8 - 1 Measuring Returns Converting Dollar Returns to Percentage Returns An investor receives the following dollar returns a stock investment of $25: $1.00

8 - 1

Measuring ReturnsConverting Dollar Returns to Percentage Returns

An investor receives the following dollar returns a stock investment of $25:

$1.00 of dividendsShare price rise of $2.00

The capital gain (or loss) return component of total return is calculated: ending price – minus beginning price, divided by beginning price

%808.$25

$25-$27 return (loss)gain Capital

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PORTFOLIO RISK AND RETURN

Portfolio’s Risk and Return

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

the economy will be growing rapidly or be in recession.

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

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

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

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

on stocks A and BState Probability Return On Return On Stock A Stock B 1 20% 5% 50% 2 30% 10% 30% 3 30% 15% 10% 4 20% 20% -10%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 growth economy. The probability reflects how likely it is that the state will occur. The sum of the probabilities must equal 100%. The last two columns present the returns or outcomes for stocks A and B that will occur in each of the four states.

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Expected Return

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

N

E[R] = S (piRi) i=1

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

Ri = the return on the stock in state i.

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Expected Return

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

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

Now you try calculating the expected return for stock B!

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Expected Return

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

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

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

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

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

6

Measures of Risk

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

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

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

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

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Measures of Risk

Probability Distribution:

State Probability Return On Return On

Stock A Stock B 1 20% 5% 50% 2 30% 10% 30% 3 30% 15% 10% 4 20% 20% -10%E[R]A = 12.5%

E[R]B = 20%

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Measures of Risk

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

N

Var(R) = s2 = S pi(Ri – E[R])2

i=1

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

Ri = the return on the stock in state iE[R] = the expected return on the stock

9

Measures of Risk

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

SD(R) = s = s2 = (s2)1/2 = (s2)0.5

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Measures of Risk

The variance and standard deviation for stock A is calculated as follows:

s2A = 0.2(.05 -.125)2 + 0.3(.1 -.125)2 + 0.3(.15 -.125)2 + 0.2(.2 -.125)2

= .002625

sA = (.002625)0.5 = .0512 = 5.12%

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

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

11

Measures of Risk

If you didn’t get the correct answer, here is how to get it:

s2B = .2(.50 -.20)2 + .3(.30 -.20)2 + .3(.10 -.20)2 + .2(-.10 - .20)2 = .042

sB = (.042)0.5 = .2049 = 20.49%

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

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

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Portfolio Risk and Return

Most investors do not hold stocks in isolation.Instead, they choose to hold a portfolio of

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

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

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

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The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio.

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

This can be expressed as follows: N

E[Rp] = S wiE[Ri] i=1

Where: E[Rp] = the expected return on the portfolio N = the number of stocks in the portfolio wi = the proportion of the portfolio invested in stock i E[Ri] = the expected return on stock i

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

E[Rp] = w1E[R1] + w2E[R2]

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

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

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

s2p = (wA)2s2

A + (wB)2s2B + 2wAwBrA,B sAsB

OR

s2p = (wA)2s2

A + (wB)2s2B + 2wAwB sA,B

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

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Portfolio Risk and ReturnThe Covariance between the returns on two

stocks can be calculated as follows:

N

Cov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB]) i=1

Where:sA,B = the covariance between the returns on stocks A and B N = the number of states pi = the probability of state i RAi = the return on stock A in state i E[RA] = the expected return on stock A RBi = the return on stock B in state iE[RB] = the expected return on stock B

17


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

Cov(RA,RB)/sA,B

Corr(RA,RB) = rA,B = sAsB = SD(RA)SD(RB)

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

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

sA=the standard deviation on stock A, and

sB=the standard deviation on stock B

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

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

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

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

-.0105rA,B = (.0512)(.2049) = -1.00

19


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

s2p = (wA)2s2

A + (wB)2s2B + 2wAwBrA,B sAsB

OR

s2p = (wA)2s2

A + (wB)2s2B + 2wAwB sA,B

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

20


Let’s calculate the variance and standard deviation of a portfolio comprised of 75% stock A and 25% stock B:

s2p =(.75)2(.0512)2+(.25)2(.2049)2+2(.75)(.25)(-1)(.0512)(.2049)= .00016

sp = .00016 = .0128 = 1.28%

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

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

21

(1) (2) (3) (4)=(2)×(1)

State of the EconomyProbability of Occurrence

Possible Returns on

Stock A in that State

Weighted Possible

Returns on the Stock

Economic Expansion 25.0% 30% 7.50%Normal Economy 50.0% 12% 6.00%Recession 25.0% -25% -6.25%

Expected Return on the Stock = 7.25%

RISK AND RETURN EXAMPLE 2

7.25%

)25.0(-25%0.5)(12% .25)0(30%

)Prob(r)Prob(r )Prob(r

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8 - 24

Measuring RiskEx post Standard Deviation

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i

Measuring RiskExample Using the Ex post Standard Deviation

ProblemEstimate the standard deviation of the historical returns on investment A that were: 10%, 24%, -12%, 8% and 10%.

Step 1 – Calculate the Historical Average Return

Step 2 – Calculate the Standard Deviation

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Scenario-based Estimate of RiskExample Using the Ex ante Standard Deviation – Raw Data

State of the Economy Probability

Possible Returns on Security A

Recession 25.0% -22.0%Normal 50.0% 14.0%Economic Boom 25.0% 35.0%

GIVEN INFORMATION INCLUDES:

- Possible returns on the investment for different discrete states

- Associated probabilities for those possible returns

Scenario-based Estimate of RiskFirst Step – Calculate the Expected Return



Weighted Possible Returns

Recession 25.0% -22.0% -5.5%Normal 50.0% 14.0% 7.0%Economic Boom 25.0% 35.0% 8.8%

Expected Return = 10.3%

Determined by multiplying the probability times the possible

return.

Expected return equals the sum of the weighted possible returns.

8 - 28

Scenario-based Estimate of RiskSecond Step – Measure the Weighted and Squared Deviations




Deviation of Possible

Return from Expected

Squared Deviations

Weighted and

Squared Deviations

Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070Economic Boom 25.0% 35.0% 8.8% 24.8% 0.06126 0.01531

Expected Return = 10.3% Variance = 0.0420

Standard Deviation = 20.50%

Second, square those deviations from the mean.The sum of the weighted and square deviations is

the variance in percent squared terms.The standard deviation is the square root of

the variance (in percent terms).

First calculate the deviation of possible returns from the expected.

Now multiply the square deviations by their probability of occurrence.

Scenario-based Estimate of RiskExample Using the Ex ante Standard Deviation Formula

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Recession 25.0% -22.0% -5.5%Normal 50.0% 14.0% 7.0%Economic Boom 25.0% 35.0% 8.8%

Expected Return = 10.3%

Expected Return of a PortfolioExample

Portfolio value = $2,000 + $5,000 = $7,000rA = 14%, rB = 6%,

wA = weight of security A = $2,000 / $7,000 = 28.6%wB = weight of security B = $5,000 / $7,000 = (1-28.6%)= 71.4%

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CHAPTER 8 – Risk, Return and Portfolio Theory

8 - 31

Expected Return and Risk For PortfoliosStandard Deviation of a Two-Asset Portfolio using Covariance

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Risk of Asset A adjusted for weight in

the portfolio

Risk of Asset B adjusted for weight in

the portfolio

Factor to take into account comovement of returns. This factor can

be negative.

Grouping Individual Assets into Portfolios

• The riskiness of a portfolio that is made of different risky assets is a function of three different factors:– the riskiness of the individual assets that make up the portfolio– the relative weights of the assets in the portfolio– the degree of comovement of returns of the assets making up the portfolio

• The standard deviation of a two-asset portfolio may be measured using the Markowitz model:

BABABABBAAp wwww ,2222 2

Range of Returns in a Two Asset PortfolioE(r)A= 14%, E(r)B= 6%

A graph of this relationship is found on the following slide.

Expected return on Asset A = 14.0%Expected return on Asset B = 6.0%

Weight of Asset A

Weight of Asset B

Expected Return on the

Portfolio0.0% 100.0% 6.0%

10.0% 90.0% 6.8%20.0% 80.0% 7.6%30.0% 70.0% 8.4%40.0% 60.0% 9.2%50.0% 50.0% 10.0%60.0% 40.0% 10.8%70.0% 30.0% 11.6%80.0% 20.0% 12.4%90.0% 10.0% 13.2%100.0% 0.0% 14.0%

Expected Portfolio ReturnsExample of a Three Asset Portfolio

Relative Weight

Expected Return

Weighted Return

Stock X 0.400 8.0% 0.03Stock Y 0.350 15.0% 0.05Stock Z 0.250 25.0% 0.06 Expected Portfolio Return = 14.70%

Correlation

• The degree to which the returns of two stocks co-move is measured by the correlation coefficient (ρ).

• The correlation coefficient (ρ) between the returns on two securities will lie in the range of +1 through - 1.

+1 is perfect positive correlation-1 is perfect negative correlation

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[8-13]

Capital Asset Pricing Model (CAPM) 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 risk of an individual stock be measured?

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

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

The equation used for CAPM is as follows: Ki = Krf + bi(Km - Krf) Where:

Ki = the required return for the individual security

Krf = the risk-free rate of return

bi = the beta of the individual security

Km = the expected return on the market portfolio

(Km - Krf) is called the market risk premium

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

36

CAPM Example

Find the required return on a stock given that the risk-free rate is 8%, the expected return on the market portfolio is 12%, and the beta of the stock is 2.

Ki = Krf + bi(Km - Krf)

Ki = 8% + 2(12% - 8%)

Ki = 16%

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

37

Another CAPM Example

Find the beta on a stock given that its expected return is 12%, the risk-free rate is 4%, and the expected return on the market portfolio is 10%.

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

bi = 12% - 4%

10% - 4% bi = 1.33

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

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Documents

8 - 1 Measuring Returns Converting Dollar Returns to Percentage Returns An investor receives the following dollar returns a stock investment of $25: $1.00