by Patrick Lynch Show
In the article on portfolio theory, we saw that the motivation behind the establishment of a portfolio is that risk (the bad) can be reduced without a consequential reduction in return (the good). This was mathematically evident when the portfolios' expected return was equal to the weighted average of the expected returns on the individual investments, while the portfolio risk was normally less than the weighted average of the risk of the individual investments. The portfolio's total risk (as measured by the standard deviation of returns) consists of unsystematic and systematic risk. We saw the dramatic risk reduction effect of diversification (see Example 1). If an investor invests in just 15 companies in different sectors (a well-diversified portfolio), it is possible to virtually eliminate unsystematic risk. The only risk affecting a well-diversified portfolio is therefore systematic. As a result, an investor who holds a well-diversified portfolio will only require a return for systematic risk. In this article, we explain how to measure an investment's systematic risk. Learning Objectives
The measurement of systematic risk The first term is the average variance of the individual investments (unsystematic risk). As N becomes very large, the first term tends towards zero. Thus, unsystematic risk can be diversified away. The second term is the covariance term and it measures systematic risk. As N becomes large, the second term will approach the average covariance. The risk contributed by the covariance (the systematic risk) cannot be diversified away. Systematic risk reflects market-wide factors such as the country's rate of economic growth, corporate tax rates, interest rates etc. Since these market-wide factors generally cause returns to move in the same direction they cannot cancel out. Therefore, systematic risk remains present in all portfolios. Some investments will be more sensitive to market factors than others and will therefore have a higher systematic risk. Remember that investors who hold well-diversified portfolios will find that the risk affecting the portfolio is wholly systematic. Unsystematic risk has been diversified away. These investors may want to measure the systematic risk of each individual investment within their portfolio, or of a potential new investment to be added to the portfolio. A single investment is affected by both systematic and unsystematic risk but if an investor owns a well-diversified portfolio then only the systematic risk of that investment would be relevant. If a single investment becomes part of a well-diversified portfolio the unsystematic risk can be ignored. The systematic risk of an investment is measured by the covariance of an investment's return with the returns of the market. Once the systematic risk of an investment is calculated, it is then divided by the market risk, to calculate a relative measure of systematic risk. This relative measure of risk is called the ‘beta' and is usually represented by the symbol b. If an investment has twice as much systematic risk as the market, it would have a beta of two. There are two different formulae for beta. The first is: You must commit both formulae to memory, as they are not given on the exam formulae sheet. The formula that you need to use in the exam will be determined by the information given in the question. If you are given the covariance, use the first formula or if you are given the correlation coefficient, use the second formula. Example 2 Calculate the beta value: be = Example 3 Calculate the beta value: be = Investors make investment decisions about the future. Therefore, it is necessary to calculate the future beta. Obviously, the future cannot be foreseen. As a result, it is difficult to obtain an estimate of the likely future co-movements of the returns on a share and the market. However, in the real world the most popular method is to observe the historical relationships between the returns and then assume that this covariance will continue into the future. You will not be required to calculate the beta value using this approach in the exam. The CAPM Formula The calculation of the required return The shares in B plc have a beta value of 0.5 The shares in C plc have a beta value of 1.0 The shares in D plc have a beta value of 2.0 Obviously, with hindsight there was no need to calculate the required return for C plc as it has a beta of one and therefore the same level of risk as the market and will require the same level of return as the market, ie the RM of 11%. The systematic risk-return relationship is graphically demonstrated by the security market line. See Example 4. Example 4 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 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 Example 5 Required: Answer: 2. Stock market investment decisions Example 6 Beta values Expected
returns The market return is 15% and the risk-free return is 5%. Required: Answer: Alpha table 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 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 Required: Answer: b(A + B) = (1.6 × .80) + (1.1 × .20) R portfolio (A + B) = 6% + (15% - 6%) 1.5 = 19.50% Alpha table The Alpha Value 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 One period model Assumes the stock market is a perfect capital market
Evidence Estimation of future b based on past b Data input problems 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 model
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 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 Required: Estimate the risk and return of the two portfolios using the principles of both portfolio theory and CAPM and decide which one should be selected. Answer
to part (a) Alpha table Portfolio 1 is chosen because it has the largest positive alpha. Portfolio theory calculations Therefore, the formula for a multi-asset portfolio with no correlation between the returns is: Summary table The portfolio with the highest return also has the highest level of risk. Therefore, neither portfolio can be said to be more efficient than the other. An objective answer cannot be reached. As the company is making decisions on behalf of its shareholders the correct way to evaluate the investments is by looking at the effect they have on a shareholders existing/enlarged portfolios. Thus, the portfolio theory decision rule will probably break down if different shareholders experience different levels of total risk or they may have different attitudes to risk. Therefore, some shareholders would prefer portfolio 1 and other shareholders portfolio 2. If the majority of Oriel's shareholders are institutional shareholders, I would recommend the use of CAPM to make the decision, as they would hold well-diversified portfolios and only be subject to systematic risk. This would be a reasonable assumption as institutional investors like pension companies and unit trust companies hold approximately 75% of all the shares that are quoted on the London stock market. Answer to part (b) sport1 = 8 × 0.1 + 10 × 0.4 + 11 × 0.3 + 9 × 0.2 = 9.9 sport2 = 7 × 0.2 + 9 × 0.4 + 12 × 0.2 + 13 × 0.2 = 10 Summary table Portfolio 1 is the most efficient portfolio as it gives us the highest return for the lowest level of risk. 10 Key Points To Remember 1. The beta is a relative measure of systematic risk. It indicates the sensitivity of the return on a share with the return on 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%. 2. We may have to calculate the beta from basic data using the following two different formulae: 3. The value of beta is normally between 0 and 2.5. 4. Ensure that you know how to calculate the required return using the CAPM formula: 5. Be able to prepare an alpha table and to give investment advice based on alpha values: Decision advice based on alpha values 6. If CAPM is a realistic model and the market is efficient, an alpha value (a temporary abnormal return) is on a journey towards zero. 7. Ensure that you are able to list the problems associated with CAPM. 8. APM suggests that a number of factors affect the risk-return relationship and in time, this model may replace CAPM when more developments take place to improve its practical application. 9. Remember that the formula for a multi-asset portfolio with no correlation between the returns is: 10. The basic exam technique required for portfolio theory is the preparation of a summary table to aid identification of the most efficient portfolio.
Similarly, the key to applying CAPM is the preparation of an alpha table to help identify the largest positive alpha value.
When you code an if statement within another if statement as in the following then the if statements are?Terms in this set (20) When you code an if statement within another if statement, the statements are nested.
Which of the following is typically used in a flowchart to indicate a decisions?Flowcharts typically use the following main symbols: A process step, usually called an activity, is denoted as a rectangular box. A decision is usually denoted as a diamond.
Which of the following data types can store the value 0 using the least amount of memory?The data type that can store the value 0 using least amount of memory is “ Byte ”. Hence, the correct answer is option “ D ”.
What is the logical structure in which one instruction occurs after another with no branching?A sequence structure is a logical structure, in which one program statement occurs after another statement without branching.
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