Arbitrage pricing theory (APT), in Finance, is a general theory of asset pricing, that has become influential in the pricing of shares. This article does not cite any references or sources. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... Valuation is the process of estimating the value of an asset or liability. ... This article does not cite any references or sources. ...

APT 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 discounted at the rate implied by model. If the price diverges, arbitrage should bring it back into line. The Beta coefficient, in terms of finance and investing, is a measure of a stock (or portfolio)’s volatility in relation to the rest of the market. ... In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash from the basis of time value of money calculations. ... In economics and finance, arbitrage is the practice of taking advantage of a price differential between two or more markets: a combination of matching deals are struck that capitalize upon the imbalance, the profit being the difference between the market prices. ...

The theory was initiated by the economist Stephen Ross in 1976. The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... Alan Greenspan, former chairman, United States Federal Reserve. ... Stephen A. Ross is the Franco Modigliani Professor of Finance and Economics at the MIT Sloan School of Management. ...

The APT model

If APT holds, then a risky asset can be described as satisfying the following relation:

where

• E(r_j) is the risky asset's expected return,
• RP_k is the risk premium of the factor,
• r_f is the risk-free rate,
• F_k is the macroeconomic factor,
• b_jk is the sensitivity of the asset to factor k, also called factor loading,
• and ε_j is the risky asset's idiosyncratic random shock with mean zero.

That is, the uncertain return of an asset j is a linear relationship among n factors. Additionally, every factor is also considered to be a random variable with mean zero. A risk premium is the minimum difference between the expected value of an uncertain bet that a person is willing to take and the certain value that he is indifferent to. ... The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ... The word linear comes from the Latin word linearis, which means created by lines. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...

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), Look up assumption in Wiktionary, the free dictionary. ... Perfect competition is an economic model that describes a hypothetical market form in which no producer or consumer has the market power to influence prices. ... In linear algebra, an n-by-n (square) matrix is called invertible or non-singular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

Arbitrage and the APT

Arbitrage is the practice of taking advantage of a state of imbalance between two (or possibly more) markets and thereby making a risk free profit; see Rational pricing. In economics and finance, arbitrage is the practice of taking advantage of a price differential between two or more markets: a combination of matching deals are struck that capitalize upon the imbalance, the profit being the difference between the market prices. ... Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be arbitraged away. This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to...

Arbitrage in expectations

The APT describes the mechanism whereby arbitrage by investors will bring an asset which is mispriced, according to the APT model, back into line with its expected price. Note that under true arbitrage, the investor locks-in a guaranteed payoff, whereas under APT arbitrage as described below, the investor locks-in a positive expected payoff. The APT thus assumes "arbitrage in expectations" - i.e. that arbi3trage by investors will bring asset prices back into line with the returns expected by the model portfolio theory.

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 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. In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash from the basis of time value of money calculations. ... The Beta coefficient, in terms of finance and investing, is a measure of a stock (or portfolio)’s volatility in relation to the rest of the market. ...

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: In finance, a long position in a security, such as a stock or a bond, or equivalently to be long a security, means the holder of the position owns the security and will profit if the price of the security goes up. ... It has been suggested that Short (finance) be merged into this article or section. ...

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: It has been suggested that Short (finance) be merged into this article or section. ...

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 pocket the difference.

It has been suggested that Short (finance) be merged into this article or section. ...

Relationship with the capital asset pricing model

The APT along with the capital asset pricing model (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. An estimation of the CAPM and the Security Market Line (purple) for the Dow Jones Industrial Average over the last 3 years for monthly data. ... Capital Market Line Modern portfolio theory (MPT) proposes how rational investors will use diversification to optimize their portfolios, and how a risky asset should be priced. ...

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 the asset's expected return, or in the case of stocks, in the firm's profitability.

On the other side, the capital asset pricing model is considered a "demand side" model. Its results, although similar to those in 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). An estimation of the CAPM and the Security Market Line (purple) for the Dow Jones Industrial Average over the last 3 years for monthly data. ...

Using the APT

Identifying the factors

As with the CAPM, the factor-specific Betas are found via a linear 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 in nature. Several a priori guidelines as to the characteristics required of potential factors are, however, suggested: In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xp, and a random term ε. The model can be written as where β1 is the intercept (constant term), the βis are the respective parameters of independent variables, and p is the... A central concept in science and the scientific method is that all evidence must be empirical, or empirically based, that is, dependent on evidence or consequences that are observable by the senses. ... The terms a priori and a posteriori are used in philosophy to distinguish between two different types of propositional knowledge. ...

1. their impact on asset prices manifests in their unexpected movements
2. they should represent undiversifiable influences (these are, clearly, more likely to be macroeconomic rather than firm-specific in nature)
3. timely and accurate information on these variables is required
4. the relationship should be theoretically justifiable on economic grounds

Chen, Roll and Ross identified the following macro-economic factors as significant in explaining security returns: Stephen A. Ross is the Franco Modigliani Professor of Finance and Economics at the MIT Sloan School of Management. ...

• surprises in inflation;
• surprises in GNP as indicted by an industrial production index;
• surprises in investor confidence due to changes in default premium in corporate bonds;
• surprise shifts in the yield curve.

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: Measures of national income and output are used in economics to estimate the value of goods and services produced in an economy. ... The US dollar yield curve as of 9 February 2005. ... Factor analysis is a statistical data reduction technique used to explain variability among observed random variables in terms of fewer unobserved random variables called factors. ...

• short term interest rates;
• the difference in long-term and short term interest rates;
• a diversified stock index such as the S&P 500 or NYSE Composite Index;
• oil prices
• gold or other precious metal prices
• Currency exchange rates

The S&P 500 is an index containing the stocks of 500 Large-Cap corporations, most of which are American. ... The NYSE Composite (NYSE: NYA) is a stock market index covering all common stock listed on the New York Stock Exchange, including American Depositary Receipts, Real Estate Investment Trusts, tracking stocks, and foreign listings. ...

APT and asset management

See also

• Rational pricing
• Fundamental theorem of arbitrage-free pricing
• Capital asset pricing model
• Efficient market hypothesis
• Modern portfolio theory
• Earnings response coefficient
• Value investing

Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be arbitraged away. This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to... In a general sense, the fundamental theorem of arbitrage/finance is a way to relate arbitrage opportunities with risk neutral measures that are equivalent to the original probability measure. ... An estimation of the CAPM and the Security Market Line (purple) for the Dow Jones Industrial Average over the last 3 years for monthly data. ... In finance, the efficient market hypothesis (EMH) asserts that financial markets are informationally efficient, or that prices on traded assets, e. ... Capital Market Line Modern portfolio theory (MPT) proposes how rational investors will use diversification to optimize their portfolios, and how a risky asset should be priced. ... There are very few or no other articles that link to this one. ... Value investing is a style of investment strategy from the so-called Graham & Dodd School. ...

References

• Burmeister E and Wall KD., The arbitrage pricing theory and macroeconomic factor measures, The Financial Review, 21:1-20, 1986
• Chen, N.F, and Ingersoll, E., Exact pricing in linear factor models with finitely many assets: A note, Journal of Finance June 1983
• Roll, Richard and Stephen Ross, An empirical investigation of the arbitrage pricing theory, Journal of Finance, Dec 1980,
• Ross, Stephen, The arbitrage theory of capital asset pricing, Journal of Economic Theory, v13, issue 3, 1976

External links

• The Arbitrage Pricing Theory Prof. William N. Goetzmann, Yale School of Management
• The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning (PDF), Richard Roll and Stephen A. Ross
• The APT, Prof. Tyler Shumway, University of Michigan Business School
• The arbitrage pricing theory, The Investment Analysts' Society of South Africa
• References on the Arbitrage Pricing Theory, Prof. Robert A. Korajczyk, Kellogg School of Management
• Chapter 12: Arbitrage Pricing Theory (APT), Prof. Jiang Wang, Massachusetts Institute of Technology.