Arbitrage pricing theory

Arbitrage pricing theory (APT) is a multifactor asset-pricing theory that explains the expected return of an asset as a function of its sensitivity to multiple systematic risk factors. It was developed by Stephen Ross and introduced in 1976.

Unlike the Capital asset pricing model (CAPM), which uses a single systematic risk factor represented by market beta, APT allows multiple economic and financial factors to influence expected returns.

APT equation

A simplified arbitrage pricing theory equation is:

$E (R_{i}) = R_{f} + \sum_{k = 1}^{K} β_{i k} λ_{k}$

where:

$E (R_{i})$ = expected return of asset $i$ ;
$R_{f}$ = risk-free rate;
$β_{i k}$ = sensitivity of asset $i$ to risk factor $k$ ;
$λ_{k}$ = risk premium associated with factor $k$ ;
$K$ = number of systematic risk factors.

The total risk premium implied by the model is therefore:

${Risk Premium}_{i} = \sum_{k = 1}^{K} β_{i k} λ_{k}$

Factor sensitivities

Under APT, an asset's expected return depends on its exposure to multiple systematic factors.

A simplified factor model for asset returns is:

$R_{i} = E (R_{i}) + \sum_{k = 1}^{K} β_{i k} F_{k} + ϵ_{i}$

where:

$R_{i}$ = realized return of asset $i$ ;
$E (R_{i})$ = expected return;
$β_{i k}$ = sensitivity to factor $k$ ;
$F_{k}$ = unexpected component of factor $k$ ;
$ϵ_{i}$ = asset-specific random component.

The factors represent systematic sources of variation that can affect multiple assets.

Risk factors

APT does not require a specific fixed set of risk factors.

Potential factors can include:

interest-rate changes;
inflation;
economic growth;
industrial production;
exchange-rate movements;
commodity prices;
market-wide financial conditions.

The factors used in an empirical APT model depend on the purpose of the analysis and the methodology used to identify systematic sources of return variation.

Risk premium

The factor risk premium $λ_{k}$ represents the additional expected return associated with exposure to systematic factor $k$ .

For an asset exposed to multiple factors:

$Total Risk Premium = β_{i 1} λ_{1} + β_{i 2} λ_{2} + \dots + β_{i K} λ_{K}$

Therefore:

$E (R_{i}) = R_{f} + β_{i 1} λ_{1} + β_{i 2} λ_{2} + \dots + β_{i K} λ_{K}$

Arbitrage principle

The central idea behind APT is that arbitrage opportunities should not persist in a well-functioning market.

Arbitrage involves constructing a portfolio that can potentially generate a riskless or near-riskless profit from inconsistent asset pricing.

If two assets or portfolios have equivalent exposures to the relevant systematic risk factors but different expected returns, investors could potentially exploit the difference.

A simplified no-arbitrage condition is:

$Equal Factor Exposure \Rightarrow Equal Expected Return$

subject to the assumptions of the model and the absence of significant market frictions.

APT and CAPM

APT and CAPM are both asset-pricing models, but they differ in their treatment of systematic risk.

CAPM uses a single market-risk factor:

$E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$

APT uses multiple systematic risk factors:

$E (R_{i}) = R_{f} + \sum_{k = 1}^{K} β_{i k} λ_{k}$

The principal distinction is therefore:

$CAPM \to Single Systematic Factor$

$APT \to Multiple Systematic Factors$

Example

Suppose an asset has exposure to three systematic factors:

factor 1 beta = 1.2 and risk premium = 3%;
factor 2 beta = 0.5 and risk premium = 2%;
factor 3 beta = -0.4 and risk premium = 1%.

Assume the risk-free rate is 4%.

The APT expected return is:

$E (R_{i}) = 4 % + (1.2) (3 %) + (0.5) (2 %) + (- 0.4) (1 %)$

Therefore:

$E (R_{i}) = 4 % + 3.6 % + 1.0 % - 0.4 % = 8.2 %$

The APT-implied expected return is therefore 8.2%.

Factor model and diversification

APT distinguishes systematic factor exposure from asset-specific effects.

A simplified representation is:

$R_{i} = E (R_{i}) + \sum_{k = 1}^{K} β_{i k} F_{k} + ϵ_{i}$

The factor-related component represents systematic exposure, while $ϵ_{i}$ represents asset-specific variation.

For a sufficiently diversified portfolio, asset-specific effects may become less important relative to systematic factor exposures.

Portfolio APT

APT can also be applied to a portfolio.

If the portfolio contains $n$ assets, its exposure to factor $k$ can be represented as:

$β_{p k} = \sum_{i = 1}^{n} w_{i} β_{i k}$

where:

$β_{p k}$ = portfolio exposure to factor $k$ ;
$w_{i}$ = portfolio weight of asset $i$ ;
$β_{i k}$ = exposure of asset $i$ to factor $k$ .

The portfolio's expected return can then be represented as:

$E (R_{p}) = R_{f} + \sum_{k = 1}^{K} β_{p k} λ_{k}$

Empirical implementation

APT does not prescribe a universal list of observable factors. In empirical applications, researchers must identify factors that appear to explain differences in asset returns.

A general regression specification can be written as:

$R_{i} - R_{f} = α_{i} + \sum_{k = 1}^{K} β_{i k} F_{k} + ϵ_{i}$

where:

$α_{i}$ = intercept;
$F_{k}$ = factor return or factor realization;
$β_{i k}$ = asset sensitivity to factor $k$ ;
$ϵ_{i}$ = residual component.

The estimated factor sensitivities can then be used to analyze the relationship between systematic risk exposures and expected returns.

Limitations

APT has several limitations.

These include:

it does not specify a universal set of risk factors;
factor identification can be difficult;
factor premiums must generally be estimated;
estimated factor sensitivities can change over time;
empirical results depend on the selected factors and data;
transaction costs and other market frictions can affect arbitrage opportunities;
the theoretical assumptions underlying arbitrage-based pricing may not fully describe actual markets.

Consequently, different implementations of APT can produce different estimates of expected returns.

Importance

Arbitrage pricing theory provides a flexible framework for relating expected asset returns to multiple sources of systematic risk. Its multifactor structure allows analysts to incorporate several economic or financial variables rather than relying exclusively on market beta. APT has influenced subsequent multifactor asset-pricing models and empirical approaches to understanding differences in asset returns.

Arbitrage pricing theory

Contents

APT equation

Factor sensitivities

Risk factors

Risk premium

Arbitrage principle

APT and CAPM

Example

Factor model and diversification

Portfolio APT

Empirical implementation

Limitations

Importance

See also