If we use our common sense, we probably agree that the risk-return relationship should be positive. However, it is hard to accept that in our complex and dynamic world that the relationship will neatly conform to a linear pattern. Indeed, there have been doubts raised about the accuracy of the CAPM.

**The meaning of beta **

The CAPM contends that shares co-move with the market. If the market moves by 1% and a share has a beta of two, then the return on the share would move by 2%. The beta indicates the sensitivity of the return on shares with the return on the market. Some companies' activities are more sensitive to changes in the market - eg luxury car manufacturers - have high betas, while those relating to goods and services likely to be in demand irrespective of the economic cycle - eg food manufacturers - have lower betas. The beta value of 1.0 is the benchmark against which all share betas are measured.

**Beta > 1 **- aggressive shares

These shares tend to go up faster then the market in a rising(bull) market and fall more than the market in a declining (bear) market.
**Beta < 1** - defensive shares

These shares will generally experience smaller than average gains in a rising market and smaller than average falls in a declining market.
**Beta = 1** - neutral shares

These shares are expected to follow the market.

The beta value of a share is normally between 0 and 2.5. A risk-free investment (a treasury bill) has a b = 0 (no risk). The most risky shares like some of the more questionable penny share investments would have a beta value closer to 2.5. Therefore, if you are in the exam and you calculate a beta of 11 you know that you have made a mistake.

**Basic exam application of CAPM**

**1. Capital investment decisions **

The calculation of Ke in the WACC calculation to enable an NPV calculation

A shareholder's required return on a project will depend on the project's perceived level of systematic risk. Different projects generally have different levels of systematic risk and therefore shareholders have a different required return for each project. A shareholder's required return is the minimum return the company must earn on the project in order to compensate the shareholder. It therefore becomes the company's cost of equity.

**Example 5**

E plc is evaluating a project which has a beta value of 1.5. The return on the FTSE All-Share Index is 15%. The return on treasury bills is 5%.

**Required: **

What is the cost of equity?

**Answer: **

5% + (15% - 5%) 1.5 = 20%

**2. Stock market investment decisions**

When we read the financial section of newspapers, it is commonplace to see analysts advising us that it is a good time to buy, sell, or hold certain shares. The CAPM is one method that may employed by analysts to help them reach their conclusions. An analyst would calculate the expected return and required return for each share. They then subtract the required return from the expected return for each share, ie they calculate the alpha value (or abnormal return) for each share. They would then construct an alpha table to present their findings.

**Example 6**

We are considering investing in F plc or G plc. Their beta values and expected returns are as follows:

**Beta values Expected returns **

F plc 1.5 18%

G plc 1.1 18%

The market return is 15% and the risk-free return is 5%.

**Required: **

What investment advice would you give us?

**Answer:**

**Alpha table **

Expected returns Required returns Alpha values F plc 18% 5% + (15% - 5%) -2%

1.5 = 20%

G plc 18% 5% + (15% - 5%) +2%

1.1 = 16%

Sell shares in F plc as the expected return does not compensate the investors for its perceived level of systematic risk, it has a negative alpha. Buy shares in G plc as the expected return more than compensates the investors for its perceived level of systematic risk, ie it has a positive alpha.

**3. The preparation of an alpha table for a portfolio **

The portfolio beta is a weighted average

A common exam-style question is a combined portfolio theory and CAPM question. A good example of this is the Oriel plc question at the end of this article where you are asked to calculate the alpha table for a portfolio.

The expected return of the portfolio is calculated as normal (a weighted average) and goes in the first column in the alpha table. We then have to calculate the required return of the portfolio. To do this we must first calculate the portfolio beta, which is the weighted average of the individual betas. Then we can calculate the required return of the portfolio using the CAPM formula.

**Example 7**

The expected return of the portfolio A + B is 20%. The return on the market is 15% and the risk-free rate is 6%. 80% of your funds are invested in A plc and the balance is invested in B plc. The beta of A is 1.6 and the beta of B is 1.1.

**Required: **

Prepare the alpha table for the Portfolio

(A + B)

**Answer:**

b(A + B) = (1.6 × .80) + (1.1 × .20)

= 1.5

R portfolio (A + B) = 6% + (15% - 6%) 1.5 = 19.50%

**Alpha table **

**Expected return Required return Alpha value **

Portfolio (A + B) 20% 19.50% 0.50%

**The Alpha Value **

If the CAPM is a realistic model (that is, it correctly reflects the risk-return relationship) and the stock market is efficient (at least weak and semi-strong), then the alpha values reflect a temporary abnormal return. In an efficient market, the expected and required returns are equal, ie a zero alpha. Investors are exactly compensated for the level of perceived systematic risk in an investment, ie shares are fairly priced. Arbitrage profit taking would ensure that any existing alpha values would be on a journey towards zero.

Remember in Example 6 that the shares in G plc had a positive alpha of 2%. This would encourage investors to buy these shares. As a result of the increased demand, the current share price would increase (which if you recall from the portfolio theory article is the denominator in the expected return calculation) thus the expected return would fall. The expected return would keep falling until it reaches 16%, the level of the required return and the alpha becomes zero.

The opposite is true for shares with a negative alpha. This would encourage investors to sell these shares. As a result of the increased supply, the current share price would decrease thus the expected return would increase until it reaches the level of the required return and the alpha value becomes zero.

It is worth noting that when the share price changes, the expected return changes and thus the alpha value changes. Therefore, we can say that alpha values are as dynamic as the share price. Of course, alpha values may exist because CAPM does not perfectly capture the risk-return relationship due to the various problems with the model.

**Problems with CAPM**

**Investors hold well-diversified portfolios **

CAPM assumes that all the company's shareholders hold well-diversified portfolios and therefore need only consider systematic risk. However, a considerable number of private investors in the UK do not hold well-diversified portfolios.

**One period model**

CAPM is a one period model, while most investment projects tend to be over a number of years.

**Assumes the stock market is a perfect capital market**

This is based on the following unrealistic assumptions:

- no individual dominates the market
- all investors are rational and risk-averse
- investors have perfect information
- all investors can borrow or lend at the risk-free rate
- no transaction costs.

**Evidence **

CAPM does not correctly express the risk-return relationship in some circumstances. To cite a number of these circumstances they are, for small companies, high and low beta companies, low PE companies, and certain days of the week or months of the year.

**Estimation of future b based on past b**

A scatter diagram is prepared of the share's historical risk premium plotted against the historical market risk premium usually over the last five years. The slope of the resulting line of best fit will be the b value. The difficulty of using historic data is that it assumes that historic relationships will continue into the future. This is questionable, as betas tend to be unstable over time.

**Data input problems**

Richard Roll (1977) criticised CAPM as untestable, because the FTSE All-Share Index is a poor substitute for the true market, ie all the risky investments worldwide. How can the risk and return of the market be established as a whole? What is the appropriate risk-free rate? However, despite the problems with CAPM, it provides a simple and reasonably accurate way of expressing the risk-return relationship. Quite simply, CAPM is not perfect but it is the best model that we have at the moment.

Additionally, some critics believe that the relationship between risk and return is more complex than the simple linear relationship defined by CAPM. Another model may possibly replace CAPM in the future. The most likely potential successor to CAPM is the arbitrage pricing model (APM).

**The Arbitrage Pricing Model - APM**

The CAPM contends that the only reason the return of a share moves is because the return on the market moves. The magnitude of a share's co-movement with the market is measured by its beta. If a share has a beta of two and the market increases by 1%, we would expect the share's return to increase by 2%. If the market increases by 5% we would expect the share's return to increase by 10%. Remember that the market only gives a return for systematic risk. Therefore, any changes in the market return are due to a large number of macro-economic factors.

**The model**

The arbitrage pricing model, developed by Stephen Ross in 1976, attempts to identify all of the macro-economic factors and then specifies how each factor would affect the return of a particular share. The APM is therefore more sophisticated than CAPM in that it attempts to identify the specific macro-economic factors that influence the return of a particular share. Commonly invoked factors are:

- inflation
- industrial production
- market risk premiums
- interest rates
- oil prices.

Each share will have a different set of factors and a different degree of sensitivity (beta) to each of the factors. To construct the APM for a share we require the risk premiums and the betas for each of the relevant factors.

Return on a share = RF + Risk premium F1.b1 + Risk premium F2.b2 + Risk premium F3.b3 + . . .

**Example 8**

beta 1 = the effect of changes in interest rates on the returns from a share

beta 2 = the effect of oil prices on the returns from a share

A share in a retail furniture company may have a high beta 1 and a low beta 2 whereas a share in a haulage company may have a low beta 1 and a high beta 2. Under the APM, these differences can be taken into account. However, despite its theoretical merits, APM scores poorly on practical application. The main problem is that it is extremely difficult to identify the relevant individual factors and the appropriate sensitivities of such factors for an individual share. This has meant that APM has not been widely adopted in the investment community as a practical decision-making tool despite its intuitive appeal.

Before we conclude the articles on the risk-return relationship, it is essential that we see the practical application of both portfolio theory and CAPM in an exam-style question. Indeed, it is quite common to have both topics examined in the same question as demonstrated in Oriel plc below.

**Exam Style Question (including the multi-asset portfolio exam trick)**

**Oriel plc**

Oriel plc is considering investing in one of two short-term portfolios of four short-term financial investments. The correlation between the returns of the individual investments is believed to be negligible (zero/independent/no correlation). See Portfolio 1 and Portfolio 2. The market return is estimated to be 15%, and the risk free rate 5%