 # Arbitrage Price Theory vs. Capital Asset Pricing

## Arbitrage Price Theory vs. Capital Asset Pricing

Risk is inevitable for all types of assets, but the risk level for assets can vary. It all depends on the specific investment itself. Fortunately, even though no one can truly determine risk in an unpredictable market, there are ways to calculate the level of risk that comes naturally with a particular asset.

There are two popular methods you can use:

• Capital asset pricing model (CAPM)
• Arbitrage pricing model (APM)

Both are based on cost against the rate of return and have their own uses and downsides. The theorems are a bit complicated to understand at first, but taking your time with them will help you get an idea of how they are applied in real life.

### Capital Asset Pricing Model

An asset’s risk level, when calculated using the CAPM, uses the formula:

ERi = Rf +βi(ERm-Rf)

• ERi  is the expected return of investment
• Rf  is the risk-free rate (Rf)
• Bi is the beta figure of the investment
• ERm is the return expected of the market

An analyst determines the Rf, Rm, and βi figures, but investors usually use a beta figure provided by a third party. Analysts and investors use CAPM mostly to calculate an asset’s fair price during arbitrage.

In arbitrage, two transactions are carried out at the same time in two separate markets. The investor takes advantage of the price differences between the two. Investors generally consider this risk-free. However, typical changing market conditions may decrease profit immensely when they conduct CAPM evaluations.

Issues with the CAPM are mainly due to the several assumptions it makes. These assumptions include:

• Zero taxes and other transaction costs
• All assets are dividable as well as marketable
• Investments in assets can be made freely without restrictions
• Investors maximize their utility
• Prices are not affected by investors

Another drawback is that CAPM calculations are made for just one period, with the formula being too linear. The biggest issue, though, is that calculations are not even consistent with empirical or actual results.

### Arbitrage Pricing Theory

The Arbitrage Pricing Theory (APT) is an alternative to CAPM. With APT, each asset’s payoff will come out as a weighted average of all the rest in a portfolio. The APT formula is:

E(rj) = rf + bj1RP1 +bj2 RP2 +……bjnRPn

• E(rj) is the asset’s expected rate of return
• Rf is the risk-free rate
• Bj is the asset’s return sensitivity
• RP is the risk premium

The idea behind APT is that an asset’s return depends on two key factors: the macroeconomic environment (inflation, interest rate fluctuations, etc.) and the possibility that the asset will move according to environmental factors.

APT assumes that returns will follow the formula and that investors are risk-averse and think the same way. It also supposes that there are no transaction costs nor restrictions on asset availability or short sales and that arbitrage is impossible in equilibrium.

The main problem with APT, however, is that it tries to accurately measure the risk for all assets. While you can determine a “factor portfolio” (reflecting very similar risks), the risk level is still essentially influenced by macroeconomic factors.

### Comparing CAPM and APT

Initially, you can easily assume that the CAPM and APT formulas are the same, but there is only one factor and one beta involved in the CAPM model. In contrast, the APT formula has several, including non-company factors that call for the asset’s beta as per every independent factor. The APT does not offer information as to what these factors might be, though, which means APT users should examine all factors that could possibly impact the asset’s returns. On the other hand, the CAPM relies on the difference between the expected and the risk-free rate of return.

APT is reliable for the medium to long term but is often inaccurate for short-term calculations. The opposite is true for CAPM. APT concentrates more on risk factors instead of assets. This gives it an advantage over CAPM simply because you do not have to create a similar portfolio for risk assessment.

While CAPM assumes that assets have a straightforward relationship, APT assumes a linear connection between risk factors. That means if there is no linear relationship, the models cannot accurately determine outcomes.

Still, both models are unrealistic in assuming that assets are unlimited in demand and availability, that you can get these assets for free, and that investors arrive at the same conclusions.

This seems conflicting, considering the most successful investors are probably those who can appreciate largely unseen potential in the market. Investors who adopt the same outlook can create a bubble that minimizes the asset’s inherent risks once asset price increases. In this case, measuring an asset’s risk according to the market’s temperament can be riskier than using CAPM or APT.

Theoretically speaking, CAPM or APT analysis may lead to lower risk as investors use set mathematical formulae. When analysts come up with risk projections, their subjective decisions can make the picture even more complex. And while they may be rational and objective when studying risk levels, their opinions will reduce the quality of their mathematical projections.

In short, the calculation is only as good as the professional who decides the factors that lead to the results.

### Wrapping Up

Even as CAPM and APT help assess market risks, they both remain static and rely on too few factors to forecast risk in an extremely complicated market. They may use mathematical principles to work, but they are still basically subjective. The analyst behind the calculation can use whatever factors they feel apply to every case.

Overall, the APT model is designed for efficiency and works to estimate the rate of return of risky assets. The rate of return using the APT model can come in handy in terms of assessing whether or not stocks are priced appropriately. Empirical tests have proven the APT formula is more reliable compared to CAPM. But in many instances, you can find similar outcomes using the CAPM model, which is comparatively simpler.