Arbitrage price theory

The arbitrage pricing theory or English Arbitrage Pricing Theory (APT) describes a method for determining the cost of equity and the expected return on securities . It was largely developed by Stephen Ross . Ross also used the term Arbitrage Pricing Model (APM).

General

In contrast to the Capital Asset Pricing Model (CAPM), the APT no longer requires a market equilibrium , but only an arbitrage-free securities market. It assumes that the future return on securities is affected by market factors ${\ displaystyle n}$

{\ displaystyle F_ {1}, F_ {2}, \ ldots, F_ {n}}

can be explained. The following then applies to the return on the security ${\ displaystyle r_ {j}}$ ${\ displaystyle j}$

{\ displaystyle r_ {j} = E (r_ {j}) + \ beta _ {1, j} \, F_ {1} + \ beta _ {2, j} \, F_ {2} + \ cdots + \ beta _ {n, j} \, F_ {n} + \ epsilon _ {j}}

,

where is the expected return and security-specific coefficients that measure the sensitivity of the return to the -th factor. is an unobservable (disruptive) term that represents the security-specific, unsystematic risk. ${\ displaystyle E (r_ {j})}$ ${\ displaystyle \ beta _ {n, y}}$ ${\ displaystyle n}$ ${\ displaystyle \ epsilon _ {j}}$

Ross now shows that, assuming no arbitrage and a perfect capital market, each factor must be assigned a risk premium , so that the expected return on each security turns out to be ${\ displaystyle F_ {i}}$ ${\ displaystyle \ lambda _ {i}}$

{\ displaystyle E (r_ {j}) = r + \ beta _ {1, j} \, \ lambda _ {1} + \ beta _ {2, j} \, \ lambda _ {2} + \ cdots + \ beta _ {n, j} \, \ lambda _ {n}}

results. Where is the interest rate of a risk-free investment. The APT thus also demands a linear risk-return relationship. ${\ displaystyle r}$

In contrast to the CAPM, the APT basically allows several factors. A special case is the single index model , which, like the CAPM, is limited to one factor.

Examples of factors

There are basically two approaches to identify the factors of APT. On the one hand, you can use macroeconomic indicators. Empirical studies have shown that five factors explain the cost of equity with sufficient accuracy:

the index of industrial production
the short-term real interest rate
the short-term inflation rate
the long-term inflation rate
the general risk of failure

Other possible factors are the amount of money , the price of oil , and gross domestic product . The relationship between factors and the expected returns is then determined using a linear regression .

A second approach is to use factor analysis to derive synthetic factors from the empirically observed returns on various securities.

motivation

The arbitrage price model was developed because of practical problems with CAPM. The CAPM makes very strict assumptions about market equilibrium. It is assumed that all investors hold shares in a market portfolio, i.e. that the relative amount of each individual security in each securities account is the same. This market equilibrium results in an equilibrium price for each security, which depends on its correlation to the market portfolio. The CAPM would therefore be the special case of an APT that knows the price of the market portfolio as the only factor.

These assumptions have turned out to be impractical. On the one hand, the market portfolio is very difficult to identify; on the other hand, the need for econometric models with more than one factor grew .

Difference to the CAPM

Compared to the CAPM, the APT does not make three very critical assumptions:

Unlike with CAPM, investors do not have to make decisions according to the Bernoulli or (μ, σ) principle.
With the APT, no statement about the risk attitude of the investors is necessary.
The assumption of the capital market equilibrium in the CAPM is weakened at the APT: there must be no arbitrage opportunities on the capital market. Therefore, knowledge of a market portfolio is no longer necessary at APT.

In deviation from the CAPM:

Security returns follow a factor model.
There are “very many” or “infinitely many” securities on the capital market, so that investors can create portfolios without unsystematic risk through suitable diversification.

The crucial assumption of the CAPM that the market is in equilibrium and that all investors are only allowed to hold shares in the market portfolio has been weakened and replaced by the assumption that the market no longer offers any opportunities for arbitrage.

It is a model that has its theoretical background in quantitative analysis (see also financial mathematics , arbitrage ). This means that the APM can also be formulated not only in terms of relative returns, but also in terms of absolute value developments. In this formulation, the final capital is a function of the influencing factors and the initial capital. The curve is similar to the CAPM securities market line. The difference is that in the CAPM the slope of the securities market line can be calculated from the characteristics of the market portfolio. The slope of the curve from the arbitrage pricing model can only be determined empirically. It is a measure of factor risk. The associated beta is a measure of a company's sensitivity to risk. The valuation relationship can also be applied to portfolios. ${\ displaystyle \ mu - \ sigma}$

literature

Tom Copeland: Company Value . Campus Verlag, Frankfurt 2000 ( summary ).
Stephen Ross: The Arbitrage Theory of Capital Asset Pricing . In: Journal of Economic Theory . 1976, p. 341-360 (English).