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CHAPTER 8Risk 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:

($1,100 - $1,000) / $1,000 = 10%.

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What is investment 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.

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Selected Realized Returns, 1990 – 2007

Average Standard

Return Deviation

Small-company stocks 17.5% 33.1%Large-company stocks 12.4

20.3L-T corporate bonds 6.2 8.6

Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Bursa Malaysia, 2008)

8-5

Investment alternatives & Rate of Returns( %)

Economy

Prob. T-Bill HT Coll USR MP

Recession

0.1 5.5% -27.0%

27.0% 6.0% -17.0%

Below avg

0.2 5.5% -7.0% 13.0% -14.0%

-3.0%

Average 0.4 5.5% 15.0% 0.0% 3.0% 10.0%

Above avg

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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Investment alternatives & Rate of Returns( %)

T-bills will return the promised 5.5%, regardless of the economy- risk-free return

Do T-bills promise a completely risk-free return?

T-bills do not provide a completely risk-free return, as they are still exposed to inflation..

T-bills are also risky in terms of reinvestment rate risk.

However, T-bills are risk-free in the default sense of the word.

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“Reinvestment Risk”

5.5%

CF

5.5%5.5% 5.5% 5.5%

CFCF CFCF

“reinvest cash inflow at going market rates”Thus, if market rates , may experience income reduction

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“inflation risk”

If inflation rate = 8%- funds deposited loses the purchasing power- “inflation risk”

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How do the returns of HT and Coll. behave in relation to the market?

HT – Moves with the economy, and has a positive correlation. This is typical.

Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.

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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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12.4HT’s expected return

Expected returns & Standard deviation

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Calculating standard deviation

deviation Standard

2Variance

i2

N

1ii P)r(rσ

ˆ

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(-27-12.4)2(0.1) + (-7-12.4)2(0.2) (15-12.4)2(0.4) + (30-12.4)2(0.2) (45-12.4)2(0.1)

1/2

6HT =

6HT = 20%

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-7.6% 12.4% 32.4%

Expected returns & Standard deviation

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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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Comparing risk and return

Security Expected return, r

Risk, 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%

^

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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 riskless asset, which serves as compensation for investors to hold riskier securities.

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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.

A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets.

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Calculating portfolio expected return

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

rw r

:average weighteda is r

p

^

N

1i

i

^

ip

^

HT

^

HT

i

= expected return

= 12.4%

= 1.0%

^

ir

colli

^

r

HTi

^

r

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An alternative method for determining portfolio expected return

Economy Prob.

HT Coll Port.Port.

Recession

0.1 -27.0%

27.0% 0.0%0.0%*A*A

Below avg

0.2 -7.0% 13.0% 3.0%3.0%*B*B

Average 0.4 15.0% 0.0% 7.5%7.5%

Above avg

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

^

*A: 0.5(-27%) + 0.5(27) = 0%*B: 0.5(-7%) + 0.5(13%) = 3%

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Calculating portfolio standard deviation

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

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%).

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

Why? Negative correlation between stocks.

Combining stocks in a portfolio generally lowers risk.

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

-10

15 15

25 2525

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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Illustrating diversification effects of a stock portfolio

# Stocks in Portfolio10 20 30 40 2,000+

Diversifiable Risk

Market Risk

20

0

Stand-Alone Risk, p

p (%)35

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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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Breaking down sources of risk

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

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

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Beta Measures a stock’s market risk, and

shows a stock’s volatility relative to the market.

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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Calculating betas Well-diversified investors are primarily

concerned with how a stock is expected to move relative to the market in the future.

A typical approach to estimate beta is to run a regression of the security’s past returns against the past returns of the market.

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

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Illustrating the calculation of beta

.

.

.ri

_

rM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:

ri = -2.59 + 1.44 rM^ ^

Year rM ri

1 15% 18%

2 -5 -10

3 12 16

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Comparing expected returns and beta coefficients

Security Expected Return Beta HT 12.4% 1.32Market 10.5 1.00USR 9.8 0.88T-Bills 5.5 0.00Coll. 1.0 -0.87

Riskier securities have higher returns, so the rank order is OK.

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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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Beta coefficients for HT, Coll, and T-Bills

ri

_

kM

_

-20 0 20 40

40

20

-20

HT: b = 1.30

T-bills: b = 0

Coll: b = -0.87

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Capital Asset Pricing Model (CAPM)

Model linking risk and required returns. CAPM suggests that there is a Security Market Line (SML) that states that a stock’s required return equals the risk-free return plus a risk premium.

ri = rRF + (rM – rRF) bi

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The Security Market Line (SML):Calculating required rates of return

SML: ri = rRF + (rM – rRF) bi

ri = rRF + (RPM) bi

Assume the rRF = 5.5% and RPM = 5.0%.

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What is the market risk premium (RPM)?

Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.

Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year.

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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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Illustrating 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

8-42

Expected vs. Required returns

r) r( ervalued Und1.2 1.0 Coll.

r) r( uedFairly val 5.5 5.5 bills-T

r) r( ervalued Und9.9 9.8 USR

r) r( uedFairly val 10.5 10.5 Market

r) r( Overvalued 12.1% 12.4% HT

r r

^

^

^

^

^

^

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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.

bP = wHT bHT + wColl bColl

bP = 0.5 (1.32) + 0.5 (-0.87)

bP = 0.225

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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.

rP = wHT rHT + wColl rColl

rP = 0.5 (12.10%) + 0.5 (1.2%)

rP = 6.6% Or, using the portfolio’s beta, CAPM can be used

to solve for expected return.

rP = rRF + (RPM) bP

rP = 5.5% + (5.0%) (0.225)

rP = 6.6%

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Factors that change the SML What if investors raise inflation expectations

by 3%, what would happen to the SML?

SML1

ri (%)SML2

0 0.5 1.0 1.5

13.510.5

8.5 5.5

I = 3%

Risk, bi

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Y = C + mx

ri = rf + (Rm - Rf)b RPm

SML line

- Inflation is captured by rf - Inflation - rf

- Risk factor is captured by (Rm – Rf) - Risk factor - (Rm – Rf)

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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?

SML1

ri (%) SML2

0 0.5 1.0 1.5

13.510.5

5.5

RPM = 3%

Risk, bi