# Arbitrage pricing theory

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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 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 the model. If the price diverges, arbitrage should bring it back into line. The theory was proposed by the economist Stephen Ross in 1976. The linear factor model structure of the APT is used as the basis for many of the commercial risk systems employed by asset managers. Finance is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

A theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking. Depending on the context, the results might, for example, include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several related meanings.

In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspondingly, these stem from general equilibrium asset pricing or rational asset pricing, the latter corresponding to risk neutral pricing.

## Model

Risky asset returns are said to follow a factor intensity structure if they can be expressed as:

$r_{j}=a_{j}+b_{j1}F_{1}+b_{j2}F_{2}+\cdots +b_{jn}F_{n}+\epsilon _{j}$ where
• $a_{j}$ is a constant for asset $j$ • $F_{n}$ is a systematic factor
• $b_{jn}$ is the sensitivity of the $j$ th asset to factor $n$ , also called factor loading,
• and $\epsilon _{j}$ 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:

$E\left(r_{j}\right)=r_{f}+b_{j1}RP_{1}+b_{j2}RP_{2}+\cdots +b_{jn}RP_{n}$ where
• $RP_{n}$ is the risk premium of the factor,
• $r_{f}$ is the risk-free rate,

For an individual, a 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. It is positive if the person is risk averse. Thus it is the minimum willingness to accept compensation for the risk.

That is, the expected excess return of an asset j is a linear function of the asset's 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).

In economics, specifically general equilibrium theory, a perfect market is defined by several idealizing conditions, collectively called perfect competition. In theoretical models where conditions of perfect competition hold, it has been theoretically demonstrated that a market will reach an equilibrium in which the quantity supplied for every product or service, including labor, equals the quantity demanded at the current price. This equilibrium would be a Pareto optimum.

## 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.

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. When used by academics, an arbitrage is a transaction that involves no negative cash flow at any probabilistic or temporal state and a positive cash flow in at least one state; in simple terms, it is the possibility of a risk-free profit after transaction costs. For example, an arbitrage opportunity is present when there is the opportunity to instantaneously buy something for a low price and sell it for a higher price.

### 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.

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 n correctly priced assets (one per risk-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 portfolio2 buy the mispriced asset with the proceeds.At the end of the period: 1 sell the mispriced asset2 use the proceeds to buy back the portfolio3 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 asset2 buy the portfolio with the proceeds.At the end of the period: 1 sell the portfolio2 use the proceeds to buy back the mispriced asset3 pocket the difference.

## 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.

A disadvantage of APT is that the selection and the number of factors to use in the model is ambiguous. Most academics use three to five factors to model returns, but the factors selected have not been empirically robust. In many instances the CAPM, as a model to estimate expected returns, has empirically outperformed the more advanced APT. 

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).

## Implementation

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:

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 (1986) identified the following macro-economic factors as significant in explaining security returns:

• surprises in inflation;
• surprises in GNP 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 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:

• 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;
• oil prices
• gold or other precious metal prices
• Currency exchange rates

## Related Research Articles

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In finance, the beta of an investment indicates whether the investment is more or less volatile than the market as a whole.

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The single-index model (SIM) is a simple asset pricing model to measure both the risk and the return of a stock. The model has been developed by William Sharpe in 1963 and is commonly used in the finance industry. Mathematically the SIM is expressed as:

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In asset pricing and portfolio management the Fama–French three-factor model is a model designed by Eugene Fama and Kenneth French to describe stock returns. Fama and French were professors at the University of Chicago Booth School of Business, where Fama still resides. In 2013, Fama shared the Nobel Memorial Prize in Economic Sciences. The three factors are (1) market risk, (2) the outperformance of small versus big companies, and (3) the outperformance of high book/market versus small book/market companies. However, the size and book/market ratio themselves are not in the model. For this reason, there is academic debate about the meaning of the last two factors. The low-volatility anomaly is the observation that portfolios of low-volatility stocks have higher risk-adjusted returns than portfolios with high-volatility stocks in most markets studied. The capital asset pricing model made some predictions of return versus beta. First, return should be a linear function of beta, and nothing else. Also, the return of a stock with average beta should be the average return of stocks. Second, the intercept should be equal to the risk-free rate. Then the slope can be computed from these two points. Almost immediately these predictions were challenged on the grounds that they are empirically not true. Studies find that the correct slope is either less than predicted, not significantly different from zero, or even negative. Also, additional factors are predictive of return independent of beta.

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In portfolio management the Carhart four-factor model is an extension of the Fama–French three-factor model including a momentum factor for asset pricing of stocks, proposed by Mark Carhart. It is also known in the industry as the MOM factor. Momentum in a stock is described as the tendency for the stock price to continue rising if it is going up and to continue declining if it is going down.

Nontraded assets are assets that are not traded on the market. Human capital is the most important nontraded assets. Other important nontraded asset classes are private businesses, claims to government transfer payments and claims on trust income.

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Factor investing is an investment approach that involves targeting quantifiable firm characteristics or “factors” that can explain differences in stock returns. Over the last 50 years, academic research has identified hundreds of factors that impact stock returns. Security characteristics that may be included in a factor-based approach includes size, value, momentum, asset growth, profitability, leverage, term and carry.

1. French, Jordan (1 March 2017). "Macroeconomic Forces and Arbitrage Pricing Theory". Journal of Comparative Asian Development. 16 (1): 1–20. doi:10.1080/15339114.2017.1297245.