2-3-1

Part 2-3 評價報酬與風險

2-3-2

Outlines Statistical calculations of risk and

return measures Risk Aversion Systematic and firm-specific risk Efficient diversification The Capital Asset Pricing Model Market Efficiency

2-3-3

Rates of Return: Single Period HPR = Holding Period Return

P0 = Beginning price P1 = Ending price D1 = Dividend during period one

PDPPHPR

0

101

2-3-4

Rates of Return: Single Period ExampleEnding Price = 48Beginning Price = 40Dividend = 2

HPR = (48 - 40 + 2 )/ (40) = 25%

2-3-5

Return for Holding Period – Zero Coupon Bonds Zero-coupon bonds are bonds that

are sold at a discount from par value.

Given the price, P (T ), of a Treasury bond with $100 par value and maturity of T years

1)(

100)( TP

Trf

2-3-6

Example - Zero Coupon Bonds Rates of ReturnHorizon,

T Price, P(T) [100/P(T)]-1Risk-free

Return for Given Horizon

Half-year $97.36 100/97.36-1 = .0271 rf(.5) = 2.71%

1 year $95.52 100/95.52-1 = .0580 rf(1) = 5.80%

25 years $23.30 100/23.30-1 = 3.2918

rf(25) = 329.18%

2-3-7

Effective annual rates, EARs

Annual percentage rates, APRs

Formula for EARs and APRs 1)(1

1 Tf TrEAR

TEARAPR

T

f TrT

1)1()(1

2-3-8

Table - Annual Percentage Rates (APR) and Effective Annual Rates (EAR)

2-3-9

Continuous Compounding Continuous compounding, CC

rCC is the annual percentage rate for the continuously compounded case e is approximately 2.71828

CCrT

TTeAPRTEAR

1

00]1[)1( limlim

2-3-10

Characteristics of Probability Distributions Mean

most likely value Variance or standard deviation Skewness

2-3-11

Subjective returns

ps = probability of a state rs = return if a state occurs

Mean Scenario or Subjective Returns

n

sss rprE

1

)(

2-3-12

Subjective or Scenario Standard deviation = [variance]1/2

ps = probability of a state rs = return if a state occurs

Variance or Dispersion of Returns

n

sss rErp

1

22

2-3-13

Deviations from Normality Skewness

Kurtosis3

3)]()([

rEsrESkew

3)]()([4

4

rEsrEKurtosis

2-3-14

Figure - The Normal Distribution

2-3-15

Figure - Normal and Skewed (mean = 6% SD = 17%)

2-3-16

Figure - Normal and Fat Tails Distributions (mean = .1 SD =.2)

2-3-17

Spreadsheet - Distribution of HPR on the Stock Index Fund

2-3-18

Mean and Variance of Historical Returns Arithmetic average or rates of return

Variance

Average return is arithmetic average

n

ss

n

sssA rn

rpr11

1

n

sAs rr

n 1

2)(112

2-3-19

Geometric Average Returns Geometric Average Returns

TV = Terminal Value of the Investment rG = geometric average rate of return

nrrrrTV

G

nn

)1()1()1)(1( 21

1/1 TV nGr

2-3-20

Spreadsheet - Time Series of HPR for the S&P 500

2-3-21

Example - Arithmetic Average and Geometric Average

%58.9095844.1)15.1()20.1()95(.)10.1(

)1()1()1()1()1(4

43214

g

g

r

RRRRr

%104

%15%20%5%104

4321

RRRRrA

Year 1 2 3 4Return 10% -5% 20% 15%

2-3-22

Measurement of Risk with Non-Normal Distributions Value at Risk, VaR Conditional Tail Expectation, CTE Lower Partial Standard Deviation, LPSD

2-3-23

Figure - Histograms of Rates of Return for 1926-2005

2-3-24

Table - Risk Measures for Non-Normal Distributions

2-3-25

Risk Averse Reject investment portfolios that are

fair games or worse Risk Neutral

Judge risky prospects solely by their expected rates of return

Risk Seeking Engage in fair games and gamble

Investor’s View of Risk

2-3-26

Fair Games and Expected Utility Assume a log utility function

A simple prospect

)ln()( WWU

2-3-27

Fair Games and Expected Utility (cont.)

2-3-28

Diversification and Portfolio Risk Sources of uncertainty

Come from conditions in the general economy Market risk, systematic risk, nondiversifiable risk

Firm-specific influences Unique risk, firm-specific risk, nonsystematic risk, diversifiable risk

2-3-29

Diversification and Portfolio Risk Example

Normal Year for Sugar Abnormal Year

Bullish Stock Market

Bearish Stock Market

Sugar Crisis

.5 .3 .2Best

Candy25% 10% -25%

SugarKane 1% -5% 35%

T-bill 5% 5% 5%

2-3-30

Diversification and Portfolio Risk Example (cont.)

%73.14,%6)(%9.18,%5.10)(

SugarSugar

BestBest

rErE

Portfolio Expected Return

Standard Deviation

All in Best 10.50% 18.90%Half in T-bill 7.75% 9.45%Half in Sugar 8.25% 4.83%

2-3-31

Components of Risk Market or systematic risk

Risk related to the macro economic factor or market index.

Unsystematic or firm specific risk Risk not related to the macro factor or

market index. Total risk = Systematic +

Unsystematic

2-3-32

Figure - Portfolio Risk as a Function of the Number of Stocks in the Portfolio

2-3-33

Figure - Portfolio Diversification

2-3-34

Two-Security Portfolio: Return Consider two mutual fund, a bond

portfolio, denoted D, and a stock fund, E

)()()(1

EEDDP

ED

EEDDP

rEwrEwrEww

rwrwr

2-3-35

Two-Security Portfolio: Risk The variance of the portfolio, is not a weighted average of the individual asset variances

The variance of the portfolio is a weighted sum of covariances),(2

),(),(),(222222

EDED

EEEEDDDD

EDEDEEDDP

rrCovwwrrCovwwrrCovwwrrCovwwww

2-3-36

Table - Computation of Portfolio Variance from the Covariance Matrix

2-3-37

Covariance and Correlation Coefficient The covariance can be computed

from the correlation coefficient

ThereforeEDDEED rrCov ),(

DEEDEDEEDDP wwww 222222

2-3-38

Example - Descriptive Statistics for Two Mutual Funds

2-3-39

Portfolio Risk and Return Example Apply this analysis to the data as

presented in the previous slide

2

22

22222

144400144

3.201222012

138)(

PP

EDED

EDEDP

EDP

wwww

wwww

wwrE

2-3-40

Table - Expected Return and Standard Deviation with Various Correlation Coefficients

2-3-41

Figure - Portfolio Opportunity Set

2-3-42

Figure - The Minimum-Variance Frontier of Risky Assets

2-3-43

Figure - Capital Allocation Lines with Various Portfolios from the Efficient Set

2-3-44

Capital Allocation and the Separation Property A portfolio manager will offer the

same risky portfolio, P, to all clients regardless of their degree of risk aversion

Separation property Determination of the optimal risky

portfolio Allocation of the complete portfolio

2-3-45

Capital Asset Pricing Model (CAPM) It is the equilibrium model that underlies all modern financial theory. Derived using principles of diversification with simplified assumptions. Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.

2-3-46

Figure - The Efficient Frontier and the Capital Market Line

2-3-47

Slope and Market Risk Premium Market risk premium

Market price of risk, Slope of the CML

fM rrE )(

M

fM rrE

)(

2-3-48

The Security Market Line Expected return – beta relationship

The security’s risk premium is directly proportional to both the beta and the risk premium of the market portfolio

All securities must lie on the SML in market equilibrium

])([)( fMifi rrErrE )()(

2,

M

Mii R

RRCov

2-3-49

Figure - The Security Market Line

2-3-50

Sample Calculations for SML Suppose that the market return is

expected to be 14%, and the T-bill rate is 6% Stock A has a beta of 1.2

If one believed the stock would provide an expected return of 17%

%6.15%)6%14(2.1%6)( ArE

%4.1%6.15%17

2-3-51

Do security prices reflect information ?

Why look at market efficiency? Implications for business and

corporate finance Implications for investment

Efficient Market Hypothesis (EMH)

2-3-52

Random Walk Stock prices are random Randomly evolving stock prices are

the consequence of intelligent investors competing to discover relevant information

Expected price is positive over time Positive trend and random about the

trend

Random Walk and the EMH

2-3-53

Random Walk with Positive Trend

Security Prices

Time

2-3-54

Why are price changes random? Prices react to information Flow of information is random Therefore, price changes are

random

Random Price Changes

2-3-55

Figure - Cumulative Abnormal Returns before Takeover Attempts: Target Companies

2-3-56

EMH and Competition Stock prices fully and accurately

reflect publicly available information.

Once information becomes available, market participants analyze it.

Competition assures prices reflect information.

2-3-57

Forms of the EMH Weak form EMH Semi-strong form EMH Strong form EMH

2-3-58

Information Sets