Arbitrage Pricing Theory (APT) & CAPM Explained: SML, Valuation and Examples

Arbitrage Pricing Theory 

Arbitrage pricing theory (APT) is a well-known method of estimating the price of an asset. The theory assumes an asset's return is dependent on various macroeconomic, market and security-specific factors.

Definition 

1. An asset pricing model based on the idea that an asset's returns can be predicted using the relationship between that same asset and many common risk factors. Created in 1976 by Stephen Ross, this theory predicts a relationship between the returns of a portfolio and the returns of a single asset through a linear combination of many independent macro-economic variables.
2. The arbitrage pricing theory (APT) describes the price where a mispriced asset is expected to be. It is often viewed as an alternative to the capital asset pricing model (CAPM), since the APT has more flexible assumption requirements. Whereas the CAPM formula requires the market's expected return, APT uses the risky asset's expected return and the risk premium of a number of macro-economic factors. Arbitrageurs use the APT model to profit by taking advantage of mispriced securities. A mispriced security will have a price that differs from the theoretical price predicted by the model. By going short an overpriced security, while concurrently going long the portfolio the APT calculations were based on, the arbitrageur is in a position to make a theoretically risk-free profit.

How It Works/Example

APT is an alternative to the capital asset pricing model (CAPM). Stephen Ross developed the theory in 1976.

The APT formula is: E(r_j) = r_f + b_j1RP₁ + bj2RP2 + b_j3RP₃ + b_j4RP₄ + ... + b_jnRP_n

where: 

• E(r_j) = the asset's expected rate of return;
• r_f = the risk-free rate; 
• b_j = the sensitivity of the asset's return to the particular factor; 
• RP = the risk premium associated with the particular factor.

The general idea behind APT is that two things can explain the expected return on a financial asset:

1. macroeconomic/security-specific influences
2. The asset's sensitivity to those influences. This relationship takes the form of the linear regression formula above.

There are an infinite number of security-specific influences for any given security including inflation, production measures, investor confidence, exchange rates, market indices or changes in interest rates. It is up to the analyst to decide which influences are relevant to the asset being analyzed. Once the analyst derives the asset's expected rate of return from the APT model, he or she can determine what the "correct" price of the asset should be by plugging the rate into a discounted cash flow model.

Note that: APT can be applied to portfolios as well as individual securities. After all, a portfolio can have exposures and sensitivities to certain kinds of risk factors as well.

Why It Matters

The APT was a revolutionary model because it allows the user to adapt the model to the security being analyzed. And as with other pricing models, it helps the user decide whether a security is undervalued or overvalued and so he or she can profit from this information. APT is also very useful for building portfolios because it allows managers to test whether their portfolios are exposed to certain factors.

APT may be more customizable than CAPM, but it is also more difficult to apply because determining which factors influence a stock or portfolio takes a considerable amount of research. It can be virtually impossible to detect every influential factor much less determine how sensitive the security is to a particular factor. But getting "close enough" is often good enough; in fact, studies find that four or five factors will usually explain most of a security's return: surprises in inflation, GNP, investor confidence and shifts in the yield curve.

Capital Asset Pricing Model (CAPM)

A model that describes the relationship between risk and expected return and that is used in the pricing of risky securities.

rₐ = r_f + βₐ ( r_m − r_f )

Where:

• r_f = Risk-free rate 
• βₐ = Beta of the security 
• r_m = Expected market return

The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk measure (beta) that compares the returns of the asset to the market over a period of time and to the market premium (Rm-rf).

The CAPM says that the expected return of a security or a portfolio equals the rate on a risk-free security plus a risk premium. If this expected return does not meet or beat the required return, then the investment should not be undertaken. The security market line plots the results of the CAPM for all different risks (betas). Using the CAPM model and the following assumptions, we can compute the expected return of a stock in this CAPM example: if the risk-free rate is 3%, the beta (risk measure) of the stock is 2 and the expected market return over the period is 10%, the stock is expected to return 17% (3%+2(10%-3%)).

Assumptions for Capital Asset Pricing Model (CAPM) 

The use of CAPM and its assumptions can be helpful in estimating the expected return of a stock. The basic assumptions of CAPM include:

1. The model aims to maximize economic utilities.
2. The results are risk-averse and rational.
3. The results are price takers. This implies that they cannot influence prices.
4. The model can lend and borrow unlimited amounts under the risk-free rate of interest.
5. The model presumes that all info is available at the same time to all investors.
6. The model trades without taxation or transaction costs.
7. The model deals with securities all of which are highly divisible into small parcels.
8. The results are widely diversified across a range of investments.

Question  

Determine the expected return on new co’ stock using the capital assets pricing model. New co.’s beta is 1.2. Assume the expected return on the market is 12% and the risk-free rate is 4%.

• Answer – E(R) - 4% + 1.2 (12% - 4%) = 13.6%.

The security market line (SML) 

The SML is derived from the CAPM, solving for expected return. However, the level of risk used is the beta, the slope of the security market line.

Beta – Bet is the measure of stock’s sentivity of return of charges in the market. It is measuring a systematic risk. 

Beta= Covariance of stock to the market ÷ Variance of the market.

Question 

Example of beta – assume the covariance between new co.’s stock and the market is 0.001 and variance market is 0.0008 what is the beta of new co.’s stock?

• Answer – Beta new co.’s= 0.001 / 0.0008 = 1.25

Determine whether a security is under over a properly valued 

The SML line can be derived using CAPM, solving for the expected return using beta as the measure of risk. Given that interpretation and a beta value for a specific security. We can than determine the expected return of the security with CAPM. Then using the expected return of the securities derived from the CAPM, an investor can determine whether a security is overvalued, undervalued or properly valued.

Question 

An investor anticipates New co.’s security will reach $ 30 by the end of the year new co.’s beta is 1.3 assume the return on the market is expected return of new co.’s stock in one year and determine whether the stock is undervalued, overvalued or properly valued, overvalued or properly valued with current value of $ 25.

•  Answer – E (R ) new co = 4% + 1.3 (16%-4%) = 20%

Given the expected return of new co stock using CAPM is 20 % and the investor anticipated a 20 % return, the security would be properly valued.

• If the expected return using the CAPM is the higher than the investor required return, the security is undervalued and the investor should buy it.
• If the expected return using the CAPM is lower than the investor required return, the security is overvalued and should be sold.

Characteristic Line 

A line formed using regression analysis that summarizes a particular security or portfolio systematic risk and rate of return. The rate of return is dependent on the standard deviation of the assets return and slope of the characteristic line, which is represented by the assets beta. A characteristic line of stock is the same as the security market line and is very useful when employing the CAPM, or when using modern portfolio formation techniques. the slope of the line which measure of systematic risk, determine the risk-return trade off. According to this metric, the more risk you take on as measure by variability in returns – the higher the returns you can expect to earn. There is considerable controversy regarding the use of beta as measure of risk & return.

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Sandeep Ghatuary

Finance & Accounting blogger simplifying complex topics.

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