Risk and return matters

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.

2. 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

3. 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.

4. 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

5. 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.

6. 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

7. 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

8. 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.

9. 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
  

10. 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
              
=
=

11. 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.

12. 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

13. 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

14. 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