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CHAPTER 8Risk and Rates of Return

Outline Stand-alone return and risk

ReturnExpected returnStand-alone risk

Portfolio return and riskPortfolio returnPortfolio riskBeta

Link Risk & return: CAPM Security Market Line

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I-1: Return: What is my reward of investing?

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

The rate of return on an investment can be calculated as follows:

(Amount received – Amount invested) Return = ________________________

Amount invested

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Rates of Return: stocksHPR P P D

P

1 0 1

0

HPR = Holding Period Return

P1 = Ending price

P0 = Beginning price

D1 = Dividend during period one

Define return?

Your gain per dollar investment

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Rates of Return: ExampleEnding Price = 24Beginning Price = 20Dividend = 1

HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

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I-2: Expected return: describe the uncertainty

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Calculating expected return Two scenarios and the concept of

expected return Extending to more than two

scenarios

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Investment alternatives

Economy Prob. T-Bill HT Coll USR MPRecession

0.1 5.5% -27.0%

27.0% 6.0% -17.0%

weak 0.2 5.5% -7.0% 13.0% -14.0%

-3.0%

normal 0.4 5.5% 15.0% 0.0% 3.0% 10.0%strong 0.2 5.5% 30.0% -

11.0%41.0% 25.0%

Boom 0.1 5.5% 45.0% -21.0%

26.0% 38.0%

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Calculating the expected return

12.4% (0.1) (45%) (0.2) (30%) (0.4) (15%)

(0.2) (-7%) (0.1) (-27%) r

P r r

return of rate expected r

HT^

N

1iii

^

^

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Summary of expected returns

Expected returnHT 12.4%Market 10.5%USR 9.8%T-bill 5.5%Coll. 1.0%

HT has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

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I-3. Stand-alone risk

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Calculating standard deviation

deviation Standard

2Variance

i2

N

1ii P)r(rσ

ˆ

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Standard deviation for each investment

15.2% 18.8% 20.0% 13.2% 0.0%

(0.1)5.5) - (5.5 (0.2)5.5) - (5.5 (0.4)5.5) - (5.5

(0.2)5.5) - (5.5 (0.1)5.5) - (5.5

P )r (r

M

USRHT

CollbillsT

2

22

22

billsT

N

1ii

2^i

21

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Comparing standard deviations

USR

Prob.T - bill

HT

0 5.5 9.8 12.4 Rate of Return (%)

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Comments on standard deviation as a measure of risk Standard deviation (σi) measures

total, or stand-alone, risk. The larger σi is, the lower the

probability that actual returns will be closer to expected returns.

Larger σi is associated with a wider probability distribution of returns.

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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 a risk free asset, which serves as compensation for investors to hold riskier securities.

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Comparing risk and returnSecurity Expected

return, rRisk, σ

T-bills 5.5% 0.0%HT 12.4% 20.0%Coll* 1.0% 13.2%USR* 9.8% 18.8%Market 10.5% 15.2%* Seem out of place.

^

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Selected Realized Returns, 1926 – 2001

Average Standard Return Deviation

Small-company stocks17.3% 33.2%Large-company stocks 12.7 20.2L-T corporate bonds 6.1 8.6

Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2002 Yearbook (Chicago: Ibbotson Associates, 2002), 28.

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

r return Expecteddeviation Standard CV ˆ

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Risk rankings, by coefficient of variation

CVT-bill 0.0HT 1.6Coll. 13.2USR 1.9Market 1.4

Collections has the highest degree of risk per unit of return.

HT, despite having the highest standard deviation of returns, has a relatively average CV.

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II: Risk and return in a portfolio

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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.

Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.

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II-1. Portfolio return

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Calculating portfolio expected returnEconomy Prob

.HT Coll Port.Port.

Recession

0.1 -27.0%

27.0%

weak 0.2 -7.0% 13.0%normal 0.4 15.0% 0.0% 7.5%7.5%strong 0.2 30.0% -

11.0%Boom 0.1 45.0% -

21.0%6.7% (12.0%) 0.10 (9.5%) 0.20 (7.5%) 0.40 (3.0%) 0.20 (0.0%) 0.10 rp

^

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Calculating portfolio expected returnEconomy Prob

.HT Coll Port.Port.

Recession

0.1 -27.0%

27.0% 0.0%0.0%

weak 0.2 -7.0% 13.0% 3.0%3.0%normal 0.4 15.0% 0.0% 7.5%7.5%strong 0.2 30.0% -

11.0%9.5%9.5%

Boom 0.1 45.0% -21.0%

12.0%12.0%6.7% (12.0%) 0.10 (9.5%) 0.20

(7.5%) 0.40 (3.0%) 0.20 (0.0%) 0.10 rp^

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An alternative method for determining portfolio expected return

6.7% (1.0%) 0.5 (12.4%) 0.5 r

rw r

:average weighted a is r

p^

N

1ii

^ip

^

p^

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II-2. Portfolio risk and beta

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Calculating portfolio standard deviation and CV

0.51 6.7%3.4% CV

3.4%

6.7) - (12.0 0.10 6.7) - (9.5 0.20 6.7) - (7.5 0.40 6.7) - (3.0 0.20 6.7) - (0.0 0.10

p

21

2

2

2

2

2

p

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Comments on portfolio risk measures σp = 3.4% is much lower than the σi of

either stock (σHT = 20.0%; σColl. = 13.2%). σp = 3.4% is lower than the weighted

average of HT and Coll.’s σ (16.6%). Therefore, the portfolio provides the

average return of component stocks, but lower than the average risk.

Why? Negative correlation between stocks.

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Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)

-10

15 15

40 4040

15

0

-10

Stock W

0

Stock M

-10

0

Portfolio WM

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Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)

Stock M

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

0

15

25

-10

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Creating a portfolio:Beginning with one stock and adding randomly selected stocks to portfolio σp decreases as stocks added, because

they would not be perfectly correlated with the existing portfolio.

Expected return of the portfolio would remain relatively constant.

Eventually the diversification benefits of adding more stocks dissipates (after about 10 stocks), and for large stock portfolios, σp tends to converge to 20%.

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Illustrating diversification effects of a stock portfolio

# Stocks in Portfolio10 20 30 40 2,000+

Company-Specific Risk

Market Risk20

0

Stand-Alone Risk, p

p (%)35

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Breaking down sources of risk

Stand-alone risk = Market risk + Firm-specific risk

Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta.

Firm-specific risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

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Beta Measures a stock’s market risk, and

shows a stock’s volatility relative to the market.

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

Portfolio beta is a weighted average of its individual securities’ beta

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Calculating betas Run a regression of past returns of

a security against past returns on the market.

The slope of the regression line is defined as the beta coefficient for the security.

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Comments on beta If beta = 1.0, the security is just as risky

as the average stock. If beta > 1.0, the security is riskier than

average. If beta < 1.0, the security is less risky

than average. Most stocks have betas in the range of

0.5 to 1.5.

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III: CAPM

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What risk do we care? Stand alone? Risk that can not be diversified?

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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.

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Capital Asset Pricing Model (CAPM)

Model linking risk and required returns. CAPM suggests that a stock’s required return equals the risk-free return plus a risk premium that reflects the stock’s risk after diversification.

ri = rRF + (rM – rRF) bi Risk premium RP: additional return to take

additional risk The market (or equity) risk premium is (rM – rRF)

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Calculating required rates of return

rHT = 5.5% + (5.0%)(1.32)= 5.5% + 6.6%= 12.10%

rM = 5.5% + (5.0%)(1.00) = 10.50% rUSR = 5.5% + (5.0%)(0.88) = 9.90% rT-bill = 5.5% + (5.0%)(0.00) = 5.50% rColl = 5.5% + (5.0%)(-0.87)= 1.15%

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Applying CAPM

Portfolio beta: Beta of a portfolio is a weighted average of its individual securities’ betas.

Computing other variables: risk free rate, market return, market risk premium

Computing the difference of return between two stocks.

Computing price in the future when current price is given

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CAPM in a graph: the Security Market Line

..Coll.

.HT

T-bills

.USR

SML

rM = 10.5

rRF = 5.5

-1 0 1 2

.

SML: ri = 5.5% + (5.0%) bi

ri (%)

Risk, bi

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Applying CAPM in real world(optional)

Total Risk vs. Beta. An experiment The difference between commonly

referred risk and beta (Are these high beta stocks really high beta)

High risk( total risk), low beta stock can hedge your portfolio (reduce portfolio risk)

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Problems with CAPM (optional)

Measurement error of beta Empirical relationship between

beta and return is weak Size and Book-to-market factors Momentum

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Optional: diversification in real world

Stock Index ETF Style: Value vs. Growth Style: Small vs. Big Performance, Risk, Expense(0.1% is low,

0.5% is about average) Examples:

Vanguard Small Cap Value ETF  VBR Small growth: VBK Large value: VTV Large growth: VUG

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diversification in real world Foreign ETF:RBL

Pros: More diversification Low PE ratio

cons Higher risk Higher expense: 0.6% vs. 0.1% Higher spread Poor prior performance