Download 財務報表分析與 產業競爭分析

Risk and Return
The objective to learn risk and return:


The appropriate discount rate used in capital
budgeting is the firm’s average cost of capital, that is,
the weighted average of costs of debt and equity.
While cost of debt is easy to estimate, the cost of
equity is difficult to determine.

The realized cost of equity does not always apply.
 We need to learn how equity holders determine their
required rate of return when investing equity.
 The force of capital market would make the equity holders’
required rate of return equal to the true cost of equity.
Risk and Return
D1 P1  P0 D1
R1 


g
P0
P0
P0

Due to the return will be realized by the end of
the period, so the investor will not know the
return for sure, so the rate of return is only a
random variable to the investor. The investor
needs to estimate what will be the return.
The subjective way to describe random
variable – Probability Distribution
Outcomes
Probability T-bill
Corp.
Bond
Common
Stock
Serious Recession
0.05
8%
12%
-3%
Mild recession
0.20
8%
10%
6%
stable
0.50
8%
9%
11%
Mild prosperity
0.20
8%
8.5%
14%
Extreme
prosperity
0.05
8%
8%
19%
Expected Return
8%
9.2%
10.3%
Standard Deviation
0%
0.84%
4.39%
Expected Return
R j Possible outcome
n
E ( Ri )   R j  Pj
j 1
Pj probability
Standard deviation
i 
n
2
(
R

E
(
R
))
 Pj
 j
i
j 1
The difficulty in using the subjective
Probability Distribution
It is difficult to describe the probability
distribution of an individual.
 It is not certain that an individual’s
probability distribution is in accordance
with the probability distribution of investors
of market as a whole, which is more
relevant in determining cost of equity.

The objective way to describe
random variable – historical data

We can measure risk and return employing
historical returns. We basically believe that
history will repeat itself.
T
Expected Return
E ( Ri )  (  Rit ) / T
t 1
T
Standard Deviation
i 
(R
it
 E ( Ri )) 2
t 1
T 1
The problems in using historical
data in estimating risk and return
Some of firms, start-up or private firms,
may lack of stock trades information in
estimating realized risk and return.
 This historical estimation sometimes may
not be representative for future
expectations.

Risk and return for two assets
Ａ
Expected return
Standard deviation
Investment ratio
Ｂ
E ( RA )
E ( RB )
A
B
wA
wB
Expected return for the portfolio E ( Rp )  wA E ( RA )  wB E ( RB )
Standard deviation for the portfolio
 p  ( w A  A  wB  B  2 w A wB  AB )1/ 2
2
N
2
2
2
 AB   [ R Aj  E ( R A )][RBj  E ( RB )] Pj   A  B  AB
j 1
The definition of Covariance
The covariance is to measure the comovement of two random variable.
 When covariance is positive, then the two
random variables tend to move into same
direction.
 When covariance is negative, then the two
random variables tend to more to opposite
directions.

Risk and return for three assets
Expected return
Standard deviation
Investment ratio
Ａ
E ( RA )
Ｂ
E ( RB )
Ｃ
E ( RC )
wA
wB
wC
A
B
C
Expected return for the portfolio
E ( R p )  w A E ( R A )  wB E ( RB )  wC E ( RC )
Standard deviation for the portfolio
 p  ( w A 2  A 2  wB 2  B 2  wC 2  C 2  2 w A wB  AB  2 w A wC  AC  2 wB wC  BC )1/ 2
Risk and return for N assets
N
Expected return
Standard deviation
E ( RP )   wk E ( Rk )
k 1
N
2
i 1
1
w1  w2  w3 ...  wn 
n
p
2
N
N
 p  ( wi  i   wi w j ij )1/ 2 , i  j,
2
1 2 N 2 1 2 N N
 ( )   i  ( )   ij , i  j,
n i 1
n i 1 j 1
1 2
1 2
2
 ( )  n   i  ( )  (n 2  n) ij
n
n
1 2
1
  i  (1  ) ij ,
n
n
i 1 j 1
Risk Diversification

When number of assets increases, the
majority of risk for the portfolio comes from
the co-movement among assets. The
distinctive risk comes from each asset
becomes less important.
How diversification works
Ａ
Expected return
Standard deviation
Investment ratio
10%
4%
50%
Ｂ
14%
6%
50%
 AB   A B  AB , 1   AB  1
E ( R p )  wA E ( RA )  wB E ( RB )  10%*50%+14%*50%=12%
 AB  1
 p  ( wA 2  A 2  wB 2  B 2  2 wA wB  AB )1/ 2  ( wA 2  A 2  wB 2  B 2  2 wA wB  A B )1/ 2  [( wA A  wB  B ) 2 ]1/ 2
( 0=.52  62  0.52  4 2  2  0.5  0.5  6  4  1)1/ 2  ( 0.5  6%  0.5  4%)1/ 2  5%
 AB  0
 p  ( w A 2  A 2  wB 2  B 2  2 w A wB  AB )1/ 2 =
(0.52  6 2  0.52  4 2  2  0.5  0.5  6  4  0)1/ 2  13 %
 AB  1
 p  ( w A 2  A 2  wB 2  B 2  2 w A wB  AB )1/ 2 =
( 0.52  62  0.52  4 2  2  0.5  0.5  6  4  ( 1))1/ 2  ( 0.5  6%  0.5  4%)  1%
Combining Stocks with Different
Returns and Risk
Assets may differ in expected rates of
return and individual standard deviations
 Negative or small positive correlations
reduce portfolio risk
 Combining two assets with -1.0 correlation
is able reduces the portfolio standard
deviation to zero.

Constant Correlation with Changing Weights
Asset
E(R i )

1
.10
.07
2
.20
.10
Case
W1
W2
f
g
h
i
j
k
l
0.00
0.20
0.40
0.50
0.60
0.80
1.00
1.00
0.80
0.60
0.50
0.40
0.20
0.00
rij = 1.00
E(Ri )
0.20
0.18
0.16
0.15
0.14
0.12
0.10
E(F port)
0.1000
0.0940
0.0880
0.0850
0.0820
0.0760
0.0700
Portfolio Risk-Return Plots for Different Weights
E(R)
0.20
0.15
0.10
0.05
With two perfectly
correlated assets, it
is only possible to
create a two asset
portfolio with riskreturn along a line
between either
single asset
2
Rij = +1.00
1
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
Constant Correlation with Changing Weights
Asset
E(R i )

1
.10
.07
2
.20
.10
rij = 0.00
Case
W1
W2
E(Ri )
E(Fport)
f
g
h
i
j
k
l
0.00
0.20
0.40
0.50
0.60
0.80
1.00
1.00
0.80
0.60
0.50
0.40
0.20
0.00
0.20
0.18
0.16
0.15
0.14
0.12
0.10
0.1000
0.0812
0.0662
0.0610
0.0580
0.0595
0.0700
Portfolio Risk-Return Plots for
Different Weights
E(R)
0.20
0.15
0.10
f
g
2
With uncorrelated
h
assets it is possible
i
j
to create a two
Rij = +1.00
asset portfolio with
k
lower risk than
1
either single asset
Rij = 0.00
0.05
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
Portfolio Risk-Return Plots for
Different Weights
E(R)
0.20
0.15
0.10
f
g
2
With correlated
h
assets it is possible
i
j
to create a two
Rij = +1.00
asset portfolio
k
Rij = +0.50
between the first
1
two curves
Rij = 0.00
0.05
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
Portfolio Risk-Return Plots for
Different Weights
E(R) With
0.20 negatively
correlated
assets it is
0.15
possible to
create a two
0.10 asset portfolio
with much
0.05 lower risk than
either single
asset
Rij = -0.50
f
2
g
h
j
k
i
Rij = +1.00
Rij = +0.50
1
Rij = 0.00
-
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
Constant Correlation with Changing Weights
E(R i )

1
.10
.07
2
.20
.10
Asset
Case
W1
W2
f
g
h
i
j
k
l
0.00
0.20
0.40
0.50
0.60
0.80
1.00
1.00
0.80
0.60
0.50
0.40
0.20
0.00
rij = -1.00
E(Ri )
0.20
0.18
0.16
0.15
0.14
0.12
0.10
E(F port)
0.1000
0.0660
0.0320
0.0150
0.0020
0.0360
0.0700
Portfolio Risk-Return Plots for
Different Weights
E(R)
0.20
Rij = -0.50
Rij = -1.00
f
2
g
h
0.15
0.10
0.05
-
j
k
i
Rij = +1.00
Rij = +0.50
1
Rij = 0.00
With perfectly negatively correlated
assets it is possible to create a two asset
portfolio with almost no risk
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
A zero standard deviation for a two
asset portfolio with correlation of -1

WA 
B
 A B
WB 
A
 A B
When
and
, the investor is
able to form a zero standard deviation
(that is, zero risk) portfolio.
The Efficient Frontier
The efficient frontier represents that set of
portfolios, within all possibilities, with the
highest rate of return for every given level of
risk, or with the lowest risk for every given
level of return
 Efficient Frontier would be composed of
portfolios rather than individual securities

 Exceptions
being the asset with the highest
return and the asset with the lowest risk
Efficient Frontier for Alternative Portfolios
E(R)
Efficient
Frontier
A
B
C
Standard Deviation of Return
The Efficient Frontier and an
Investor’s Utility
The optimal portfolio chosen has the
highest utility for a given investor
 The optimal portfolio lies at the point of
tangency between the efficient frontier and
the utility curve with the highest possible
utility

Selecting an Optimal Risky Portfolio
E(R port )
U3’
U2’
U1’
Y
U3
X
U2
U1
E( port )
Summary:





The efficient frontier derived from risky assets will
be a curve, rather than a linear equation.
Investors with different risk preferences will choose
different points on efficient frontier to be their
optimal choices.
So the equilibrium will be on the whole efficient
frontier.
Problem: it is difficult to describe and apply a nonlinear relationship of risk and return.
Solution: introduce a risk-free asset into the
model…
Combining a Risk-Free Asset
with a Risky Portfolio
Since both the expected return and the
standard deviation of return for such a
portfolio are linear combinations for that
risky portfolio and the risk free asset, a
graph of possible portfolio returns and risks
looks like a straight line between the two
assets.
Risk-Free Asset
An asset with zero standard deviation
 Zero correlation with all other risky assets
 Provides the risk-free rate of return (RFR)
 Will lie on the vertical axis of a portfolio
graph

Risk-Free Asset
Covariance between two sets of returns is
n
Cov ij   [R i - E(R i )][R j - E(R j )]/n
i 1
Because the returns for the risk free asset are certain,
 RF  0
Thus Ri = E(Ri), and Ri - E(Ri) = 0
Consequently, the covariance of the risk-free asset with any
risky asset or portfolio will always equal zero. Similarly the
correlation between any risky asset and the risk-free asset
would be zero.
Combining a Risk-Free Asset
with a Risky Portfolio
Expected return
the weighted average of the two returns
E(R port )  WRF (RFR)  (1 - WRF )E(R i )
This is a linear relationship
Combining a Risk-Free Asset
with a Risky Portfolio
Standard deviation
E(
E(
2
port
2
port
)  w   w   2w 1 w 2 r1,2 1 2
2
1
)w 
2
RF
2
1
2
RF
2
2
2
2
 (1  w RF )   2w RF (1 - w RF )rRF,i RF i
2
2
i
Since we know that the variance of the risk-free asset is
zero and the correlation between the risk-free asset and any
risky asset i is zero we can adjust the formula
2
E( port
)  (1  w RF ) 2  i2
Capital Market Line (CML)
E ( Rp )  R f 
E(R port )
E ( Rm )  R f
 ( Rm )
 ( Rp )
D
M
C
RFR
B
A
E( port )
Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient Frontier
E(R port )
M
RFR
E( port )
The Security Market Line (SML)
The relevant risk measure for an individual
risky asset is its covariance with the market
portfolio (Covi,m)
 This is shown as the risk measure
 The return for the market portfolio should be
consistent with its own risk, which is the
covariance of the market with itself - or its
2
variance:  m

Graph of Security Market Line (SML)
E(R i )
SML
Rm
RFR

2
m
Cov im
The Security Market Line (SML)
The equation for the risk-return line is
E(R i )  RFR 
 RFR 
We then define
R M - RFR

2
M
Cov i,M

2
M
Cov i,M
 M2
(Cov i,M )
(R M - RFR)
as beta
( i )
E(R i )  RFR   i (R M - RFR)
Graph of SML with
Normalized Systematic Risk
E(R i )
SML
Rm
Negative
Beta
RFR
0
1.0
Beta(
 im
m
2
)
Factors that influences the shape of SML
Inflation risk premium (inflation expectation)
 Investors’ attitude toward risk (risk aversion)

Inflation risk premium
E ( Rp )
SML
SML
i
Investors’ attitude toward risk (risk aversion)
E ( Rp )
SML
SML
i
Capital Market equilibrium

The required rate of return:
E(R i )  RFR   i (R M - RFR)

D1
g
The expected rate of return: Ri 
P0

When expected return > required return
 Then

P0 increases, expected return decreases
When expected return < required return
 Then
P0 decreases, expected return increases
E ( Rp )
.A
SML
.B
i