Risk

CHAPTER 5 Risk and Rates of Return   

Stand-alone risk Portfolio risk Risk & return: CAPM / SML

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Investment returns The rate of return on an investment can be calculated as follows: (Amount received – Amount invested)

Return =

________________________ Amount invested

For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is: 5-2 ($1,100 - $1,000) / $1,000 = 10%.

Returns 





Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc. k Required Return - the return that an investor requires on an asset given its risk. k Realized Return – the return that was _



actually earned during some past period k Average Return - _ the average

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Probability distributions 



A listing of all possible outcomes, and the probability of each occurrence. Can be shown graphically. Firm X

Firm Y -70

0

15

Expected Rate of Return

100

Rate of Return (%)

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Expected Return State of Return Economy Y

Recession -10% Normal 14% Boom

Probability (P)

X

.20

4%

.50

10%

.30

14% 5-5

Expected Return State of Return Economy Y

Recession -10% Normal 14% Boom

Probability (P)

X

.20

4%

.50

10%

.30

14% 5-6

Expected Return State of Return Economy Y

Recession -10% Normal 14% Boom

Probability (P)

X

.20

4%

.50

10%

.30

14% 5-7

Expected Return State of Return Economy Y

Recession -10% Normal 14% Boom

Probability (P)

X

.20

4%

.50

10%

.30

14% 5-8

Based only on your expected return calculations, which stock would you prefer?

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Have you considered

RISK?

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RISK How to measure risk (variance, standard deviation, beta)  How to reduce risk (diversification) 

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What is investment risk?   

Risk is an uncertain outcome or chance of an adverse outcome. Concerned with the riskiness of cash flows from financial assets. Two types of investment risk  





Stand-alone risk Portfolio risk

Investment risk is related to the probability of earning a low or negative actual return. The greater the chance of lower than expected or negative returns, the riskier the investment. 5-12



Stand Alone Risk: Single Asset  relevant

risk measure is the total risk of expected cash flows measured by standard deviation .



Portfolio Context: A group of assets. Total risk consists of:  Diversifiable

Risk (company-specific, unsystematic)  Market Risk (non-diversifiable, systematic) 5-13

13

How do we Measure Risk? A more scientific approach is to examine the stock’s STANDARD DEVIATION of returns.  Standard deviation is a measure of the dispersion of possible outcomes.  The greater the standard deviation, the greater the uncertainty, and therefore , the 5-14 

Standard Deviation

σ

n

Σ

= i=1 P(ki)

2

(ki - k)

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σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company X.

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σ

=

nn

Σ

2

(ki - k) P(ki)

i=1 i=1

Company X ( 4% - 10%)2 (.2) = 7.2

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σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company X ( 4% - 10%)2 (.2) = 7.2 (10% - 10%)2 (.5) = 0

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σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company X ( 4% - 10%)2 (10% - 10%)2 (14% - 10%)2

(.2) = 7.2 (.5) = 0 (.3) = 4.8

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σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company X ( 4% - 10%)2 (10% - 10%)2 (14% - 10%)2 Variance

(.2) = 7.2 (.5) = 0 (.3) = 4.8 = 12 5-20

σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company X ( 4% - 10%)2 (.2) = (10% - 10%)2 (.5) = (14% - 10%)2 (.3) = Variance = Stand. dev. = 12 3.46%

7.2 0 4.8 12 = 5-21

σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company Y (-10% - 14%)2 (.2) = 115.2

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σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company Y (-10% - 14%)2 (.2) = 115.2 (14% - 14%)2 (.5) = 0 5-23

σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company Y (-10% - 14%)2 115.2 (14% - 14%)2 0 (30% - 14%)2 76.8

(.2) = (.5) = (.3) = 5-24

σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company Y (-10% - 14%)2 115.2 (14% - 14%)2 0 (30% - 14%)2 76.8

(.2) = (.5) = (.3) = 5-25

σ

=

n

Σ

2

(ki - k) P(ki)

i=1

Company y (-10% - 14%)2 115.2 (14% - 14%)2 0 (30% - 14%)2 76.8

(.2) = (.5) = (.3) = 5-26

Which stock would you prefer? How would you decide?

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Summary Company Company X Expected Return 14% Standard Deviation

Y 10%

3.46%

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It depends on your tolerance for risk! Return

Risk

Remember there’s a tradeoff

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Coefficient of variation Coefficient of variation (CV): A standardized measure of dispersion about the expected value, that shows the amount of risk per unit of return.

Standard deviation s CV = = Expected return rˆ 5-30

Portfolio construction: Risk and return Assume a two-stock portfolio is created with $50,000 invested in both HT and Collections.  Expected return of a portfolio is a weighted average of each of the component assets of the portfolio.

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Calculating portfolio expected return ^

k p is a weighted average : ^

n

^

k p = ∑ wi k i i=1

^

k p = 0.5 (17.4%) + 0.5 (1.7%) = 9.6% 5-32

Portfolios Combining several securities in a portfolio can actually reduce overall risk.  How does this work? 

5-33

Diversification Investing in more than one security to reduce risk.  If two stocks are perfectly positively correlated, diversification has no effect on risk.  If two stocks are perfectly negatively correlated, the 

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Some risk can be diversified away and some can not. Market Risk is also called Nondiversifiable risk. This type of risk can not be diversified away.  Firm-Specific risk is also called diversifiable risk. This type of risk can be 

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Market Risk Unexpected changes in interest rates.  Unexpected changes in cash flows due to tax rate changes, foreign competition, and the overall business cycle. 

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Firm-Specific Risk A company’s labor force goes on strike.  A company’s top management dies in a plane crash.  A huge oil tank bursts and floods a company’s production area. 

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Portfolio Risk sp (%) 35

Diversifiable Risk Stand-Alone Risk, sp

20 Market Risk 0

10

20

30

40

2,000+

# Stocks in Portfolio 5-38

Investor Attitude Towards Risk 

Investors are assumed to be risk averse.



Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.



Risk premium – the difference between the return on a risky asset and a riskless asset, which serves as compensation for investors to hold 5-39

Investor attitude towards risk 



Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities. Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities. 5-40

Capital Asset Pricing Model (CAPM) 



Model based upon concept that a stock’s required rate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification. Primary conclusion: The relevant riskiness of a stock is its contribution to the riskiness of a well-diversified portfolio. 5-41

Well-diversified Portfolio  Large Portfolio (10-15 assets) eliminates diversifiable risk for the most part.  Interested in Market Risk which is the risk that cannot be diversified away.  The relevant risk measure is Beta which measures the riskiness of an individual asset in relation to the 5-42

42

Failure to diversify 

If an investor chooses to hold a one-stock portfolio (exposed to more risk than a diversified investor), would the investor be compensated for the risk they bear?   

 

NO! Stand-alone risk is not important to a well-diversified investor. Rational, risk-averse investors are concerned with σp, which is based upon market risk. There can be only one price (the market return) for a given security. No compensation should be earned for5-43

Beta Beta: a measure of market risk.

Measures a stock’s market risk, and shows a stock’s volatility relative to the market. or

It’s a measure of the “sensitivity” of an individual stock’s returns to changes in the market. 

Indicates how risky a stock is if the stock is held in a well-diversified portfolio. 5-44

The market’s beta is 1 A firm that has a beta = 1 has average market risk. The stock is no more or less volatile than the market.  A firm with a beta > 1 is more volatile than the market (ex: computer firms).  A firm with a beta < 1 is less 5-45 

What is the market risk premium? 





Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk. Its size depends on the perceived risk of the stock market and investors’ degree of risk aversion. Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year. 5-46

Required rate of = return

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Required Risk-free rate of = rate of return return

+

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Required Risk-free rate of = rate of return return

Risk + Premium

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Required Risk-free rate of = rate of return return

Risk + Premium

Market Risk 5-50

Required Risk-free rate of = rate of return return

Market Risk

Risk + Premium

Firm-specific Risk 5-51

Required Risk-free rate of = rate of return return

Market Risk

Risk + Premium

Firm-specific Risk can be diversified 5-52 away

The CAPM equation: kj = krf β+

j

(km - krf)

where:

kj = the Required Return on security j,

k βrf = the risk-free rate of interest,

j = the beta of security j,

and

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This linear relationship between risk and required return is known as the Capital Asset Pricing Model (CAPM).

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Requir ed rate of return

Let’s try to graph this relationship!

Beta 5-55

Requir ed rate of return

12%

.

Risk-free rate of return (6%)

1

Beta 5-56

Requir ed rate of return

12%

.

security market line (SML)

1

Beta 5-57

Risk-free rate of return (6%)

Requir ed rate of return

Is there a riskless (zero beta) security?

SML

.

12%

Risk-free rate of return (6%)

0

1

Beta 5-58

Requir ed rate of return

Is there a riskless (zero beta) security?

.

12%

Risk-free rate of return (6%)

0

1

SML

Treasury securities are as close to riskless as possible.

Beta 5-59

Can the beta of a security be negative? 





Yes, if the correlation between Stock i and the market is negative (i.e., ρi,m < 0). If the correlation is negative, the regression line would slope downward, and the beta would be negative. However, a negative beta is highly unlikely.

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Factors that change the SML 

What if investors raise inflation expectations by 3%, what would happen to the SML? ki (%)

∆ I = 3%

18 15

SML2 SML1

11 8 Risk, βi 0

0.5

1.0

1.5

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Factors that change the SML What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML? ki (%) SML2 ∆ RP = 3% 

M

SML1

18 15 11 8

Risk, βi 0

0.5

1.0

1.5

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Verifying the CAPM empirically 





The CAPM has not been verified completely. Statistical tests have problems that make verification almost impossible. Some argue that there are additional risk factors, other than the market risk premium, that must be considered. 5-63

More thoughts on the CAPM 

Investors seem to be concerned with both market risk and total risk. Therefore, the SML may not produce a correct estimate of ki. ki = kRF + (kM – kRF) βi + ???



CAPM/SML concepts are based upon expectations, but betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness. 5-64

Example: Suppose the Treasury bond rate is 6%, the average return on the S&P 500 index is 12%, and Walt Disney has a beta of 1.2.  According to the CAPM, what should be the required rate of return on Disney stock? 5-65 

kj = krf + β (km - krf) kj = .06 + 1.2 (.12 - .06) kj = .132 = 13.2% According to the CAPM, Disney stock should be priced to give a 13.2% return. 5-66





An example: Equally-weighted two-stock portfolio

Create a portfolio with 50% invested in HT and 50% invested in Collections. The beta of a portfolio is the weighted average of each of the stock’s betas. βP = wHT βHT + wColl βColl βP = 0.5 (1.30) + 0.5 (-0.87) βP = 0.215

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Calculating portfolio required returns 

The required return of a portfolio is the weighted average of each of the stock’s required returns. kP = wHT kHT + wColl kColl kP = 0.5 (17.1%) + 0.5 (1.9%) kP = 9.5%



Or, using the portfolio’s beta, CAPM can be used to solve for expected return. kP = kRF + (kM – kRF) βP kP = 8.0% + (15.0% – 8.0%) (0.215)

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