Part 2-3
評價
報酬與風險
2-3-1
Outlines






Statistical calculations of risk and return
measures
Risk Aversion
Systematic and firm-specific risk
Efficient diversification
The Capital Asset Pricing Model
Market Efficiency
2-3-2
Rates of Return: Single Period

HPR = Holding Period Return
P
1  P0  D1
HPR 
P0



P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
2-3-3
Rates of Return: Single Period
Example
Ending Price
Beginning Price
Dividend
=
=
=
48
40
2
HPR = (48 - 40 + 2 )/ (40) = 25%
2-3-4
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
100
rf (T ) 
1
P(T )
2-3-5
Example - Zero Coupon Bonds
Rates of Return
Horizon, T
Price, P(T)
[100/P(T)]-1
Risk-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%
2-3-6
Formula for EARs and APRs

Effective annual rates, EARs
EAR  1  r (T )
f

1
T
1
Annual percentage rates, APRs
1
(
1

EAR
)
1
APR   rf (T ) 
T
T
T
2-3-7
Table - Annual Percentage Rates (APR)
and Effective Annual Rates (EAR)
2-3-8
Continuous Compounding

Continuous compounding, CC
lim (1  EAR)  lim [1  T  APR]
T 0


1
T
e
rCC
T 0
rCC is the annual percentage rate for the
continuously compounded case
e is approximately 2.71828
2-3-9
Characteristics of Probability
Distributions

Mean



most likely value
Variance or standard deviation
Skewness
2-3-10
Mean Scenario or Subjective
Returns

Subjective returns
n
E (r)   ps rs
s 1


ps = probability of a state
rs = return if a state occurs
2-3-11
Variance or Dispersion of
Returns

Subjective or Scenario

Standard deviation = [variance]1/2
n

   ps  r s  E  r 
2
s 1



2
ps = probability of a state
rs = return if a state occurs
2-3-12
Deviations from Normality

Skewness
Skew 

E[r ( s )  E ( r )]
3

3
Kurtosis
Kurtosis 
E[r ( s )  E ( r )]

4
4
3
2-3-13
Figure - The Normal
Distribution
2-3-14
Figure - Normal and Skewed
(mean = 6% SD = 17%)
2-3-15
Figure - Normal and Fat Tails
Distributions (mean = .1 SD =.2)
2-3-16
Spreadsheet - Distribution of
HPR on the Stock Index Fund
2-3-17
Mean and Variance of
Historical Returns


Arithmetic average or rates of return
n
1 n
rA   ps rs   rs
n s 1
s 1
Variance
n
1
2
2
(
r

r
)
s
A
 n 1 
s 1

Average return is arithmetic average
2-3-18
Geometric Average Returns

Geometric Average Returns
TV
n
 (1  r1)(1  r 2)(1  r n )
 (1  rG )n
rG  TV


1/ n
1
TV = Terminal Value of the Investment
rG = geometric average rate of return
2-3-19
Spreadsheet - Time Series of
HPR for the S&P 500
2-3-20
Example - Arithmetic Average
and Geometric Average
Year
1
2
3
4
Return
10%
-5%
20%
15%
R1  R2  R3  R4 10%  5%  20%  15%
rA 

 10%
4
4
(1  rg )4  (1  R1 )  (1  R2 )  (1  R3 )  (1  R4 )
rg  4 (1.10)  (.95)  (1.20)  (1.15)  1  .095844  9.58%
2-3-21
Measurement of Risk with
Non-Normal Distributions



Value at Risk, VaR
Conditional Tail Expectation, CTE
Lower Partial Standard Deviation, LPSD
2-3-22
Figure - Histograms of Rates of
Return for 1926-2005
2-3-23
Table - Risk Measures for NonNormal Distributions
2-3-24
Investor’s View of Risk

Risk Averse


Risk Neutral


Reject investment portfolios that are fair
games or worse
Judge risky prospects solely by their
expected rates of return
Risk Seeking

Engage in fair games and gamble
2-3-25
Fair Games and Expected
Utility

Assume a log utility function
U (W )  ln( W )

A simple prospect
2-3-26
Fair Games and Expected
Utility (cont.)
2-3-27
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
2-3-28
Diversification and Portfolio
Risk Example
Normal Year for Sugar
Abnormal
Year
Sugar Crisis
Bullish Stock
Market
Bearish Stock
Market
Best Candy
.5
25%
.3
10%
.2
-25%
SugarKane
1%
-5%
35%
T-bill
5%
5%
5%
2-3-29
Diversification and Portfolio
Risk Example (cont.)
E ( rBest )  10.5% ,  Best  18.9%
E ( rSugar )  6%
Portfolio
,  Sugar  14.73%
All in Best
Expected
Return
10.50%
Standard
Deviation
18.90%
Half in T-bill
7.75%
9.45%
Half in Sugar
8.25%
4.83%
2-3-30
Components of Risk

Market or systematic risk


Unsystematic or firm specific risk


Risk related to the macro economic factor
or market index.
Risk not related to the macro factor or
market index.
Total risk = Systematic + Unsystematic
2-3-31
Figure - Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
2-3-32
Figure - Portfolio
Diversification
2-3-33
Two-Security Portfolio: Return

Consider two mutual fund, a bond
portfolio, denoted D, and a stock fund,
E
rP  wD rD  wE rE
wD  wE  1
E ( rP )  wD E ( rD )  wE E ( rE )
2-3-34
Two-Security Portfolio: Risk

The variance of the portfolio, is not a
weighted average of the individual asset
variances
 P2  wD2  D2  wE2 E2  2 wD wE Cov( rD , rE )
 wD wDCov( rD , rD )  wE wE Cov( rE , rE )
 2 wD wE Cov( rD , rE )

The variance of the portfolio is a weighted sum of
covariances
2-3-35
Table - Computation of Portfolio
Variance from the Covariance Matrix
2-3-36
Covariance and Correlation
Coefficient

The covariance can be computed from
the correlation coefficient
Cov(rD , rE )   DE D E

Therefore
  w   w   2wD wE D E  DE
2
P
2
D
2
D
2
E
2
E
2-3-37
Example - Descriptive Statistics
for Two Mutual Funds
2-3-38
Portfolio Risk and Return
Example

Apply this analysis to the data as
presented in the previous slide
E ( rP )  8wD  13wE
 P2  122 wD2  202 wE2  2  12  20  .3  wD wE
 144 wD2  400wE2  144 wD wE
 P   P2
2-3-39
Table - Expected Return and Standard
Deviation with Various Correlation
Coefficients
2-3-40
Figure - Portfolio Opportunity
Set
2-3-41
Figure - The Minimum-Variance
Frontier of Risky Assets
2-3-42
Figure - Capital Allocation Lines with
Various Portfolios from the Efficient Set
2-3-43
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
2-3-44
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.
2-3-45
Figure - The Efficient Frontier and
the Capital Market Line
2-3-46
Slope and Market Risk
Premium

Market risk premium
E ( rM )  rf

Market price of risk, Slope of the CML
E ( rM )  rf
M
2-3-47
The Security Market Line

Expected return – beta relationship
E ( ri )  rf   i [ E ( rM )  rf ]


i 
Cov( Ri , RM )
 2 ( RM )
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
2-3-48
Figure - The Security Market
Line
2-3-49
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
E(rA )  6%  1.2  (14%  6%)  15.6%

If one believed the stock would provide an
expected return of 17%
  17%  15.6%  1.4%
2-3-50
Efficient Market Hypothesis
(EMH)


Do security prices reflect information ?
Why look at market efficiency?


Implications for business and corporate
finance
Implications for investment
2-3-51
Random Walk and the EMH

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
2-3-52
Random Walk with Positive
Trend
Security
Prices
Time
2-3-53
Random Price Changes
Why are price changes random?
 Prices react to information
 Flow of information is random
 Therefore, price changes are random
2-3-54
Figure - Cumulative Abnormal Returns
before Takeover Attempts: Target
Companies
2-3-55
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.
2-3-56
Forms of the EMH



Weak form EMH
Semi-strong form EMH
Strong form EMH
2-3-57
Information Sets
2-3-58