2. 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.
3. Risk and Return
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.
g
P
D
P
P
P
P
D
R
0
1
0
0
1
0
1
1
4. 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%
5. Expected Return
i j i j
j
n
R E R P
( ( ))2
1
E R R P
i j j
j
n
( )
1
Rj Possible outcome
Pj probability
Standard deviation
6. 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.
7. 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.
Expected Return E R R T
i it
t
T
( ) ( ) /
1
Standard Deviation
i
it i
t
T
R E R
T
( ( ))2
1
1
8. 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.
9. Risk and return for two assets
E RA
( ) E RB
( )
A B
wA
wB
A B
Expected return
Standard deviation
Investment ratio
Expected return for the portfolio E R w E R w E R
p A A B B
( ) ( ) ( )
Standard deviation for the portfolio
p A A B B A B AB
w w w w
( ) /
2 2 2 2 1 2
2
AB Aj A
j
N
Bj B j A B AB
R E R R E R P
[ ( )][ ( )]
1
p A A B B A B AB
w w w w
( ) /
2 2 2 2 1 2
2
10. The definition of Covariance
The covariance is to measure the co-
movement 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.
11. Risk and return for three assets
E RA
( ) E RB
( ) E RC
( )
A B C
wA wB
wC
A B C
Expected return
Standard deviation
Investment ratio
Expected return for the portfolio
E R w E R w E R w E R
p A A B B C C
( ) ( ) ( ) ( )
Standard deviation for the portfolio
p A A B B C C A B AB A C AC B C BC
w w w w w w w w w
( ) /
2 2 2 2 2 2 1 2
2 2 2
12. Risk and return for N assets
E R w E R
P k k
k
N
( ) ( )
1
2 2 1/ 2
1 1 1
( ) , ,
N N N
p i i i j ij
i i j
w w w i j
w w w w
n
n
1 2 3
1
...
2 2
2 2
1 1 1
2
2 2 2
2
1 1
( ) ( ) , ,
1 1
( ) ( ) ( )
1 1
(1 ) ,
N N N
p i ij
i i j
i ij
i ij
i j
n n
n n n
n n
n n
Expected return
Standard deviation
13. 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.
14. How diversification works
A B
Expected return 10% 14%
Standard deviation 4% 6%
Investment ratio 50% 50%
, 1 1
AB A B AB AB
( ) ( ) ( ) 10%*50%+14%*50%=12%
p A A B B
E R w E R w E R
15. 1
AB
p A A B B A B AB A A B B A B A B A A B B
w w w w w w w w w w
( ) ( ) [( ) ]
/ / /
2 2 2 2 1 2 2 2 2 2 1 2 2 1 2
2 2
( . . . . ) ( . % . %) %
/ /
0 5 6 0 5 4 2 0 5 0 5 6 4 1 0 5 6 0 5 4 5
2 2 2 2 1 2 1 2
0
AB
p A A B B A B AB
w w w w
( ) /
2 2 2 2 1 2
2
%
13
)
0
4
6
5
.
0
5
.
0
2
4
5
.
0
6
5
.
0
( 2
/
1
2
2
2
2
1
AB
p A A B B A B AB
w w w w
( ) /
2 2 2 2 1 2
2 =
( . . . . ( )) ( . % . %) %
/
0 5 6 0 5 4 2 0 5 0 5 6 4 1 0 5 6 0 5 4 1
2 2 2 2 1 2
=
=
16. 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.
17. Constant Correlation with Changing Weights
i
Asset E(R )
1 .10 .07 rij = 1.00
2 .20 .10
Case W1 W2 E(Ri) E(Fport)
f 0.00 1.00 0.20 0.1000
g 0.20 0.80 0.18 0.0940
h 0.40 0.60 0.16 0.0880
i 0.50 0.50 0.15 0.0850
j 0.60 0.40 0.14 0.0820
k 0.80 0.20 0.12 0.0760
l 1.00 0.00 0.10 0.0700
18. Portfolio Risk-Return Plots for Different Weights
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = +1.00
1
2
With two perfectly
correlated assets, it
is only possible to
create a two asset
portfolio with risk-
return along a line
between either
single asset
19. Constant Correlation with Changing Weights
Case W1 W2 E(Ri) E(F
port)
f 0.00 1.00 0.20 0.1000
g 0.20 0.80 0.18 0.0812
h 0.40 0.60 0.16 0.0662
i 0.50 0.50 0.15 0.0610
j 0.60 0.40 0.14 0.0580
k 0.80 0.20 0.12 0.0595
l 1.00 0.00 0.10 0.0700
i
Asset E(R )
1 .10 .07 rij = 0.00
2 .20 .10