Arbitrage pricing theory
Encyclopedia
In finance
, arbitrage pricing theory (APT) is a general theory
of asset pricing that holds that the expected return
of a financial asset can be modeled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient
. The model-derived rate of return will then be used to price the asset correctly - the asset price should equal the expected end of period price discount
ed at the rate implied by the model. If the price diverges, arbitrage
should bring it back into line.
The theory
was initiated by the economist
Stephen Ross
in 1976.
Idiosyncratic shocks are assumed to be uncorrelated across assets and uncorrelated with the factors.
The APT states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities:
That is, the expected return of an asset j is a linear
function of the assets sensitivities to the n factors.
Note that there are some assumptions and requirements that have to be fulfilled for the latter to be correct: There must be perfect competition
in the market, and the total number of factors may never surpass the total number of assets (in order to avoid the problem of matrix singularity),
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.
on investors’ asset demand. For example:
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 cash flows discount
ed 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 asset here 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 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:
(CAPM) is one of two influential theories on asset pricing. The APT differs from the CAPM in that it is less restrictive in its assumptions. It allows for an explanatory (as opposed to statistical) model of asset returns. It assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". In some ways, the CAPM can be considered a "special case" of the APT in that the securities market line represents a single-factor model of the asset price, where beta is exposed to changes in value of the market.
Additionally, the APT can be seen as a "supply-side" model, since its beta coefficients reflect the sensitivity of the underlying asset to economic factors. Thus, factor shocks would cause structural changes in assets' expected returns, or in the case of stocks, in firms' profitabilities.
On the other side, the capital asset pricing model
is considered a "demand side" model. Its results, although similar to those of the APT, arise from a maximization problem of each investor's utility function, and from the resulting market equilibrium (investors are considered to be the "consumers" of the assets).
of historical security returns on the factor in question. Unlike the CAPM, the APT, however, does not itself reveal the identity of its priced factors - the number and nature of these factors is likely to change over time and between economies. As a result, this issue is essentially empirical
in nature. Several a priori
guidelines as to the characteristics required of potential factors are, however, suggested:
Chen, Roll
and Ross
(1986) identified the following macro-economic factors as significant in explaining security returns:
As a practical matter, indices or spot or futures market prices may be used in place of macro-economic factors, which are reported at low frequency (e.g. monthly) and often with significant estimation errors. Market indices are sometimes derived by means of factor analysis
. More direct "indices" that might be used are:
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, arbitrage pricing theory (APT) is a general theory
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...
of asset pricing that holds that the expected return
Expected return
The expected return is the weighted-average outcome in gambling, probability theory, economics or finance.It isthe average of a probability distribution of possible returns, calculated by using the following formula:...
of a financial asset can be modeled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient
Beta coefficient
In finance, the Beta of a stock or portfolio is a number describing the relation of its returns with those of the financial market as a whole.An asset has a Beta of zero if its returns change independently of changes in the market's returns...
. The model-derived rate of return will then be used to price the asset correctly - the asset price should equal the expected end of period price discount
Discounts and allowances
Discounts and allowances are reductions to a basic price of goods or services.They can occur anywhere in the distribution channel, modifying either the manufacturer's list price , the retail price , or the list price Discounts and allowances are reductions to a basic price of goods or services.They...
ed at the rate implied by the model. If the price diverges, arbitrage
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
should bring it back into line.
The theory
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...
was initiated by the economist
Economist
An economist is a professional in the social science discipline of economics. The individual may also study, develop, and apply theories and concepts from economics and write about economic policy...
Stephen Ross
Stephen Ross (economist)
Stephen Alan "Steve" Ross is the inaugural Franco Modigliani Professor of Financial Economics at the MIT Sloan School of Management. He is known for initiating several important theories and models in financial economics...
in 1976.
The APT model
Risky asset returns are said to follow a factor structure if they can be expressed as:- where
-
is a constant for asset 
-
is a systematic factor -
is the sensitivity of the 
th asset to factor 
, also called factor loading, - and
is the risky asset's idiosyncratic random shock with mean zero.
-
Idiosyncratic shocks are assumed to be uncorrelated across assets and uncorrelated with the factors.
The APT states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities:
- where
-
is the risk premiumRisk premiumA risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset, in order to induce an individual to hold the risky asset rather than the risk-free asset...
of the factor, -
is the risk-free rate,
-
That is, the expected return of an asset j is a linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
function of the assets sensitivities to the n factors.
Note that there are some assumptions and requirements that have to be fulfilled for the latter to be correct: There must be perfect competition
Perfect competition
In economic theory, perfect competition describes markets such that no participants are large enough to have the market power to set the price of a homogeneous product. Because the conditions for perfect competition are strict, there are few if any perfectly competitive markets...
in the market, and the total number of factors may never surpass the total number of assets (in order to avoid the problem of matrix singularity),
Arbitrage and the APT
ArbitrageArbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
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.
Arbitrage in expectations
The capital asset pricing model and its extensions are based on specific assumptionson investors’ asset demand. For example:
- Investors care only about mean return and variance.
- Investors hold only traded assets.
Arbitrage 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 cash flows discount
Discount
Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date...
ed 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
Beta coefficient
In finance, the Beta of a stock or portfolio is a number describing the relation of its returns with those of the financial market as a whole.An asset has a Beta of zero if its returns change independently of changes in the market's returns...
.
A correctly priced asset here 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 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
Long (finance)
In finance, a long position in a security, such as a stock or a bond, or equivalently to be long in a security, means the holder of the position owns the security and will profit if the price of the security goes up. Going long is the more conventional practice of investing and is contrasted with...
the asset and short
Short selling
In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
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
Short sellingIn finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
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
Short sellingIn finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
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 pocket the difference.
Relationship with the capital asset pricing model (CAPM)
The APT along with the capital asset pricing modelCapital asset pricing model
In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
(CAPM) is one of two influential theories on asset pricing. The APT differs from the CAPM in that it is less restrictive in its assumptions. It allows for an explanatory (as opposed to statistical) model of asset returns. It assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". In some ways, the CAPM can be considered a "special case" of the APT in that the securities market line represents a single-factor model of the asset price, where beta is exposed to changes in value of the market.
Additionally, the APT can be seen as a "supply-side" model, since its beta coefficients reflect the sensitivity of the underlying asset to economic factors. Thus, factor shocks would cause structural changes in assets' expected returns, or in the case of stocks, in firms' profitabilities.
On the other side, the capital asset pricing model
Capital asset pricing model
In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
is considered a "demand side" model. Its results, although similar to those of the APT, arise from a maximization problem of each investor's utility function, and from the resulting market equilibrium (investors are considered to be the "consumers" of the assets).
Identifying the factors
As with the CAPM, the factor-specific betas are found via a linear regressionLinear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
of historical security returns on the factor in question. Unlike the CAPM, the APT, however, does not itself reveal the identity of its priced factors - the number and nature of these factors is likely to change over time and between economies. As a result, this issue is essentially empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
in nature. Several a priori
A priori and a posteriori (philosophy)
The terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...
guidelines as to the characteristics required of potential factors are, however, suggested:
- their impact on asset prices manifests in their unexpected movements
- they should represent undiversifiable influences (these are, clearly, more likely to be macroeconomic rather than firm-specific in nature)
- timely and accurate information on these variables is required
- the relationship should be theoretically justifiable on economic grounds
Chen, Roll
Richard Roll
Richard Roll is an American economist, best known for his work on portfolio theory and asset pricing, both theoretical and empirical....
and Ross
Stephen Ross (economist)
Stephen Alan "Steve" Ross is the inaugural Franco Modigliani Professor of Financial Economics at the MIT Sloan School of Management. He is known for initiating several important theories and models in financial economics...
(1986) identified the following macro-economic factors as significant in explaining security returns:
- surprises in inflationInflationIn economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
; - surprises in GNPGNPGross National Product is the market value of all products and services produced in one year by labor and property supplied by the residents of a country...
as indicated by an industrial production index; - surprises in investor confidence due to changes in default premium in corporate bonds;
- surprise shifts in the yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
.
As a practical matter, indices or spot or futures market prices may be used in place of macro-economic factors, which are reported at low frequency (e.g. monthly) and often with significant estimation errors. Market indices are sometimes derived by means of factor analysis
Factor analysis
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables...
. More direct "indices" that might be used are:
- short term interest rates;
- the difference in long-term and short-term interest rates;
- a diversified stock index such as the S&P 500S&P 500The S&P 500 is a free-float capitalization-weighted index published since 1957 of the prices of 500 large-cap common stocks actively traded in the United States. The stocks included in the S&P 500 are those of large publicly held companies that trade on either of the two largest American stock...
or NYSE Composite Index; - oil prices
- gold or other precious metal prices
- Currency exchange rateExchange rateIn finance, an exchange rate between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency...
s
APT and asset management
The linear factor model structure of the APT is used as the basis for many of the commercial risk systems employed by asset managers. These include MSCI Barra, APT, Northfield and Axioma.See also
- Beta coefficientBeta coefficientIn finance, the Beta of a stock or portfolio is a number describing the relation of its returns with those of the financial market as a whole.An asset has a Beta of zero if its returns change independently of changes in the market's returns...
- Capital asset pricing modelCapital asset pricing modelIn finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
- Cost of capitalCost of capitalThe cost of capital is a term used in the field of financial investment to refer to the cost of a company's funds , or, from an investor's point of view "the shareholder's required return on a portfolio of all the company's existing securities"...
- Earnings response coefficientEarnings response coefficient-Introduction:The earnings response coefficient, or ERC, is the estimated relationship between equity returns and the unexpected portion of companies' earnings announcements....
- Efficient-market hypothesis
- Fundamental theorem of arbitrage-free pricingFundamental theorem of arbitrage-free pricingThe fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...
- Investment theoryInvestment theoryInvestment theory encompasses the body of knowledge used to support the decision-making process of choosing investments for various purposes. It includes portfolio theory, the Capital Asset Pricing Model, Arbitrage Pricing Theory, and the Efficient market hypothesis.-References:*...
- Roll's critique
- Rational pricingRational pricingRational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away"...
- Modern portfolio theoryModern portfolio theoryModern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
- Post-modern portfolio theoryPost-modern portfolio theoryPost-modern portfolio theory is an extension of the traditional modern portfolio theory...
- Value investingValue investingValue investing is an investment paradigm that derives from the ideas on investment and speculation that Ben Graham and David Dodd began teaching at Columbia Business School in 1928 and subsequently developed in their 1934 text Security Analysis...
External links
- The Arbitrage Pricing Theory Prof. William N. Goetzmann, Yale School of ManagementYale School of ManagementThe Yale School of Management is the graduate business school of Yale University and is located on Hillhouse Avenue in New Haven, Connecticut, United States. The School offers Master of Business Administration and Ph.D. degree programs. As of January 2011, 454 students were enrolled in its MBA...
- The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning (PDFPortable Document FormatPortable Document Format is an open standard for document exchange. This file format, created by Adobe Systems in 1993, is used for representing documents in a manner independent of application software, hardware, and operating systems....
), Richard Roll and Stephen A. RossStephen Ross (economist)Stephen Alan "Steve" Ross is the inaugural Franco Modigliani Professor of Financial Economics at the MIT Sloan School of Management. He is known for initiating several important theories and models in financial economics... - The APT, Prof. Tyler Shumway, University of Michigan Business School
- The arbitrage pricing theory Investment Analysts Society of South Africa
- References on the Arbitrage Pricing Theory, Prof. Robert A. Korajczyk, Kellogg School of ManagementKellogg School of ManagementThe Kellogg School of Management is the business school of Northwestern University in Evanston, Illinois, downtown Chicago, Illinois and Miami, Florida. Kellogg offers full-time, part-time, and executive programs, as well as partnering programs with schools in China, India, Hong Kong, Israel,...
- Chapter 12: Arbitrage Pricing Theory (APT), Prof. Jiang Wang, Massachusetts Institute of TechnologyMassachusetts Institute of TechnologyThe Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
.