estimateFrontier
Estimate specified number of optimal portfolios on the efficient frontier
Description
[
estimates the specified number of optimal portfolios on the efficient frontier for
pwgt
,pbuy
,psell
]
= estimateFrontier(obj
)Portfolio
, PortfolioCVaR
, or
PortfolioMAD
objects. For details on the respective workflows when using
these different objects, see Portfolio Object Workflow, PortfolioCVaR Object Workflow, and PortfolioMAD Object Workflow.
Examples
Create a Portfolio
Object and Determine Efficient Portfolios
Create efficient portfolios:
load CAPMuniverse p = Portfolio('AssetList',Assets(1:12)); p = estimateAssetMoments(p, Data(:,1:12),'missingdata',true); p = setDefaultConstraints(p); plotFrontier(p);
pwgt = estimateFrontier(p, 5); pnames = cell(1,5); for i = 1:5 pnames{i} = sprintf('Port%d',i); end Blotter = dataset([{pwgt},pnames],'obsnames',p.AssetList); disp(Blotter);
Port1 Port2 Port3 Port4 Port5 AAPL 0.017926 0.058247 0.097816 0.12955 0 AMZN 0 0 0 0 0 CSCO 0 0 0 0 0 DELL 0.0041906 0 0 0 0 EBAY 0 0 0 0 0 GOOG 0.16144 0.35678 0.55228 0.75116 1 HPQ 0.052566 0.032302 0.011186 0 0 IBM 0.46422 0.36045 0.25577 0.11928 0 INTC 0 0 0 0 0 MSFT 0.29966 0.19222 0.082949 0 0 ORCL 0 0 0 0 0 YHOO 0 0 0 0 0
Create a Portfolio
Object with BoundType
and MaxNumAssets
Constraints and Determine Efficient Portfolios
Create a Portfolio
object for 12 stocks based on CAPMuniverse.mat
.
load CAPMuniverse p0 = Portfolio('AssetList',Assets(1:12)); p0 = estimateAssetMoments(p0, Data(:,1:12),'missingdata',true); p0 = setDefaultConstraints(p0);
Use setMinMaxNumAssets
to define a maximum number of 3 assets.
p1 = setMinMaxNumAssets(p0, [], 3);
Use setBounds
to define a lower and upper bound and a BoundType
of 'Conditional'
.
p1 = setBounds(p1, 0.1, 0.5,'BoundType', 'Conditional'); pwgt = estimateFrontier(p1, 5);
The following table shows that the optimized allocations only have maximum 3 assets invested, and small positions less than 0.1 are avoided.
result = table(p0.AssetList', pwgt)
result=12×2 table
Var1 pwgt
________ ___________________________________________________________________
{'AAPL'} 0 0 0 0.14232 0
{'AMZN'} 0 0 0 0 0
{'CSCO'} 0 0 0 0 0
{'DELL'} 0 0 0 0 0
{'EBAY'} 0 0 0 0 0.5
{'GOOG'} 0.16891 0.29534 0.42177 0.5 0.5
{'HPQ' } 0 0 4.996e-15 -2.7756e-17 0
{'IBM' } 0.49968 0.43657 0.37326 0.35768 0
{'INTC'} 0 0 0 0 0
{'MSFT'} 0.3314 0.2681 0.20496 4.6838e-17 0
{'ORCL'} 0 0 0 0 0
{'YHOO'} 0 0 0 0 0
The estimateFrontier
function uses the MINLP solver to solve this problem. Use the setSolverMINLP
function to configure the SolverType
and options.
p1.solverTypeMINLP
ans = 'OuterApproximation'
p1.solverOptionsMINLP
ans = struct with fields:
MaxIterations: 1000
AbsoluteGapTolerance: 1.0000e-07
RelativeGapTolerance: 1.0000e-05
NonlinearScalingFactor: 1000
ObjectiveScalingFactor: 1000
Display: 'off'
CutGeneration: 'basic'
MaxIterationsInactiveCut: 30
ActiveCutTolerance: 1.0000e-07
IntMainSolverOptions: [1x1 optim.options.Intlinprog]
NumIterationsEarlyIntegerConvergence: 30
ExtendedFormulation: 0
NumInnerCuts: 10
NumInitialOuterCuts: 10
Construct a PortfolioCVaR Object and Determine Efficient Portfolios
Create efficient portfolios:
load CAPMuniverse p = PortfolioCVaR('AssetList',Assets(1:12)); p = simulateNormalScenariosByData(p, Data(:,1:12), 20000 ,'missingdata',true); p = setDefaultConstraints(p); p = setProbabilityLevel(p, 0.95); plotFrontier(p);
pwgt = estimateFrontier(p, 5); pnames = cell(1,5); for i = 1:5 pnames{i} = sprintf('Port%d',i); end Blotter = dataset([{pwgt},pnames],'obsnames',p.AssetList); disp(Blotter);
Port1 Port2 Port3 Port4 Port5 AAPL 0.010223 0.073393 0.11939 0.13137 0 AMZN 0 0 0 0 0 CSCO 0 0 0 0 0 DELL 0.02301 0 0 0 0 EBAY 0 0 0 0 0 GOOG 0.20389 0.38068 0.56253 0.75919 1 HPQ 0.041396 0.009472 0 0 0 IBM 0.44369 0.36472 0.26247 0.10944 0 INTC 0 0 0 0 0 MSFT 0.27779 0.17174 0.055611 0 0 ORCL 0 0 0 0 0 YHOO 0 0 0 0 0
Create a PortfolioMAD
Object and Determine Efficient Portfolios
Create efficient portfolios:
load CAPMuniverse p = PortfolioMAD('AssetList',Assets(1:12)); p = simulateNormalScenariosByData(p, Data(:,1:12), 20000 ,'missingdata',true); p = setDefaultConstraints(p); plotFrontier(p);
pwgt = estimateFrontier(p, 5); pnames = cell(1,5); for i = 1:5 pnames{i} = sprintf('Port%d',i); end Blotter = dataset([{pwgt},pnames],'obsnames',p.AssetList); disp(Blotter);
Port1 Port2 Port3 Port4 Port5 AAPL 0.029643 0.075874 0.11335 0.13405 0 AMZN 0 0 0 0 0 CSCO 0 0 0 0 0 DELL 0.0086367 0 0 0 0 EBAY 0 0 0 0 0 GOOG 0.16177 0.35217 0.54489 0.74913 1 HPQ 0.056891 0.023419 0 0 0 IBM 0.45916 0.37921 0.29376 0.11682 0 INTC 0 0 0 0 0 MSFT 0.2839 0.16933 0.048005 0 0 ORCL 0 0 0 0 0 YHOO 0 0 0 0 0
Obtain the Default Number of Efficient Portfolios for a Portfolio Object
Obtain the default number of efficient portfolios over the entire range of the efficient frontier.
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; p = Portfolio; p = setAssetMoments(p, m, C); p = setDefaultConstraints(p); pwgt = estimateFrontier(p); disp(pwgt);
0.8891 0.7215 0.5540 0.3865 0.2190 0.0515 0 0 0 0 0.0369 0.1289 0.2209 0.3129 0.4049 0.4969 0.4049 0.2314 0.0579 0 0.0404 0.0567 0.0730 0.0893 0.1056 0.1219 0.1320 0.1394 0.1468 0 0.0336 0.0929 0.1521 0.2113 0.2705 0.3297 0.4630 0.6292 0.7953 1.0000
Obtain Purchases and Sales for Portfolios on the Efficient Frontier for a Portfolio Object
Starting from the initial portfolio, the estimateFrontier
function returns purchases and sales to get from your initial portfolio to each efficient portfolio on the efficient frontier. Given an initial portfolio in pwgt0
, you can obtain purchases and sales.
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; p = Portfolio; p = setAssetMoments(p, m, C); p = setDefaultConstraints(p); pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ]; p = setInitPort(p, pwgt0); [pwgt, pbuy, psell] = estimateFrontier(p); display(pwgt);
pwgt = 4×10
0.8891 0.7215 0.5540 0.3865 0.2190 0.0515 0 0 0 0
0.0369 0.1289 0.2209 0.3129 0.4049 0.4969 0.4049 0.2314 0.0579 0
0.0404 0.0567 0.0730 0.0893 0.1056 0.1219 0.1320 0.1394 0.1468 0
0.0336 0.0929 0.1521 0.2113 0.2705 0.3297 0.4630 0.6292 0.7953 1.0000
display(pbuy);
pbuy = 4×10
0.5891 0.4215 0.2540 0.0865 0 0 0 0 0 0
0 0 0 0.0129 0.1049 0.1969 0.1049 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0.0521 0.1113 0.1705 0.2297 0.3630 0.5292 0.6953 0.9000
display(psell);
psell = 4×10
0 0 0 0 0.0810 0.2485 0.3000 0.3000 0.3000 0.3000
0.2631 0.1711 0.0791 0 0 0 0 0.0686 0.2421 0.3000
0.1596 0.1433 0.1270 0.1107 0.0944 0.0781 0.0680 0.0606 0.0532 0.2000
0.0664 0.0071 0 0 0 0 0 0 0 0
Obtain the Default Number of Efficient Portfolios for a PortfolioCVaR Object
Obtain the default number of efficient portfolios over the entire range of the efficient frontier.
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; m = m/12; C = C/12; rng(11); AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioCVaR; p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); p = setProbabilityLevel(p, 0.95); pwgt = estimateFrontier(p); disp(pwgt);
0.8445 0.6841 0.5148 0.3534 0.1897 0.0303 0 0 0 0 0.0609 0.1429 0.2302 0.3171 0.3987 0.4742 0.3524 0.1803 0 0 0.0458 0.0640 0.0945 0.1081 0.1340 0.1590 0.1738 0.1918 0.2211 0 0.0488 0.1090 0.1606 0.2215 0.2776 0.3365 0.4738 0.6280 0.7789 1.0000
The function rng
() resets the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.
Obtain Purchases and Sales for Portfolios on the Efficient Frontier for a PortfolioCVaR Object
Starting from the initial portfolio, the estimateFrontier
function returns purchases and sales to get from your initial portfolio to each efficient portfolio on the efficient frontier. Given an initial portfolio in pwgt0
, you can obtain purchases and sales.
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; m = m/12; C = C/12; rng(11); AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioCVaR; p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); p = setProbabilityLevel(p, 0.95); pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ]; p = setInitPort(p, pwgt0); [pwgt, pbuy, psell] = estimateFrontier(p); display(pwgt);
pwgt = 4×10
0.8445 0.6841 0.5148 0.3534 0.1897 0.0303 0 0 0 0
0.0609 0.1429 0.2302 0.3171 0.3987 0.4742 0.3524 0.1803 0 0
0.0458 0.0640 0.0945 0.1081 0.1340 0.1590 0.1738 0.1918 0.2211 0
0.0488 0.1090 0.1606 0.2215 0.2776 0.3365 0.4738 0.6280 0.7789 1.0000
display(pbuy);
pbuy = 4×10
0.5445 0.3841 0.2148 0.0534 0 0 0 0 0 0
0 0 0 0.0171 0.0987 0.1742 0.0524 0 0 0
0 0 0 0 0 0 0 0 0.0211 0
0 0.0090 0.0606 0.1215 0.1776 0.2365 0.3738 0.5280 0.6789 0.9000
display(psell);
psell = 4×10
0 0 0 0 0.1103 0.2697 0.3000 0.3000 0.3000 0.3000
0.2391 0.1571 0.0698 0 0 0 0 0.1197 0.3000 0.3000
0.1542 0.1360 0.1055 0.0919 0.0660 0.0410 0.0262 0.0082 0 0.2000
0.0512 0 0 0 0 0 0 0 0 0
The function rng
() resets the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.
Obtain the Default Number of Efficient Portfolios for a PortfolioMAD Object
Obtain the default number of efficient portfolios over the entire range of the efficient frontier.
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; m = m/12; C = C/12; rng(11); AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioMAD; p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); pwgt = estimateFrontier(p); disp(pwgt);
0.8817 0.7150 0.5488 0.3811 0.2173 0.0503 0 0 0 0 0.0435 0.1290 0.2130 0.2987 0.3821 0.4668 0.3614 0.1751 0 0 0.0385 0.0600 0.0826 0.1061 0.1242 0.1477 0.1780 0.2101 0.2267 0 0.0363 0.0960 0.1556 0.2141 0.2764 0.3352 0.4605 0.6148 0.7733 1.0000
The function rng
() resets the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.
Obtain Purchases and Sales for Portfolios on the Efficient Frontier for a PortfolioMAD Object
Starting from the initial portfolio, the estimateFrontier
function returns purchases and sales to get from your initial portfolio to each efficient portfolio on the efficient frontier. Given an initial portfolio in pwgt0
, you can obtain purchases and sales.
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; m = m/12; C = C/12; rng(11); AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioMAD; p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ]; p = setInitPort(p, pwgt0); [pwgt, pbuy, psell] = estimateFrontier(p); display(pwgt);
pwgt = 4×10
0.8817 0.7150 0.5488 0.3811 0.2173 0.0503 0 0 0 0
0.0435 0.1290 0.2130 0.2987 0.3821 0.4668 0.3614 0.1751 0 0
0.0385 0.0600 0.0826 0.1061 0.1242 0.1477 0.1780 0.2101 0.2267 0
0.0363 0.0960 0.1556 0.2141 0.2764 0.3352 0.4605 0.6148 0.7733 1.0000
display(pbuy);
pbuy = 4×10
0.5817 0.4150 0.2488 0.0811 0 0 0 0 0 0
0 0 0 0 0.0821 0.1668 0.0614 0 0 0
0 0 0 0 0 0 0 0.0101 0.0267 0
0 0 0.0556 0.1141 0.1764 0.2352 0.3605 0.5148 0.6733 0.9000
display(psell);
psell = 4×10
0 0 0 0 0.0827 0.2497 0.3000 0.3000 0.3000 0.3000
0.2565 0.1710 0.0870 0.0013 0 0 0 0.1249 0.3000 0.3000
0.1615 0.1400 0.1174 0.0939 0.0758 0.0523 0.0220 0 0 0.2000
0.0637 0.0040 0 0 0 0 0 0 0 0
The function rng
() resets the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.
Input Arguments
obj
— Object for portfolio
object
Object for portfolio, specified using Portfolio
,
PortfolioCVaR
, or PortfolioMAD
object. For more
information on creating a portfolio object, see
Data Types: object
NumPorts
— Number of points to obtain on efficient frontier
value from hidden property defaultNumPorts
(default
value is 10
) (default) | scalar integer
Number of points to obtain on the efficient frontier, specified as a scalar integer.
Note
If no value is specified for NumPorts
, the default value is obtained
from the hidden property defaultNumPorts
(default value is
10
). If NumPorts
= 1
, this
function returns the portfolio specified by the hidden property
defaultFrontierLimit
(current default value is
'min'
).
Data Types: double
Output Arguments
pwgt
— Optimal portfolios on efficient frontier with specified number of portfolios spaced equally from minimum to maximum portfolio return
matrix
Optimal portfolios on the efficient frontier with specified number of portfolios spaced
equally from minimum to maximum portfolio return, returned as a
NumAssets
-by-NumPorts
matrix. pwgt
is returned for a Portfolio
, PortfolioCVaR
, or
PortfolioMAD
input object (obj
).
pbuy
— Purchases relative to initial portfolio for optimal portfolios on efficient frontier
matrix
Purchases relative to an initial portfolio for optimal portfolios on the efficient
frontier, returned as NumAssets
-by-NumPorts
matrix.
Note
If no initial portfolio is specified in obj.InitPort
, that value is
assumed to be 0
such that pbuy = max(0, pwgt)
and
psell = max(0, -pwgt)
.
pbuy
is returned for a Portfolio
,
PortfolioCVaR
, or PortfolioMAD
input object
(obj
).
psell
— Sales relative to initial portfolio for optimal portfolios on efficient frontier
matrix
Sales relative to an initial portfolio for optimal portfolios on the efficient frontier,
returned as a NumAssets
-by-NumPorts
matrix.
Note
If no initial portfolio is specified in obj.InitPort
, that value is
assumed to be 0
such that pbuy = max(0, pwgt)
and
psell = max(0, -pwgt)
.
psell
is returned for Portfolio
,
PortfolioCVaR
, or PortfolioMAD
input object
(obj
).
Tips
You can also use dot notation to estimate the specified number of optimal portfolios over the entire efficient frontier.
[pwgt, pbuy, psell] = obj.estimateFrontier(NumPorts);
When introducing transaction costs and turnover constraints to the
Portfolio
,PortfolioCVaR
, orPortfolioMAD
object, the portfolio optimization objective contains a term with an absolute value. For more information on how Financial Toolbox™ handles such cases algorithmically, see References.
References
[1] Cornuejols, G., and R. Tutuncu. Optimization Methods in Finance. Cambridge University Press, 2007.
Version History
Introduced in R2011a
See Also
estimateFrontierByReturn
| estimateFrontierByRisk
| estimateFrontierLimits
| setBounds
| setMinMaxNumAssets
Topics
- Estimate Efficient Portfolios for Entire Efficient Frontier for Portfolio Object
- Estimate Efficient Frontiers for Portfolio Object
- Estimate Efficient Portfolios for Entire Frontier for PortfolioCVaR Object
- Estimate Efficient Frontiers for PortfolioCVaR Object
- Estimate Efficient Portfolios Along the Entire Frontier for PortfolioMAD Object
- Estimate Efficient Frontiers for PortfolioMAD Object
- Portfolio Optimization Examples Using Financial Toolbox
- Bond Portfolio Optimization Using Portfolio Object
- Portfolio Optimization Theory
- Choose MINLP Solvers for Portfolio Problems
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