Arbitrage pricing theory
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Arbitrage pricing theory (APT) holds that the expected return of a financial asset can be modelled as a linear function of various macro-economic factors, 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 theory was initiated by the economist Stephen Ross in 1976.
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The APT model
If APT holds, then a risky asset can be described as satisfying the following relation:
- <math>E\left(r_j\right) = r_f + b_{j1}RP_1 + b_{j2}RP_2 + ... + b_{jn}RP_n<math>
- <math>r_j = E\left(r_j\right) + b_{j1}F_1 + b_{j2}F_2 + ... + b_{jn}F_n + \epsilon_j<math>
- where
- <math>E(r_j)<math> is the risky asset's expected return,
- <math>RP_k<math> is the risk premium of the factor,
- <math>r_f<math> is the risk free rate,
- <math>F_k<math> is the macroeconomic factor,
- <math>b_{jk}<math> is the sensitivity of the asset to factor <math>k<math>, also called factor loading,
- and <math>\epsilon_j<math> is the risky asset's idiosyncratic random shock with mean zero.
That is, the uncertain return of an asset <math>j<math> is a linear relationship among <math>n<math> factors. Additionally, every factor is also considered to be a random variable with mean zero.
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 assets may never surpass the total number of factors (in order to avoid the problem of matrix singularity), respectively.
Derivation
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.
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 arbitrage by investors will bring asset prices back into line with the returns expected by the model.
Arbitrage mechanics
In the APT context, arbitrage consists of trading in two assets – one which is mispriced and one which is correctly priced. The arbitrageur sells the asset which is too expensive and uses the proceeds to buy one which is correctly priced (or sells a correctly priced asset and uses the proceeds to buy the asset which is 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 macro-economic factors, and sensitivity to changes in each factor is represented by a factor specific beta coefficient.
The correctly priced asset here, is, 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. 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 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 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
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 exposure 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 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).
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:
- 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)
- timeous and accurate information on these variables is required
- 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:
- surprises in inflation;
- surprises in GNP;
- surprises in investor confidence;
- surprise shifts in the yield curve.
APT and asset management
See also
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 pricing, Journal of Economic Theory, v13, 1976
External links
- The Arbitrage Pricing Theory (http://viking.som.yale.edu/will/finman540/classnotes/class6.html) Prof. William N. Goetzmann, Yale School of Management
- The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning (http://www.cfapubs.org/faj/issues/v51n1/pdf/f0510122a.pdf) (PDF), Richard Roll and Stephen A. Ross
- The arbitrage pricing theory (http://www.moneymax.co.za/articles/displayarticlewide.asp?ArticleID=273656), The Investment Analysts' Society of South Africa
- Arbitrage Pricing Theory (http://www.nuim.ie/academic/economics/tflavin/FlavinT.html) (PDF), Thomas Flavin, National University of Irelandes:Teorķa del arbitraje