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RISK AND RETURN

Risk:    a hazard, a peril, exposure to loss or injury. The chance that some unfavorable event will occur.

Risk & Return relationship shows that no investment will be made unless the expected rate of return is high enough to compensate the investor for the perceived risk of the investment.

Investment Risk: the probability of actually earning significantly less than the expected return.

Probability Distribution:    A listing of all possible outcomes, or events, with a probability  assigned to each outcome. [Outcome and Probability]

State of the Economy Probability of this state of occurring (P) Rate of Return Designer Phones  (k^) Rate of Return Standard Phones (k^) Expected Rate of Return Designer Phones  (Piki)                   Expected Rate of Return Standard Phones (Piki)
Boom 0.3 100% 20% 0.3 0.06
Normal 0.4 15% 15% 0.06 0.06
Recession 0.3 (70%) 10% (0.21) 0.03
        0.15 0.15

Expected rate of Return ( k^ ) :    Multiply each possible outcome by its probability of occurrence and then sum all products, we get weighted average of outcomes (the mean of distribution) is called expected rate of return, k^ of a probability distribution.

                                   

The rate of return expected to be realized from an investment; mean value of the probability distribution of possible outcomes.

k^ - Designer Phones: k^ = P1(K1) + P2(k2) + P3(k3)    => 0.3(100%) + 0.4(15%) + 0.3(-70%) = 15%

k^ - Standard Phones: k^ = P1(K1) + P2(k2) + P3(k3)    => 0.3(20%) + 0.4(15%) + 0.3(10%) = 15%

We can calculate the expected Cash flows for each of the two projects. The procedure is the same, only change is that we would have estimated cash flows for each of the state of the economy and would then substitute CFi with ki, as follow:

Continuous Probability Distributions:

Earlier we discussed 3 states of the economy, recession, normal and boom. It is very much possible that economy ranges from deep depression to a fantastic boom and there is an unlimited number of possibilities in between (continuous probability distributions). Suppose we had the time and patience to assign a probability to each possible state of the economy and to estimate a rate of return on each project for each state. By using equation 12 - 1 except that our weighted average would now have more entries (due to more outcomes, however, total equal to 1.0). By plotting the data on graph, we find as follow:

 

The tighter or more peaked, the probability distribution, the more likely it is that the actual outcome will be close to the expected value, and consequently, the less likely it is that the actual return will end up far below the expected return. Thus the tighter the probability distribution the lower the risk of the project. Since the Standard Line has a relatively tight probability distribution, its actual return is likely to be closer to the a5% expected return than is the actual return of the Designer Line.

Measuring Risk:    The Standard Deviation

A statistical measure of the variability of a set of observations. It measures the tightness (risk) of the probability distribution. The smaller the standard deviation, the tighter the probability distribution and consequently the lower the riskiness of the project.

The calculations of the standard deviation follows these steps:

1.    Calculate the expected rate of return, by using following equation:

2.    Subtract the expected rate of return from each possible outcome to obtain a set of deviations about the expected rate of      return, k^:

Deviationi = ki - k^

3.    Square each deviation, multiply the squared deviation by the probability of occurrence for its related outcome, and sum these products to obtain the variance of the probability distribution:

4.  Find the standard deviation by taking the square root of the variance:

We can illustrate these procedures by calculating the standard deviation for both the Designer and Standard lines:

  

We can summarize as follows. The standard deviation gives us an idea of how far above or below the expected value (the mean) actual outcomes are likely to fall; it is a measure of dispersion around the mean. From the calculations above, we can conclude that the Designer line, with its standard deviation of 65.84 percent has more risk than the Standard line where standard deviation is much lower at 3.87 percent.

If we have a continuous probability distribution which is also a normal, bell-shaped distribution, the actual return will lie within ±1 standard deviation of the expected return about 68% (68.26%) of the time.  

Assume that 2 projects in fact have continuous normal distribution, and that we have found k^ = 15% for both and standard deviation to be 65.84% for Designer line and only 3.87% for the Standard line.

Thus for Designer line there is 68.26% probability that the actual return will be in the range of 15% ± 65.84%, or from -50.84% to 80.84%.

For the Standard line there is 68.26% probability that the actual return will be in the range of 15% ± 3.87%, or from 11.13% to 18.87%.

With such a small standard deviation for the standard line, we can conclude that there will be little chance of significant loss from investing in that project, so is not very risky.

If a choice has to be made between two projects, which have the same expected returns (k^) but different standard deviations, most people would choose the project with the lower standard deviation and therefore, lower risk. On the other hand, given a choice between two investments with the same risk (standard deviations) but different returns, investors would tend to prefer the project with the higher expected return.

How do we decide when we have to pick one of two projects and neither the expected returns (k^) nor the standard deviations are the same. The answer is that we need yet another measure of risk, which will be discussed next.

Measuring Risk:    The Coefficient of Variation (CV)

The coefficient of variation shows the risk per unit of return, whether that unit is a % of in amount. It provides a more meaningful comparison when the expected returns and standard deviations on two alternatives are not the same.

The coefficient of variation is generally a better measure of risk than the standard deviation alone.

Since both above projects have the same expected return (k^), the calculation of the coefficient of variation is not really necessary in this case, because result is obvious; the project with the larger standard deviation, the Designer line, will have the larger coefficient of variation. In fact the Designer line is almost 17 times as risky as the Standard line based on this measure of risk:

CVDesigner Line = 65.84% / 15% = 4.39

CVStandard Line = 3.87% / 15% = 0.26

Consider following two  projects,  A and B, which have different expected rates of return as well s different standard deviations. 

  Project A Project B
Expected Return (k^) 45% 20%
Standard Deviation 15% 10%
Coefficient of Variation (CV) 0.33 0.50

We see that Project B actually has more risk per unit of return than project A, even though Project A's standard deviation is larger. When such differences occur, the CV is generally a better measure of risk than the SD alone.

Risk Aversion:    A dislike for risk. Risk averse investors require higher rates of return on higher-risk  investments. It indicates that individuals or businesses will take on risk only if a stock's or project's expected rate of return is sufficiently high to justify taking the risk.

What are the implications of risk aversion for securities prices and rates of return? 

Other things held constant, the higher a security's risk, (1) the lower its price and (2) the higher its required rate of return.

Risk Premium, RP:    The difference between the required (and expected) rate of return on a given risky asset and that on a less risky asset, which compensates investors from taking additional risk by investing in risky asset. For example, two firms Sanaulla (Clothing manufacturers) and Oratech's data is as follow:

  Expected Dividend Expected Rate of Return (k^) Stock Price
Sanaulla Rs. 11.25 15% ......... 10% Rs. 75..... 112.50
Oratech Rs. 11.25 15% ........  20% Rs. 75 .....  56.25

Sanaulla's k^ is less risky as compare to Oratech which is a high tech company and in case of failure it might go down to zero and in case of success it might go to Rs. 150. Investors are risk averse, there preference would be to buy less risky shares of Sanaulla. Oratech stockholders would also sell their shares and purchase Sanaulla's shares. Buying pressure would drive up the price of Sanaulla's stock, and selling pressure would simultaneously cause Oratech's stock price to decline. The price changes would, in turn, cause changes in the expected rates of return on the two securities. The new expected rates may be 10% on Sanaullah and 20% on Oratech. The difference in rates, 20 - 10 = 10% is called RISK PREMIUM, RP

Portfolio: A collection of investments.

Capital Asset Pricing Model (CAPM):     [Professor Harry Markowitz & William.F Sharpe - 1990 Nobel Prize winners]

A model based on the proposition that any stock's required rate of return is equal to the risk-free rate of return + a risk premium, where risk reflects the effect of diversification.

In CAPM, we shall emphasis on expected rate of return (k^) on the PORTFOLIO and the portfolio's risk. Risk and Return of an individual security should be analyzed in terms of how that security affects the risk and return of the portfolio in which it is held.

Portfolio Returns (k^p): The expected return on  portfolio, k^p,  is simply the weighted average of the expected returns on the individual stocks in the portfolio, with the weights being the fraction of the total portfolio invested in each stock:

Here k^i = expected returns on the individual stocks; Wi's are the weights, and there are n stocks in the portfolio.

Note (1) Wi is the proportion of the portfolio's Rupees value invested in Stock i ( Rs. in stock i  ÷ Rs. value of total Portfolio); (2) Wi's must sum to 1.0.

Assume that in January 2001, a securities analyst had estimated the following returns on 4 large companies:

Script k^ Amount w
ICI 11% Rs.25,000 0.25
PTCL 20% Rs.25,000 0.25
Engro 13% Rs.25,000 0.25
HUBCO 18% Rs.25,000 0.25
Rs.100,000 1.0

If we formed a Rs. 100,000 portfolio, investing Rs. 25,000 in each stock, the expected portfolio return would be 15.50%.

k^p = w1k^1 +w2k^2 + w3k^3 + w4k^4 => 0.25(11%) + 0.25(20%) + 0.25(13%) + 0.25(18%) = 15.50%

Realized Rate of Returns (k¯ ): The return that is actually earned. Its usually different from the expected return (k^).

After a year later, the realized rates of return, k¯, on the individual stock is certainly different from K^p = 15.50%. For example, ICI stock might double in price and provide a return of +100%, whereas PTCL might have a terrible year, fall sharply, and have a return of -75%. Note, though, that those two events would be somewhat offsetting, so the portfolio's return might still be close to its expected return, even though the individual stocks' actual returns were fall from their expected returns.

Portfolio Risk :     

Banks pension funds, insurance fund, mutual funds, and other financial institutions are required by law to hold diversified portfolios.

The portfolios risk will be smaller than the weighted average of the stocks.

The tendency of two variables to move to gather is called correlation and the correlation coefficient, r, measures this tendency.

When stocks are perfectly negatively correlated (r = -1.0) this risk will be diversified away, but when stocks are perfectly positively correlated diversification does no good whatsoever.

In reality, most stocks are positively correlated, but not perfectly so. On average, the correlation coefficient for the return on two randomly selected stocks would be about +0.6, and for most pairs of stocks, r would lie in in the range of +0.5 to +0.7. Under such conditions combing stocks into portfolios reduces risk but does not eliminate it.

The extent to which adding stocks to a portfolio reduces its risk depends to the degree of correlation among the stocks. The smaller the positive correlation coefficient the lower the risk in a large portfolio.

If we could set a set of stocks whose correlation was zero or negative, all risk would be eliminated .In the typical case, where the correlation among the individual stocks are positive but not less than +1.0 some, but not all risks are eliminated.

To test your understanding, would you expect to find higher correlations between the returns on two companies in the same or in different industries? For example: would the correlation of returns on Toyota and Suzuki Motors's stocks be higher, or would the correlation coefficient be higher between either Toyota or Suzuki and IBM, and how would those correlations affect the risk of portfolios containing them?

    Answer:    Toyota and Suzuki's returns have a correlation coefficient of about 0.9 with one another because both are affected by auto sales, but only about 0.6 with those of IBM.

    Implications:    A two-stock portfolio consisting of Toyota and Suzuki would be riskier than a two-stock portfolio consisting of either Toyota or Suzuki, plus IBM. Thus to minimize risk, portfolio should be diversified across industries.

Almost half of the riskiness inherent in an average individual stock can be eliminated if the stock is held in a reasonably well-diversified portfolio, which is one containing 40 or more stocks. Some risk is always remains, however, due to the fact that it is virtually impossible to diversify away the effects of broad stock market movements that affect almost all stocks.

Company Specific Risk :    That part of the risk of a stock, which can be eliminated, is called diversifiable, or company-specific, or unsystematic risk. Such things as lawsuits, strikes, successful and unsuccessful marketing programs cause company risk, the winning and losing of major contracts and other events are unique to a particular firm. Since these events essentially random. Their effects on a portfolio can be eliminated by diversification- bad events in one firm will be offset by good events in another.

Market Risk : That part of the risk of a stock, which cannot be eliminated is called non-diversifiable or market or systematic risk.  It stems from factors which systematically effect the firm such as war inflation recession and high interest rates. Since most stocks will tend to be negatively effected systematic risk cannot be eliminated by diversification.

Relevant Risk: The risk of a security that cannot be diversified away, or its market risk. This reflects a security's contribution to the risk of a portfolio.

The risk that remains after diversification is market risk, or risk that in inherent in the market, and it can be measured by the degree to which a given stock’s returns tend to move up and down with returns on the market.

CAPM tells us the relevant riskiness of an individual stock's contribution to the riskiness of a well-diversified portfolio.

Concept of Beta:     A measure of the extent to which the returns on a given stock move with the stock market. Betas are regression coefficients and betas for literally thousands of companies are calculated and published by Merril Lynch, Value Line, and numerous other organizations.

If a relatively high-beta stock (beta > 1.0) is added to an average-risk (b=1.0) portfolio, both the portfolio's beta and the riskiness of the portfolio will increase. Conversely, if a relatively low-beta stock (b<1.0) is added to an average-risk portfolio, the portfolio's beta and risk will decline. Thus, because a stock's beta measures its contribution to the riskiness of any portfolio, beta is the appropriate measure of the stock's riskiness. Therefore, a stock's market risk is measured by its beta coefficient, and we use the terms "market risk" and "beta risk" interchangeably.

Portfolio Beta:   the weighted average of the betas of the specific securities in that portfolio:                                                   

                               

bp is the beta of the portfolio, and it reflects how volatile the portfolio is in relation to the market; wi is the fraction of the portfolio invested in the ith stock; and bi is the beta coefficient of the ith stock.

For example:    If an investor holds a Rs.100,000 portfolio consisting of Rs.25,000 in one stock, Rs.55,000 in second and Rs.20,000 in third stock, with betas 0.7, 0.9 and 0.5 respectively, then the portfolio's beta will be:

                                 bp =    0.25(0.7) + 0.55(0.9) + 0.20(0.5) = 0.175  + 0.495 + 0.1 = 0.77

  Such a portfolio will be less risky than the market; it should experience relatively narrow price swings and have relatively small rate-of-return fluctuations. Adding a low beta stock would reduce the riskiness of the portfolio; and, conversely, adding a high-beta stock would increase the riskiness.

Summary:

  1. A stock's risk consists of two components, market, or beta, risk and company-specific risk.
  2. Company-specific risk can be eliminated by diversification. Most investors indeed diversify either by holding large portfolios or by purchasing shares in a mutual fund. Market risk is the only relevant risk to a rational, diversified investor.
  3. Investors must be compensated by bearing risk -  the greater the riskiness of a stock, the higher its required return. However, compensation is required only for risk which cannot be eliminated by diversification. If risk premium is existed on stock with high diversifiable risk, well-diversified investors would start buying these securities and bidding up their prices, and their final (equilibrium) expected returns would reflect only non-diversifiable market risk.                    Suppose, half of stock A's risk is market risk (it occurs because Stock A moves up and down with the market). The other half of A's risk is diversifiable. You plan to buy only stock A, so you would be exposed to all of its risk. As compensation for bearing so much risk, you want a risk premium of 8% over the 7% T-bond rate (kA = 7 + 8 = 15%). But suppose other investors are well diversified, and also hold stock A, but they have eliminated its diversifiable risk and thus are exposed to only half as much risk as you (kA = 7 + 4 = 11%). If the stock were yielding  > 11% in the market, other investors, would buy it. If it were yielding = 15%, you would be willing to buy it, but well-diversified investors would bid its prices up and yield down, and keep you away from getting it. In the end, you would have to accept an 11% return if you still wanted the stock. Thus, risk premiums in a market populated with rational investors will reflect only market risk.
  4. The market risk of a stock is measured by its beta coefficient, which is an index of the stock's relative volatility. Some benchmark betas follow:    b = 0.5; Stock's returns are only half as volatile, or risky as returns of the avg. risk stock. b=1.0; Stock is of average risk.    b =  2.0; Stock's returns are twice as risky as returns of the average stock.
  5. Since a stock's beta coefficient determines how the stock affects the riskiness of a diversified portfolio, beta is the most relevant measure of a stock's risk.

The Relationship Between Risk And Rates of Return:

We may specify the relationship between risk and return for a given level of beta. What rate of return will investors require on a stock in order to compensate them for assuming the risk? The Security Market Line (SML) is the line that shows the relationship between risk as measured by beta and the required rate of return for individual securities; and is given by the following equation: 

             ki = kRF + (kM - kRF)bi ............................................. SML Equation

ki =  

required rate of return on the ith stock. Note that k^i is the expected rate of return on the ith stock. If k^i < ki, you would not purchase the stock or sell if you owned it.If k^i > ki you would want to buy the stock and you would be indifferent if k^i = ki.

kRF =   risk-free rate of return
bi =   beta coefficient of the ith stock. The beta of an average stock, by definition is bA = 1.0.
kM =   required rate of return on a portfolio consisting of all stocks, which is the market portfolio. kM is also the required rate of return on an average (bA = 1.0) stock.
RPM = (kM - kRF) =   market risk premium. This is the additional return over the risk-free rate required to compensate an average investor for assuming an average amount of risk. Average risk means bA = 1.0.
RPi = (kM - kRF)bi = risk premium on the ith stock. The stock's risk premium <, =, > the premium on an average stock, depending on whether its beta <, =, > 1.0. If bi = bA = 1.0, then RPi = RPM

Suppose, Treasury bonds yield kRF =  9% and an average share of stock has a required rate of return of kM = 13%. Therefore, the market risk premium is 4% [RPM = (kM - kRF) = 13% - 9% = 4% ]

The Market Risk Premium , RPM, depends on the degree of aversion that investors, on average, have to risk. We can measure a stock's relative riskiness by its beta coefficient. If we know the market risk premium, RPM, and the stock's risk as measured by its beta coefficient, bi we can find its risk premium as the product (RPM)bi. e.g.: if bi = 0.5 and RPM = 4% then RPi = 2%:

Risk Premium for Stock i = RPi = (RPM)bi    => (4%)(0.5) = 2%

The required return for any investment can be expressed in general terms as: 

Required return = Risk free return + premium for risk

Using the SML Equation [ki = kRF + (kM - kRF)bi] the required rate of return for stock i can be found as:

ki = kRF + (kM - kRF)bi => 9% + (13% - 9%)(0.5) => 9% + 4%(0.5) => 9% + 2% => 11%

Stock i is relatively low risk stock and, therefore,  has a lower than average required return; its risk premium has been scaled down. On the other other hand, if another stock, j, were four times riskier than stock i and had bi = 2.0 (stock i had bi=0.5) then:

ki = kRF + (kM - kRF)bi => 9% + (13% - 9%)(2.0) => 9% + 4%(2.0) => 9% + 8% => 17%

An average stock, with b = 1.0, would naturally have a required rate of return of 13%, the same as the market return, kM:

ki = kRF + (kM - kRF)bi => 9% + (13% - 9%)(1.0) => 9% + 4%(1.0) => 9% + 4% => 13%

 

Above figure shows the following points:

  1. Required rates of return are shown on the vertical axis, while risk measured by beta is shown on the horizontal axis.
  2. Riskless securities have bi = 0; therefore, kRF appears as the vertical axis intercept.
  3. The slope of the SML reflects the degree of risk aversion in the economy; the greater the average investor's aversion to risk, (1) the steeper the slope of the line, (2) the greater the risk premium for any stock, and (3) the higher the required rate f return on stocks. 
  4. The values we worked out for stocks with bi = 0.5, bi = 1.0, and bi = 2.0 agree with the values shown on the graph for kLOW, kA and kHIGH.

Changes in the Security Market Line and Betas: 

Both the security Market Line and company’s position on it change over time because of changes in inflation expectations, investors’ risk aversion, and individual companies’ betas. Such changes are discussed in the following sections.

Changes in Expected Inflation:

We know that Risk free interest rate kRF = k* + IP. However, as the expected rate of inflation increases, a premium must be added to the real risk-free rate to compensate investors for the loss of purchasing power that results from inflation. Therefore, the 9% kRF shown in above figure 12-8 might be thought of as consisting of a 3% real risk-free rate of return + a 6% inflation premium: kRF = k* + IP = 3% + 6% = 9%.

[NOTE: Students sometimes confuse beta with the slope of the SML. This is a mistake. The slope of any line is equal to the "RISE" divided by the "RUN," or ( y1 - y0 ) / (x1 - x0 ). Consider figure 12-8. If we let y = k and x = beta, and we go from the origin to b = 1.0, we see that the slope is (kM - kRF) / (betaM - betaRF) = ( 13 - 9 ) / (  1-0 ) = 4. Thus, the slope of the SML is equal to (kM - kRF), the market risk premium. To put it another way, the slope coefficient of the SML is the numerical value of the market risk premium, in this case, 4.]

If the expected  rate of INFLATION rose by 2%  to 8%, this would cause kRF to rise to 11% (3 + 8 = 11%).

 

Under the CAPM, the increase in kRF causes an equal increase in the rate of return on all risky assets because the inflation premium is built into the required rate of return of both risk-free and risky assets. [Recall that the inflation premium for any asset is equal to the average expected rate of inflation over the life of the asset. Thus, in this analysis we must assume either that all securities plotted on the SML graph have the same life or else that the expected rate of future  inflation is constant] e.g.:

In the figure, the rate of return on an average stock, kM, increase from 13% to 15%. Other risky securities' returns also rise by 2%; the new Security Market Line, SML2 is parallel to SML1, but is 2% higher. On the other hand, if inflation expectations decline, and if other things are equal,  SML2 will be at a new lower level, again parallel to the original Security Market Line, SML1.  

Changes in Risk Aversion:

The slope of SML reflects the extent to which investors are averse to risk.; the greater the average investor's risk aversion, the steeper the slope of the SML. If investors were indifferent to risk, and if kRF was 9% , then risky assets would also sell to provide an expected return of 9%. If there were no risk aversion, there would be no risk premium, so the SML would be horizontal. As risk aversion increases,  so does the risk premium, and, thus, the slope of the SML.

The figure 12-10 illustrates an increase in risk aversion. The market risk premium rises from 4% to 6%, and kM rises from 13% to 15%. The returns on other risky assets also rise, and the effect of this shift in risk aversion is greater for riskier securities. For example, the required return on a stock with bi = 0.5 increase by only 1% point from 11% to 12%, but that on a stock with bi = 1.5 increases by 3% points from 15% to 18%. Instead of the parallel upward shift due to increased inflation expectations in earlier case, here we see an upward pivoting of the SML with the risk-free rate acting as an "anchor"; the risk-free rate, by definition, will not be influenced by changes in investors risk aversion, but the required rates on all stocks will. Offcourse, the opposite could also occur: a downward pivoting of the SML, if investors became less reluctant to take on risk.

Changes in a Stock's Beta Coefficient:

The beta f a stock reflects the firm's industry characteristics and its own management policies. For example, a firm can affect its beta risk through changes in the composition of its assets as well as through its use of debt financing. A company's beta can also be changed as a result of external factors such as increased competition in its industry, the expiration of basic patents, major changes in technology etc. When such changes occur, the required rate of return also changes, which will affect the price of the firm's stock. For example: in context of figure 12-8, due to some external factors Company's beta is increased from  1.0 to 1.5, now the company's required rate of return would increase from:

ki = kRF + (kM - kRF)bi   => 9% + (13% - 9%)1.0 = 13%    to

                                            k2 = 9% + (13 - 9%)1.5 = 15%

Any change which affects the required rate of return on a security, such a change in its beta coefficient, in its expected inflation, or in investors risk aversion, will an impact on the price of the security.