Solution to Portfolio Optimization: Parametric Approach1
Consider a portfolio composed of three stocks. Let the random variable si be the
annual return on stock i, with expected value μi, variance  i2 , and standard
deviation  i .
The correlation between si and sj, for any pair of stocks i and j, is ρij.
The covariance between si and sj, for any pair of stocks i and j, is σiσjρij.
The means are: μ1 = 0.14, μ2 = 0.11, μ3 = 0.10.
The variances are:  12 = 0.20,  22 = 0.08,  32 = 0.18.
The correlations are: ρ12. = 0.8, ρ13. = 0.7, and ρ23. = 0.9.
(a) Determine the minimum-variance portfolio that attains an expected
annual return of at least 0.12, with no shorting of stocks allowed.
Managerial Formulation
Decision Variables
We need to allocate 100% of the available funds to the three stocks.
Objective
We want to minimize risk.
Constraints
The expected return on the portfolio must be at least 12% (or 0.12).
All of the money must be invested.
No shorting is allowed.
Based on 7-49 (p. 393) and 9-24 (p. 471) in Practical Management Science (2nd ed., Winston and
Albright, 2001 Duxbury Press). Solution by David Juran, 2002.
1
Mathematical Formulation
Decision Variables
Define xi to be the proportion of the portfolio allocated to stock i.
The decision variables are three proportions: x1, x2, and x3.
The expected return on the portfolio is:
  x1 1  x2  2  x3  3
The variance of the return on the portfolio is:
 2   12 x 12   22 x 22   32 x 32  2 x 1 x 2 COV1 , 2  2 x 2 x 3COV2 , 3  2 x 1 x 3COV1 , 3
The standard deviation of the return on the portfolio is:
   2   12 x 12   22 x 22   32 x 32  2 x 1 x 2 COV1 , 2  2 x 2 x 3 COV2 , 3  2 x 1 x 3 COV1 , 3
Objective
Minimize Z =    12 x 12   22 x 22   32 x 32  2 x 1 x 2 COV1 , 2  2 x 2 x 3COV2 , 3  2 x 1 x 3COV1 , 3
Constraints
  x 1  1  x 2  2  x 3  3  0.12
3
x
i 1
i
(1)
 1 .0
(2)
For all i, x i  0
B60.2350
(3)
2
Prof. Juran
Solution Methodology
Here’s the spreadsheet model:
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Mean return
Variance of return
StDev of return
B
Stock 1
0.140
0.200
0.447
C
Stock 2
0.110
0.080
0.283
D
Stock 3
0.100
0.180
0.424
E
F
G
H
I
J
=HLOOKUP($G8,$B$1:$D$4,4)*B8*HLOOKUP(H$7,$B$1:$D$4,4)
Correlations
Stock 1
Stock 2
Stock 3
Covariances
Stock 1
1.00
0.80
0.70
Stock 2
0.80
1.00
0.90
Stock 3
0.70
0.90
1.00
Stock 1
0.2000
0.1012
0.1328
Stock 2
0.1012
0.0800
0.1080
Stock 3
0.1328
0.1080
0.1800
=SUM(B14:D14)
Investment decision
Fractions to invest
Stock 1
Stock 2
Stock 3
Stock 1
1.000
Stock 2
0.000
Stock 3
0.000
Total
1
=
Required
1
=SUMPRODUCT(B2:D2,B14:D14)
Expected portfolio return
Actual
0.140
>=
Required
0.120
{=MMULT(B14:D14,MMULT(H8:J10,TRANSPOSE(B14:D14)))}
Portfolio variance
Portfolio stdev
0.200
0.447
=SQRT(B20)
Notes:
The decision variables are in B14:D14.
Cell D18 keeps track of constraint (1).
Cell B14 keeps track of constraint (2).
We can comply with constraint (3) by selecting “assume nonnegative” in the
Solver Options box.
The range H8:J10 uses the HLOOKUP Excel function to calculate the covariances.
Cell B20 uses two of Excel’s matrix functions, MMULT and TRANSPOSE, to
calculate the portfolio variance. If you use these functions, you will want to learn
more about working with “arrays” in Excel, an advanced topic beyond the scope
of this course. In this case, you need to know not to type in the “curly brackets”;
instead, type in the rest of the function and then Ctrl+Shift+Enter. The curly
brackets will appear automatically. You will get a #VALUE! Error message if you
do this wrong.
B60.2350
3
Prof. Juran
Here is the optimized spreadsheet:
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Mean return
Variance of return
StDev of return
B
Stock 1
0.140
0.200
0.447
C
Stock 2
0.110
0.080
0.283
D
Stock 3
0.100
0.180
0.424
Stock 1
1.00
0.80
0.70
Stock 2
0.80
1.00
0.90
Stock 3
0.70
0.90
1.00
Stock 1
0.333
Stock 2
0.667
Stock 3
0.000
Actual
0.120
>=
Required
0.120
E
F
Correlations
Stock 1
Stock 2
Stock 3
G
H
I
J
Stock 1
0.2000
0.1012
0.1328
Stock 2
0.1012
0.0800
0.1080
Stock 3
0.1328
0.1080
0.1800
Covariances
Stock 1
Stock 2
Stock 3
Investment decision
Fractions to invest
Total
1
=
Required
1
Expected portfolio return
Portfolio variance
Portfolio stdev
0.103
0.321
Conclusions
The least risky way to invest these three stocks while having an expected return
of at least 12% is to invest 33.3% in Stock 1 and 66.7% in Stock 2. This portfolio
will have an expected return of 12% and a standard deviation of return of 32.1%.
B60.2350
4
Prof. Juran
(b) Construct a trade-off curve between the chosen portfolio's expected return
and risk (standard deviation), with risk on the horizontal axis. In finance
terminology this trade-off curve is called the efficient frontier.
We need to solve the model repeatedly for different levels of required return; a
perfect situation for SolverTable.
B60.2350
5
Prof. Juran
Here’s the SolverTable output:
A
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
B60.2350
0.100
0.101
0.102
0.103
0.104
0.105
0.106
0.107
0.108
0.109
0.110
0.111
0.112
0.113
0.114
0.115
0.116
0.117
0.118
0.119
0.120
0.121
0.122
0.123
0.124
0.125
0.126
0.127
0.128
0.129
0.130
0.131
0.132
0.133
0.134
0.135
0.136
0.137
0.138
0.139
0.140
0.141
0.142
0.143
0.144
0.145
0.146
0.147
0.148
0.149
0.150
B
Stock 1
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.033
0.067
0.100
0.133
0.167
0.200
0.233
0.267
0.300
0.333
0.367
0.400
0.433
0.467
0.500
0.533
0.567
0.600
0.633
0.667
0.700
0.733
0.767
0.800
0.833
0.867
0.900
0.933
0.967
1.000
Not feasible
Not feasible
Not feasible
Not feasible
Not feasible
Not feasible
Not feasible
Not feasible
Not feasible
Not feasible
6
C
Stock 2
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.967
0.933
0.900
0.867
0.833
0.800
0.767
0.733
0.700
0.667
0.633
0.600
0.567
0.533
0.500
0.467
0.433
0.400
0.367
0.333
0.300
0.267
0.233
0.200
0.167
0.133
0.100
0.067
0.033
0.000
D
Stock 3
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
E
Return
0.110
0.110
0.110
0.110
0.110
0.110
0.110
0.110
0.110
0.110
0.110
0.111
0.112
0.113
0.114
0.115
0.116
0.117
0.118
0.119
0.120
0.121
0.122
0.123
0.124
0.125
0.126
0.127
0.128
0.129
0.130
0.131
0.132
0.133
0.134
0.135
0.136
0.137
0.138
0.139
0.140
F
Risk
0.283
0.283
0.283
0.283
0.283
0.283
0.283
0.283
0.283
0.283
0.283
0.285
0.288
0.292
0.295
0.299
0.303
0.307
0.311
0.316
0.321
0.326
0.331
0.336
0.342
0.347
0.353
0.359
0.365
0.371
0.378
0.384
0.391
0.398
0.404
0.411
0.418
0.425
0.433
0.440
0.447
Prof. Juran
Here’s a picture of the efficient frontier, as well as the three individual stocks.
Note that there is no way to get a portfolio that is less risky than Stock 2, nor is it
possible to get a portfolio with a higher expected return than Stock 1.
Efficient Frontier
Expected Return
16%
15%
14%
Stock 1
13%
12%
11%
Stock 2
10%
Stock 3
9%
8%
20%
25%
30%
35%
40%
45%
50%
Risk
(c) Determine the minimum-variance portfolio that attains an expected
annual return of at least 0.12, with shorting of stocks allowed.
All we need to do here is remove the nonnegativity constraint and re-run
SolverTable.
We are able to achieve a lower risk than Stock 2 (or a higher return than Stock 1)
by shorting in various combinations (see the SolverTable output on the next
page).
Efficient Frontier
Expected Return
16%
15%
14%
13%
Stock 1
12%
11%
Stock 2
10%
9%
8%
20%
Stock 3
25%
30%
35%
40%
45%
50%
Risk
B60.2350
7
Prof. Juran
A
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
B60.2350
0.100
0.101
0.102
0.103
0.104
0.105
0.106
0.107
0.108
0.109
0.110
0.111
0.112
0.113
0.114
0.115
0.116
0.117
0.118
0.119
0.120
0.121
0.122
0.123
0.124
0.125
0.126
0.127
0.128
0.129
0.130
0.131
0.132
0.133
0.134
0.135
0.136
0.137
0.138
0.139
0.140
0.141
0.142
0.143
0.144
0.145
0.146
0.147
0.148
0.149
0.150
0.151
0.152
0.153
0.154
0.155
0.156
0.157
0.158
0.159
B
Stock 1
-0.244
-0.244
-0.244
-0.244
-0.244
-0.244
-0.244
-0.244
-0.244
-0.240
-0.212
-0.185
-0.158
-0.130
-0.103
-0.075
-0.048
-0.021
0.007
0.034
0.061
0.089
0.116
0.144
0.171
0.198
0.226
0.253
0.280
0.308
0.335
0.363
0.390
0.417
0.445
0.472
0.499
0.527
0.554
0.582
0.609
0.636
0.664
0.691
0.719
0.746
0.773
0.801
0.828
0.855
0.883
0.910
0.938
0.965
0.992
1.020
1.047
1.074
1.102
1.129
8
C
Stock 2
1.861
1.861
1.861
1.861
1.861
1.861
1.861
1.861
1.861
1.859
1.849
1.840
1.830
1.821
1.811
1.802
1.792
1.783
1.773
1.764
1.754
1.745
1.735
1.726
1.716
1.707
1.697
1.688
1.678
1.669
1.659
1.650
1.640
1.631
1.621
1.612
1.602
1.592
1.583
1.573
1.564
1.554
1.545
1.535
1.526
1.516
1.507
1.497
1.488
1.478
1.469
1.459
1.450
1.440
1.431
1.421
1.412
1.402
1.393
1.383
D
Stock 3
-0.616
-0.616
-0.616
-0.616
-0.616
-0.616
-0.616
-0.616
-0.616
-0.619
-0.637
-0.655
-0.673
-0.691
-0.709
-0.726
-0.744
-0.762
-0.780
-0.798
-0.816
-0.834
-0.851
-0.869
-0.887
-0.905
-0.923
-0.941
-0.959
-0.976
-0.994
-1.012
-1.030
-1.048
-1.066
-1.084
-1.102
-1.119
-1.137
-1.155
-1.173
-1.191
-1.209
-1.227
-1.244
-1.262
-1.280
-1.298
-1.316
-1.334
-1.352
-1.369
-1.387
-1.405
-1.423
-1.441
-1.459
-1.477
-1.494
-1.512
E
Return
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.110
0.111
0.112
0.113
0.114
0.115
0.116
0.117
0.118
0.119
0.120
0.121
0.122
0.123
0.124
0.125
0.126
0.127
0.128
0.129
0.130
0.131
0.132
0.133
0.134
0.135
0.136
0.137
0.138
0.139
0.140
0.141
0.142
0.143
0.144
0.145
0.146
0.147
0.148
0.149
0.150
0.151
0.152
0.153
0.154
0.155
0.156
0.157
0.158
0.159
F
Risk
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.241
0.241
0.242
0.244
0.245
0.247
0.249
0.252
0.254
0.257
0.260
0.264
0.267
0.271
0.275
0.279
0.283
0.288
0.292
0.297
0.302
0.307
0.313
0.318
0.323
0.329
0.335
0.341
0.347
0.353
0.359
0.365
0.371
0.377
0.384
0.390
0.397
0.404
0.410
0.417
0.424
0.431
0.438
0.445
0.452
0.459
0.466
0.473
0.480
Prof. Juran