1 risk and return learning module. 2 expected return the future is uncertain. the future is...

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Risk and ReturnRisk and Return

Learning ModuleLearning Module

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Expected ReturnExpected Return

The future is uncertain. The future is uncertain. Investors do not know with certainty whether the Investors do not know with certainty whether the

economy will be growing rapidly or be in economy will be growing rapidly or be in recession.recession.

Investors do not know what rate of return their Investors do not know what rate of return their investments will yield.investments will yield.

Therefore, they base their decisions on their Therefore, they base their decisions on their expectations concerning the future.expectations concerning the future.

The The expected rate of returnexpected rate of return on a stock on a stock represents the mean of a probability distribution represents the mean of a probability distribution of possible future returns on the stock.of possible future returns on the stock.

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Expected ReturnExpected Return The table below provides a probability distribution for the The table below provides a probability distribution for the

returns on stocks returns on stocks AA and and BBState Probability Return On Return OnState Probability Return On Return On

Stock A Stock BStock A Stock B

1 20% 5% 50%1 20% 5% 50%

2 30% 10% 30%2 30% 10% 30%

3 30% 15% 10%3 30% 15% 10%

4 20% 20% -10%4 20% 20% -10% The state represents the state of the economy one period in The state represents the state of the economy one period in

the future i.e. state 1 could represent a recession and state 2 a the future i.e. state 1 could represent a recession and state 2 a growth economy.growth economy.

The probability reflects how likely it is that the state will occur. The probability reflects how likely it is that the state will occur. The sum of the probabilities must equal 100%.The sum of the probabilities must equal 100%.

The last two columns present the returns or outcomes for The last two columns present the returns or outcomes for stocks stocks AA and and BB that will occur in each of the four states. that will occur in each of the four states.

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Given a probability distribution of returns, the Given a probability distribution of returns, the expected return can be calculated using the expected return can be calculated using the following equation: following equation:

NN

E[R] = E[R] = (p(piiRRii)) i=1i=1

Where:Where: E[R] = the expected return on the stock E[R] = the expected return on the stock N = the number of statesN = the number of states ppii = the probability of state i = the probability of state i

RRii = the return on the stock in state i. = the return on the stock in state i.

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In this example, the expected return for In this example, the expected return for stock A would be calculated as follows:stock A would be calculated as follows:

E[R]E[R]AA = .2(5%) + .3(10%) + .3(15%) + .2(20%) = = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%12.5%

Now you try calculating the expected return Now you try calculating the expected return for stock B! for stock B!

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Did you get 20%? If so, you are correct.Did you get 20%? If so, you are correct.

If not, here is how to get the correct answer:If not, here is how to get the correct answer:

E[R]E[R]BB = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = 20%20%

So we see that So we see that Stock BStock B offers a higher offers a higher expected return than expected return than Stock A.Stock A.

However, that is only part of the story; we However, that is only part of the story; we haven't considered risk.haven't considered risk.

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Measures of RiskMeasures of Risk

Risk reflects the chance that the actual return Risk reflects the chance that the actual return on an investment may be different than the on an investment may be different than the expected return.expected return.

One way to measure risk is to calculate the One way to measure risk is to calculate the variance and standard deviation of the variance and standard deviation of the distribution of returns. distribution of returns.

We will once again use a probability We will once again use a probability distribution in our calculations.distribution in our calculations.

The distribution used earlier is provided again The distribution used earlier is provided again for ease of use.for ease of use.

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Probability Distribution:Probability Distribution:

State Probability Return On Return OnState Probability Return On Return On

Stock A Stock BStock A Stock B

1 20% 5% 50%1 20% 5% 50%

2 30% 10% 30%2 30% 10% 30%

3 30% 15% 10%3 30% 15% 10%

4 20% 20% -10%4 20% 20% -10% E[R]E[R]A A = 12.5%= 12.5%

E[R]E[R]BB = 20% = 20%

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Given an asset's expected return, its variance Given an asset's expected return, its variance can be calculated using the following equation:can be calculated using the following equation:

NN

Var(R) = Var(R) = 22 = = p pii(R(Rii – E[R]) – E[R])22

i=1i=1

Where:Where: N = the number of states N = the number of states ppii = the probability of state i = the probability of state i

RRii = the return on the stock in state i = the return on the stock in state i E[R] = the expected return on the stockE[R] = the expected return on the stock

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The standard deviation is calculated as the The standard deviation is calculated as the positive square root of the variance:positive square root of the variance:

SD(R) = SD(R) = = = 2 2 = = ((22))1/21/2 = ( = (22))0.50.5

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Measures of RiskMeasures of Risk The variance and standard deviation for stock A is The variance and standard deviation for stock A is

calculated as follows:calculated as follows:

22AA = .2(.05 -.125) = .2(.05 -.125)22 + .3(.1 -.125) + .3(.1 -.125)22 + .3(.15 -.125) + .3(.15 -.125)22 + .2(.2 -.125) + .2(.2 -.125)2 2

= .002625= .002625

Now you try the variance and standard deviation for Now you try the variance and standard deviation for stock B!stock B!

If you got .042 and 20.49% you are correct.If you got .042 and 20.49% you are correct.

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Measures of RiskMeasures of Risk If you didn’t get the correct answer, here is how to If you didn’t get the correct answer, here is how to

get it:get it:

22BB = .2(.50 -.20) = .2(.50 -.20)22 + .3(.30 -.20) + .3(.30 -.20)22 + .3(.10 -.20) + .3(.10 -.20)22 + .2(-.10 - .20) + .2(-.10 - .20)22 = .042= .042

Although Although Stock BStock B offers a higher expected return offers a higher expected return than than Stock AStock A, it also is riskier since its variance and , it also is riskier since its variance and standard deviation are greater than standard deviation are greater than Stock AStock A's.'s.

This, however, is still only part of the picture because This, however, is still only part of the picture because most investors choose to hold securities as part of a most investors choose to hold securities as part of a diversified portfolio.diversified portfolio.

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Portfolio Risk and ReturnPortfolio Risk and Return Most investors do not hold stocks in isolation.Most investors do not hold stocks in isolation. Instead, they choose to hold a portfolio of Instead, they choose to hold a portfolio of

several stocks.several stocks. When this is the case, a portion of an individual When this is the case, a portion of an individual

stock's risk can be eliminated, stock's risk can be eliminated, i.e.,i.e., diversified diversified away.away.

From our previous calculations, we know that:From our previous calculations, we know that: the expected return on Stock A is 12.5%the expected return on Stock A is 12.5% the expected return on Stock B is 20%the expected return on Stock B is 20% the variance on Stock A is .00263the variance on Stock A is .00263 the variance on Stock B is .04200the variance on Stock B is .04200 the standard deviation on Stock A is 5.12%the standard deviation on Stock A is 5.12% the standard deviation on Stock B is 20.49%the standard deviation on Stock B is 20.49%

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Portfolio Risk and ReturnPortfolio Risk and Return The The Expected Return on a PortfolioExpected Return on a Portfolio is computed as is computed as

the weighted average of the expected returns on the the weighted average of the expected returns on the stocks which comprise the portfolio.stocks which comprise the portfolio.

The weights reflect the proportion of the portfolio The weights reflect the proportion of the portfolio invested in the stocks.invested in the stocks.

This can be expressed as follows:This can be expressed as follows: NN

E[RE[Rpp] = ] = w wiiE[RE[Rii]] i=1i=1

Where:Where: E[RE[Rpp] = the expected return on the portfolio] = the expected return on the portfolio N = the number of stocks in the portfolioN = the number of stocks in the portfolio wwii = the proportion of the portfolio invested in stock i = the proportion of the portfolio invested in stock i E[RE[Rii] = the expected return on stock i] = the expected return on stock i

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Portfolio Risk and ReturnPortfolio Risk and Return

For a portfolio consisting of two assets, the For a portfolio consisting of two assets, the above equation can be expressed as: above equation can be expressed as:

E[RE[Rpp] = w] = w11E[RE[R11] + w] + w22E[RE[R22]]

If we have an equally weighted portfolio of If we have an equally weighted portfolio of stock A and stock B (50% in each stock), then stock A and stock B (50% in each stock), then the expected return of the portfolio is:the expected return of the portfolio is:

E[RE[Rpp] = .50(.125) + .50(.20) = 16.25%] = .50(.125) + .50(.20) = 16.25%

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Portfolio Risk and ReturnPortfolio Risk and Return The variance/standard deviation of a portfolio reflects The variance/standard deviation of a portfolio reflects

not only the variance/standard deviation of the stocks not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together.the stocks which comprise the portfolio vary together.

Two measures of how the returns on a pair of stocks Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation vary together are the covariance and the correlation coefficient.coefficient. Covariance is a measure that combines the variance of a stock’s Covariance is a measure that combines the variance of a stock’s

returns with the tendency of those returns to move up or down returns with the tendency of those returns to move up or down at the same time other stocks move up or down.at the same time other stocks move up or down.

Since it is difficult to interpret the magnitude of the covariance Since it is difficult to interpret the magnitude of the covariance terms, a related statistic, the correlation coefficient, is often terms, a related statistic, the correlation coefficient, is often used to measure the degree of co-movement between two used to measure the degree of co-movement between two variables. The correlation coefficient simply standardizes the variables. The correlation coefficient simply standardizes the covariance.covariance.

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Portfolio Risk and ReturnPortfolio Risk and Return The The CovarianceCovariance between the returns on two stocks between the returns on two stocks

can be calculated as follows:can be calculated as follows: NN

Cov(RCov(RAA,R,RBB) = ) = A,BA,B = = p pii(R(RAi Ai - E[R- E[RAA])(R])(RBi Bi - E[R- E[RBB])]) i=1i=1

Where:Where: = the covariance between the returns on stocks A and B = the covariance between the returns on stocks A and B N = the number of states N = the number of states ppii = the probability of state i = the probability of state i RRAiAi = the return on stock A in state i = the return on stock A in state i E[RE[RAA] = the expected return on stock A ] = the expected return on stock A RRBiBi = the return on stock B in state i = the return on stock B in state i E[RE[RBB] = the expected return on stock B ] = the expected return on stock B

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The The Correlation CoefficientCorrelation Coefficient between the returns on between the returns on two stocks can be calculated as follows:two stocks can be calculated as follows:

A,BA,B Cov(R Cov(RAA,R,RBB))

Corr(RCorr(RAA,R,RBB) = ) = A,BA,B = = AAB B = SD(R= SD(RAA)SD(R)SD(RBB))

Where:Where: A,BA,B=the correlation coefficient between the returns on stocks =the correlation coefficient between the returns on stocks

A and BA and B A,BA,B=the covariance between the returns on stocks A and B, =the covariance between the returns on stocks A and B,

AA=the standard deviation on stock A, and =the standard deviation on stock A, and

BB=the standard deviation on stock B=the standard deviation on stock B

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The covariance between stock A and stock B is as The covariance between stock A and stock B is as follows:follows:

A,B A,B = .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) + = .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) +

.3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105.3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105

The correlation coefficient between stock A and The correlation coefficient between stock A and stock B is as follows:stock B is as follows:

-.0105-.0105

A,BA,B = (.0512)(.2049) = -1.00 = (.0512)(.2049) = -1.00

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Using either the correlation coefficient or the Using either the correlation coefficient or the covariance, the covariance, the Variance on a Two-Asset Variance on a Two-Asset PortfolioPortfolio can be calculated as follows: can be calculated as follows:

22pp = (w = (wAA))2222

AA + (w + (wBB))2222BB + 2w + 2wAAwwBBA,B A,B AABB

OROR

22pp = (w = (wAA))2222

AA + (w + (wBB))2222BB + 2w + 2wAAwwB B A,BA,B

The The Standard Deviation of the PortfolioStandard Deviation of the Portfolio equals equals the positive square root of the the variance.the positive square root of the the variance.

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Portfolio Risk and ReturnPortfolio Risk and Return Let’s calculate the variance and standard deviation of a Let’s calculate the variance and standard deviation of a

portfolio comprised of 75% stock A and 25% stock B:portfolio comprised of 75% stock A and 25% stock B:

22pp =(.75) =(.75)2222+(.25)+(.25)22(.2049)(.2049)22+2(.75)(.25)(-1)(.0512)+2(.75)(.25)(-1)(.0512)(.2049)= .00016(.2049)= .00016

pp = .00016 = .0128 = 1.28% = .00016 = .0128 = 1.28%

Notice that the portfolio formed by investing 75% in Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A.portfolio has a higher expected return than Stock A.

This is the purpose of diversification; by forming This is the purpose of diversification; by forming portfolios, some of the risk inherent in the individual portfolios, some of the risk inherent in the individual stocks can be eliminated.stocks can be eliminated.

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Capital Asset Pricing Model Capital Asset Pricing Model (CAPM) (CAPM)

If investors are mainly concerned with the risk of their portfolio rather If investors are mainly concerned with the risk of their portfolio rather than the risk of the individual securities in the portfolio, how should the than the risk of the individual securities in the portfolio, how should the risk of an individual stock be measured?risk of an individual stock be measured?

In important tool is the CAPM.In important tool is the CAPM. CAPM concludes that the relevant risk of an individual stock is its CAPM concludes that the relevant risk of an individual stock is its

contribution to the risk of a well-diversified portfolio.contribution to the risk of a well-diversified portfolio. CAPM specifies a linear relationship between risk and required return.CAPM specifies a linear relationship between risk and required return.

The equation used for CAPM is as follows:The equation used for CAPM is as follows: KKi i = K= Krf rf + + ii(K(Kmm - K - Krfrf)) Where:Where:

KKi i = the required return for the individual security= the required return for the individual security KKrf rf = the risk-free rate of return= the risk-free rate of return ii = the beta of the individual security= the beta of the individual security KKm m = the expected return on the market portfolio= the expected return on the market portfolio (K(Kmm - K - Krfrf) is called the market risk premium) is called the market risk premium

This equation can be used to find any of the variables listed above, This equation can be used to find any of the variables listed above, given the rest of the variables are known.given the rest of the variables are known.

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CAPM ExampleCAPM Example Find the required return on a stock given that the risk-Find the required return on a stock given that the risk-

free rate is 8%, the expected return on the market free rate is 8%, the expected return on the market portfolio is 12%, and the beta of the stock is 2.portfolio is 12%, and the beta of the stock is 2.

KKi i = K= Krf rf + + ii(K(Kmm - K - Krfrf))

KKi i = 8%= 8% + 2(12% - 8%)+ 2(12% - 8%)

KKi i = 16%= 16%

Note that you can then compare the required rate of return to Note that you can then compare the required rate of return to the expected rate of return. You would only invest in stocks the expected rate of return. You would only invest in stocks where the expected rate of return exceeded the required rate of where the expected rate of return exceeded the required rate of return.return.

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Another CAPM ExampleAnother CAPM Example Find the beta on a stock given that its expected return is Find the beta on a stock given that its expected return is

12%, the risk-free rate is 4%, and the expected return on 12%, the risk-free rate is 4%, and the expected return on the market portfolio is 10%. the market portfolio is 10%.

12%12% = 4%= 4% + + ii(10% - 4%)(10% - 4%)

i i = 12% - 4%= 12% - 4%

10% - 4% 10% - 4% i i = 1.33= 1.33

Note that beta measures the stock’s volatility (or risk) Note that beta measures the stock’s volatility (or risk) relative to the market. relative to the market.

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