In Previous Chapter , the return from holding an investment was defined in Equation as:
Distributions include interest on debt, dividends on common stock, rent on real property, and so on. Recall in Previous Chapter that risk was defined as the possibility that actual future returns will deviate from expected returns. In other words, it represents the variability of returns. From the perspective of security analysis or the analysis of an investment in some project (such as the development of a new product line), risk is the possibility that actual cash flows (returns) will be different from forecasted cash flows (returns).
An investment is said to be riskfree if the dollar returns from the initial investment are known with certainty. Some of the best examples of riskfree investments are U.S. Treasury securities. There is virtually no chance that the Treasury will fail to redeem these securities at maturity or that the Treasury will default on any interest payments owed.As a last resort, the Treasury can always print more money.
In contrast, R.J. Reynolds Tobacco Company (RJR) bonds constitute a risky investment because it is possible that the company will default on one or more interest payments and will lack sufficient funds to redeem the bonds at face value at maturity. In other words, the possible returns from this investment are variable, and each potential outcome can be assigned a probability.
Probability of Default on RJR Bonds
If, for example, you were considering investing in RJR bonds, you might assign the probabilities shown in to the three possible outcomes of this investment. These probabilities are interpreted to mean that an 80 percent chance exists that the bonds will not be in default over their life and will be redeemed at maturity, a 15 percent chance of interest default during the life of the bonds, and a 5 percent chance that the bonds will not be redeemed at maturity.
Hence, from an investment perspective, risk refers to the chance that returns from an investment will be different from those expected.We can define risk more precisely, however, by introducing some probability concepts.
Probability Distributions
The probability that a particular outcome will occur is defined as the percentage chance (or likelihood) of its occurrence. A probability distribution indicates the percentage chance of occurrence of each of the possible outcomes. Probabilities may be determined either objectively or subjectively. An objective determination is based on past occurrences of similar outcomes, whereas a subjective one is merely an opinion made by an individual about the likelihood that a given outcome will occur. In the case of projects that are frequently repeated— such as the drilling of developmental oil wells in an established oil field—reasonably good objective estimates can be made about the success of a new project. Similarly, good objective estimates can often be made about the expected returns of an RJR bond.
However, the expected returns from securities of new, small firms are often much more difficult to estimate objectively. Hence, highly subjective estimates regarding the likelihood of various returns are necessary. The fact that many probability estimates in business are at least partially subjective does not diminish their usefulness.
Summary of Notation
Before examining specific measures of risk and return, it is useful to summarize the basic elements of notation used throughout the chapter.
r = single rate of return on a given security; a subscript denotes the rate of return on a particular security (or portfolio of securities), such as rf , described next, and a hat (^) symbol denotes an expected rate of return
rf = nominal riskless (riskfree) rate of return; the return offered on shortterm U.S. Treasury securities
rp = rate of return on a portfolio of securities
rm = rate of return on the Market Portfolio; a broadbased security market index, such as the Standard & Poor’s 500 Market Index or the New York Stock Exchange Index, is normally used as a measure of total market returns
p = probability of occurrence of a specific rate of return
s = standard deviation of the rate of return on a security (or portfolio of securities); the square root of the variance of returns
sp = standard deviation of the rate of return on a portfolio of securities
sm = standard deviation of the rate of return on the Market Portfolio
v = coefficient of variation
z = number of standard deviations that a particular value of a random variable (such as rate of return) is from its expected value
r = correlation coefficient between the returns on two securities
w = portion (weight) of funds invested in a given security within a portfolio
kj = required rate of return on a given security
uj = risk premium required by investors on a given security
bj = measure of the volatility (or risk) of a security’s returns relative to the returns on the Market Portfolio
bp = measure of risk of a portfolio of securities
Expected Value
Suppose an investor is considering an investment of $100,000 in the stock of either Duke Energy, a public utility firm, or Texas Instruments TI), a maker of electronic equipment. By investing in the stock of either of these firms, an investor expects to receive dividend payments plus stock price appreciation.We will assume that the investor plans to hold the stock for one year and then sell it. Over the coming year, the investor feels there is a 20 percent chance for an economic boom, a 60 percent chance for a normal economic environment, and a 20 percent chance for a recession. Given this assessment of the economic environment over the next year, the investor estimates the probability distribution of returns from the investment in Duke and TI
From this information, the expected value of returns (or expected return) from investing in the stock of Duke and TI can be calculated. The expected value is a statistical measure of the mean or average value of the possible outcomes. Operationally, it is defined as the weighted average of possible outcomes, with the weights being the probabilities of occurrence. Algebraically, the expected value of the returns from a security or project may be defined as follows:
where rˆ is the expected return; rj is the outcome for the jth case, where there are n possible outcomes; and pj is the probability that the jth outcome will occur. The expected returns for Duke and TI are computed in Table. The expected return is 18 percent for both Duke and TI.
Probability Distribution of Returns from Duke and TI
Standard Deviation: An Absolute Measure of Risk
The standard deviation is a statistical measure of the dispersion of possible outcomes about the expected value. It is defined as the square root of the weighted average squared deviations of possible outcomes from the expected value and is computed as follows:
where s is the standard deviation.
The standard deviation can be used to measure the variability of returns from an investment. As such, it gives an indication of the risk involved in the asset or security. The larger the standard deviation, the more variable are an investment’s returns and the riskier is the investment. A standard deviation of zero indicates no variability and thus no risk. Table shows the calculation of the standard deviations for the investments in Duke and TI. As shown in the calculations in Table, TI appears riskier than Duke because possible returns from TI are more variable, measured by its standard deviation of 13.91 percent, than those from Duke, which have a standard deviation of only 5.06 percent.
This example deals with a discrete probability distribution of outcomes (returns) for each firm; that is, a limited number of possible outcomes are identified, and probabilities are assigned to them. In reality, however, many different outcomes are possible for the investment in the stock of each firm—ranging from losses during the year to returns in excess of TI’s 40 percent return. To indicate the probability of all possible outcomes for these investments, it is necessary to construct a continuous probability distribution. This is done by developing a table similar to Table, except that it would have many more possible outcomes and their associated probabilities. The detailed table of outcomes and probabilities can be used to develop the expected value of returns from Duke and TI, and a continuous curve would be constructed to approximate the probabilities associated with each outcome. Figure illustrates continuous probability distributions of returns for investments in the stock of Duke and TI.
Expected Return Calculation for Investment in Duke and TI
Computation of Standard Deviations of Return for Duke and TI
As seen in this figure, the possible returns for Duke have a tighter probability distribution, indicating a lower variability of returns, whereas the TI possible returns have a flatter distribution, indicating higher variability and, by extension, more risk.
Normal Probability Distribution
The possible returns from many investments tend to follow a normal probability distribution. The normal probability distribution is characterized by a symmetrical, belllike curve. If the expected continuous probability distribution of returns is approximately normal, a table of the standard normal probability distribution (that is, a normal distribution with a mean equal to 0.0 and a standard deviation equal to 1.0, such as Table V at the end of the text) can be used to compute the probability of occurrence of any particular outcome.
From this table, for example, it is apparent that the actual outcome should be between plus or minus 1 standard deviation from the expected value 68.26 percent of the time,3 between plus or minus 2 standard deviations 95.44 percent of the time, and between plus or minus 3 standard deviations 99.74 percent of the time. This is illustrated in Figure. The number of standard deviations, z, that a particular value of r is from the expected value, rˆ, can be computed as follows:
Continuous Probability Distributions for the Expected Returns from
Areas Under the Normal Probability Distribution
Equation , along with Table V, can be used to compute the probability of a return from an investment being less than (or greater than) some particular value. For example, as part of the analysis of the risk of an investment in TI stock, suppose we are interested in determining the probability of earning a negative rate of return, that is, a return less than 0 percent. This probability is represented graphically in Figure as the area to the left of 0 (that is, the shaded area) under the TI probability distribution. The number of standard deviations that 0 percent is from the expected return (18 percent) must be calculated. Substituting the expected return and the standard deviation from Tables into Equation yields the following:
0% – 18%z = 13.91% = –1.29In other words, the return of 0 percent is 1.29 standard deviations below the mean. From Table V, the probability associated with 1.29 standard deviations is 0.0985. Therefore, there is a 9.85 percent chance that TI will have returns below 0 percent. Conversely, there is a 90.15 percent 100 – 9.85) chance that the return will be greater than 0 percent.
Coefficient of Variation: A Relative Measure of Risk
The standard deviation is an appropriate measure of total risk when the investments being compared are approximately equal in expected returns and the returns are estimated to have symmetrical probability distributions. Because the standard deviation is an absolute measure of variability, it is generally not suitable for comparing investments with different expected returns. In these cases, the coefficient of variation provides a better measure of risk. It is defined as the ratio of the standard deviation, s, to the expected return, rˆ:
The coefficient of variation is a relative measure of variability, since it measures the risk per unit of expected return.As the coefficient of variation increases, so does the risk of an asset. Consider, for example, two assets, T and S.Asset T has expected annual returns of 25 percent and a standard deviation of 20 percent, whereas Asset S has expected annual returns of 10 percent and a standard deviation of 18 percent. Although Asset T has a higher standard deviation than Asset S, intuition tells us that Asset T is less risky, because its relative variation is smaller. The coefficients of variation for Assets T and S are computed as follows using Equation:
v =Asset T 20% / 25% = 0.8
v =Asset S 18% / 10% = 1.8
Asset T’s returns have a lower coefficient of variation than Asset S’s; therefore Asset T is the less risky of the two investments.
In general, when comparing two equalsized investments, the standard deviation is an appropriate measure of total risk.When comparing two investments with different expected returns, the coefficient of variation is the more appropriate measure of total risk.
Risk as an Increasing Function of Time
Most investment decisions require that returns be forecasted several years into the future. The riskiness of these forecasted returns may be thought of as an increasing function of time. Returns that are generated early can generally be predicted with more certainty than those that are anticipated farther out into the future.
Consider the risk facing the Radio Shack in its decision to market a new line of stereo speakers through its retail stores. This project is expected to generate cash flows to Radio Shack of $2 million per year over the 7year life of the project. Even though the expected annual cash flows are equal for each year, it is reasonable to assume that the riskiness of these flows increases over time as more and more presently unknown variables have a chance to affect the project’s cash flows. Figure illustrates this situation. The distribution is relatively tight in year 1, because the factors affecting that year’s cash flows (e.g., demand and costs) are reasonably well known.
By year 7, however, the distribution has become relatively flat, indicating a considerable increase in the standard deviation, which is caused by increased uncertainty about the factors that affect cash flows. For example, competitors may introduce similar (or improved) products, which would cause demand to decline for the Radio Shack speakers.
Some types of cash flows are not subject to increasing variability. These include, for example, contractual arrangements, such as lease payments, in which the expected cash flows remain constant (or change at some predefined rate) over the life of the contract. In spite of the exceptions, it is reasonable to conclude that the riskiness of the cash flows from most investment projects gradually increases over time. Similarly, the riskiness of returns from most securities increases the farther into the future these returns are being considered. For instance, the interest return from the purchase of General Motors Corporation (GM) bonds is nearly guaranteed for the next year.However, projecting the interest returns to be received 10 years in the future is much more difficult due to the potential impact of competition, new technology, and other factors that might cause GM to have to default on its interest payment on its bonds.
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