriced asset
2 buy the portfolio with the proceeds.
At the end of the period:
1 sell the portfolio
2 use the proceeds to buy back the
mispriced asset
3 pocketthe difference.
Arbitrage in the APT
The APT suggests that the returns on assets follow a linear pattern. An investor can leverage
deviations in returns from the linear pattern using the arbitrage strategy. Arbitrage is a practice
of the simultaneous purchase and sale of an asset, taking advantage of slight pricing
discrepancies to lock in a risk-free profit for the trade.
However, the APT’s conceptof arbitrage is different from the classic meaning of the term. In
the APT, arbitrage is not a risk-free operation – but it does offer a high probability of success.
What the arbitrage pricing theory offers traders is a model for determining the theoretical fair
market value of an asset. Having determined that value, traders then look for slight deviations
from the fair market price, and trade accordingly. For example, if the fair market value of stock
A is determined, using the APT pricing model, to be $13, but the market price briefly drops to

5. $11, then a trader would buy the stock, based on the belief that further market price action will
quickly “correct”the market price back to the $13 a share level.
Mathematical Model of the APT
The Arbitrage Pricing Theory can be expressed as a mathematical model:
Where:
E(rj) – Expected return on asset
rf – Risk-free rate
ßn – Sensitivity of the asset price to macroeconomic factor n
RPn – Risk premium associated with factor n
The beta coefficients in the APT are estimated by using linear regression. In general, historical
securities returns are regressed on the factor to estimate its beta.
Factors in the APT
The APT provides analysts and investors with a high degree of flexibility regarding the factors
that can be applied to the model. The factors, and how many of them are used to analyze a given
security, are subjective choices made by the individual market analyst or investor. Therefore,
two different investors using the APT to analyze the same security may have widely varying
results when it comes to their actual trading. Even among the most devoted advocates of the
theory, there is no consensus agreement of finance professionals and academics on which
factors are best for predicting returns on securities.
However, Ross suggests that there are some specific macroeconomic factors that have proven
most reliable as price predictors. These include suddenchanges in inflation and GNP, corporate
bond premiums, and shifts in the yield curve. Some other commonly used factors in the APT
are GDP, commodities prices, market indices levels, and currency exchange rates.
Although a bit complex to work with, and something that requires time and practice to become
adept at using, the Arbitrage Pricing Theory is an analytical toolthat investors can use to
evaluate their portfolio holdings from a basic value investing perspective, looking to identify
securities that may be temporarily mispriced, well below or above their fair market value.

6. To learn more about evaluating securities in regard to risk vs. return, you may wish to take a
look at the following related resources from CFI.
A two-factor version of the arbitrage pricing theory formula is as follows:
r = E(r) + B1F1 + B2F2 + e
r = return on the security
E(r) = expected return on the security
F1 = the first factor
B1 = the security’s sensitivity to movements in the first factor
F2 = the second factor
B2 = the security’s sensitivity to movements in the second factor
e = the idiosyncratic componentof the security’s return
Example of How Arbitrage Pricing Theory Is Used
For example, the following four factors have been identified as explaining a stock's return and
its sensitivity to each factor and the risk premium associated with each factor have been
calculated:
 Gross domestic product(GDP) growth: ß = 0.6, RP = 4%
 Inflation rate: ß = 0.8, RP = 2%
 Gold prices: ß = -0.7, RP = 5%
 Standard and Poor's 500 index return: ß = 1.3, RP = 9%
 The risk-free rate is 3%
Using the APT formula, the expected return is calculated as:
 Expected return = 3% + (0.6 x 4%) + (0.8 x 2%) + (-0.7 x 5%) + (1.3 x 9%) = 15.2%
Factor Model : Example
 Ri = ai + biF1 + ei
 Example :
Factor-1 is predicted rate of growth in industrial
production
i mean Ri bi
Stock 1 15% 0.9
Stock 2 21% 3.0
Stock 3 12% 1.8

7. Arbitrage Portfolio
 Arbitrage portfolio requires no ‘own funds’
– Assume there are 3 stocks : 1, 2 and 3
– Xi denotes the change in the investors holding
(proportion) of security i, then X1 + X2 + X3 = 0
– No sensitivity to any factor, so that b1X1 + b2X2 +
b3X3 = 0
– Example : 0.9 X1 + 3.0 X2 + 1.8 X3 = 0
– (assumes zero non factor risk)
Arbitrage Portfolio
(Cont.)
 Let X1 be 0.1.
 Then
 0.1 + X2 + X3 = 0
 0.09 + 3.0 X2 + 1.8 X3 = 0
– 2 equations, 2 unknowns.
– Solving this system gives
 X2 = 0.075
 X3 = -0.175
Arbitrage Portfolio
(Cont.)
 Expected return
X1 ER1 + X2 ER2 + X3 ER3 > 0
Here 15 X1 + 21 X2 + 12 X3 > 0 (= 0.975%)
 Arbitrage portfolio is attractive to investors
who
– Wants higher expected returns
– Is not concerned with risk due to factors other than
F1

8. Portfolio Stats / Portfolio
Weights (Example)
Weights Old Portf. Arbitr. Portf. New Portf.
X1 1/3 0.1 0.433
X2 1/3 0.075 0.408
X3 1/3 -0.175 0.158
Properties
ERp 16% 0.975% 16.975%
bp 1.9 0.00 1.9
sp 11% small approx 11%
Pricing Effects
 Stock 1 and 2
– Buying stock 1 and 2 will push prices up
– Hence expected returns falls
 Stock 3
– Selling stock 3 will push price down
– Hence expected return will increase
 Buying/selling stops if all arbitrage possibilities are
eliminated.
 Linear relationship between expected return and
sensitivities
ERi = l0 + l1bi
where bi is the security’s sensitivity to the factor.
Interpreting the APT
 ERi = l0 + l1bi
l0 = rf
l1 = pure factor portfolio, p* that has unit
sensitivity to the factor
 For bi = 1
ERp* = rf + l1
or l1 = ERp* - rf (= factor risk premium)

9. MULTI FACTOR MODELS
A multi-factor model is a financial model that employs multiple factors in its calculations to
explain market phenomena and/or equilibrium asset prices. The multi-factor model can be used
to explain either an individual security or a portfolio of securities. It does so by comparing two
or more factors to analyze relationships between variables and the resulting performance.
Multi-factor models are used to constructportfolios with certain characteristics, such as risk, or
to track indexes. When constructing a multi-factor model, it is difficult to decide how many and
which factors to include. Also, models are judged on historical numbers, which might not
accurately predict future values.
As used in investments, a factoris a variable or a characteristic with which individual asset
returns are correlated. Models using multiple factors are used by asset owners, asset managers,
investment consultants, and risk managers for a variety of portfolio construction, portfolio
management, risk management, and general analytical purposes. In comparisonto single-factor
models (typically based on a market risk factor), multifactor models offer increased explanatory
power and flexibility. These comparative strengths of multifactor models allow practitioners to
 build portfolios that replicate or modify in a desired way the characteristics of a particular
index;
 establish desired exposures to one or more risk factors, including those that express specific
macro expectations (such as views on inflation or economic growth), in portfolios;
 perform granular risk and return attribution on actively managed portfolios;
 understand the comparative risk exposures of equity, fixed-income, and other asset class
returns;
 identify active decisions relative to a benchmark and measure the sizing of those decisions;
 and ensure that an investor’s aggregate portfolio is meeting active risk and return objectives
commensurate with active fees.
Multifactor models have come to dominate investment practice, having demonstrated
their value in helping asset managers and asset owners address practical tasks in
measuring and controlling risk. This reading explains and illustrates the various practical
uses of multifactor models.
The reading is organized as follows. Section 2 describes the modern portfolio theory
background of multifactor models. Section 3 describes arbitrage pricing theory and
provides a general expression for multifactor models. Section 4 describes the types of
multifactor models, and Section 5 describes selected applications. Section 6 summarizes
major points.

10. Learning Outcomes
The candidate should be able to:
a. describe arbitrage pricing theory (APT), including its underlying assumptions and its
relation to multifactor models;
b. define arbitrage opportunity and determine whether an arbitrage opportunity exists;
c. calculate the expected return on an asset given an asset’s factor sensitivities and the factor
risk premiums;
d. describe and comparemacroeconomic factor models, fundamental factor models, and
statistical factor models;
e. explain sources of active risk and interpret tracking risk and the information ratio;
f. describe uses of multifactor models and interpret the output of analyses based on
multifactor models;
g. describe the potential benefits for investors in considering multiple risk dimensions when
modeling asset returns.
Multifactor models permit a nuanced view of risk that is more granular than the single-
factor approachallows.
Multifactor models describe the return on an asset in terms of the risk of the asset with
respect to a set of factors. Such models generally include systematic factors, which
explain the average returns of a large number of risky assets. Suchfactors represent
priced risk—risk for which investors require an additional return for bearing.
The arbitrage pricing theory (APT) describes the expected return on an asset (or
portfolio) as a linear function of the risk of the asset with respect to a set of factors. Like
the CAPM, the APT describes a financial market equilibrium, but the APT makes less
strong assumptions.
The major assumptions of the APT are as follows:
a. Asset returns are described by a factor model.
b. There are many assets, so asset-specific risk can be eliminated.
c. Assets are priced such that there are no arbitrage opportunities.
Multifactor models are broadly categorized according to the type of factor used as
follows:
d. Macroeconomic factor models
e. Fundamental factor models
f. Statistical factor models
In macroeconomic factor models, the factors are surprises in macroeconomic variables that
significantly explain asset class (equity in our examples) returns. Surprise is defined as

11. actual minus forecasted value and has an expected value of zero. The factors can be
understood as affecting either the expected future cash flows of companies or the interest
rate used to discount these cash flows back to the present and are meant to be uncorrelated.
In fundamental factormodels, the factors are attributes of stocks orcompanies that are
important in explaining cross-sectional differences in stock prices. Among the
fundamental factors are book-value-to-price ratio, market capitalization, price-to-earnings
ratio, and financial leverage.
In contrast to macroeconomic factor models, in fundamental models the factors are
calculated as returns rather than surprises. In fundamental factor models, we generally
specify the factor sensitivities (attributes) first and then estimate the factor returns
through regressions, in contrast to macroeconomic factor models, in which we first
develop the factor (surprise) series and then estimate the factor sensitivities through
regressions. The factors of most fundamental factor models may be classified as company
fundamental factors, company share-related factors, or macroeconomic factors.
In statistical factor models, statistical methods are applied to a set of historical returns to
determine portfolios that explain historical returns in one of two senses. In factor analysis
models, the factors are the portfolios that best explain (reproduce)historical return
covariances. In principal-components models, the factors are portfolios that best explain
(reproduce) the historical return variances.
Multifactor models have applications to return attribution, risk attribution, portfolio
construction, and strategic investment decisions.
A factor portfolio is a portfolio with unit sensitivity to a factor and zero sensitivity to
other factors.
Active return is the return in excess of the return on the benchmark.
Active risk is the standard deviation of active returns. Active risk is also called tracking
error or tracking risk. Active risk squared can be decomposed as the sum of active factor
risk and active specific risk.
The information ratio (IR) is mean active return divided by active risk (tracking error).
The IR measures the increment in mean active return per unit of active risk.
Factormodels have uses in constructing portfolios that track market indexes and in
alternative index construction.
Traditionally, the CAPM approachwould allocate assets between the risk-free asset and a
broadly diversified index fund. Considering multiple sources of systematic risk may
allow investors to improve on that result by tilting away from the market portfolio.

12. Generally, investors would gain from accepting aboveaverage (below average) exposures
to risks that they have a comparative advantage (comparative disadvantage) in bearing.
CategoriesofMulti-FactorModels
Multi-factor models can be divided into three categories: macroeconomic models, fundamental
models and statistical models. Macroeconomic models compare a security's return to such
factors as employment, inflation and interest. Fundamental models analyze the relationship
between a security's return and its underlying financials, such as earnings. Statistical models are
used to compare the returns of different securities based on the statistical performance of each
security in and of itself.
Beta
The beta of a security measures the systemic risk of the security in relation to the overall
market. A beta of 1 indicates that the security theoretically experiences the same degree of
volatility as the market and moves in tandem with the market. A beta greater than 1 indicates
the security is theoretically more volatile than the market. Conversely, a beta less than 1
indicates the security is theoretically less volatile than the market.
Multi-FactorModel Formula
Factors are compared using the following formula:
Ri = ai + _i(m) * Rm + _i(1) * F1 + _i(2) * F2 +...+_i(N) * FN + ei
Where:
Ri is the return of security i
Rm is the market return
F(1, 2, 3 ... N) is each of the factors used
_ is the beta with respect to each factor including the market (m)
e is the error term
a is the intercept
Fama and French Three-Factor Model
One widely used multi-factor model is the Fama and French three-factor model. The Fama and
French model has three factors: size of firms, book-to-market values and excess return on the
market. In other words, the three factors used are SMB (small minus big), HML (high minus
low) and the portfolio's return less the risk free rate of return. SMB accounts for publicly traded
companies with small market caps that generate higher returns, while HML accounts for value
stocks with high book-to-market ratios that generate higher returns in comparison to the market.

13. BEHAVIORAL FINANCE
Behavioral finance, a sub-field of behavioral economics, proposespsychology-based theories to
explain stockmarket anomalies, such as severe rises or falls in stockprice. The purposeis to
identify and understand why people make certain financial choices. Within behavioral finance,
it is assumed the information structure and the characteristics of market participants
systematically influence individuals' investment decisions as well as market outcomes.
The efficient market hypothesis (EMH) proposes that at any given time in a liquid
market, prices reflect all available information. There have been many studies, however, that
document long-term historical phenomena in securities markets that contradict the efficient
market hypothesis and cannot be captured plausibly in models based on perfect investor
rationality. Many traditional models are based on the belief that market participants always act
in a rational and wealth-maximizing manner, severely limiting these models' ability to make
accurate or detailed predictions.
Behavioral finance attempts to fill this void by combining scientific insights into cognitive
reasoning with conventional economic and financial theory. More specifically, behavioral
finance studies different psychological biases that humans possess. Thesebiases, or mental
shortcuts, while having their place and purposein nature, lead to irrational investment
decisions. This understanding, at a collective level, gives a clearer explanation of
why bubbles and panics occur. Also, investors and portfolio managers have a vested interest in
understanding behavioral finance, not only to capitalize on stockand bond market fluctuations
but to also be more aware of their own decision-making process.
BehavioralFinance Concepts
Behavioral finance encompasses many concepts, butfour are key: mental accounting, herd
behavior, anchoring, and high self-rating. Mental accounting refers to the propensity for people
to allocate money for specific purposes. Herd behavior states that people tend to mimic the
financial behaviors of the majority, or herd. Anchoring refers to attaching a spending level to a
certain reference, such as spending more money on what is perceived to be a better item of
clothing. Lastly, high self-rating refers to a person's tendency to rank him/herself better than
others or higher than an average person. Forexample, an investor may think that he is an
investment guru when his investment performs optimally but will dismiss his contributions to
an investment performing poorly.

14. Biases Studiedin BehavioralFinance
Of the four concepts, two (herd instinct and self-rating/self-attribution) are biases that
significantly affect financial decisions. A prominent psychological bias is the herd instinct,
which leads people to follow popular trends without any deep thought of their own. Herding is
notorious in the stockmarket as the cause behind dramatic rallies and sell-offs. The herd
instinct is correlated closely with the empathy gap, which is an inability to make rational
decisions under emotional strains, such as anxiety, anger, or excitement.
The self-attribution bias, a habit of attributing favorable outcomes to expertise and unfavorable
outcomes to bad luck or an exogenous event, is also closely studied within behavioral finance.
George Soros, ahighly successfulinvestor, is known to accountfor this tendency by keeping a
journal log of his reasoning behind every investment decision. Many other tendencies are
studied within behavioral finance, including loss aversion, confirmation bias, availability
bias, disposition effect, and familiarity bias.

16. CONCLUSION
Risk is the energy that drives markets. A robustmodel of market risk must take accountof the
comovement of asset returns both in normal times and when markets are stressed. SunGard
APT provides a risk management solution based on the conceptof“three pillars of risk
management”: risk measurement, risk attribution, and scenario analysis. Flexibility in both
attribution and scenario analysis is the key to effective risk management. To achieve that
flexibility, APT constructs statistical factor models of risky asset markets and then allows users
to select the explanatory factors that match their own investment process when carrying out
attribution and scenario analysis; this approachbased on explanatory factors is called RiskScan.
However, for robust risk measurement, APT has always relied on statistical factor modeling
methodologies.
Statistical factor models are economically motivated and consistent with asset pricing theory
and the observed effects of arbitrage across markets. Theoretical work on arbitrage pricing
theory was pioneered by Ross1 and Roll and Ross,2and extended by Connor and
Korajczyk;3 early applications to risk management for investment were documented by Blin,
Bender, and Guerard.4Recent evidence for the robustness of the approach for optimized
strategies is provided by ...

17. BIBLIOGRAPHY
References
From the book“stockexchange trading in india” written by “maachi raju”
From the book”security analysis and portfolio management” written by
S Kevin
Web references
www.investopedia.com
www.wikipidia.com
www.moneycontrol.com
TOTAL NO OF SLIDE--16

18. SUBMITTED BY
GROUP NO-7
PRIYA SINGH
KUMUD SAGAR
PARUL SACHAN
RAJESH BABU KATIYAR
M.B.A (F.C) 4TH SEM. SESSION: 2017-19