Risk and Return Concepts

Readings: Chapter 5

At the end of this unit students should be able to: 

1. Define dollar return and rate of return.

2. Define risk and calculate the expected rate of return, average rate of return, standard deviation, and coefficient of variation for a probability distribution.

3. Specify how risk aversion influences required rates of return.

4. Graph diversifiable risk and market risk; explain which of these is relevant to a well-diversified investor.

5. Explain how and why a portfolio’s risk may be reduced.

6. State the basic proposition of the Capital Asset Pricing Model (CAPM).

7. Explain the significance of a stock’s beta coefficient, and use the Security Market Line to calculate a stock’s required rate of return.

8. List changes in the market or within a firm that would cause the required rate of return on the firm’s stock to change.

9. Identify concerns about beta and the CAPM.

Definitions and Computations

As was mentioned in our introductory lecture, a factor which affects the goal of shareholders' wealth maximization is the element of risk. Risk reflects the uncertainty surrounding the outcome of future events. In the case of a security, a risky security is one in which the expected outcome (pay-off/return) may be different from the actual outcome. The greater the difference, the more risky the security.

Measuring Risk

Researchers have developed the use of the term, standard deviation to measure risk (This risk element as measured by the standard deviation is also known as 'stand alone' or 'total' risk, and refers to the risk inherent in an investment/asset if that investment/asset were held in isolation). Now once we speak about future events, of which we are never certain, we have to incorporate the likelihood of the outcomes materializing. This likelihood is accounted for by the use of probabilities. However, before we evaluate the riskiness of an expected outcome we need to first determine the expected outcome/return itself.

Expected Return or Mean:

Note - the result obtained will be in the same denomination as the individual returns (i.e. $ then $ or % then %)

Where P   = the probability of state i occurring and k = the expected return if state i occurs and n = the number of different states of occurrences..

Having determined the expected return, we can now find the standard deviation, which is given by :

Note - the result obtained will also be in the same denomination as the individual returns (i.e. $ then $ or % then %)

Illustration 1

Stock A may provide you with the following returns based on the state of nature that exists: State of Nature                   Probability of state occurring               Return           Boom                              0.20                                          70%           Stagnation                       0.50                                           30%           Recession                       0.30                                          -10%

The expected return is: 0.2 x 70 = 14 0.5 x 30 = 15 0.3 x -10 = -3 = 26 %   The standard deviation is:

This stock has an expected return of 26% and a standard deviation of 28%.

Illustration 2 Stock B may provide you with the following returns based on the state of nature that exists: State of Nature                   Probability of state occurring               Return           Boom                              0.30                                          21%           Stagnation                      0.40                                          11%           Recession                       0.30                                          -2%

The expected return is:

The standard deviation is:

• This stock has an expected return of 10.1% and a standard deviation of 9%.

Because of the smaller range of variability, Stock B is less risky as illustrated below in the graph:

Graph 4.1

.

Which stock is the preferred option?

The preferred option depends on your attitude towards risk: If you are risk averse, then you hate risk and initially may decide to choose option 2. If you love risk you may decide to choose option A, because of the higher expected return. However, as in this situation, once the expected returns are different for each option, a more scientific approach to determining the better option is to calculate what is called the "Coefficient of Variation (CV)"

This is found by the formula:

The result which is expressed in absolute terms, gives a figure which expresses the amount of risk involved per unit of return.

For Stock A,

  For Stock B, 

In finance this is the preferred approach, hence Stock B is better as it provides only 0.891% of risk for every 1% of return as opposed to Stock A which is 1.077% of risk for every 1% of return.

Note, in reference to the last point - As it relates to the usage of the CV, always compute it to decide upon investments if a decision cannot be based solely upon returns or upon standard deviations. In other words we assume in finance that every individual is risk averse.  

Risk aversion and rates of returns

As just mentioned above, an individual who is risk averse hates risk. Consequently, he or she will demand a greater rate of return for a risky investment to compensate themselves for this risk element. Therefore, in summary, riskier investments will attract greater rate of returns.

Realized returns/historical data

In our previous discussion we looked at future events. It is common for historical data to be used to analyze the return and riskiness involved with a particular investment.

Illustration 3

Stock C had the following prices and issued the following dividends throughout the last 6 months:

 Month                             Price $                  Dividends $                    

Jan                                  13.75                              -

Feb                                 15.25                              3.00

Mar                                 17.85                              1.00

Apr                                 16.50                              -

May                                 15.00                              4.00

Jun                                  18.05                              2.00

 A shareholder bought this stock in January for $13.75

If we recall from the introductory lecture, the wealth of a shareholder is effectively improved if the price of each of his shares increases. It is also improved if he received any dividends. We can always quantify this improvement by calculating his return for the periods in which he held this share. Using the illustration, his return for the period Jan-Feb can be computed using the following two steps;

(a)      computing his capital gains yield (change in price)  

(b)             computing his dividend yield (his cash inflow as % of the original price) 

The sum of these two will give his total yield/return for the period:

In general the holding period return for any security for time t can be found by the following formula:

Let's complete the other periods: 

Feb-Mar:

Mar-Apr:

Apr-May: 

May - Jun:  

We now have the returns for the five periods. We can now calculate the shareholder's average return for the 6-month period.

Note - we use the term average because the returns have already taken place (as opposed to expected).

Average Return: (Note the change in the formula)

Therefore the average return is :

To determine the standard deviation we must note that - the formula changes as well

Standard deviation:

Therefore the standard deviation is:

The formula for the CV remains the same, therefore the CV is:

Therefore the stock provided 0.87% risk per 1% of return.

Portfolio Risk  

Broadly speaking risks in Finance can be classified into three types of risks - Business Risk, Financial Risk and Portfolio Risk 

Business Risk refers to the uncertainty a company has with regard to its operating income.

Financial Risk  refers to the extent to which a company is financed by debt and the volatility of its net income caused by the corresponding interest expenses .

Our focus, however will be on Portfolio Risk.   

As previously mentioned, risk that is estimated by the standard deviation is known as 'stand alone' or 'total risk' which is comprised of

(1)    Non-diversifiable risk/ market risk/ systematic risk

(2)    Diversifiable risk/firm-unique risk/company specific risk/un-systematic risk

Non-diversifiable risk - is that quantity of risk which cannot be diversified away by combining assets into portfolios. It results from general macro-economic factors such as inflation, interest rates, war etc.

Diversifiable risk is that component of risk which results from factors which are unique to the individual firm e.g. the type of product it produces, the industry it is in or the level of gearing (amount debt in its capital structure). This type of risk can be reduced or totally removed by appropriately combining individual assets with other assets in a portfolio. So then, diversification is the process whereby individual assets are combined with each other in a portfolio in an attempt to increase returns without a commensurate increase in risks.

Graph 4.2: 

The reduction of portfolio risk by diversification is only possible if assets which are not perfectly positively correlated with the existing portfolio are added.

Correlation

Correlation in this context refers to the degree to which one asset moves with another asset. Correlation between two assets is measured by what is called the correlation coefficient,  'r'. The values for 'r' ranges from   +1   to  -1 .  

The ideal objective of diversification of a portfolio is to add assets which are perfectly negatively correlated to the existing assets in the portfolio. 

Therefore, decreases in the returns of some will be offset by increases in the returns of others. In reality, one never finds assets which are perfectly negatively correlated, however the closer the correlation coefficient is to -1, then the more the portfolio risk will be reduced.

Expected Return and standard deviation of a portfolio

For a n-asset portfolio, the expected return is given by:

Where w_i is the weight of asset i. (That is, the proportion of funds invested in asset i) 

For a 2-asset portfolio (stocks a and b), the standard deviation is given by:

Where r_ab is the correlation coefficient between stocks a and b.

Note - as mentioned, this is the standard deviation formula for a 2-asset portfolio. The formula for a portfolio which exceeds 2 assets is extremely complex and will not be explored. However, the formula for the expected return applies to any number of assets in the portfolio.

Illustration 4

A company is considering two stocks A and B, in which it will invest 60% and 40% of funds available, respectively. The correlation coefficient is -0.7 and other related information is presented below:

                                           Stock A                Stock B

Expected Return                 10%                      12%

Standard deviation               2%                        4%   

Coefficient of variation         0.2                       0.33

The expected return is :

 The standard deviation is:

The coefficient of variation is:

Analysis: The expected return of the portfolio is greater than that of stock A although it is lower than that of stock B. The risk ( as per the standard deviation) of the portfolio is lower than either of the two stocks. The question is now this...Is stock B preferred to the portfolio of stocks A and B? The answer rests with the CV. The portfolio has the lower CV, hence it is the preferred investment. 

Capital Asset Pricing Model(CAPM)

 It is always assumed that the more risky assets are, then the greater the return should be to compensate for this risk. But how does one know what level of return that is appropriate for a given level of risk?       

In answering this question, the model (CAPM) was developed with the aim of finding the required return on an asset given a particular risk element contained in it. This particular risk element is Beta (b) which measures the level of riskiness of the returns of the asset/security with respect to that of the market. In other words, Beta shows how risky a stock is if the stock is held in a well-diversified portfolio .In essence then, b measures non-diversifiable risk i.e. market risk. (Note - if the stock is not held in a well diversified portfolio then beta is not relevant, but stand-alone risk is.)

 As previously mentioned the rate of return of an asset is equal to a risk-free portion plus a risk premium. The CAPM formula expresses this relationship in a similar manner as follows:

Where:

                   k_s = the required return of the security

                   k_rf = the nominal risk free rate

                   k_m = the average return on the market

                   b_s = Beta of the security

Also note that:

(k_m - k_rf)b_s   = the risk premium

(k_m - k_rf) = the market premium, i.e. the premium that the market provides over the risk free rate

A security with a beta of 1 indicates that the security is equally as risky as the market. It also indicates that the scrutiny is similar to the average security on the market and thus will have a return similar to that of the market return.

This can be seen by placing b=1 into the equation: 

A security with a beta of 0 indicates that the security is risk-less. It also indicates that the scrutiny moves independent of the market and thus will have a return similar to that of the risk free return. When we place b=0 into the equation we see: 

However, a beta of 2 indicates that the security is twice as volatile as the market but it does not mean that the required rate of return of this security will be twice that of the market.

The graph of the CAPM known as the Security Market Line (SML): Graph 4.3:  (This graph shows the relationship between an asset's required rate of return and its beta.)

Illustration 5

If the treasury bill rate is 7% and the return on the market is 10%, then what is the required rate of return on a security whose beta is 0.8?  Draw the SML, and label accordingly.

 Solution:

Graph 4.4:

• Note - the SML equation is often times used to find the required rate of return used in calculating the price of bonds, stocks etc. We will revisit this later when we look at Bonds and Stocks in detail.

Factors that cause the required rate of return of a stock to change

As previously mentioned the SML shows the required rate of return of a particular asset given its Beta. Generally, the required rate of return of an asset is equal to its  expected rate of return, in which case the asset is said to be in equilibrium. However, there may be occasions when the required rate of return will change and therefore will not equal to the expected rate of return. The factors that would lead to such a change are:

1. Changes in inflation which would lead to changes in the risk-free rate.

2. Changes in external factors and a company's financial structure which would lead to changes in beta.

3. Changes in an investor's risk aversion which would lead to changes in the market premium.

Incorrectly priced assets (those not in equilibrium) 

Looking at the previous example, let's assume that the security had an expected return of 15%. Its position on the SML would be as follows:  

Graph 4.5:

Here the expected return > the required return i.e. the expected return is above the SML. "What does this mean?"

The required return is only 9.4% but it is expected to have a return of 15%, therefore the asset is worth purchasing.

An alternate analysis is that assets which have high returns are usually very risky and consequently lowly priced, as in this case. By extension, this asset would be priced at an amount lower than it should really be (undervalued). It is therefore an attractive asset to purchase.

Note - The converse also holds, i.e. if the expected return falls below the SML =>highly priced/over priced     

SML for a portfolio

A SML can also be constructed for a portfolio of stocks, using the same CAPM equation. However, we would need to determine the beta of the portfolio, which is given by the formula::

where b_i is the beta of stock i.

Concerns about beta and the CAPM ^[1]

The CAPM is more than just an abstract theory described in textbooks - it is also widely used by analysts, investors, and corporations. However, despite the CAPM's intuitive appeal, a number of recent studies have raised concerns about its validity. In particular, a recent study by Eugene Fama of the University of Chicago and Kenneth French of Yale found no historical relationship between stocks' returns and their market betas, confirming a position long held by a number of professors and stock market analysts.

If beta does not determine returns, what does? Fama and French found two variables that are consistently related to sock returns: (1) the firm's size and (2) its market/book ratio. After adjusting for other factors, they found that smaller firms have provided relatively high returns, and that returns are higher on stocks with low market/book value ratios. By contrast, they found no relationship between a stock's beta and its return.

^[1]This section has been taken directly from the text, page 204 ( the first two paragraphs of the section with the same heading)