Risk in a portfolio context

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This section introduces the basic concepts of risk and return. High risk almost always accompanies high expected returns. The investor must know how to measure both – risk and rate of return, which will eventually help him to choose investments with the most favorable combination. According to Brealey R. A., Marcus A. J., Myers S. C. (2011) the following should be taken into consideration before starting to deal with risk and rate of return: 1. All financial assets are expected to produce cash flows, and the riskiness of the asset is judged in terms of its cash flow. 2. The riskiness of an asset can be considered in two ways: (1) on a stand-alone basis, where the asset’s cash flows are analyzed individually, or (2) in a portfolio context, where the cash flows from a number of assets are combined. There is an important difference between stand-alone and portfolio risk, because an asset which has a great deal of risk if held alone may be much less risky if it is held as part of a large portfolio. 3. In a portfolio context, an asset’s risk can be divided into two components: (1) a diversifiable risk component, which can be diversified away and hence is of little concern to diversified investors, and (2) market risk component, which is caused by broad market movements that cannot be eliminated by diversification and which therefore does concern investors. Only market risk is relevant – diversifiable risk is irrelevant because it can be eliminated. 4. An asset with a high degree of relevant risk must provide a relatively high expected rate of return to attract investors. Investors in general are averse to risk, so they will not buy risky assets unless they have high expected returns. (Brealey R. A., Marcus A. J., Myers S. C. (2011)) In this section, we focus on financial assets such as stocks and bonds, but the concepts discussed here also apply to physical assets such as machines, trucks, or even whole plants.

Stand - alone risk In general, risk is the uncertainty surrounding the expected rate of return on investment, and quantitative measures of risk reflect the degree of uncertainty. An according to Brealey R. A., Marcus A. J., Myers S. C. (2011) asset’s risk can be analyzed in two ways: (1) on a stand-alone basis, where the asset is considered in isolation, and (2) on a portfolio basis, where the asset is held as one of a number of assets in a portfolio. Thus, an asset’s stand-alone risk is the risk an investor would face if he or she held only this one asset. Using an example of Brigham, E., Houston, J. (2008) suppose an investor buys 100 000 CZK of short-term Treasury bills with an expected return of 5 percent. In this case, the rate of return on investment, 5 %, can be estimated quite precisely, and the investment is defined as being essentially risk-free. However, if the 100 000 CZK were invested in the stock of a company just being established for oil prospecting, then the investment’s return could not be estimated precisely. One might analyze the situation and conclude that the expected rate of return, in a statistical sense, is 20 percent, but the investor should also recognize that the actual rate of return could range from, say +1000 % to –100 %. Because there is a significant danger of actually earning considerably less than the expected return, the stock would be described as being relatively risky. Brigham, E., Houston, J. (2008) state that no investment will be undertaken unless the expected rate of return is high enough to compensate for the perceived risk of the investment. In this example it is clear that few if any investors would be willing to buy the oil company’s stock if its expected return were the same as that of Treasury bills. Investment risk is then related to the probability of actual profits – the greater the chance of low or negative returns, the riskier the investment. Suppose the risky investment in Oil company has two possible outcomes. If the outcome is favourable, the investor receives 400 000 CZK. If the outcome is unfavourable, the project generates no cash and the investor loses his entire 100 000 CZK investment. Thus, the net payoff is either 300 000 CZK or –100 000 CZK. Assume the outcome occurs almost immediately, it is possible to disregard the time value of money.

Probability distribution Suppose the probability of each outcome is 50 %, with the information given, we can obtain a probability distribution of the possible payoffs. A probability distribution is a list of all possible payoffs from the investment, together with their respective probabilities. See Table 1. Probability distribution of payoffs for Oil company Outcome

Probability

Payoff

Favourable

0,50

300 000 CZK

Unfavourable

0,50

-100 000 CZK

Source: Brigham, E., Houston, J. (2008) Of course, if the investment is made only once, the investor will either gain 300 000 CZK, or lose 100 000 CZK. It is interesting to know how much an investor can expect to earn or lose on average if he makes a similar investment repeatedly. In this example, we expect him to gain 300 000 CZK half of the time and lose 100 000 CZK the other half of the time. This suggests that the expected value of payoff variable, E(Payoff), can be given as follows: E(Payoff)=300 000 * 0,5 + (-100 000) * 0,5 = 100 000 In general, if a random variable r can have n possible outcomes ri, where i = 1,2,…,n, and each outcome has probability pi, then the expected value of r is given by: E(r) = r1 p1 + r2 p2 + ... + rn pn The above equation can be re-written as follows: Er= ∑i=1nripi The sum of all probabilities must add to 1. This is true because we know with certainty that some outcome will be observed. In our equation, we have: ∑i=1npi=1

Variance and standard deviation Now that we know how to compute the expected payoff of the investment, we have to quantify its risk. A simplified definition of the riskiness of a random variable is according to Brigham, E., Houston, J. (2008) the possibility that the actual outcome will be different from the expected. In general, the greater the difference, the greater the riskiness of the random variable. Thus, the notation of risk is associated with the dispersion of possible outcomes. A common way to measure the dispersion of any random variable, r, around its mean is to calculate its variance, sigma2 , as follows:

or, using the simplified equation, 2=∑i=1nri-E(r)2pi Considering the example of Oil company from Brigham, E., Houston, J. (2008), the variance of the payoffs is equal to:

Notice that if the unit for the random variable is crowns, the unit for the variance is crowns squared, making the variance hard to interpret. Because of this difficulty, the standard deviation is often used as an alternative measure of risk. The standard deviation of a random variable is the square root of its variance, and is denoted by . Thus, it is defined by the following relationship: =2 =40 000 000 000 = 200 000 Kč The standard deviation measures the dispersion of a random variable around its mean. For the 100 000 CZK investment, the standard deviation of the random payoffs is 200 000. This measure gives a range of values around the mean that are likely to occur more frequently. Since the Oil company has an expected payoff of l00 000 CZK, the standard deviation of 200 000 indicates that we can expect most payoffs to be between –100 000 CZK and 300 000 CZK. In fact, in this example the standard deviation gives the entire range of possible payoffs: -100 000 CZK and 300 000 CZK. Calculate the mean, variance, and standard deviation for a stock with the probability distribution outlined in the following table:

Outcome

Probability

Stock Return

Recession

10%

-40%

Expansion

60%

20%

Boom

30%

50%

To continue the example of Brigham, E., Houston, J. (2008), lets assume that another firm, e.g. a Gas company, offers the same payoff distribution as the Oil company. It would be interesting to know if there is any benefit in investing 50 000 in Oil and the other 50 000 in Gas. Because the investor invests half as much as before in each firm, he will receive a payoff of 150 000 CZK from one firm if its outcome is favourable, and he will lose 50 000 CZK for the firm with an unfavourable outcome. Let us assume that Oil and Gas are independent investments; that is, the outcome of one will not influence the outcome of the other. Because the payoffs of the two firms are independent, the probability distribution of the combined investment will show the following outcomes. Probability distribution of payoffs from Oil and Gas company Oil company outcome

Gas company Probability Outcome

Oil payoff CZK

Gas payoff CZK

Total payoff CZK

Favorable

Favorable

0,25

150 000

150 000

300 000

Favorable

Unfavorable

0,25

150 000

-50 000

100 000

Unfavorable

Favorable

0,25

-50 000

150 000

100 000

Unfavorable

Unfavorable

0,25

-50 000

-50 000

-100 000

Source: Brigham, E., Houston, J. (2008) Under the new probability distribution of returns, there is a 50 percent chance of obtaining a payoff of l00 000 CZK because two of the four possible outcomes have this payoff. Also, the probability of each extreme outcome has been reduced to 25 percent. Knowing the probability distribution of this two-investment strategy, we can, according to Brigham, E., Houston, J. (2008), compute its expected payoff as follows: E(Payoff) = (300 000 * 0.25) + (100 000 * 0.25) + (100 000 * 0.25) + (-100 000 * 0.25) = 100 000 CZK We can conclude that, dividing money into two identical independent investments provides the same expected payoff as putting all the money into one of the investments. However, consider what happens to the risk of the combined investment. We compute the variance of the combined investment as follows: 2 = [(300 000 – l00 000)2 * 0.25] + [(l00 000 – 100 000)2 * 0.25] + [(l00 000 – l00 000)2 * 0.25] + [(-100 000 –l00 000)2 * 0.25] =200 000

Notice that the variance from investing equally in Oil and Gas is half that of investing the entire 100 000 CZK in Oil alone. Since the variance is a measure of risk, the combined investment is less risky than investing entirely in only one of the firms. Most importantly, no cost is associated with risk reduction because the expected payoff is unchanged. Knowing the variance, we can now compute the standard deviation of the two independent identical investments: =20 000 000 000= 141 421 Kč This simple example with two independent identical investments illustrates the following general result. Brigham, E., Houston, J. (2008) state that by investing in equal proportions in n identical independent projects, the expected payoff will be the same as that of investing all the money in only one project. Calculate the expected return, variance, and standard deviation for the stocks in the table below. Next, form an equally weighted portfolio of all three stocks and calculate its mean, variance and standard deviation.

Probability

Return

BMW

Atomic

Nestlé

Boom

20%

-10%

50%

40%

Expansion

50%

10%

30%

40%

Recession

30%

50%

-20%

-30%

Risk in a portfolio context In the preceding section, we considered the riskiness of assets held in isolation. Now we analyze the riskiness of assets held in portfolios. According to Brealey R. A., Marcus A. J., Myers S. C. (2011) an asset held as part of a portfolio is usually less risky than the same asset held in isolation. According to Graham, J., Scott S., Megginson, W. (2009) combining different stocks into a portfolio reduces total risk as measured by standard deviation. The standard deviation becomes smaller as it reflects the reduced variability in portfolio returns and greater certainty about the expected return. To illustrate the decline in total risk with increased diversification, we first divide total risk into two parts: Total risk = Diversifiable risk + Nondiversifiable risk Diversifiable risk is also known as unsystematic risk or company-specific risk and nondiversifiable risk is also known as systematic risk or market risk. Diversifiable risk approaches zero in portfolios of several stocks because it arises from company-specific factors whose effects on individual stock returns offset each other. The following unexpected events are according to Ehrhardt, M., Brigham, E. (2013)diversifiable because investors can offset adverse effects by diversifying their investments among stocks of several different companies: 1. 2. 3. 4.

A company's labour force unexpectedly goes on strike. A company unexpectedly hires a new management team. A company's top management dies in a plane crash. An oil tank bursts and floods a company's production area.

The above list stresses unexpected changes because stock prices already reflect the changes that investors expect. Expected events do not cause returns and stock prices to change because investors have already acted on their expectations by buying or selling certain stocks. Unexpected changes, however, cause returns and stock prices to change in ways investors cannot anticipate. Portfolios need not be large to reduce diversifiable risk significantly. On average only 15 randomly selected stocks with equal investments would rid the investor of most of the diversifiable risk. (Brealey R. A., Marcus A. J., Myers S. C. (2011)) because diversifiable risk constitutes more than 50 percent of the total risk of a typical common stock, getting rid of it through diversification is clearly beneficial. Although a well-diversified portfolio virtually eliminates diversifiable risk, it nevertheless contains nondiversifiable risk, and its actual return may still differ from what the investor expects. Consider, for example, the performance of Standard & Poor's 500 Index. It is a well diversified portfolio containing 500 stocks and therefore has very little diversifiable risk. Despite being a portfolio of 500 stocks, the S&P 500 varies considerably over time and is difficult for investors to predict. Several economic influences contribute to nondiversifiable risk, causing swings in returns on diversified portfolios (Ehrhardt, M., Brigham, E. (2013)). Notice in the following examples of nondiversifiable events that each factor affects all stocks taken collectively, even though each individual stock may respond differently: 1. Unexpected changes in interest rates - interest rate risk. 2. Unexpected changes in inflation - inflation risk. 3. Unexpected changes in cash flows such as those resulting from tax rate changes, foreign competition, and the business cycle. Classify each of the following events as a source of systematic or unsystematic risk:

1. Prime minister retires and Paris Hilton is appointed to take his place. 2. CEO of the analyzed firm is convicted of insider trading and is sentenced to prison. 3. An Ukrainian crisis raises the world market price of oil. Nondiversifiable risk is a relevant concept of risk because simply diversifying a portfolio cannot eliminate it. Based on this logic, nondiversifiable risk is the only type of risk needed to determine required rates of return on stocks. . (Ehrhardt, M., Brigham, E. (2013)).Figure 3 shows that nondiversifiable risk is the irreducible component of total risk: Nondiversifiable risk = Total risk - Diversifiable risk Increasing the number of different stocks in a portfolio enables an investor to reduce diversifiable risk. For well-diversified portfolios, nondiversifiable risk approximately equals total risk as diversifiable risk approaches zero (Ehrhardt, M., Brigham, E. (2013)).

Measuring Nondiversifiable Risk According to the capital asset pricing model, the marketplace rewards investors with a percentage premium for incurring nondiversifiable risk only; it does not reward investors for risk that can be easily eliminated by diversification. To estimate the rate of return required on a common stock, we must have a method for measuring nondiversifiable risk. Figure 1 illustrates how historical (ex post) rates of return are used to measure nondiversifiable risk. The slope of the line in Figure 1 provides an estimate of how a particular stock's historical rates of return relate to returns on a market index. A slope of 0,95 means that the stock's rate of return is slightly less volatile than the market's rate of return. Graph of historical rates of return for estimating stock’s beta

Source: Graham, J., Scott S., Megginson, W. (2009) When the market earns a positive 10 percent return for a particular period, this stock tends to produce a positive 9,5 percent return (0,95 x 10). A 10 percent decline in the market index (say, Standard & Poor's 500 Index) tends to be associated with a -9.5 percent return on the stock. The slope of the line in Figure 4 is the beta coefficient, or simply beta 0. Beta is the measure of nondiversifiable risk. Technically, beta is a regression coefficient defined as follows (Graham, J., Scott S., Megginson, W. (2009)): =cov (rj,rm)2(rm) Where cov(rj,rm) is the covariance between the stock return (rj) and the market portfolio return (rm), 2 (rm) is the variance of the market portfolio return. Alternatively, beta can be defined in terms of the correlation coefficient between rj and rm: =jmj,m Where j is the standard deviation of rj and m is the standard deviation of rm. Calculate the Beta of particular stocks.

Probability

Returns in Each State of the Economy

A.

B.

C.

MARKET

Boom

20%

40%

20%

20%

27%

Expansion

50%

10%

10%

40%

20%

Recession

30%

-20%

-10%

-30%

-20%

According to Graham, J., Scott S., Megginson, W. (2009) by looking at the historical relationship, a financial manager hopes to gain some idea of how a stock's rate of return will react to market movements in the future. A beta of 1.0 indicates that a common stock has average nondiversifiable risk. Stated differently, a beta of 1,0 tells us that the stock has nondiversifiable risk equal to that of the market portfolio of all stocks (as represented by the market index used to estimate beta). Percentage changes in the price of the stock tend to be the same as those of the market index. Volatility of the stock price and therefore its rate of return tend to be equivalent to that of the market index. (Graham, J., Scott S., Megginson, W. (2009)) Betas different from 1,0 mean that nondiversifiable risk differs from the market portfolio's. Graham, J., Scott S., Megginson, W. (2009) give the example, that betas greater than 1,0 indicate nondiversifiable risk greater than that of the market generally. For example, a beta of 2,0 indicates that a stock has twice the volatility in return as does the market index. If the market return is negative 10 percent, then the price of this stock would tend to decline by 20 percent. Betas less than 1,0 indicate nondiversifiable risk less than those of the market, and therefore a risk level lower than average. For example, a stock with a beta of 0,80 has 80 percent of the volatility of the market index. If market return is negative 10 percent, then the price of this stock would tend to decline by 8 percent. In Figure 1 all of the dots, denoting rates of return on the stock and the market index, do not fall on the line. According to Brealey R. A., Marcus A. J., Myers S. C. (2011) dispersion of the dots around the line measures diversifiable risk, and the slope of the line measures nondiversifiable risk. The effects of unexpected diversifiable events, measured by the distances of the dots from the line, can be diversified away by including the stock in a portfolio. Nondiversifiable risk measured by beta still remains, however, after diversification; unexpected market-wide events would still cause returns and stock prices to change in ways investors cannot anticipate. In general, a portfolio beta (p ) is the weighted average of the stock betas in the portfolio, with the weights assigned according to the amount invested in 1 stock as a percentage of the total invested: p = w11 + w22 + … + w33 Where w = weight reflecting each stock's value as a percentage of portfolio value, i = beta for each stock (1, 2, ..., n) . Built equally weighted portfolio and calculate the Beta of it.

Probability

Returns in Each State of the Economy

A.

B.

C.

MARKET

Boom

20%

40%

20%

20%

27%

Expansion

50%

10%

10%

40%

20%

Recession

30%

-20%

-10%

-30%

-20%

The beta of a stock measures its contribution to the riskiness of a portfolio. Brealey R. A., Marcus A. J., Myers S. C. (2011) claim that adding high-beta stocks to the portfolio increases its risk, and adding low-beta stocks lowers its risk. Investors can adjust their exposure to portfolio risk by buying and selling stocks with differing betas. Many analysts estimate and use betas for common stock but not for other securities. Theoretically, all securities have betas, but estimation problems have precluded their widespread use for securities other than common stock. (Brealey R. A., Marcus A. J., Myers S. C. (2011))

The Capital Asset Pricing Model For stocks held in a well-diversified portfolio, the risk of each stock is appropriately represented by beta, which measures the risk contributed by each stock to the portfolio. The capital asset pricing model (CAPM) incorporates beta as the relevant measure of risk and relates it to required rate of return. (Brealey R. A., Marcus A. J., Myers S. C. (2011)). According to the CAPM, risk-averse investors require higher rates of return (pay smaller prices, other things being equal) on stocks with larger betas. To induce investors to forgo the risk-free rate of return (Rf), stocks must provide the expectation of a return in excess of Rf. The general idea is as follows: Required rate of return = Rf + Risk premium The risk-free return accounts for the time value of money, and the risk premium is the additional percentage points necessary to compensate investors for risk. The primary focus of the capital asset pricing model is on the risk premium added to the risk-free return to derive the required return. Reekie, W. D., Crook, J. N. (1995) state that prior to development of the model in 1964, risk premiums were based largely on judgment and intuition. According to the CAPM, the risk premium consists of two parts multiplied together: Risk premium = Market price of risk * Nondiversifiable risk Nondiversifiable risk is measured with beta as discussed in the preceding section. Reekie, W. D., Crook, J. N. (1995) explain that the market price of risk is the reward per unit of risk measured in market terms. It is measured with the required rate of return on the average-risk common stock minus the risk-free rate of return (Rf). The required rate of turn on the average-risk common stock is equivalent to the required rate of return on the collection of all common stocks (rm). The collection of all common stocks trading in the secondary market is called the market portfolio. Putting together these ideas yields the following measure of the percentage points for the risk premium in the required rate of return: Risk premium = market price of risk * nondiversifiable risk = (rm – Rf) Adding this risk premium to the risk-free rate of return yields the capital asset pricing model: rc = Rf + (rm – Rf) where rc is the required rate of return on a common stock. Each company's common stock has its own particular value for nondiversifiable risk (): higher values indicate higher risk levels and correspondingly higher required rates of return (rc). Values for the risk-free return (Rf) and the required return on the market portfolio (rm) are the same for all companies (Brealey R. A., Marcus A. J., Myers S. C. (2011)). Graphic portrayal of the capital asset pricing model

Source: Brealey R. A., Marcus A. J., Myers S. C. (2011) Figure 2 is a graphic representation of the capital asset pricing model, with required rate of return (rc) on the vertical axis and beta on the horizontal axis. The security market line (SML) shows the relationship between nondiversifiable risk and the rate of return investors require on common stock. According to Kolb, A. B., Demong, R. F. (1988) the trade-off between risk and return determines the slope of the security market line - to obtain a greater return, an investor must be willing to incur greater nondiversifiable risk. Added risk leads to a quantifiable increase in the required rate of return. The graph makes it easier to visualize and understand the capital asset pricing model. Calculate the stock’s beta if the risk-free rate equals four percent, the expected return on the market is ten percent and stock’s expected return is 13 percent.



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Risk in a portfolio context

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