Arbitrage Pricing Theory: Opportunities And Limitations

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Imagine travelling and observing that apples in one region of the country are cheaper compared to the prices charged in your home region. Given that scenario, a rational human being will seek to maximize their “level of…utility [as they] would rather be better off than worse off” (Kenton, 2018). Subsequently, the rational individual will purchase the apples from the cheaper location to sell in the more costly area earning a risk free profit, ceteris paribus. This is known as arbitrage. Arbitrage is when an individual makes “risk-free gains with zero initial outlay” (Bailey, 2005) as a result, gaining ‘something for nothing’. This is an example of a market failure due to imperfect information, thus the market being inefficient. According to Adam Smith, “led by the invisible hand” (Smith, 1998) the market will self-correct and clear at the equilibrium. (Bailey, 2005) stated that “arbitrage principle [is] the assertion that arbitrage opportunities vanish in market equilibrium”; hence the market is efficient at this point. The purpose of this paper is to analyse the role of the arbitrage principle in the analysis of the asset price.

Arbitrage is said to be a keynote in finance especially financial markets. Following on from the scenario in the introduction, assume a frictionless market i.e. absence of transaction costs and institutional trading restraints. Arbitrage will ultimately equalise the prices of the apples in both markets due to the excess demand in one market leading to a higher price and the excess supply in the other market leading to a fall in prices. Eventually the prices in both locations will converge as arbitrage generalises the law of one price (LoOP). This states that identical assets with identical returns should have the same prices. Subsequently, in its simplest form, this is how arbitrage links asset prices in different geographic locations. In the absence of arbitrage opportunities (AoAO), there are zero payoffs.

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The diagrams above show the simple case of arbitrage. The arbitrageur (investor) purchases the cheaper apples from market A priced at £8 and simultaneously sells them in market B for the price of £10. This earns him an arbitrage profit of £2. Overtime, more arbitrageurs will also do the same leading an increase in demand in market A; leading to a shift along the supply curve from D to D’. This will lead to an increase in price to £9 whilst simultaneously doing the same to market B where increased supply, S to S’ leads to a fall in price from £10 to £9. Hence, showcasing how arbitrage can lead to the LoOP. Ultimately, arbitrage leads to an increase in the quantity supplied in both markets. Market A’s quantity increases from 5 apples to 7 apples. Similarly, market B’s quantity rises from 6apples to 8 apples

AoAO is important in the analysis of asset prices as it makes one assumption about investors; that “more wealth is preferred to less” (Bailey, 2005). This is the first of the three main propositions that allow the AoAO to hold. The first proposition states that the arbitrage principle holds in a frictionless market if and only if, there is an investor who prefers more wealth to less and for whom an optimal portfolio can be constructed. This essentially means, that the rational investor is willing to part with their money in order for the arbitrageur to exploit the arbitrage opportunities. In pursuit of more wealth “the investor would seek to magnify this portfolio…to an unbound extent” (Bailey, 2005), this means that he would prefer to hold more assets in his portfolio which will require more investment funds. The second proposition states that the arbitrage principle is equivalent to the existence of positive state prices, q1, q2, …, ql such that pj=q1v1j +q2v2j +…+qlvl . This is known as linear pricing rule. A state price is a price of an asset with payoff of “one unit of wealth” (Bailey, 2005) in a given state, and zero in all other states. Assets with this property are known as Arrow securities. Arrow securities are a constructed “portfolio of assets in such a way as to yield the required payoff” (Bailey, 2005). The linear pricing rule alludes to the risk-neutral valuation relationship (RNVR) often referred to as the martingale probabilities. The RNVR leads to the third and final proposition which states that the linear pricing rule is equivalent to the existence of (1) a risk free rate of return, r0, with associated discount factor , δ≡1/(1+r0); and (2) probabilities, π1,π2,…, πl, one for each state, such that pj= δE*[vj] j=1,2,…, n. This means that in the absence of arbitrage opportunities, state prices are present and vice versa.

Due to the competitive nature of the markets, the existence of many traders, means that they are all price takers therefore, no one trader is large enough to dictate the price. The arbitrage principle has helped financial markets become more efficient as it erodes away price discrimination. This has been due to the process of globalisation leading to technological improvements. Globalisation is the process by which countries become more integrated and interdependent thus raising their GDP levels. These improvements in technology can be seen in telecommunications and internet speeds. This has allowed for perfect information and quick transfer of data hence investors in different geographical locations receive the same information simultaneously. Consequently, no one investor can legitimately gain an unfair advantage with scope of exploiting an arbitrage opportunity. The innovation in equity trading such as electronic trading systems have led to linked computer systems which have narrowed the spread between the buying and selling prices thus reducing the likelihood of arbitrage. Also, the development of high frequency trading where computer based algorithms are used to trade, have made the likelihood of arbitrage opportunities almost non-existent. This is due to the fact that these supercomputers are able to process large volumes of data and new information to make trades quickly (microseconds). Subsequently, they can exploit prices differences across platforms. If an arbitrage opportunity arises, it is consumed within microseconds.

AoAO has also allowed for assets to be fairly priced across the plane subsequently individuals can invest with confidence and expect to be fairly compensated for their investments.

In order to gain arbitrage profit, an investor has to work diligently to uncover arbitrage opportunities thus requires vast sums of capital to make worthwhile gains. Subsequently, (Shleifer and Vishny, 1997), observed that the “brains and resources are separated by an agency relationship”, this division, shifts power to the professional investors. The pursuit to obtain larger than proportionate returns is likely to lead to the investors taking risky trades which are not in the best interest of their clients. Outside investors are aware that there is likelihood that they could lose all their funds so do the professional investors. Therefore, due to the knowledge by both parties, the professional investors are likely to increase their exposure to risk thus moral hazard as they are insured. There is also asymmetric information in the sense that “money managers window dress their portfolios to impress investors” (Shleifer and Vishny, 1997), this means that investors are defrauded as they ‘prefer more wealth to less’. Thus make an adverse selection which ultimately leads to market inefficiency. Also, the complexity of the processes employed in exploiting arbitrage opportunities means that “investors have no information… [and]… do not know the trading strategy employed” ” (Shleifer and Vishny, 1997), this shows that they are blindingly parting with their money with expectations of a better future payoff. However, it would not be fair to blame the investors as “arbitrageurs do not share their information…and cultivate secrecy to protect their knowledge from imitation”,” (Shleifer and Vishny, 1997). This almost indicates the niche market that arbitrageurs operate within as they essentially carve out the market. As a consequence of their lack of knowledge (Shleifer and Vishny, 1997), argued that “past performance of the arbitrageurs… determine the resources they…manage, regardless of the actual opportunities available in their market”: because of this, arbitrageurs are judged on their past achievements even when there are very few.

Factor models are expressed in terms of rates of return as asset prices depend on a small number of factors. Factors are the determinants of assets rates of return. Arbitrage pricing theory (APT) is the combination of factor models and the arbitrage principle as its function is “to bring prices to fundamental values and to keep markets efficient” (Shleifer and Vishny, 1997). APT is a valuation concept that relies on annualized yield hence allowing investors to take calculated risks. It has experienced a surge in popularity in recent times due to its ability to provide a fair evaluation about assets. It states that asset prices are influenced by various factors namely common risk and independent risk. Common risk is a risk that is perfectly correlated and affects all assets. It is commonly known as systematic risk. On the other hand, an independent risk is a risk that is uncorrelated and affects just a particular asset. APT is used to assess the expected rate of return of a stock and is illustrated by the following equation

rj = bj0 + bj1F1 + εj for all j bj0,bj1 are parameters

A single factor model

In a single factor model, F1 measure the factor. Also there is a single separate line for each asset. This means that bj0 and bj1 are different assets.

APT makes certain assumptions namely:

  1. Indentifying a portfolio with zero funds required from the offset.
  2. Remove the systematic risk
  3. Approximately remove the unsystematic risk
  4. And lastly, artbitrage portfolio return has to approximate to zero.

The APT prediction then becomes;

μj= λ0 + λ1bj1 ,for all j where

λ0=ro if a risk free asset exists

λ1= risk premium of factor 1

Therefore, asset j’s risk premium stands as: μj – λ0 or μj – ro.

Multiple Factor models

APT predicts the k-factor to be,

μj= λ0 + λ1bj1 + λ2bj2+…+ μj= λ0 + λkbjk , j=1,2,…,n

The risk premium can correspond either to an asset or to a factor.

APT prediction in a single factor model

The risk premium for unsystematic risk is zero, so investors are compensated for holding unsystematic risk. If the unsystematic risk off stocks were to be compensated with an additional risk premium, the investors would buy the stocks, earn the additional premium and simultaneously diversify and eliminate the risk. By doing so, investors could earn an additional premium without taking on additional risk. This arbitrage opportunity would be exploited and eliminated. Because can eliminate unsystematic risk for free by diversifying their portfolio, they will not require or earn a reward or risk premium for holding it.

The risk premium of a securities is determined by its systematic risk and does not depend on it unsystematic risk. This implies that a stock volatility which is the measure of total risk is not especially useful in determining the risk premium that investors will earn.

APT is similar to the efficient market hypothesis (EMH) which states that assets are fairly priced based on future returns as all available information is reflected in the asset prices without delay. This means that there are no arbitrage opportunities as all information is already reflected in the asset prices. Although this doesn’t suggest that prices are always right as investors might under or overestimate the value of the news. New information about an asset has the ability to increase its value in turn reducing it perceived risk. This means that the market reacts to the news. Thus there is no opportunity to earn profits as all information is already reflected in the market prices. However, this doesn’t mean that the prices are correct in every case as investors may under or overestimate the value of the news. When many assets are combined in a large portfolio, the firm specific risks for each asset will be averaged out and be diversified. The systematic risk will affect all firms and will not be diversified.

The random walk hypothesis states that the EMH has implications for the evolution of stock prices. This suggests the market prices fully incorporate existing information and expectations. Then price changes reflect new information only. Given that news is random, it is impossible to forecast from past events. Subsequently, price changes are random and can not be forecast.


To conclude, the arbitrage principle is a keynote of financial markets. It has led to improvements in the markets which have ultimately led to the near elimination of arbitrage opportunities.


  1. Bailey, R. (2005). The Economics of Financial Markets. Cambridge: Cambridge University Press, pp.166-182.
  2. Kenton, W. (2018). Rational Behaviour. [online] Investopedia. Available at: [Accessed 16 Jan. 2019].
  3. Smith, A. (1998). An inquiry into the nature and causes of the wealth of nations. [Milano]: Cofide,.p. 456.
  4. Shleifer, A. and Vishny, R. (1997). The limits of Arbitrage. The journal of Finance, [online] LII(1), pp.35-55. Available at: [Accessed 18 Jan. 2019].


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