Risk and Return - College of...

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Copyright 2005 by Larry C. Holland

Risk and Return

Financial Management Class

Expected Return, Risk, CAPM, Beta, Systematic and Unsystematic risk,

Portfolio diversification

This presentation is about risk and return. A definition of risk might be uncertainty of a return from a project or a financial asset, with a possibility of a loss. Return identifies the reward one might receive as a result of investing in a project or a financial asset. In this presentation, we’ll discuss the concepts of expected return, the relationship of expected return to risk, the Capital Asset Pricing Model, Beta as a measure of risk, differentiating between systematic and unsystematic risk, and the idea behind portfolio diversification.

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Historical Returns

First, it might be helpful to look at what has happened in the past regarding various financial assets. This graph shows the value today of one dollar invested in 1925, in various securities. For example, if one were to have invested in U.S. Treasury Bills, over the last 80 years, that one dollar would have increased in value to $16.56. And if one had invested in long-term U.S. Treasury Bonds, one dollar would have increased in value to $48.86. However, if one had invested in large company stocks, one dollar invested in 1925 would be worth $2,586 today. And finally, if one had been wise enough to have invested in small company stocks, one dollar invested in 1925 would be worth $6,402 today. After looking at this graph, the obvious question is why everybody did not invest in stocks, since they seem to have had the highest return. And why would anyone ever invest in U.S. Treasury Bills or Treasury Bonds since the average returns here seem so low compared to stocks.

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Year-to-Year Total Returns

Large-Company Stock Returns

Long-Term Government Bond Returns

U.S. Treasury Bill Returns

Of course, the answer to this question relates directly to the risk involved in each of these investments. This becomes more obvious when we look at year-to-year total returns.

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Large Company Stocks

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

1926

1929

1932

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Year

Tota

l Ret

urn

For example, this is a chart showing the average annual returns for large company stocks year by year from 1925. Notice the large amount of fluctuation in the annual returns. The average annual return is about 13%. However, the historical returns ranged from -45% in one year to +55% in one year. There is so much fluctuation in the annual returns that there is a great deal of uncertainty as to the return one might receive in any one year, and there is certainly the possibility that one could lose a great deal of money in any one year. We would identify this uncertainty in returns with a possibility of a loss as “risk”. In other words, there is a considerable amount of risk in investing in large company stocks, even though the average return is around 13% per year.

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Long-Term Government Bonds

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

1926

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The year to year returns on long-term U.S. government bonds shows a somewhat different pattern. Since 1925, the range in annual returns is from a low of about -8% in one year to a high of about 43%. There also seems to have been fewer years in which a loss occurred. With a lower amount of fluctuation in the annual returns, and a lesser chance of losing money, we might say that U.S. Treasury Bonds have less risk than large company stocks. However, we might also notice that the average annual return is considerably lower than large company stocks, an average of about 5.7% per year.

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U.S. Treasury Bills

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

1926

1929

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Finally, U.S. Treasury Bills never had an annual loss in any year since 1925. The returns also appear to be more uniform and predictable. One might expect that the return in any one year from Treasury Bills would be less uncertain because of less fluctuation in returns from year to year. We would therefore refer to Treasury Bills as having less risk than either large company stocks or U.S. Treasury Bonds. However, the average return in any one year appears to be quite low, at an average of about 3.9% per year.

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Average Returns

3.2%Inflation

3.9%U.S. Treasury Bills

5.7%Long-term Government Bonds

6.0%Long-term Corporate Bonds

17.3%Small Stocks

13.0%Large stocks

Average ReturnInvestment

So we can see that there is a significant difference in each of these investments in both the average return and the amount of risk in each of these securities. The riskier securities seem to have a higher average return compared to the less risky securities. This table seems to indicate that there is a positive relationship between risk and average return – the higher the risk, the higher the average return. If we notice that U.S. Treasury Bills have the least amount of risk (perhaps near zero default risk), we might be willing to assume that they are nearly a risk-free asset, or at least close enough to being risk-free for all practical purposes. Actually, Treasury Bills are short-term securities. Often we stretch a little and treat U.S. Treasury Bonds as risk-free because they are longer term securities, which can be compared more appropriately with the returns from stocks. If Treasury Bonds are considered risk-free, then anything that has a higher level of risk would require a higher average return. The additional average return above a risk-free asset needed to compensate an investor for the amount of risk in the security is called the risk-premium.

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Risk Premiums• The “extra” return earned for taking on risk• Long Term Government bonds are assumed to

be risk-free• The risk premium is the return over and above

the risk-free rate

The risk premium is the extra return earned for taking on risk. Often analysts assume that long-term U.S. Treasury Bonds are close to being risk-free. Then the historical risk premium is the average return for an asset over and above the risk-free rate, which we assume to be the average for U.S. Treasury Bonds.

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the risk-free rate

fiRRemiumPrRiskHistorical −−−−====

In this case, the historical risk premium would be the average return of an asset over some period of time minus the average return of long-term U.S. Treasury bonds.

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the risk-free rate

fi

fi

R)R(EemiumPrRisk

RRemiumPrRiskHistorical

−−−−====

−−−−====

Most of the time, we are really interested in identifying the risk premium that will occur in the future. In this case, the expected risk premium would be the expected return from a security minus the risk-free rate.

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Historical Risk Premiums

• Large stocks: 13.0 – 5.7 = 7.3%

• Small stocks: 17.3 – 5.7 = 11.6%

Often we are willing to make some sort of estimate about the future by observing what has occurred in the past. For example, the historical risk premium for large company stocks is the average return for large stocks minus an assumed risk-free rate, which in this case is long-term U.S. Treasury bonds. This shows a historical risk premium of 7.3% for large stocks. Likewise, small stocks show a larger historical risk premium of 11.6%. The larger risk premium for small stocks is a result of a higher level of risk.

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Figure 12.10

Actually, risk can be identified by the degree to which the annual returns fluctuate about the average annual return, which can be measured as the standard deviation. For example, this chart shows the average annual return for each security, as well as the standard deviation of returns around the mean. The higher the standard deviation, the more the observed data fluctuates around the mean, which indicates a higher level of risk. This is also shown graphically in the accompanying distribution of returns on the right of this chart. Notice that the distribution of small stock returns is wider than all the other distributions. This is also indicated by the highest standard deviation. As the standard deviation gets larger, the returns are much more uncertain, and thus have a higher level of risk.

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Figure 12.11

If the returns are normally distributed, we can even make a statistical statement about the returns for large company stocks. For example, we can say that 68% of the time, the annual returns were within plus or minus one standard deviation from the mean, or between -7.2% and +33.2%. Likewise, 95% of the time, the annual returns were within plus or minus two standard deviations from the mean, or between -27.4% and +53.4%.

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Expected Returns• Expected returns are based on the probabilities

of possible outcomes• In this context, “expected” means average if

the process is repeated many times• The “expected” return does not even have to

be a possible return

�=

=n

iii RpRE

1

)(

Instead of working with historical returns, sometimes we are more interested in working with expected future returns. One way to estimate future returns is to determine several scenarios of the future and establish an estimate of the probability of each outcome. In this context, “expected” means an average of estimated future outcomes. To calculate the expected return, we simply find a probability weighted average. We’ll show this calculation in an example problem in the next few slides.

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Example Problem

State Probability RA RB

Boom 0.4 30% -5%Bust 0.6 -10% 25%

Suppose we have two stocks, Stock A and Stock B. If a boom in the economy occurs, Stock A is estimated to have a 30% return and Stock B is estimated to have a -5% return. If a bust in the economy occurs, Stock A is estimated to have a -10% return and Stock B is estimated to have a 25% return.

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Example Problem

State Prob. RA Prob. x RA

Boom 0.4 30% 12%Bust 0.6 -10% - 6%

E(RA) = 6%

To find the expected return for Stock A, we must calculate the probability weighted average of the returns for all the scenarios. For a boom in the economy, we multiply the probability of 0.4 times the return of 30% for Stock A for this scenario, which yields 12%. Likewise, for a bust in the economy, we multiply the probability of 0.6 times the return of -10%, which yields -6%. Adding these probability weighted numbers yields an expected return of 6% for Stock A.

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Example Problem

State Prob. RB Prob. x RB

Boom 0.4 -5% -2%Bust 0.6 25% 15%

E(RB) = 13%

Following a similar procedure, the expected return for Stock B, 0.4 times -5% is -2%, and 0.6 times 25% is 15%. The sum of -2% and plus 15% yields an expected return of 13% for Stock B.

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Variance and Standard Deviation• Variance and standard deviation still measure

the volatility of returns• Using unequal probabilities for the entire

range of possibilities• Weighted average of squared deviations

�=

−=n

iii RERp

1

22 ))((�

The variance and standard deviation measure the volatility of returns. We can calculate the variance as the probability weighted squared deviation from the mean. Then the standard deviation is the square root of the variance.

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Example Problem

State Prob. RA Prob. x [RA – E(RA)]2

Boom 0.4 30% 0.4 [.30-.06]2 = .0230Bust 0.6 -10% 0.6[-.10-.06]2 = .0154

�A2 = .0384

�A = 19.6%

To find the variance of returns for Stock A, we must calculate the probability weighted squared deviation from the mean for all the scenarios. For a boom in the economy, we multiply the probability of 0.4 times the return of 30% for Stock A for this scenario minus the expected return of 6% squared, which yields .0230. Likewise, for a bust in the economy, we multiply the probability of 0.6 times the return of -10% minus the expected return of 6% squared, which yields .0154.Adding these probability weighted numbers yields a variance of .0384 for Stock A. Taking the square root of the variance yields a standard deviation of 19.6% for Stock A.

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Example Problem

State Prob. RB Prob. x [RB – E(RB)]2

Boom 0.4 -5% 0.4 [-.05-.13]2 = .0130Bust 0.6 25% 0.6[.25-.13]2 = .0086

�B2 = .0216

�B = 14.7%

In a similar manner, to find the variance of returns for Stock B, for a boom economy, we multiply the probability of 0.4 times the return of -5% minus the expected return of 13% squared, which yields .0130. Likewise, for a bust in the economy, we multiply the probability of 0.6 times the return of 25% minus the expected return of 13% squared, which yields .0086. Adding these probability weighted numbers yields a variance of .0216 for Stock B. Taking the square root of the variance yields a standard deviation of 14.7% for Stock B.

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Portfolios• A portfolio is a collection of assets• An asset’s risk and return is important in how

it affects the risk and return of the portfolio• The risk-return trade-off for a portfolio is

measured by the portfolio expected return and standard deviation, just as with individual assets

We can also create a portfolio of stocks. A portfolio is just a collection of assets. However, the risk and return for each individual asset is important in how it affects the risk and return of the portfolio. Of course, portfolios have the same risk-return trade-off as individual assets. And the expected return and standard deviation of returns are calculated using the same procedure as with individual assets.

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Portfolio Expected Returns• The expected return of a portfolio is the weighted

average of the expected returns for each asset in the portfolio

• You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

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=m

jjjP REwRE

1

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You could also calculate the expected return of the portfolio by finding the weighted average of the expected returns for each stock in the portfolio. However, we are going to calculate the expected return the same way we did in the previous example problem.

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Portfolio Variance• Compute the portfolio return for each state:

RP = w1R1 + w2R2 + … + wmRm

• Compute the expected portfolio return using the same formula as for an individual asset

• Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

First, we find the portfolio return in each possible scenario, and then calculate the expected value and standard deviation as we did for each of the individual stocks. We’ll continue the example problem for a portfolio including both Stock A and Stock B.

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Example: Portfolio Variance• Consider the following information

– Invest 50% of your money in Stock A and 50% in Stock B

– State Probability A B Portfolio– Boom .4 30% -5% 12.5%– Bust .6 -10% 25% 7.5%

• What is the expected return and standard deviation for the portfolio?

For example, suppose you invest 50% of your money in Stock A, and 50% of your money in Stock B. For a boom in the economy, half of your money would earn 30% and the other half would earn -5%. Therefore, the portfolio would earn .5 times 30% plus .5 times -5%, or 12.5%. Likewise, for a bust in the economy, the portfolio would earn .5 times -10% plus .5 times 25%, or 7.5%.

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Example Problem

State Prob. RP Prob. x RP

Boom 0.4 12.5% 5.0%Bust 0.6 7.5% 4.5%

E(RP) = 9.5%

As we did before, to find the expected return for the portfolio, for a boom in the economy, we multiply the probability of 0.4 times the return of 12.5%, which yields 5%. Likewise, for a bust in the economy, we multiply the probability of 0.6 times the return of 7.5%, which yields 4.5%. Adding these probability weighted numbers yields an expected return of 9.5% for the portfolio.

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Example Problem

State Prob. RP Prob. x [RP – E(RP)]2

Boom 0.4 12.5% 0.4 [.125-.095]2 = .00036Bust 0.6 7.5% 0.6[.075-.095]2 = .00024

�P2 = .00060

�P = 2.45%

To find the variance of returns for the portfolio, we again calculate the probability weighted squared deviation from the mean for all the scenarios. For a boom in the economy, we multiply the probability of 0.4 times the return of 12.5% minus the expected return of 9.5% squared, which yields .00036. Likewise, for a bust in the economy, we multiply the probability of 0.6 times the return of 7.5% minus the expected return of 9.5% squared, which yields .00024. Adding these probability weighted numbers yields a variance of .00060 for the Portfolio. Taking the square root of the variance yields a standard deviation of 2.45% for the Portfolio.

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Example Problem

Stock A Stock B Portfolio

Expected Return 6.0% 13.0% 9.5%Standard Deviation 19.6% 14.7% 2.4%

This table summarizes the results for the example problem. Notice that the expected return of the portfolio is just a weighted average of the expected returns of the stocks in the portfolio. Thus, a 9.5% expected return is between the expected return for stock A and the expected return for stock B. However, the standard deviation of returns for the portfolio is not between the standard deviations of the individual stocks in the portfolio. In fact, the portfolio standard deviation is significantly smaller than the standard deviation of either of the stocks within the portfolio. In words, the risk of the portfolio is significantly smaller than the risk of the individual stocks within the portfolio. This is a very interesting result. Why would the risk of the total portfolio be smaller than the individual components? The answer is that there is some cancellation of risk going on inside the portfolio. In other words, when there are losses on one stock, there is a gain on the other stock. This tends to smooth the returns that are realized for the portfolio compared to the individual stocks within the portfolio. This benefit of reducing the risk while maintaining an average expected return is a direct effect of diversification.

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Expected versus Unexpected Returns

There is an expected component and an unexpected component.

a. At any point in time, the unexpected return can be either positive or negative

b. Over time, the unexpected components will tend to cancel out.

Ri = E(Ri) + Unexpected Return

In order to fully understand this concept, we need to understand the difference between the expected part of returns and the unexpected part. Although not very insightful, the equation on this slide shows the relationship between a realized return and an expected return. Suppose I buy some IBM stock with an expected return of 11% per year. And then suppose I hold the stock for one year and actually realize a 13% return. Then this equation says that the 13% return that I actually received is equal to the 11% that I expected plus a 2% return that I did not expect. In general, at any point in time, the unexpected return can be either positive or negative. And therefore, over time, the average of the unexpected component tends to cancel out. This unexpected component of returns will be explained further in the next few slides.

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Announcements and News

• Announcements and news contain both an expected component and a surprise component

• It is the surprise component that affects a stock’s price and therefore its return

• This is very obvious when we watch how stock prices move when an unexpected announcement is made or earnings are different than anticipated

But first we need to explain the effect of announcements by a company and news about a company. In either an announcement or news, there is again an expected component and a surprise component. When an announcement tells us what we already expected to hear, then there would be no effect on the stock price of that company, because we already expected that information about the company. However, if the announcement includes a surprise that was not expected, then the stock price would be affected. So again, any unexpected information tends to have an effect, positive or negative, on a stock price and therefore on stock returns. And of course, we identify the fluctuations in stock returns as a result of unexpected new information as the risk in a stock. There are two types of risk that can occur.

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Systematic Risk

• Risk factors that affect a large number of assets

• Also known as non-diversifiable risk or market risk

• Includes such things as changes in GDP, inflation, interest rates, etc.

First, there is systematic risk. Systematic risk occurs from factors that affect a large number of assets at the same time. Since the effect tends to be in the same direction for all assets at the same time, building a portfolio of stocks does not help in reducing this type of risk. It is therefore also known as non-diversifiable risk, or market risk. Systematic risk includes such things as changes in economic growth, inflation, interest rates, and other total market related events.

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Unsystematic Risk

• Risk factors that affect a limited number of assets

• Also known as unique risk and asset-specific risk

• Includes such things as labor strikes, part shortages, etc.

The second type of risk is unsystematic risk. Unsystematic risk occurs from factors that affect only one asset, or a limited number of assets. Since this risk only affects one stock, it is also known as unique risk and firm-specific risk. Unsystematic risk includes such things as a labor strike against a company, a local parts shortage, a warehouse burning down, losing a key executive, or other events that have an effect limited to a single company. Since the unexpected event can be either positive or negative, building a portfolio of assets provides a diversification effect of reducing risk because there is some cancellation of the positives and negatives.

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Ri = E(Ri) + Unexpected Return

Unexpected Return:mi = market related riskei = unsystematic risk

Ri = E(Ri) + mi + ei

So now, we can return to the equation that the total realized return is equal to the expected return plus the unexpected return. The unexpected return then has two parts – the systematic portion (which is pervasive across the market) and the unsystematic portion (which is unique to a particular firm). Therefore, the equation at the bottom of the slide shows that the total realized return is equal to the expected return plus the systematic risk portion plus the unsystematic risk portion. These two components have a much different affect when we add stocks to a portfolio. The unsystematic portion of the risk tends to be randomly positive and negative, and thus, in a portfolio, tends to cancel out to zero. However, the systematic portion of risk tends to be in the same direction for all stocks, and therefore does not cancel out in a portfolio.

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Diversification

• Portfolio diversification is the investment in several different asset classes or sectors

• Diversification is not just holding a lot of assets

• For example, if you own 50 internet stocks, you are not diversified

• However, if you own 50 stocks that span 20 different industries, then you are diversified

This leads to a more developed concept of diversification. Portfolio diversification is the investment in several different asset classes or stocks, not just holding a portfolio of stocks. For example, a portfolio of 50 internet stocks is not diversified because the returns for these stocks tend to move in the same direction (systematic) rather than cancel out. However, if you own 50 stocks across 20 different industries, then you have a more diversified portfolio.

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Table 13.7

This table shows the relative amount of risk contained in portfolios that include various numbers of stock. For example, an average portfolio with only one stock typically has a standard deviation of returns around 49%. However, an average portfolio of 10 stocks will typically have a standard deviation of less than half of that. And as you add more and more stocks to a portfolio, the additional benefits become less and less. The data in this table show that most of the benefits from diversification are captured with as few as 20 stocks. Adding more stocks to a portfolio will not increase the benefits much more. The data in this table is from the mid 1970s. More recent research indicates that you need at least 50 stocks to capture most of the benefits of diversification.

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The Principle of Diversification

• Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns

• This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another

• However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion

We can now state the principle of diversification. Diversification means adding a sufficient number of disimilar stocks to a portfolio. This can substantially reduce the variability of returns without an equivalent reduction in expected returns. The reduction in risk occurs when unsystematic events offset one another. However, there is a minimum level of risk that cannot be diversified away – this is the systematic portion.

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Figure 13.1

This chart shows the effects of diversification in graphical form. The y-axis shows the portfolio risk measured as the average annual standard deviation of returns for a portfolio. The x-axis shows the number of stocks in a portfolio. Again, this chart shows that there are large benefits from creating a diversified portfolio. However, after a given number of stocks are contained within a portfolio, most of the unsystematic risk has been diversified away, and there are only minimal benefits to adding more stocks to the portfolio. The systematic risk does not go away no matter how many stocks are in the portfolio.

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Diversifiable Risk

• The risk that can be eliminated by combining assets into a portfolio

• Often considered the same as unsystematic, unique or asset-specific risk

• If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away

Therefore, the benefits from diversification are a result of the elimination of diversifiable risk. This unsystematic risk can be eliminated by combining assets into a portfolio. Equivalent names for this diversifiable risk are unsystematic, unique, asset-specific, or firm-specific risk. This illustrates the if we hold only one stock, or stocks in the same industry, then we are exposing ourselves to unnecessary risk that could be diversified away.

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Total Risk

• Total risk = systematic risk + unsystematic risk• The standard deviation of returns is a measure of

total risk• For well diversified portfolios, unsystematic risk

is very small• Consequently, the total risk for a diversified

portfolio is essentially equivalent to the systematic risk

To summarize, the total risk in an investment is the sum of the systematic and the unsystematic risk. The standard deviation of returns is a measure of total risk. However, for well diversified portfolios (of 50 stocks or more), the unsystematic risk portion is very small. In this case, the total risk for a fully diversified portfolio is essentially equivalent to the systematic risk.

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Systematic Risk Principle

• There is a reward for bearing risk• There is not a reward for bearing risk

unnecessarily• The expected return on a risky asset

depends only on that asset’s systematic risk since unsystematic risk can be diversified away at no cost through a portfolio of assets

The systematic risk principle is that there is a reward for bearing risk, but not a reward for bearing risk unnecessarily. Earlier, we noted that there is a trade-off between expected return and risk. Now we can be a little more precise in this statement. The expected return on a risky asset depends only on that asset’s systematic risk, since unsystematic risk can be diversified away at no cost by including that asset in a portfolio of assets. Thus, only the systematic risk for a stock is important in terms of a higher expected return. This means that measuring the standard deviation of returns for a single stock can be misleading because only the systematic part of the total risk relates to expected return. For a single stock, we need to be able to measure just the systematic part of risk, rather than the total risk as measured by standard deviation.

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Measuring Systematic Risk

• How do we measure systematic risk?• We use the beta coefficient to measure systematic

risk• What does beta tell us?

– A beta of 1 implies the asset has the same systematic risk as the overall market

– A beta < 1 implies the asset has less systematic risk than the overall market

– A beta > 1 implies the asset has more systematic risk than the overall market

As it turns out, there is a way to measure systematic risk. We use the beta coefficient to measure systematic risk. Beta is a standardized measurement of systematic risk. A beta of 1 implies the asset has the same systematic risk as the overall market. A beta less than one implies less risk than the market, and a beta greater than one implies more risk than the market.

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Table 13.8

This table shows the beta for a sample of companies. For example, American Electric Power is a utility with a relatively low level of systematic risk (that is, a beta less than 1). And AOL-Time Warner is an internet and entertainment company with a relatively high level of systematic risk (that is, a beta greater than 1).

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Total versus Systematic Risk

• Consider the following information:Standard Deviation Beta

– Security C 20% 1.25– Security K 30% 0.95

• Which security has more total risk?• Which security has more systematic risk?• Which security should have the higher

expected return?

This slide shows a contrast between total risk and systematic risk. Consider the two securities, C and K. Which security has more total risk and which has more systematic risk? Security K has the higher total risk, because this is measured with standard deviation of returns. However, Security C has the higher systematic risk, because that is measured by beta.

And finally, which security should have the higher expected return? Security C should have the higher expected return, because expected return is related directly to the systematic risk of an asset, which is measured by beta.

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Find Beta on the Internet

• Many sites provide betas for companies• Yahoo Finance provides beta, plus a lot of

other information under its profile link• Click on the web surfer to go to Yahoo

Finance at http://finance.yahoo.com– Enter a ticker symbol and get a basic quote– Click on profile

Where can you find the beta for a particular company? Many sites on the internet provide betas for companies. Yahoo Finance provides beta plus a lot of other financial information. It would be beneficial to go to Yahoo Finance on the web, enter a ticker symbol for a stock, and clidk on profile. There is a wealth of information about a company at this site.

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Example: Portfolio Betas

• Consider the previous example with the following four securities– Security Weight Beta– DCLK .133 3.69– KO .2 0.64– INTC .167 1.64– KEI .4 1.79

• What is the portfolio beta?• .133(3.69) + .2(.64) + .167(1.64) + .4(1.79) = 1.61

If you know the betas for each stock in a portfolio, you can find the beta for the portfolio by calculating a weighted average of the betas within a portfolio. The weight to be used in this calculation is simply the percent invested in each stock. At the bottom of this slide is a calculation of the beta for this portfolio.

Given these four stocks in a portfolio,

Which security has the highest systematic risk? DCLK, because it has the highest beta at 3.69

Which security has the lowest systematic risk? KO, because it has the lowest beta at 0.64

Is the systematic risk of the portfolio more or less than the market? More, because the portfolio beta is 1.61

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Beta and the Risk Premium

• Remember that the risk premium = expected return – risk-free rate

• The higher the beta, the greater the risk premium should be

• Can we define the relationship between the risk premium and beta so that we can estimate the expected return?– YES!

Earlier we stated that there is a positive relationship between the expected risk premium and risk. If risk is properly measured by Beta, then there should be a relationship between the risk premium and beta. Recall that the risk premium is defined as the expected return minus the risk-free rate. A positive relationship should mean that the higher the beta risk, the greater the risk premium. However, can we be more precise in defining the relationship between the risk premium and beta? The answer is YES. And this leads us to a discussion of the Capital Asset Pricing Model, generally referred to as CAPM.

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B

fB

A

fAR)R(ER)R(E

ββββ−−−−


Suppose we have two stocks that we are evaluating for possible investment. In order to compare the two stocks, we decide to look at the expected risk premium per unit of risk. This ratio is often called the reward to risk ratio, because the numerator is the reward (that is, the risk premium), and the denominator is a measure of the risk.

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B

fB

A

fAR)R(ER)R(E



5% 3%

Suppose that Stock A offers a 5% risk premium per unit of beta risk, and Stock B offers a 3% risk premium per unit of beta risk. Which stock would be the best investment? Clearly, if Stock A is offering a higher reward per unit of risk than Stock B, everyone would prefer Stock A. At the same time, no one would choose Stock B as long as they could choose the more lucrative Stock A. Of course, if everyone buys Stock A and no one buys Stock B, the price of Stock A will increase and the price of Stock B will decrease. However, a higher price for Stock A will lower the Expected future return for Stock A. Likewise, a lower price for Stock B will increase the Expected return for Stock B.

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B

fB

A

fAR)R(ER)R(E

ββββ−−−−====


This process will continue until, at equilibrium, the prices of Stock A and Stock B will adjust so that the reward to risk ratio of the two stocks will be equal.

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i

fi

B

fB

A

fAR)R(ER)R(ER)R(E


ββββ−−−−====


We can also include any other investment, say Stock i.

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i

fi

B

fB

A

fAR)R(ER)R(ER)R(E

ββββ−−−−====

ββββ−−−−====


and the price of Stock i will also adjust until the reward to risk ratio of Stock i equals the reward to risk ratio of Stock A and Stock B.

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m

fm

i

fi

B

fB

A

fAR)R(ER)R(ER)R(ER)R(E


ββββ−−−−====

ββββ−−−−====


At equilibrium, if all individual stocks must have the same reward to risk ratio, this must also be true of portfolios of stocks, such as the Market portfolio.

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m

fm

i

fi

B

fB

A


ββββ−−−−====

ββββ−−−−====

ββββ−−−−====


Which means that the reward to risk ratio of the Market portfolio must also equal the reward to risk ratio of every stock.

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m

fm

i

fi

B

fB

A


ββββ−−−−====

ββββ−−−−====

ββββ−−−−====


In equilibrium, all assets and portfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market

So, in equilibrium, all assets and portfolios must have the same reward to risk ratio, and they all must equal the reward to risk ratio for the market.

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m

fm

i

fiR)R(ER)R(E

ββββ−−−−====


Let’s look at the last two examples – any stock i and the Market portfolio.

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m

fm

i

fiR)R(ER)R(E

ββββ−−−−====


Looking at the right side of this equation, recall that beta measures risk relative to the Market. Therefore, the beta of the Market portfolio must equal one.

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1R)R(ER)R(E

fm

i

fi−−−−====


So we can replace the Beta of the Market with 1. Of course, dividing by one does not change the expression,

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fm

i

fi R)R(ER)R(E

−−−−====ββββ

−−−−

so the right side of the equation is simply the expected return on the Market minus the risk-free rate. If we then multiply both sides by the beta for Stock i,

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[[[[ ]]]]fmifi

fm

i

fi

R)R(ER)R(E

R)R(ER)R(E

−−−−ββββ====−−−−

−−−−====ββββ

−−−−

We get this second equation. And if we add the Risk-free rate to both sides of the equation,

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[[[[ ]]]]

[[[[ ]]]]fmifi

fmifi

fm

i

fi

R)R(ER)R(E

R)R(ER)R(E

R)R(ER)R(E

−−−−ββββ++++====

−−−−ββββ====−−−−

−−−−====ββββ

−−−−

We arrive at the third equation in this slide. This equation shows a positive relationship between risk and expected return. As the risk measured by beta increases on the right side of the equation, the Expected return on the left side also increases.

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[[[[ ]]]]fmifi

R)R(ER)R(E −−−−ββββ++++====

Capital Asset Pricing Model(CAPM)

This equation is known as the Capital Asset Pricing Model, or CAPM for short.

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[[[[ ]]]]

[[[[ ]]]])MRP(ER)R(E

R)R(ER)R(E

ifi

fmifi

ββββ++++====

−−−−ββββ++++====

Capital Asset Pricing Model(CAPM)

Since the Expected return on the Market minus the risk-free rate is equal to the Expected Market Risk Premium, the CAPM is also sometimes shown as this second equation. Therefore, the Expected return of a security is equal to the Risk-free rate plus Beta times the Expected Market Risk Premium.

Notice that the equation for CAPM shows a linear relationship between the Expected Return for an asset and the Beta for that asset. This linear relationship can also be shown graphically.

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0%

5%

10%

15%

20%

25%

0 0.5 1 1.5 2 2.5

Beta

Exp

ecte

d R

etur

n

E(RM)

The Security Market Line(SML)

This slide shows the Security Market Line as a red line in this chart, which is simply a graphical representation of the CAPM equation. Notice that the intercept is the risk free rate. This is the expected return when there is zero risk. The slope of the security market line is simply the rise over the run. Using the expected return of the market as a guide, the rise would be the expected return of the market minus the risk free rate, which is the expected market risk premium. The run would be a beta of one minus a beta of zero, or simply one. The rise over the run would therefore be the expected market risk premium divided by one. Therefore, the slope of the Security Market Line is the Market Risk Premium.

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The Capital Asset Pricing Model (CAPM)

• The capital asset pricing model defines the equilibrium relationship between risk and expected return

• If we know an asset’s systematic risk, we can use the CAPM to determine its expected return

• E(RA) = Rf + βA(E(RM) – Rf)

CAPM defines the equilibrium relationship between risk, measured as beta, and expected return. In fact, if we know an asset’s systematic risk, or beta, we can use the CAPM to determine its expected return, simply by plugging into the CAPM equation.

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Example Using CAPMConsider the betas for each of the stocks given earlier. If the risk-free rate is 4.5% and the market risk premium is 8.5%, what is the expected return for each stock?

4.5 + 1.79(8.5) = 19.7%1.79KEI

4.5 + 1.64(8.5) = 18.4%1.64INTC

4.5 + .64(8.5) = 9.9%.64KO

4.5 + 3.69(8.5) = 35.9%3.69DCLK

Expected ReturnBetaSecurity

Given the four stocks we looked at earlier, we can calculate an expected return for each stock, if we assume that the CAPM is accurate and true. For example, if the risk-free rate is 4.5% and the expected market risk premium is 8.5%, the expected return for DCLK is 35.9%. Likewise, the expected return for KO is 9.9%, INTC is 18.4%, and KEI is 19.7%. Just as a word of caution, the estimates for beta, the expected Market Risk Premium, and the Risk-Free rate are all subject to forecasting errors. So even if CAPM is close to being true (which some people doubt) errors in the input data can mean significant errors in an expected return. Therefore, it would be wise to use the results from CAPM with caution.

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Risk and Return1. Expected Return2. Risk3. Relationship between E(R) and Risk4. Portfolio Diversification5. Systematic and Unsystematic risk6. Beta7. CAPM.

To summarize a little, in this presentation we explored the concepts of Expected Return, Risk, the relationship between Expected Return and Risk, Portfolio Diversification, further defining risk as Systematic and Unsystematic Risk, Beta as a measure of Systematic Risk, and the CAPM equation. These concepts will be useful in further understanding the financial management of a firm. This concludes the presentation on Risk and Return.

Risk and Return - College of...

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