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In this chapter, we examine the factors that determine rates of return (discount rates) in the capital markets. We are particularly interested in the relationship between risk and rates of return. We look at risk both in terms of the riskiness of an individual security and that of a portfolio of securities.
A. The expected benefits or returns to be received from an investment come from the cash flows the investment generates.
B. The rate of return earned from an investment can be calculated as the ratio of the dollar gain divided by the amount of the investment at the beginning of the period. We can formalize these calculations as follows:
Holding-period dollar gain
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C. In an uncertain world, we must use expected cash flow, CF , for computing expected gains and rates of return. We compute an expected cash flow as follows:
where n = the number of possible states of the economy
CFi = the cash flow in the ith state of the economy
Pbi = the probability of the ith state of the economy
D. Similar to the calculation of expected cash flow, the expected rate of return can be calculated as follows:
where n = the number of possible states of the economy
ri = the rate of return in the ith state of the economy
Pbi = the probability of the ith state of the economy
A. Risk can be defined as the potential variability in expected future cash flows
B. Statistically, risk may be measured by the standard deviation about the expected cash flow.
1. The standard deviation of returns is the square root of the variance of returns, a statistical measure of the dispersion of returns from their expected value. Variance is calculated as follows:
where n = the number of possible states of the economy
ri = the rate of return in the ith state of the economy
r = the expected rate of return
Pbi = the probability of the ith state of the economy
2. The standard deviation of returns is the square room of the variance of returns, calculated as follows:
deviation in =
rates of return where n = the number of possible states of the economy
ri = the rate of return in the ith state of the economy
r = the expected rate of return
Pbi = the probability of the ith state of the economy
A. Data have been compiled by Ibbotson Associates on the actual returns for various portfolios of securities from 1926–2014.
B. The following portfolios, plus the inflation rate, were studied:
1. Common stocks of large firms
2. Common stocks of small firms
3. Corporate bonds
4. Intermediate U.S. government bonds
5. U.S. Treasury bills
6. Inflation rate
C. From studying the average returns and standard deviations of the above portfolios, we learn that investors historically have received greater returns for greater risk taking
D. All of these portfolios earned returns exceeding the inflation rate. However, the portfolio that consistently earned the highest rate of return, on average, has been a portfolio made up of common stocks.
A. The market rewards diversification. We can lower risk without sacrificing expected return, or we can increase expected return without having to assume more risk by holding a diversified portfolio of investments
B Total variability of returns can be divided into the following:
1. The variability of returns unique to the security (diversifiable or unsystematic risk)
2. The risk related to market movements (nondiversifiable or systematic risk)
C By diversifying, the investor can eliminate the “unique” security risk. Systematic risk, however, cannot be diversified away. A portfolio of 20–25 different securities can essentially eliminate diversifiable risk from a portfolio of stocks.
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D. For a sample of observed holding period returns, the average holding period return and standard deviation of returns are calculated below.
Average holding = return in period 1 + return in period 2 + + return in period n (6-8) period return number of periodic returns
rates of return (n – 1)
where n = the number of possible states of the economy
ri = the rate of return in the ith period
r = the expected rate of return
1. The characteristic line tells us the average movement in a firm's stock price in response to a movement in the general market, measured by a broad market portfolio such as the S&P 500 Index. The slope of the characteristic line, which has come to be called beta, is a measure of a stock's systematic or market risk. The slope of the line is merely the ratio of the “rise” of the line relative to the “run” of the line, that is, the change in a security's returns relative to the change in returns in a market portfolio.
2. If a security's beta equals one, a 10 percent increase (decrease) in market returns will produce, on average, a 10 percent increase (decrease) in the security's returns. If a security's beta is less than one, a 10 percent increase (decrease) in market returns will produce, on average, a less than 10 percent increase (decrease) in the security's returns. If a security's beta is greater than one, a 10 percent increase (decrease) in market returns will produce, on average, a greater than 10 percent increase (decrease) in the security's returns.
3. A security having a higher beta is more volatile and thus more risky than a security having a lower beta value.
F. A portfolio's beta is equal to the average of the betas of the stocks in the portfolio, with the weights being equal to the proportion of the portfolio invested in each security
G Diversifying among different kinds of assets is called asset allocation. Compared to diversification within the different asset classes, the benefits received are far greater through effective asset allocation.
H Asset allocation between stocks and bonds clearly affects the expected returns of the portfolios, as does how long you hold the portfolio. The more bonds in your portfolio and the longer you hold the portfolio, the less will be the variability of returns.
A. The required rate of return is the minimum rate necessary to compensate an investor for accepting the risk he or she associates with the purchase and ownership of an asset.
B. Two factors determine the required rate of return for the investor:
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1. The risk-free rate of return, which recognizes the time value of money
2. The risk premium, which considers the riskiness (variability of returns) of the asset and the investor's attitude toward risk
3. These factors can be expressed in the equation below.
Investor's required = risk-free rate + risk (6-11) rate of return of return premium
(a) The risk-free rate of return is the rate of return on risk-free investments such as short-term U. S. government securities.
(b) The risk premium is the additional return expected for assuming risk. It is calculated as follows:
Risk = investor's required risk-free rate of (6-12) premium rate of return, r return, rf
C. Capital Asset Pricing Model CAPM
1. The Capital Asset Pricing Model (CAPM) is a model that states that the expected rate of return of an investment is a function of (1) the risk-free rate, (2) the investment's systematic risk, and (3) the expected risk premium for the market portfolio comprised of all risky securities.
2. The required rate of return for a given security can be expressed as
3. Security market line
a. The security market line is a graphical representation of the CAPM
b. The graph shows a security's appropriate rate of return given its level of systematic risk, beta, as defined according to the CAPM.
c. Traditionally, the security's expected return is on the vertical axis, and the level of systematic risk, beta, is on the horizontal axis.
6-1. a. An investor's required rate of return is the minimum rate of return necessary to attract an investor to purchase or hold a security.
b. Risk is the potential variability in returns on an investment. Thus, the greater the uncertainty as to the exact outcome, the greater is the risk. Risk may be measured in terms of the standard deviation of rates of return or by the variance of rates of return, which is simply the standard deviation squared.
c. A large standard deviation of the returns indicates greater riskiness associated with an investment. Future cash flows have a greater potential variation. However, whether the standard deviation is large relative to the returns has to be examined with respect to other investment opportunities. Alternatively, probability analysis is a meaningful approach to capture greater understanding of the significance of a standard deviation figure. However, we have chosen not to incorporate such an analysis into our explanation of the valuation process.
6-2. a. Unique risk is the variability in a firm's stock price that is associated with the specific firm and not the result of some broader influence. An employee strike is an example of a company-unique influence.
b. Systematic risk is the variability in a firm's stock price that is the result of general influences within the industry or resulting from overall market or economic influences. A general change in interest rates charged by banks is an example of systematic risk.
6-3. Beta indicates the responsiveness of a security's return to changes in the market return. According to the CAPM, beta is multiplied by the market risk premium and added to the risk-free rate of return to calculate a required rate of return.
6-4. The security market line is a graphical representation of the risk-return trade-off that exists in the market. The line indicates the minimum acceptable rate of return for investors given the level of systematic risk of a security.
6-5. The beta for a portfolio is equal to the weighted average of the betas of individual stocks, weighted by the percentage invested in each stock.
6-6. If a stock has a great amount of variability about its characteristic line (the line of best fit in the graph of the stock's returns against the market's returns), then it has a high amount of unsystematic or company-unique risk. If, however, the stock's returns closely follow the market movements, then there is little unsystematic risk.
6-7. Data have been compiled by Ibbotson Associates, Inc. on the actual returns for the following portfolios of securities, plus the inflation rate, from 1926–2014
1. Common stocks of large firms
2. Common stocks for small firms
3. Corporate bonds
Investors historically have received greater returns for greater risk-taking with the exception of the U.S. government bonds. All portfolios generated returns that exceeded the inflation rate. The portfolio that, on average, has consistently generated the highest rate of return has been a portfolio made up of common stocks.
6-8. Through diversification, we can potentially accomplish one of two results: (1) We can decrease the variability in returns without lowering the expected rate of return of the portfolio, or (2) we can increase the expected rate of return without increasing the variability in returns. The extent of these effects is in part determined by the types of assets in the portfolio. For instance, diversification has greater effect when investing in different types of assets, such as government securities and stocks, rather than just investing in different stocks.
From our studies in statistics, we know that if the distribution of returns were normal, then Universal Corporation could expect a return of 13 percent between 1.78 percent (13% – 11.22%) and 24.22 percent (13% + 11.22%) of the time On average, the firm can expect a return of 13 percent. However, it is apparent from the probabilities that the distribution is not normal.
Investment A is better. It has a higher expected return with less risk.
6-6. This problem has students visit websites that extend their knowledge about risk and return. The students' answers will vary, depending on the information entered. It should be a fun exercise that lets the students see the practical aspects of this chapter.
6-7. This problem has students visit websites that extend their knowledge about risk and return. The students' answers will vary, depending on the information entered. It should be a fun exercise that lets students see the practical aspects of this chapter.
6-8. This problem has students visit websites that extend their knowledge about risk and return. The students' answers will vary, depending on the information entered. It should be a fun exercise that lets students see the practical aspects of this chapter.
b. A holding-period return indicates the rate of return you would earn if you bought a security at the beginning of a period and sold it at the end of the period, such as the end of the month or year.
return = Risk-free rate + [Beta × (Market return – Risk-free rate)]
25.6%
c. Zemin's historical return of 20 percent is below what we would consider a fair return of 25.6 percent, given the stock's high level of systematic risk. 6-12. Holding Period Gain = (12 × $25.68) – (12 × $24.22) = $17.52
Period Return = $17.52 ÷ (12 × $24.22) = 6%
6-13. This problem has students visit websites that extends their knowledge about risk and return. The students' answers will vary, depending on the information entered. It should be a fun exercise that lets students see the practical aspects of this chapter.
c. Using the Excel function, Slope =linest(Nike returns,S&P returns), we find that the slope of the characteristic line to be 0.471.
d. Nike's returns are positively correlated to the S&P 500 returns with the characteristic line having a slope of 0.471, which suggests that the systematic risk for Nike is less than the general market. Moreover, the variability of returns around the characteristic line reflects the relatively low level of
risk for Nike's stock.
August
d. Calculate the slope using the Excel function =linst as follows:
Slope: = linest(Citigroup return,S&P return) = 1.733
e. Citigroup's returns are positively correlated to the S&P 500 returns with the characteristic line having a slope of 1.733, which suggests that the systematic risk for Nike is much greater than the general market. Moreover, the variability of returns around the characteristic line reflects the unsystematic risk for Citigroup's stock.
6-16 Keown/Martin/Petty Instructor's Manual with Solutions
6-17. a. Portfolio expected return:
(0.4 × 12%) + (0.25 × 11%) + (0.35 × 15%) = 12.8%
Portfolio beta:
(0.4 × 1.0) + (0.25 × 0.75) + (0.35 × 1.3) = 1.04
b.
a.
The greater the level of systematic risk, the higher the expected return is.
12.8%
b. The 12.8 percent “fair rate” predicted by the CAPM compensates the investor for the time value of money and for assuming risk. However, only nondiversifiable (systematic) risk is being considered, which is appropriate.
6-19 Eye-balling the characteristic line for the problem, the rise relative to the run is about 0.5. That is, when the S&P 500 return is 8 percent, Aram's expected return would be about 4 percent. Thus, the beta is also approximately 0.5 (4 ÷ 8).
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A 2.75% + [(12% – 2.75%) × 1.50] = 16.63%
B 2.75% + [(12% – 2.75%) × 0.90] = 11.08%
C 2.75% + [(12% – 2.75%) × 0.70] = 9.23%
D 2.75% + [(12% – 2.75%) × 1.15] = 13.39%
6-21 Required return = Risk-Free Rate + [Beta × (Market Return – Risk-Free Rate)]
= 3% + [0.86 × (11.5% – 3%)] = 10.31%
6-22. If the expected market return is 12.8 percent and the risk premium is 9.3 percent, the risk-free rate of return is 3.5 percent (12.8% – 9.3%). Therefore;
Tasaco = 3.5% + (9.3% × 0.864) = 11.54%
LBM = 3.5% + (9.3% × 0.693) = 9.94%
Exxos = 3.5% + (9.3% × 0.575) = 8.85%
6-23 a. The portfolio expected return, rp equals a weighted average of the individual stock's expected returns.
rp = (0.20)(12%) + (0.30)(8%) + (0.15)(12%) + (0.25)(7%) + (0.10)(16%)
= 10%
b. The portfolio beta, ßp, equals a weighted average of the individual stock betas
ßp = (0.20)(1.00) + (0.30)(0.85) + (0.15)(1.20) + (0.25)(0.60) + (0.10)(1.60) = 0.95
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c. Plot the security market line and the individual stocks.
d. A “winner” may be defined as a stock that falls above the security market line, which means that these stocks are expected to earn a return exceeding what should be expected given their beta or level of systematic risk. In the above graph, these stocks include 1 and 5. “Losers” would be those stocks falling below the security market line, which are represented by stocks 2, 3, and 4. The five-stock portfolio is slightly below the security market line as well.
e. Our results are less than certain because we have problems estimating the security market line with certainty. The final estimate of beta depends on the methodology used to compute beta. The number of months used to determine beta can affect its value. We have also difficulty in exactly specifying the market portfolio. In addition, the simplifying assumptions of the CAPM may not necessarily hold for the investments being considered.
d. The returns for Walmart and Target are positively related to the market returns, but with a lot of noise, which represents unsystematic or diversifiable risk.
When we plot the two-stock portfolio of Walmart and Target together against the S&P 500, the spread of the returns is a slightly tighter, with the exception of one really high monthly return In other words, more of the variation in the two-stock portfolio can be explained by the returns of the market portfolio (S&P 500) than was the case with the individual stock returns. That is, systematic risk has been reduce somewhat.
Keown/Martin/Petty Instructor's Manual with Solutions
We see that as risk, measured by the standard deviation of returns, increased, the average returns increased the exceptions being the S&P 500 Index and Walmart. Of course, it is hard to generalize based on a 24-month period.
If our data are representative of the rates of returns an investor would receive and the variation of these returns, an investor would have a higher expected rate of return for Target, the stock that exhibits greater total risk (a larger standard deviation) and greater systematic risk (beta).