1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
A B C D
Production Rate Profit
0$0
200 $9,500
500 $22,500
800 $34,000
1,000 $40,000
$0
$5,000
$10,000
$15,000
$20,000
$25,000
$30,000
$35,000
$40,000
0 200 400 600 800 1,000
Profit
Production Rate
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
A B C D E F
Production Rate Profit
0$0
200 $9,500
500 $22,500
800 $34,000
1,000 $40,000
y = -0.0103x2+ 50.504x – 82.585
$0
$5,000
$10,000
$15,000
$20,000
$25,000
$30,000
$35,000
$40,000
0 200 400 600 800 1,000
Profit
Production Rate
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D
Level of Activity Profit
0$0
1$9
2$16
3$21
4$24
5$25
$0
$5
$10
$15
$20
$25
0 1 2 3 4 5
Profit
Level of Activity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D
Level of Activity Profit
0$0
1$6
2$14
3$24
4$36
5$50
$0
$10
$20
$30
$40
$50
0 1 2 3 4 5
Profit
Level of Activity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D
Level of Activity Profit
0$0
1$5
2$6
3$3
4$4
5$7
$0
$1
$2
$3
$4
$5
$6
$7
0 1 2 3 4 5
Profit
Level of Activity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D
Level of Activity Profit
0$0
1$9
2$16
3$21
4$24
5$25
y = -x2+ 10x
$0
$5
$10
$15
$20
$25
0 1 2 3 4 5
Profit
Level of Activity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D
Level of Activity Profit
0$0
1$6
2$14
3$24
4$36
5$50
y = x2+ 5x
$0
$10
$20
$30
$40
$50
0 1 2 3 4 5
Profit
Level of Activity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D
Level of Activity Profit
0$0
1$5
2$6
3$3
4$4
5$7
y = -0.1786x2+ 1.7214x + 1.5
$0
$1
$2
$3
$4
$5
$6
$7
0 1 2 3 4 5
Profit
Level of Activity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
A B C D E
Sales (thousands) Profit (millions)
0 0
100 15
200 18
300 13
400 4
500 1
600 6
700 30
800 70
0
10
20
30
40
50
60
70
0 100 200 300 400 500 600 700 800
Profit
($millions)
Sales (thousands)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
A B C D
Sales (thousands) Profit (millions)
0 0
100 15
200 18
300 13
400 4
500 1
600 6
700 30
800 70
y = 0.000191x2– 0.105x + 16.0
0
10
20
30
40
50
60
70
0 100 200 300 400 500 600 700 800
Profit
($millions)
Sales (thousands)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
A B C D
Sales (thousands) Profit (millions)
0 0
100 15
200 18
300 13
400 4
500 1
600 6
700 30
800 70
y = 0.00000096x3– 0.000959x2+ 0.242x – 0.101
0
10
20
30
40
50
60
70
0 100 200 300 400 500 600 700 800
Profit
($millions)
Sales (thousands)
1
2
3
4
5
6
7
8
9
10
11
12
15
16
17
24
26
27
29
30
31
32
33
34
Expected Return 19.3% >= 18.0% Total F14
Risk (Stand. Dev.) 19.1%
A B C D E F G H I J K
Portfolio Selection Problem (Nonlinear Programming)
Stock 1 Stock 2 Stock 3 Range Name Cells
Expected Return 21% 30% 8% Covar12 D9
Covar13 E9
Risk (Stand. Dev.) 25% 45% 5% Covar23 E10
Covariance C9:E11
Joint Risk (Covar.) Stock 1 Stock 2 Stock 3 ExpectedReturn C19
Stock 1 0.040 -0.005 MinExpectedReturn E19
Stock 2 -0.010 OneHundredPercent H14
Stock 3 Portfolio C14:E14
37
38
39
40
41
A B C D E F G H I J K
22% 56.2% 30.4% 13.4% 22.5% 22.0%
24% 64.2% 34.8% 1.0% 26.1% 24.0%
26% 44.4% 55.6% 0.0% 30.8% 26.0%
28% 22.2% 77.8% 0.0% 37.3% 28.0%
30% 0.0% 100.0% 0.0% 45.0% 30.0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
A B C D E F G H I J K
Portfolio Selection Problem (Nonlinear Programming)
Stock 1 Stock 2 Stock 3 Range Name Cells
Expected Return 21% 30% 8% Covar12 D9
Covar13 E9
Risk (Stand. Dev.) 25% 45% 5% Covar23 E10
Covariance C9:E11
Joint Risk (Covar.) Stock 1 Stock 2 Stock 3 ExpectedReturn C19
Stock 1 0.040 -0.005 MinExpectedReturn E19
Stock 2 -0.010 OneHundredPercent H14
Stock 3 Portfolio C14:E14
SD1 C6
Stock 1 Stock 2 Stock 3
Total SD2 D6
Portfolio 87.8% 12.2% 0.0% 100% =100% SD3 E6
<= StandDev C23
0% Stock1 C14
Stock2 D14
Minimum Stock3 E14
Expected StockExpectedReturn C4:E4
Portfolio Return StockStandDev C6:E6
Expected Return 22.1% >= 18.0% Total F14
Variance C21
Risk (Variance) 0.0598
Risk (Stand. Dev.) 24.4%
Risk Expected
Min Return Stock 1 Stock 2 Stock 3
(St. Dev.)
Return
87.8% 12.2% 0.0% 24.4% 22.1%
8% 7.1% 3.7% 89.1% 3.9% 9.7%
10% 8.1% 4.3% 87.6% 3.9% 10.0%
12% 16.2% 8.6% 75.2% 5.6% 12.0%
14% 24.2% 13.0% 62.8% 8.6% 14.0%
16% 32.2% 17.3% 50.5% 12.0% 16.0%
18% 40.2% 21.7% 38.1% 15.4% 18.0%
20% 48.2% 26.1% 25.7% 18.9% 20.0%
37
38
39
40
41
A B C D E F G H I J K
22% 56.2% 30.4% 13.4% 22.5% 22.0%
24% 64.2% 34.8% 1.0% 26.1% 24.0%
26% 44.4% 55.6% 0.0% 30.8% 26.0%
28% 22.2% 77.8% 0.0% 37.3% 28.0%
30% 0.0% 100.0% 0.0% 45.0% 30.0%
1
2
3
4
5
6
7
8
9
10
11
12
15
16
17
24
26
27
29
30
31
32
33
34
Expected Return 22.1% >= 18.0% Total F14
Risk (Stand. Dev.) 24.4%
A B C D E F G H I J K
Portfolio Selection Problem (Nonlinear Programming)
Stock 1 Stock 2 Stock 3 Range Name Cells
Expected Return 21% 30% 8% Covar12 D9
Covar13 E9
Risk (Stand. Dev.) 25% 45% 5% Covar23 E10
Covariance C9:E11
Joint Risk (Covar.) Stock 1 Stock 2 Stock 3 ExpectedReturn C19
Stock 1 0.040 -0.005 MinExpectedReturn E19
Stock 2 -0.010 OneHundredPercent H14
Stock 3 Portfolio C14:E14
37
38
39
40
A B C D E F G H I J K
35% 47.8% 17.2% 35.0% 15.6% 18.0%
40% 40.2% 21.7% 38.1% 15.4% 18.0%
45% 40.2% 21.7% 38.1% 15.4% 18.0%
50% 40.2% 21.7% 38.1% 15.4% 18.0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A B C D E F
Stock 1 Stock 2
Expected Profit (on $50,000) $12,500 $20,000
Risk (Stand. Dev. for $50,000) $5,000 $30,000
Stock 1 Stock 2 Total
Portfolio (% of $50,000) 93.3% 6.7% 100% <= 100%
$46,667 $3,333
Minimum
Expected
Portfolio Profit
Expected Profit $13,000 >= $13,000
Risk (Variance) $25,777,778
Risk (Stand. Dev.) $5,077
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A B C D E F
Stock 1 Stock 2
Expected Profit (on $50,000) $12,500 $20,000
Risk (Stand. Dev. for $50,000) $5,000 $30,000
Stock 1 Stock 2 Total
Portfolio (% of $50,000) 66.7% 33.3% 100% <= 100%
$33,333 $16,667
Minimum
Expected
Portfolio Profit
Expected Profit $15,000 >= $15,000
Risk (Variance) $111,111,111
Risk (Stand. Dev.) $10,541
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A B C D E F
Stock 1 Stock 2
Expected Profit (on $50,000) $12,500 $20,000
Risk (Stand. Dev. for $50,000) $5,000 $30,000
Stock 1 Stock 2 Total
Portfolio (% of $50,000) 40.0% 60.0% 100% <= 100%
$20,000 $30,000
Minimum
Expected
Portfolio Profit
Expected Profit $17,000 >= $17,000
Risk (Variance) $328,000,000
Risk (Stand. Dev.) $18,111