7-1 2014_4_3

CHAPTER 7 Risk and Rates of Return

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Investment returns

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

Holding Period (Amount received – Amount invested)

Rate of 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%.

7-3 2014_4_3

What is an 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.

Two types of investment risk

Stand-alone risk : unique vs. market risk

Portfolio risk

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Breaking down sources of risk

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

Market risk (non-diversifiable, systematic)– portion of a security’s stand-alone risk that cannot be eliminated through diversification. (war, inflation, recessions, etc.)

Firm-specific risk (company-specific, unsystematic) – portion of a security’s stand-alone risk that can be eliminated through proper diversification. (strikes, M&A, lawsuits)

7-5 2014_4_3

Illustrating diversification effects of a stock portfolio

# Stocks in Portfolio 10 20 30 40 2,000+

Company-Specific Risk

Portfolio’s Market Risk

20

0

Portfolio’s Stand-Alone Risk

sp (%)

35

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Probability distributions

A listing of all possible outcomes, and the probability of each occurrence.

Expected Rate of Return

Rate of

Return (%) 100 15 0 -70

Firm X

Firm Y

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Expected Rate of Return

15.0% (0.3) (-70%)

(0.4) (15%) (0.3) (100%) k

P k k

return of rate expected k

M

^

n

1i

ii

^

^

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Stand-Alone Risk: the standard deviation

2variancedeviation Standard ss

2222 )ˆ( kEkEkkE s

21

2

22

n

1i

i

2^

i

(0.3)15.0) - (-70.0

(0.4)15.0) - (15.0 (0.3)15.0) - (100.0

P )k (k

Ms

s

=65.84%

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Normal Distribution with Mean of 12%

and St Dev of 20%

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Comments on standard deviation as a measure of risk

Standard deviation (σi) measures total, or

stand-alone, risk.

Larger σi is associated with a wider probability

distribution of returns.

The larger σi is, the lower the probability that

actual returns will be close to expected returns.

Difficult to compare standard deviations,

because return has not been accounted for.

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Comparing standard deviations

F1

Prob. T - bill

F2

0 8 13.8 17.4 Rate of Return (%)

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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 lover, Risk neutral

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.

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Portfolio construction: Risk and return

Expected return of a portfolio is a weighted average of the expected return of each of the component assets in the portfolio.

Standard deviation is not a weighted average of the individual asset’s S.D. It is generally smaller than the average of the assets’ S.D.

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Calculating portfolio expected return

10.75% 0.25(9.5%)0.25(10%)

(11.5%) 0.25 (12%) 0.25 k

kw k

:average weighteda is k

p

^

n

1i

i

^

ip

^

p

^

7-15 2014_4_3

Calculating portfolio Risk

1 2 n

1 w12s1

2 w2w1 s21 wnw1 sn1

2 w1w2 s12 w22 s22 wnw2 sn2

N w1wns1n w2wn s2n wn2sn2

< Variance-Covariance Matrix >

구분 E(ki) σi wi

S1 15% 15% 30%

S2 30% 40% 50%

S3 25% 30% 20%

E(Rp)= 0.245; σp= 0.2625

Ρ12

S1 S2 S3

S1 1 0.45 -0.3

S2 1 0.7

S3 1

예)

N

i

N

j

jiijji

N

i

N

j

ijjiP

ww

ww

1 1

1 1

2

ss

ss

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투자비율 P sP

xA xB AB=+1 AB=0 AB=-1

1 0 0.30 0.4 0.4 0.4

0.75 0.25 0.25 0.35 0.3 0.25

0.5 0.5 0.2 0.3 0.22 0.1

0.25 0.75 0.15 0.25 0.18 0.05

0 1 0.1 0.2 0.2 0.2

※ 포트폴리오 효과(분산투자효과; gains from diversification)

투자안 A: A=30% sA=40%; 투자안 B: B=10% sB=20%

Portfolio Risks with different ρ

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Portfolio Risks with different ρ

A

B

Z

AB=1

AB=0

AB=-1

AB=-1

P

sP

0.3

0.4

0.1

0.2

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※ 공분산(covariance)

- 두 확률변수(수익률)의 공조성(co-movement)

- 수익률이 체계적인 관계없이 움직이면 sAB0

기대수익률의주식

수익률의주식상황에서

확률발생할상황이

jRE

BAjjiR

niip

RERRERp

RERRERpRERRERp

RERRERE

j

ij

i

BnBAnAn

BBAABBAA

BBAAAB

:)(

),( :

),,2,1( :

))())(((

))())((())())(((

)()(((

,

,,

2,2,21,1,1

s

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※ 상관계수(correlation coefficient)

- 두 수익률의 표준화된 공조성

AB>0 : 양의 상관관계 (rho)

AB=+1 : 완전 양의 상관관계

AB

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※ 공분산, 상관계수 산출

상태(s) 1 2 3 4 5

확률(ps) 0.1 0.15 0.25 0.35 0.15

수익률(RA,s) -0.4 -0.1 0.2 0.5 0.8

수익률(RB,s) -0.15 -0.05 0.2 0.25 0.15

78726.0

137727.0;35623.0;1375.0)(;29.0)(

038625.0))())(((

))())((())())(((

)()(((

,,

2,2,21,1,1

BA

ABAB

BABA

BnBAnAn

BBAABBAA

BBAAAB

RERE

RERRERp

RERRERpRERRERp

RERRERE

ss

s

ss

s

7-21 2014_4_3

Capital Asset Pricing Model (CAPM)

A 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 only the risk remaining after diversification.

concerns about only stock’s market risk, beta, not its stand-alone risk

The relevant riskiness of a stock is its contribution to the riskiness of a well-diversified portfolio.

7-22 2014_4_3

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

The line shows the relationship between risk as measured by beta and the required rate of return for individual securities and portfolios.

SML: ki = kRF + (kM – kRF) βi

Assume kRF = 8% and kM = 15%.

The market risk premium is RPM = kM – kRF = 15% – 8% = 7%. (premium investors require for bearing the risk of an average stock)

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Illustrating the Security Market Line

SML: ki = 8% + (15% – 8%) βi

.

. HT

T-bills

.

SML

kM = 15

kRF = 8

-1 0 1 2

.

ki (%)

Risk, βi

. Expected

Required

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Beta

Measures a stock’s market risk, and shows a stock’s tendency to move up and down with the market

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

Measures a stock’s contribution to the riskiness of a portfolio => measure of the stock’s riskiness

N

i

iiNNp

m

im

i wwww1

22112,

s

s

7-25 2014_4_3

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 betas in the range of 0.5 to 1.5