6-9
LG 2: Assessing Return and Risk
a.
Project 257
1.
Range: 1.00 - (-.10) = 1.10
n
Expected return: k   k i Pr i
2.
i 1
Rate of Return
Probability
Weighted Value
Expected Return
n
ki
Pri
ki x Pri
k   k i Pr i
i 1
-.10
.10
.20
.30
.40
.45
.50
.60
.70
.80
1.00
3.
.01
.04
.05
.10
.15
.30
.15
.10
.05
.04
.01
1.00
Standard Deviation:  
-.001
.004
.010
.030
.060
.135
.075
.060
.035
.032
.010
.450
n
 (k  k ) 2
i
x Pri
i 1
ki
k
-.10
.10
.20
.30
.40
.45
.50
.60
.70
.80
1.00
.450
.450
.450
.450
.450
.450
.450
.450
.450
.450
.450
ki  k
-.550
-.350
.250
-.150
-.050
.000
.050
.150
.250
.350
.550
(ki  k ) 2
Pri
(ki  k ) 2 x Pri
.3025
.1225
.0625
.0225
.0025
.0000
.0025
.0225
.0625
.1225
.3025
.01
.04
.05
.10
.15
.30
.15
.10
.05
.04
.01
.003025
.004900
.003125
.002250
.000375
-.00000'0
.000375
.002250
.003125
.004900
.003025.
.027350
Project 257 =
4.
CV 
.027350 = .165378
.165378
 .3675
.450
Project 432
1.
Range: .50 - .10 = .40
2.
Expected return: k   k i Pr i
n
i 1
Rate of Return
Probability
Weighted Value
Expected Return
n
ki
Pri
k   k i Pr i
ki x Pri
i 1
.10
.15
.20
.25
.30
.35
.40
.45
.50
3.
.05
.10
.10
.15
.20
.15
.10
.10
.05
1.00
.0050
.0150
.0200
.0375
.0600
.0525
.0400
.0450
.0250
.300
Standard Deviation:  
n
 (k  k ) 2
i
x Pri
i 1
ki
k
ki  k
(ki  k ) 2
Pri
.10
.15
.20
.25
.30
.35
.40
.45
.50
.300
.300
.300
.300
.300
.300
.300
.300
.300
-.20
-.15
-.10
-.05
.00
.05
.10
.15
.20
.0400
.0225
.0100
.0025
.0000
.0025
.0100
.0225
.0400
.05
.10
.10
.15
.20
.15
.10
.10
.05
Project 432 =
.011250 = .106066
(ki  k ) 2 x Pri
.002000
.002250
.001000
.000375
.000000
.000375
.001000
.002250
.002000
.011250
4.
b.
CV 
.106066
 .3536
.300
Bar Charts
Project 257
0.3
0.25
0.2
0.15
Probability
0.1
0.05
0
-10%
10%
20%
30%
40%
45%
50%
60%
70%
80%
100%
Rate of Return
Project 432
0.2
0.18
0.16
0.14
0.12
0.1
Probability
0.08
0.06
0.04
0.02
0
10%
15%
20%
25%
300%
35%
Rate of Return
40%
45%
50%
c.
Summary Statistics
Project 257
Range
1.100
Expected Return ( k )
0.450
Standard Deviation ( k )
0.165
Coefficient of Variation (CV) 0.3675
Project 432
.400
.300
.106
.3536
Since Projects 257 and 432 have differing expected values, the coefficient
of variation should be the criterion against which the risk of the asset is
judged. Since Project 432 has a smaller CV, it is the opportunity with
lower risk.
6-10 LG 2:
Integrative-Expected Return, Standard Deviation, and
Coefficient of Variation
n
a.
Expected return: k   ki  Pr i
i 1
Rate of Return
Probability
Weighted Value
Expected Return
n
ki
Pri
ki x Pri
k   k i Pr i
i 1
Asset F
.40
.10
.00
-.05
-.10
.10
.20
.40
.20
.10
.04
.02
.00
-.01
-.01
.04
Asset G
.35
.10
-.20
.40
.30
.30
.14
.03
-.06
.11
Asset H
.40
.20
.10
.10
.20
.40
.04
.04
.04
.00
-.20
.20
.10
.00
-.02
.10
Asset G provides the largest expected return.
b.
Standard Deviation: k 
n
 (k  k ) 2
i
x Pri
i 1
(ki  k )
Asset F .40
.10
.00
-.05
-.10
-
.04
.04
.04
.04
.04
= .36
= .06
= -.04
= -.09
= -.14
Asset G
.35 - .11
.02304
.10 - .11 = -.01
-.20 - .11 = -.31
-
.10
.10
.10
.10
.10
Pri
2
k
.1296
.0036
.0016
.0081
.0196
.10
.20
.40
.20
.10
.01296
.00072
.00064
.00162
.00196
.01790
.1338
=
.0001
.0961
(ki  k )
Asset H .40
.20
.10
.00
-.20
(ki  k ) 2
(ki  k ) 2
=
=
=
=
=
.30
.10
-.10
-.10
-.30
.0900
.0100
.0000
.0100
.0900
.24
.0576
.40
.30
.30
.00003
.02883
.05190
.2278
Pri
2
k
.10
.20
-.40
.20
.10
.009
.002
.000
.002
.009
.022
.1483
Based on standard deviation, Asset G appears to have the greatest risk,
but it must be measured against its expected return with the statistical
measure coefficient of variation, since the three assets have differing
expected values. An incorrect conclusion about the risk of the assets
could be drawn using only the standard deviation.
c.
Coefficien t of Variation =
standard deviation ()
expected value
Asset F:
CV 
.1338
 3.345
.04
Asset G:
CV 
.2278
 2.071
.11
Asset H:
CV 
.1483
 1.483
.10
As measured by the coefficient of variation, Asset F has the largest
relative risk.
6-12 LG 3: Portfolio Return and Standard Deviation
a.
Expected Portfolio Return for Each Year: kp = (wL x kL) + (wM x kM)
Year
Asset L
(wL x kL)
+
Asset M
(wM x kM)
1998
1999
2000
2001
2002
2003
(14% x.40 = 5.6%)
(14% x.40 = 5.6%)
(16% x.40 = 6.4%)
(17% x.40 = 6.8%)
(17% x.40 = 6.8%)
(19% x.40 = 7.6%)
+
+
+
+
+
+
(20% x .60 =12.0%)
(18% x .60 =10.8%)
(16% x .60 = 9.6%)
(14% x .60 = 8.4%)
(12% x .60 = 7.2%)
(10% x .60 = 6.0%)
Expected
Portfolio Return
kp
=
=
=
=
=
=
17.6%
16.4%
16.0%
15.2%
14.0%
13.6%
n
w k
j
b.
Portfolio Return: kp 
kp 
j
j1
n
17.6  16.4  16.0  15.2  14.0  13.6
 15.467  15.5%
6
( ki  k ) 2

i 1 ( n  1)
n
c.
Standard Deviation: kp 
(17.6%  15.5%) 2  (16.4%  15.5%) 2  (16.0%  15.5%) 2 

2
2
2
 (15.2%  15.5%)  (14.0%  15.5%)  (13.6%  15.5%) 
kp 
6 1
(2.1%) 2  (.9%) 2  (0.5%) 2


2
2
2
 (.3%)  (1.5%)  (1.9%) 

kp 
5
kp 
(4.41%  .81%  0.25%  .09%  2.25%  3.61%)
5
kp 
11.42
 2.284  1.51129
5
d.
The assets are negatively correlated.
e.
risk.
Combining these two negatively correlated asset reduces overall portfolio
6-13 LG 3: Portfolio Analysis
a.
Expected portfolio return:
Alternative 1: 100% Asset F
kp 
16%  17%  18%  19%
 17.5%
4
Alternative 2: 50% Asset F + 50% Asset G
Year
Asset F
(wF x kF)
+
Asset G
(wG x kG)
2001
2002
2003
2004
(16% x .50 = 8.0%)
(17% x .50 = 8.5%)
(18% x .50 = 9.0%)
(19% x .50 = 9.5%)
+
+
+
+
(17% x .50 = 8.5%)
(16% x .50 = 8.0%)
(15% x .50 = 7.5%)
(14% x .50 = 7.0%)
kp 
Portfolio Return
kp
=
=
=
=
16.5%
16.5%
16.5%
16.5%
66
 16.5%
4
Alternative 3: 50% Asset F + 50% Asset H
Asset F
Asset H
Portfolio Return
Year
(wF x kF)
+
(wH x kH)
kp
2001
2002
2003
2004
(16% x .50 = 8.0%)
(17% x .50 = 8.5%)
(18% x .50 = 9.0%)
(19% x .50 = 9.5%)
+
+
+
+
(14% x .50 = 7.0%)
(15% x .50 = 7.5%)
(16% x .50 = 8.0%)
(17% x .50 = 8.5%)
15.0%
16.0%
17.0%
18.0%
kp 
66
 16.5%
4
( ki  k ) 2

i 1 ( n  1)
n
b.
Standard Deviation: kp 
(1)
F 
F 
(16.0%  17.5%)
(-1.5%)
2
 (17.0%  17.5%) 2  (18.0%  17.5%) 2  (19.0%  17.5%) 2
4 1
2
 (.5%) 2  (0.5%) 2  (1.5%) 2
3
F 
(2.25%  .25%  .25%  2.25%)
3
F 
5
 1.667  1.291
3
(2)
FG 
FG 
(16.5%  16.5%)
(0)
2
2


 (16.5%  16.5%) 2  (16.5%  16.5%) 2  (16.5%  16.5%) 2
4 1
 (0) 2  (0) 2  (0) 2
3


FG  0
(3)
FH 
 (15.0%  16.5%)
2
 (16.0%  16.5%) 2  (17.0%  16.5%) 2  (18.0%  16.5%) 2 
4 1
FH 
FH 
(1.5%)
2
 (0.5%) 2  (0.5%) 2  (1.5%) 2
3
(2.25  .25  .25  2.25)
3
FH 
5
 1.667  1.291
3
c.
Coefficient of variation: CV =
CVF 
d.

k  k
1.291
 .0738
17.5%
CVFG 
0
0
16.5%
CVFH 
1.291
 .0782
16.5%
Summary:
kp: Expected Value
of Portfolio
kp
CVp
Alternative 1 (F)
17.5%
1.291
.0738
Alternative 2 (FG)
16.5%
-0.0
Alternative 3 (FH)
16.5%
1.291
.0782
Since the assets have different expected returns, the coefficient of variation
should be used to determine the best portfolio. Alternative 3, with positively
correlated assets, has the highest coefficient of variation and therefore is the
riskiest. Alternative 2 is the best choice; it is perfectly negatively correlated and
therefore has the lowest coefficient of variation.
6-20 LG 5: Betas and Risk Rankings
a.
Stock
Most risky
B
A
Least risky
C
Beta
1.40
0.80
-0.30
b. and c.
Increase in
Expected Impact Decrease in
Impact on
Asset Beta Market Return
Return
A 0.80
.12
B 1.40
.12
C - 0.30
.12
on Asset Return Market Return
.096
.168
-.036
-.05
-.05
-.05
Asset
-.04
-.07
.015
d.
In a declining market, an investor would choose the defensive stock, Stock
C. While the market declines, the return on C increases.
e.
In a rising market, an investor would choose Stock B, the aggressive
stock. As the market rises one point, Stock B rises 1.40 points.
n
6-21 LG 5: Portfolio Betas: bp
=
w  b
j
j
j1
a.
b.
Asset
Beta
wA
A
B
C
D
E
1.30
0.70
1.25
1.10
.90
.10
.30
.10
.10
.40
wA x bA
.130
.210
.125
.110
.360
bA = .935
wB
.30
.10
.20
.20
.20
wB x bB
.39
.07
.25
.22
.18
bB = 1.11
Portfolio A is slightly less risky than the market (average risk), while
Portfolio B is more risky than the market. Portfolio B's return will move
more than Portfolio A’s for a given increase or decrease in market risk.
Portfolio B is the more risky.
6-22 LG 6: Capital Asset Pricing Model: kj = RF + [bj x (km - RF)]
Case
kj
=
A
B
C
D
E
8.9%
12.5%
8.4%
15.0%
8.4%
=
=
=
=
=
RF + [bj x (km - RF)]
5% + [1.30 x (8% - 5%)]
8% + [0.90 x (13% - 8%)]
9% + [- 0.20 x (12% - 9%)]
10% + [1.00 x (15% - 10%)]
6% + [0.60 x (10% - 6%)]
6-23 LG 5, 6: Beta Coefficients and the Capital Asset Pricing Model
To solve this problem you must take the CAPM and solve for beta. The
resulting model is:
k  RF
Beta 
km  RF
a.
Beta 
10%  5% 5%

 .4545
16%  5% 11%
b.
Beta 
15%  5% 10%

 .9091
16%  5% 11%
c.
Beta 
18%  5% 13%

 1.1818
16%  5% 11%
d.
Beta 
20%  5% 15%

 1.3636
16%  5% 11%
e.
If Katherine is willing to take a maximum of average risk then she will only
be able to have an expected return of 16%. k = 5% + 1.0(16% - 5%) =
16%.