Efficient portfolios and Efficient frontier

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Data

Date Apple WMT MSFT Apple WMT MSFT Asset A Asset B Asset C
1/1/00 10.97 44.68 20.69 Apple WMT MSFT
2/1/00 11.63 49.42 19.85 6.02% 10.61% -4.06%
2/29/00 13.15 49.63 18.66 13.07% 0.42% -5.99%
3/31/00 12.54 47.4 19.56 -4.64% -4.49% 4.82% Table of Covariance
5/2/00 13.64 46.45 19.63 8.77% -2.00% 0.36% Asset A Asset B Asset C Things in dark grey are input parameters
5/31/00 15.82 43.76 21.37 15.98% -5.79% 8.86% Asset A 1.15% -0.02% 0.30% Things in light grey are computed intermediate things
6/30/00 15.73 44.18 21.32 -0.57% 0.96% -0.23% Asset B -0.02% 0.21% 0.05% Things in yellow are labels
8/1/00 16.77 44 20.49 6.61% -0.41% -3.89% Asset C 0.30% 0.05% 0.46% Things in red are "Solver" inputs or outputs
8/31/00 18.84 44.45 20.75 12.34% 1.02% 1.27%
9/30/00 25.48 45.05 20.99 35.24% 1.35% 1.16% Table of expected returns and risk
10/31/00 32.6 43.49 22.43 27.94% -3.46% 6.86% E(return) Standard Deviation
11/30/00 31.32 44.24 22.35 -3.93% 1.72% -0.36% Asset A 3.94% 10.74%
1/2/01 37.39 43.89 21.99 19.38% -0.79% -1.61% Asset B 0.56% 4.56%
1/31/01 43.63 43.22 21.11 16.69% -1.53% -4.00% Asset C 0.70% 6.80%
2/28/01 40.53 42.09 20.28 -7.11% -2.61% -3.93%
3/31/01 35.07 39.6 21.23 -13.47% -5.92% 4.68%
5/1/01 38.67 39.8 21.72 10.27% 0.51% 2.31%
5/31/01 35.8 40.62 20.91 -7.42% 2.06% -3.73%
6/30/01 41.48 41.59 21.56 15.87% 2.39% 3.11%
7/31/01 45.6 38.01 23.12 9.93% -8.61% 7.24%
8/31/01 52.14 37.04 21.72 14.34% -2.55% -6.06%
10/2/01 56.01 39.99 21.7 7.42% 7.96% -0.09%
10/31/01 65.96 41.05 23.44 17.76% 2.65% 8.02%
11/30/01 69.92 39.68 22.14 6.00% -3.34% -5.55%
1/2/02 73.44 39.1 23.84 5.03% -1.46% 7.68%
1/31/02 66.61 38.46 22.83 -9.30% -1.64% -4.24%
2/28/02 61 40.2 23.12 -8.42% 4.52% 1.27%
4/2/02 68.46 38.32 20.52 12.23% -4.68% -11.25%
4/30/02 58.13 41.38 19.32 -15.09% 7.99% -5.85%
5/31/02 55.7 41.14 19.87 -4.18% -0.58% 2.85%
7/2/02 66.09 38 20.52 18.65% -7.63% 3.27%
7/31/02 65.99 38.34 22 -0.15% 0.89% 7.21%
8/31/02 74.87 42.28 23.41 13.46% 10.28% 6.41%
10/1/02 78.85 42.25 24.58 5.32% -0.07% 5.00%
10/31/02 89.14 39.52 25.22 13.05% -6.46% 2.60%
11/30/02 82.51 39.73 25.65 -7.44% 0.53% 1.70%
1/2/03 83.38 41.03 26.51 1.05% 3.27% 3.35%
1/31/03 82.29 41.57 24.28 -1.31% 1.32% -8.41%
2/28/03 90.36 40.59 24.02 9.81% -2.36% -1.07%
4/1/03 97.06 41.43 25.81 7.41% 2.07% 7.45%
4/30/03 117.86 41.34 26.54 21.43% -0.22% 2.83%
5/31/03 118.69 41.79 25.49 0.70% 1.09% -3.96%
7/1/03 128.14 39.91 25.07 7.96% -4.50% -1.65%
7/31/03 134.68 38.09 24.93 5.10% -4.56% -0.56%
9/3/03 149.26 38.1 25.57 10.83% 0.03% 2.57%
9/30/03 184.74 39.47 31.95 23.77% 3.60% 24.95%
10/31/03 177.22 41.81 29.26 -4.07% 5.93% -8.42%
12/2/03 192.64 41.68 31 8.70% -0.31% 5.95%
1/1/04 131.64 44.49 28.39 -31.67% 6.74% -8.42%
1/31/04 121.59 43.48 23.78 -7.63% -2.27% -16.24%
3/2/04 139.56 46.41 24.81 14.78% 6.74% 4.33%
3/31/04 169.18 51.08 24.93 21.22% 10.06% 0.48%
4/30/04 183.57 51.09 24.85 8.51% 0.02% -0.32%
6/1/04 162.84 49.72 24.13 -11.29% -2.68% -2.90%
6/30/04 154.59 51.86 22.56 -5.07% 4.30% -6.51%
7/31/04 164.88 52.47 24.04 6.66% 1.18% 6.56%
9/1/04 110.54 53.2 23.51 -32.96% 1.39% -2.20%
9/30/04 104.64 49.58 19.67 -5.34% -6.80% -16.33%
11/2/04 90.13 49.64 17.93 -13.87% 0.12% -8.85%
11/30/04 83.01 50.02 17.24 -7.90% 0.77% -3.85%
1/1/05 87.66 42.04 15.16 5.60% -15.95% -12.06%
2/1/05 86.86 43.93 14.42 -0.91% 4.50% -4.88%
3/1/05 102.23 46.74 16.4 17.70% 6.40% 13.73%
3/31/05 122.38 45.22 18.09 19.71% -3.25% 10.30%
4/30/05 132.08 44.87 18.77 7.93% -0.77% 3.76%
5/31/05 138.52 43.7 21.36 4.88% -2.61% 13.80%
6/30/05 158.91 44.99 21.13 14.72% 2.95% -1.08%
8/2/05 163.59 46.14 22.27 2.95% 2.56% 5.40%
8/31/05 180.26 44.52 23.24 10.19% -3.51% 4.36%
9/30/05 183.33 45.06 25.06 1.70% 1.21% 7.83%
11/1/05 194.42 49.48 26.69 6.05% 9.81% 6.50%
11/30/05 204.95 48.72 27.66 5.42% -1.54% 3.63%
1/3/06 186.79 48.71 25.58 -8.86% -0.02% -7.52%
1/31/06 199 49.29 26.14 6.54% 1.19% 2.19%
2/28/06 228.55 50.97 26.71 14.85% 3.41% 2.18%
3/31/06 253.92 49.17 27.85 11.10% -3.53% 4.27%
5/2/06 249.83 46.62 23.63 -1.61% -5.19% -15.15%
5/31/06 244.63 44.32 21.08 -2.08% -4.93% -10.79%
6/30/06 250.19 47.2 23.64 2.27% 6.50% 12.14%
8/1/06 236.43 46.5 21.61 -5.50% -1.48% -8.59%
8/31/06 275.96 49.64 22.55 16.72% 6.75% 4.35%
9/30/06 292.72 50.24 24.56 6.07% 1.21% 8.91%
10/31/06 302.61 50.16 23.4 3.38% -0.16% -4.72%
11/30/06 313.71 50.29 25.86 3.67% 0.26% 10.51%
1/2/07 330.01 52.29 25.69 5.20% 3.98% -0.66%
1/31/07 343.52 48.47 24.77 4.09% -7.31% -3.58%
2/28/07 338.94 48.88 23.66 -1.33% 0.85% -4.48%
3/31/07 340.52 51.63 24.16 0.47% 5.63% 2.11%
5/1/07 338.28 52.2 23.46 -0.66% 1.10% -2.90%
5/31/07 326.46 50.23 24.39 -3.49% -3.77% 3.96%
6/30/07 379.76 49.83 25.7 16.33% -0.80% 5.37%
7/31/07 374.27 50.64 25.11 -1.45% 1.63% -2.30%
8/31/07 370.85 49.42 23.5 -0.91% -2.41% -6.41%
10/2/07 393.67 54 25.14 6.15% 9.27% 6.98%
10/31/07 371.71 56.08 24.33 -5.58% 3.85% -3.22%
11/30/07 393.88 57.25 24.69 5.96% 2.09% 1.48%
1/2/08 443.95 58.79 28.09 12.71% 2.69% 13.77%
1/31/08 527.55 56.6 30.39 18.83% -3.73% 8.19%
2/29/08 583.09 59.03 30.88 10.53% 4.29% 1.61%
4/1/08 567.95 56.82 30.65 -2.60% -3.74% -0.74%
4/30/08 561.87 63.91 28.13 -1.07% 12.48% -8.22%
5/31/08 567.97 67.7 29.48 1.09% 5.93% 4.80%
7/1/08 594 72.27 28.4 4.58% 6.75% -3.66%
7/31/08 649.76 70.88 29.9 9.39% -1.92% 5.28%
9/3/08 651.58 72.05 28.87 0.28% 1.65% -3.44%
9/30/08 581.47 73.24 27.68 -10.76% 1.65% -4.12%
10/31/08 574.27 70.31 26.03 -1.24% -4.00% -5.96%
12/2/08 522.16 66.98 26.12 -9.07% -4.74% 0.35%
1/1/09 446.92 68.67 26.85 -14.41% 2.52% 2.79%
1/31/09 435.62 69.49 27.41 -2.53% 1.19% 2.09%
2/28/09 436.86 73.93 28.21 0.28% 6.39% 2.92%
3/31/09 436.98 76.79 32.64 0.03% 3.87% 15.70%
4/30/09 446.78 74.39 34.66 2.24% -3.13% 6.19%
6/2/09 393.93 74.04 34.3 -11.83% -0.47% -1.04%
6/30/09 449.56 77.47 31.62 14.12% 4.63% -7.81%
7/31/09 487.22 72.98 33.4 8.38% -5.80% 5.63%
9/2/09 476.75 73.96 33.28 -2.15% 1.34% -0.36%
9/30/09 525.96 76.08 35.73 10.32% 2.87% 7.36%

MVP

Table of expected returns and risk
E(return) Standard Deviation
Asset A 3.94% 10.74%
Asset B 0.56% 4.56%
Asset C 0.70% 6.80%
Table of covariances
Asset A Asset B Asset C
Asset A 1.15% -0.02% 0.30%
Asset B -0.02% 0.21% 0.05%
Asset C 0.30% 0.05% 0.46%
Proporition of wealth invested in
Asset A 0.09 Wa
Asset B 0.70 Wb
Asset C 0.21 Wc
Total 1.00 (this is a CONSTRAINT for "Solver", this cell B23 must be forced to equal one)
(it is the constraint that the total proportions of our wealth must equal one)
This bit is used to find the risk of the portfolio
Asset A Asset B Asset C
Proportion 8.94% 70.40% 20.66% These numbers are a table of, for instance
Asset A 8.94% 0.0092% -0.0010% 0.0056% proportion of A * proportion of B *
Asset B 70.40% -0.0010% 0.1030% 0.0066% covariance of A and B
Asset C 20.66% 0.0056% 0.0066% 0.0197%
Expected return on the portfolio of all 3 assets
0.89% (if you are minimizing risk for a given return you should tell "Solver" that
this cell, B35, is fixed--at whatever level of return you want. If you are maximizing
return for a fixed risk, you should tell "Solver" to maximize this cell
Risk associated with the portfolio of all 3 assets
3.93% (if you are minimizing risk for a given return you should tell "Solver" to minimize
this cell, B40. If you are maximizing return for a fixed risk you should
tell "Solver" that this cell is fixed.)
Now use "solver" to either minimize the risk for a given return
maximize return for a given risk (ie, maximize B30 for a given
value of B35).
In both cases B18 should be constrained to equal 1, this is the
condition that our "total wealth" is invested in the assets.

EF

Return Risk Wa Wb Wc Total weight Efficient frontier for risky portfolio
Table of expected returns and risk 0.9285% 3.9301% 10.1323% 70.1055% 19.7622% 100.0000%
E(return) Standard Deviation 1.1368% 4.0001% 16.4903% 68.5472% 14.9625% 100.0000%
Asset A 3.94% 10.74% 1.3766% 4.2000% 23.8082% 66.7536% 9.4382% 100.0000%
Asset B 0.56% 4.56% 1.5392% 4.4001% 28.7689% 65.5378% 5.6933% 100.0000%
Asset C 0.70% 6.80% 1.6740% 4.6001% 32.8829% 64.5294% 2.5876% 100.0000%
1.7935% 4.8000% 36.5238% 63.4762% 0.0000% 100.0000%
Table of covariances 1.9013% 4.9999% 39.7109% 60.2891% 0.0000% 100.0000%
Asset A Asset B Asset C 1.9998% 5.2000% 42.6267% 57.3733% 0.0000% 100.0000%
Asset A 1.15% -0.02% 0.30% 2.0919% 5.4000% 45.3509% 54.6491% 0.0000% 100.0000%
Asset B -0.02% 0.21% 0.05% 2.1792% 5.6000% 47.9326% 52.0674% 0.0000% 100.0000%
Asset C 0.30% 0.05% 0.46% 2.2627% 5.8002% 50.4071% 49.5929% 0.0000% 100.0000%
2.3433% 6.0000% 52.7876% 47.2124% 0.0000% 100.0000%
Proporition of wealth invested in 2.4215% 6.2001% 55.1003% 44.8997% 0.0000% 100.0000%
Asset A 1.00 Wa 2.4976% 6.4001% 57.3519% 42.6481% 0.0000% 100.0000%
Asset B 0.00 Wb 2.5720% 6.6003% 59.5561% 40.4439% 0.0000% 100.0000%
Asset C 0.00 Wc 2.6449% 6.8000% 61.7112% 38.2888% 0.0000% 100.0000%
Total 1.00 (this is a CONSTRAINT for "Solver", this cell B23 must be forced to equal one) 2.7166% 7.0000% 63.8319% 36.1681% 0.0000% 100.0000%
(it is the constraint that the total proportions of our wealth must equal one) 2.8221% 7.3000% 66.9534% 33.0466% 0.0000% 100.0000%
This bit is used to find the risk of the portfolio 2.9256% 7.6003% 70.0169% 29.9831% 0.0000% 100.0000%
Asset A Asset B Asset C 3.0273% 7.9000% 73.0236% 26.9764% 0.0000% 100.0000%
Proportion 100.00% 0.00% 0.00% These numbers are a table of, for instance 3.1276% 8.2000% 75.9900% 24.0100% 0.0000% 100.0000%
Asset A 100.00% 1.1535% 0.0000% 0.0000% proportion of A * proportion of B * 3.2266% 8.5000% 78.9193% 21.0807% 0.0000% 100.0000%
Asset B 0.00% 0.0000% 0.0000% 0.0000% covariance of A and B 3.3246% 8.8000% 81.8164% 18.1836% 0.0000% 100.0000%
Asset C 0.00% 0.0000% 0.0000% 0.0000% 3.4215% 9.1000% 84.6853% 15.3147% 0.0000% 100.0000%
3.5177% 9.4000% 87.5296% 12.4704% 0.0000% 100.0000%
3.6131% 9.7000% 90.3520% 9.6480% 0.0000% 100.0000%
3.7078% 10.0000% 93.1549% 6.8451% 0.0000% 100.0000%
Expected return on the portfolio of all 3 assets 3.8020% 10.3000% 95.9403% 4.0597% 0.0000% 100.0000%
3.94% (if you are minimizing risk for a given return you should tell "Solver" that 3.8956% 10.6000% 98.7101% 1.2899% 0.0000% 100.0000%
this cell, B35, is fixed--at whatever level of return you want. If you are maximizing 3.9392% 10.7402% 100.0000% 0.0000% 0.0000% 100.0000%
return for a fixed risk, you should tell "Solver" to maximize this cell
Risk associated with the portfolio of all 3 assets
10.74% (if you are minimizing risk for a given return you should tell "Solver" to minimize
this cell, B40. If you are maximizing return for a fixed risk you should
tell "Solver" that this cell is fixed.)
Now use "solver" to either minimize the risk for a given return
maximize return for a given risk (ie, maximize B30 for a given
value of B35).
In both cases B18 should be constrained to equal 1, this is the
condition that our "total wealth" is invested in the assets.

Solve List

MPR

Table of expected returns and risk on the risky assets
E(return) Risk
Asset A 3.94% 10.74%
Asset B 0.56% 4.56%
Asset C 0.70% 6.80%
Riskfree rate 0.33% The return on the risk free asset
Table of covariances for the risky assets
Asset A Asset B Asset C Construct covariance matrix using formulae
Asset A 1.15% -0.02% 0.30% such as covar(A,B)=risk(A)*risk(B)*correl(A,B)
Asset B -0.02% 0.21% 0.05%
Asset C 0.30% 0.05% 0.46%
Proporition of wealth invested in purely risky assets
Asset A 0.7041 Wa Normally these values are set by "Solver" but it may be necessary to
Asset B 0.2959 Wb reset them to "sensible" values and then re-run "Solver" if "Solver"
Asset C 0.0000 Wc fails to find a solution (eg, B23=1/3, B24=1/3, B25=1/3 )
Total 1.0000 (this is a CONSTRAINT for "Solver", this cell B26 must be forced to equal one)
(it is the constraint that the sum of the proportions of our wealth must equal one)
This bit is used to find the risk of the portfolio of purely risky assets
Asset A Asset B Asset C
Proportion 70.41% 29.59% 0.00% These numbers are a table of, for instance
Asset A 70.41% 0.5719% -0.0032% 0.0000% proportion of A * proportion of B *
Asset B 29.59% -0.0032% 0.0182% 0.0000% covariance of A and B
Asset C 0.00% 0.0000% 0.0000% 0.0000%
Expected return on the portfolio of all 3 risky assets
2.94%
Risk associated with the portfolio of all 3 risky assets
7.64%
Market Price of Risk ( = Slope of the Capital Market Line)
0.3410881206
To find the Market Price of Risk (and hence the Capital Market Line)
use "solver" to maximize cell B39 subject to the constraint that
cell B21 is equal to one.
You access "solver" from the "Tools" menu.

Results

Efficient frontier for risky portfolio Capital market line market price of risk (Slope of CML) 0.3410881206
risk return risk return risk free rate 0.33%
3.93% 0.93% 0.00% 0.33%
4.00% 1.14% 1.00% 0.67%
4.20% 1.38% 2.00% 1.02%
4.40% 1.54% 3.00% 1.36%
4.60% 1.67% 4.00% 1.70%
4.80% 1.79% 5.00% 2.04%
5.00% 1.90% 6.00% 2.38%
5.20% 2.00% 7.00% 2.72%
5.40% 2.09% 8.00% 3.06%
5.60% 2.18% 9.00% 3.40%
5.80% 2.26% 10.00% 3.74%
6.00% 2.34% 11.00% 4.09%
6.20% 2.42% 12.00% 4.43%
6.40% 2.50% 13.00% 4.77%
6.60% 2.57% 14.00% 5.11%
6.80% 2.64% 15.00% 5.45%
7.00% 2.72% 16.00% 5.79%
7.30% 2.82% 17.00% 6.13%
7.60% 2.93% 18.00% 6.47%
7.90% 3.03% 19.00% 6.81%
8.20% 3.13% 20.00% 7.16%
8.50% 3.23% 21.00% 7.50%
8.80% 3.32% 22.00% 7.84%
9.10% 3.42% 23.00% 8.18%
9.40% 3.52% 24.00% 8.52%
9.70% 3.61% 25.00% 8.86%
10.00% 3.71% 26.00% 9.20%
10.30% 3.80% 27.00% 9.54%
10.60% 3.90% 28.00% 9.88%
10.74% 3.94% 29.00% 10.22%

Graph

efficient frontier 3.9300903154214445E-2 4.0000971950775441E-2 4.2000409276367681E-2 4.4000885477340441E-2 4.6000936146382926E-2 4.800002974337314E-2 4.9999329872095273E-2 5.1999671072641973E-2 5.3999824189192372E-2 5.599989961830662E-2 5.8002380232898255E-2 5.999996223351492E-2 6.2000853876285907E-2 6.4000660858974587E-2 6.6003234437913269E-2 6.8000411838988833E-2 7.000033036616074E-2 7.3000374683549904E-2 7.6003207940923634E-2 7.9000215666830034E-2 8.2000166504337252E-2 8.5000129862588078E-2 8.8000102220514576E-2 9.100008113444015E-2 9.4000064886853751E-2 9.700005224935504E-2 0.10000004233583047 0.10300003449733379 0.10600001427572135 0.10740249530280017 9.2854459608744284E-3 1.1368022209747919E-2 1.3766154242889555E-2 1.539182829179271E-2 1.6740050381067396E-2 1.7935391302936667E-2 1.9012715955609882E-2 1.9998341873015434E-2 2.0919177157052014E-2 2.1791877470303977E-2 2.2627312061082348E-2 2.3433008645377486E-2 2.4214753955602495E-2 2.4975867000024111E-2 2.571993319996831 E-2 2.6449413760578279E-2 2.716628233376301E-2 2.8221421251479341E-2 2.9255980762155409E-2 3.027332654788693E-2 3.1276049830214585E-2 3.2266219244892563E-2 3.3245511423684829E-2 3.4215303153075714E-2 3.5176737966139709E-2 3.6130775249408137E-2 3.707822711420948E-2 3.8019786516475279E-2 3.8956044657352841E-2 3.9392073272486811E-2 assets 0.10740249530280019 4.5588889409706755E-2 6.7954784154793599E-2 3.9392073272486818E-2 5.5893536849876187E-3 6.9560816202119766E-3

risk

return

efficient frontier 3.9300903154214445E-2 4.0000971950775441E-2 4.2000409276367681E-2 4.4000885477340441E-2 4.6000936146382926E-2 4.800002974337314E-2 4.9999329872095273E-2 5.1999671072641973E-2 5.3999824189192372E-2 5.599989961830662E-2 5.8002380232898255E-2 5.999996223351492E-2 6. 2000853876285907E-2 6.4000660858974587E-2 6.6003234437913269E-2 6.8000411838988833E-2 7.000033036616074E-2 7.3000374683549904E-2 7.6003207940923634E-2 7.9000215666830034E-2 8.2000166504337252E-2 8.5000129862588078E-2 8.8000102220514576E-2 9.100008113444015E-2 9.4000064886853751E-2 9.700005224935504E-2 0.10000004233583047 0.10300003449733379 0.10600001427572135 0.10740249530280017 9.2854459608744284E-3 1.1368022209747919E-2 1.3766154242889555E-2 1.539182829179271E-2 1.6740050381067396E-2 1.7935391302936667E-2 1.9012715955609882E-2 1.9998341873015434E-2 2.0919177157052014E-2 2.1791877470303977E-2 2.2627312061082348E-2 2.3433008645377486E-2 2.4214753955602495E-2 2.4975867000024111E-2 2.571993319996831E-2 2.6449413760578279E-2 2.716628233376301E-2 2.8221421251479341E-2 2.9255980762155409E-2 3.027332654788693E-2 3.1276049830214585E-2 3.2266219244892563E-2 3.3245511423684829E-2 3.4215303153075714E-2 3.5176737966139709E-2 3.6130775249408137E-2 3.707822711420948E-2 3.8019786516475279E-2 3.8956044657352841E-2 3.9392073272486811E-2 cap market line 0 0.01 0.02 0.03 0.04 0.05 0.06 7.0000000000000007E-2 0.08 0.09 0.1 0.11 0.12 0.13 0.14000000000000001 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28000000000000003 0.28999999999999998 3.3333333333333335E-3 6.7442145389255623E-3 1.0155095744517791E-2 1.3565976950110021E-2 1.6976858155702247E-2 2.0387739361294479E-2 2.3798620566886707E-2 2.7209501772478939E-2 3.0620382978071164E-2 3.4031264183663389E-2 3.7442145389255624E-2 4.0853026594847852E-2 4.426390780044008E-2 4.7674789006032309E-2 5.1085670211624544E-2 5.4496551417216765E-2 5.7907432622808994E-2 6.1318313828401229E-2 6.472919503399345E-2 6.8140076239585678E-2 7.1550957445177907E-2 7.4961838650770135E-2 7.8372719856362363E-2 8.1783601061954592E-2 8.519448226754682E-2 8.8605363473139048E-2 9.2016244678731277E-2 9.5427125884323519E-2 9.8838 007089915747E-2 0.10224888829550796 assets 0.10740249530280019 4.5588889409706755E-2 6.7954784154793599E-2 3.9392073272486818E-2 5.5893536849876187E-3 6.9560816202119766E-3

risk

return