In last month’s issue, we examined the basic Capital Asset Pricing Model (CAPM). We saw that the model states that the expected return on an investment is directly proportional to that investment’s volatility (or Beta, ß).

The relationship is defined as:

r = R_{f }+ ß x R_{m }– R_{f}

where:

- r is the expected return on the investment
- R
_{f }is the return on a risk-free investment (for example, a short-term government security or cash) - R
_{m }is the rate of return for the appropriate asset class or the market as a whole - ß is the volatility of the security with respect to the appropriate asset class or the market.

A number of empirical studies questioned the relationship between expected return and volatility which the CAPM predicts.

Two main concerns were identified:

- That the security market line (which shows the relationship between risk, in the form of Beta, and return) is too flat. If this is the case, there is only a marginal increase in expected return as volatility grows.
- Return varies according to size – smaller companies have been found to provide greater returns than larger ones.

Some criticisms of CAPM after empirical studies were inevitable. This is because the CAPM was developed as a predictive model to provide the expected return on an investment, whereas the empirical studies used actual observed results after the event. There are many reasons why a return is not as expected. As a predictive model, the CAPM cannot take them into account.

Moreover, it is possible to explain the concerns. For example, the return for smaller companies can be explained by the liquidity of their assets. Smaller companies’ equity will be relatively less liquid than that in larger companies. There will be higher transaction costs associated with trades in smaller companies. Investors in smaller companies will insist on a higher return to compensate for these higher costs.

#### Arbitrage Pricing Theory

As an alternative to CAPM, the Arbitrage Pricing Theory was developed. This assumes that an investment return is dependent on macroeconomic factors as well as the imperfections in the market (or ‘noise’).

A company’s share price, for instance, will be subject to two different types of risk. Firstly, there is the performance of the economy as a whole – this cannot be reduced by diversification within a portfolio. Secondly, each individual asset within a portfolio will be affected by factors specific to the issuer – these can be reduced within a diversified portfolio.

In other words, the theory argues that as long as a portfolio is diversified, an investor need only consider the macroeconomic factors when identifying expected risk.

Therefore the theory states that there are three key stages to identifying expected returns:

- Identify the relevant macroeconomic factors. These will be the factors that will most affect any cash flows associated with the investment and their value. There is, however, no agreement as to what these factors should be.
- Estimate the risk premium on each factor.
- Estimate the risk sensitivity of each investment to these factors.

These could then be used to calculate the expected return on an investment, thus:

r = b_{1}R _{factor1 }+ b_{2 }R _{factor2 }+ … + R_{f}

where:

- r is the expected return on the investment
- R
_{f }is the return on a risk-free investment (for example, a short-term government security or cash) - b
_{1, }b_{2 }, etc. are the investment’s sensitivities to each factor, and - R
_{factor1 }etc. are the risk premiums for each factor.

This result should then provide the expected return on an investment which should be related to the investment’s contribution to risk within a portfolio.

The arbitrage pricing theory is so-named because, if the relationship did not hold, then it would be possible to make a profit by selling those assets that offer a lower than expected return and purchasing those promising a higher return, given the relative exposure to risk.

However, the arbitrage pricing model is not as commonly used as the CAPM. One reason for this is the difficulty in determining the different and relative macroeconomic factors.