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Analyze Performance Attribution Using Brinson Model

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This example shows how to prepare data, create a brinsonAttribution object, and then analyze the performance attribution with respect to category (sector) weights and returns. In this example, you use the categoryAttribution, categoryReturns, categoryWeights, totalAttribution, and summary functions for the analysis. Also, you can generate plots for the results, using categoryReturnsChart, categoryWeightsChart, and attributionsChart.

Prepare Data

Load the stock price data into a table. 

T = readtable('dowPortfolio.xlsx');
MonthIdx = [1;20;39;62;81;103;125;145;168;188;210;231;251];
MonthlyPrices = T(MonthIdx,3:end);

Compute the monthly returns using tick2ret. 

MonthlyReturns = tick2ret(MonthlyPrices.Variables)';
[NumAssets,NumPeriods] = size(MonthlyReturns);

Define the asset categories (sectors) using categorical. 

Category = categorical(["Materials";"Financials";"Financials"; ...
    "Industrials";"Financials";"Industrials";"Materials"; ...
    "Communication Services";"Industrials";"Consumer Discretionary"; ...
    "Consumer Discretionary";"Industrials";"Information Technology"; ...
    "Information Technology";"Information Technology";"Health Care"; ...
    "Financials";"Consumer Staples";"Consumer Discretionary"; ...
    "Industrials";"Consumer Staples";"Health Care"; ...
    "Information Technology";"Health Care";"Consumer Staples"; ...
    "Communication Services";"Industrials";"Communication Services"; ...
    "Consumer Staples";"Energy"]);
Category = repmat(Category,1,NumPeriods);

Define Benchmark and Portfolio Weights

Define the benchmark weight and the portfolio weights, by quarter, for each of the stock assets.

BenchmarkWeight = 1./length(MonthlyReturns).*ones(NumAssets, NumPeriods);

PortfolioWeightQ1 = [0;0;0.022;0.033;0;0.044;0.022;0.011;0.065;0.033; ...
    0.055;0.072;0;0.04;0;0.05;0.08;0.042;0.03;0.043;0.055;0.036;0.111; ...
    0.036;0;0.03;0;0.05;0;0.04]*ones(1,3);

PortfolioWeightQ2 = [0;0;0.022;0.033;0;0.044;0.022;0.011;0.049;0.033; ...
    0.055;0.074;0;0.04;0;0.05;0.08;0.042;0.03;0.02;0.055;0.036;0.148; ...
    0.036;0;0.03;0;0.05;0;0.04]*ones(1,3);

PortfolioWeightQ3 = [0;0;0.022;0.033;0;0.042;0.022;0.01;0.049;0.033; ...
    0.05;0.07;0;0.04;0;0.05;0.08;0.042;0.03;0.02;0.055;0.036;0.16; ...
    0.036;0;0.03;0;0.05;0;0.04]*ones(1,3);

PortfolioWeightQ4 = [0;0;0.022;0.033;0;0.042;0.02;0.01;0.039;0.033; ...
    0.05;0.07;0;0.04;0;0.05;0.08;0.042;0.03;0.02;0.055;0.036;0.198; ...
    0.036;0;0.03;0;0.03;0;0.034]*ones(1,3);

PortfolioWeight = [PortfolioWeightQ1 PortfolioWeightQ2 ...
    PortfolioWeightQ3 PortfolioWeightQ4];

Period = (1:NumPeriods).*ones(NumAssets,1);

Name = repmat(string(MonthlyPrices.Properties.VariableNames(:)),1,NumPeriods);

Create AssetTable Input

Use table to create the AssetTable input to use when creating a brinsonAttribution object.

AssetTable = table(Period(:), Name(:), ...
MonthlyReturns(:), Category(:), ...
PortfolioWeight(:), BenchmarkWeight(:), ...
VariableNames=["Period", "Name","Return","Category",...
"PortfolioWeight","BenchmarkWeight"])
AssetTable=360×6 table
    Period     Name       Return              Category           PortfolioWeight    BenchmarkWeight
    ______    ______    ___________    ______________________    _______________    _______________

      1       "AA"         0.053621    Materials                          0            0.033333    
      1       "AIG"        -0.05964    Financials                         0            0.033333    
      1       "AXP"     -0.00019406    Financials                     0.022            0.033333    
      1       "BA"        -0.030162    Industrials                    0.033            0.033333    
      1       "C"         -0.055015    Financials                         0            0.033333    
      1       "CAT"         0.17956    Industrials                    0.044            0.033333    
      1       "DD"        -0.090708    Materials                      0.022            0.033333    
      1       "DIS"        0.037221    Communication Services         0.011            0.033333    
      1       "GE"         -0.07381    Industrials                    0.065            0.033333    
      1       "GM"          0.27273    Consumer Discretionary         0.033            0.033333    
      1       "HD"        -0.016838    Consumer Discretionary         0.055            0.033333    
      1       "HON"        0.025457    Industrials                    0.072            0.033333    
      1       "HPQ"        0.083598    Information Technology             0            0.033333    
      1       "IBM"       -0.009235    Information Technology          0.04            0.033333    
      1       "INTC"        -0.1685    Information Technology             0            0.033333    
      1       "JNJ"       -0.066351    Health Care                     0.05            0.033333    
      ⋮

Create brinsonAttribution Object

Use brinsonAttribution to create a brinsonAttribution object. 

BrinsonPAobj = brinsonAttribution(AssetTable)
BrinsonPAobj = 
  brinsonAttribution with properties:

                  NumAssets: 30
         NumPortfolioAssets: 22
         NumBenchmarkAssets: 30
                 NumPeriods: 12
              NumCategories: 9
                  AssetName: [30x1 string]
                AssetReturn: [30x12 double]
              AssetCategory: [30x12 categorical]
       PortfolioAssetWeight: [30x12 double]
       BenchmarkAssetWeight: [30x12 double]
    PortfolioCategoryReturn: [9x12 double]
    BenchmarkCategoryReturn: [9x12 double]
    PortfolioCategoryWeight: [9x12 double]
    BenchmarkCategoryWeight: [9x12 double]
            PortfolioReturn: 0.2234
            BenchmarkReturn: 0.2046
               ActiveReturn: 0.0188

Compute Category Weights

Use the brinsonAttribution object and categoryWeights to compute the average and periodic category weights for the portfolio and the benchmark, as well as, the corresponding active weights.

[AverageCategoryWeights,PeriodicCategoryWeights] = categoryWeights(BrinsonPAobj)
AverageCategoryWeights=9×4 table
           Category           AveragePortfolioWeight    AverageBenchmarkWeight    AverageActiveWeight
    ______________________    ______________________    ______________________    ___________________

    Communication Services            0.0855                        0.1                  -0.0145     
    Consumer Discretionary            0.1155                        0.1                   0.0155     
    Consumer Staples                   0.097                    0.13333                -0.036333     
    Energy                            0.0385                   0.033333                0.0051667     
    Financials                         0.102                    0.13333                -0.031333     
    Health Care                        0.122                        0.1                    0.022     
    Industrials                      0.22375                        0.2                  0.02375     
    Information Technology           0.19425                    0.13333                 0.060917     
    Materials                         0.0215                   0.066667                -0.045167     


PeriodicCategoryWeights=108×5 table
    Period           Category           PortfolioWeight    BenchmarkWeight    ActiveWeight
    ______    ______________________    _______________    _______________    ____________

      1       Communication Services         0.091                 0.1            -0.009  
      1       Consumer Discretionary         0.118                 0.1             0.018  
      1       Consumer Staples               0.097             0.13333         -0.036333  
      1       Energy                          0.04            0.033333         0.0066667  
      1       Financials                     0.102             0.13333         -0.031333  
      1       Health Care                    0.122                 0.1             0.022  
      1       Industrials                    0.257                 0.2             0.057  
      1       Information Technology         0.151             0.13333          0.017667  
      1       Materials                      0.022            0.066667         -0.044667  
      2       Communication Services         0.091                 0.1            -0.009  
      2       Consumer Discretionary         0.118                 0.1             0.018  
      2       Consumer Staples               0.097             0.13333         -0.036333  
      2       Energy                          0.04            0.033333         0.0066667  
      2       Financials                     0.102             0.13333         -0.031333  
      2       Health Care                    0.122                 0.1             0.022  
      2       Industrials                    0.257                 0.2             0.057  
      ⋮

Compute Category Returns

Use the brinsonAttribution object and categoryReturns to compute the aggregate and periodic category (sector) returns for the portfolio and the benchmark.

[AggregateCategoryReturns,PeriodicCategoryReturns] = categoryReturns(BrinsonPAobj)
AggregateCategoryReturns=9×3 table
           Category           AggregatePortfolioReturn    AggregateBenchmarkReturn
    ______________________    ________________________    ________________________

    Communication Services            0.41756                     0.42797         
    Consumer Discretionary            0.27772                     0.34718         
    Consumer Staples                  0.20572                     0.14112         
    Energy                            0.33598                     0.33598         
    Financials                        0.22678                     0.15838         
    Health Care                       0.19451                     0.20679         
    Industrials                       0.14132                     0.14253         
    Information Technology            0.14339                     0.13485         
    Materials                         0.17109                     0.10402         


PeriodicCategoryReturns=108×4 table
    Period           Category           PortfolioReturn    BenchmarkReturn
    ______    ______________________    _______________    _______________

      1       Communication Services        0.056262           0.052385   
      1       Consumer Discretionary        0.079767            0.10017   
      1       Consumer Staples              -0.01485         -0.0033588   
      1       Energy                        0.073093           0.073093   
      1       Financials                  -0.0021179          -0.029374   
      1       Health Care                   0.012124           0.022297   
      1       Industrials                  0.0018686          0.0088589   
      1       Information Technology        0.033481          -0.011315   
      1       Materials                    -0.090708          -0.018543   
      2       Communication Services        0.069253           0.078033   
      2       Consumer Discretionary       -0.023091          -0.036416   
      2       Consumer Staples             0.0027749         0.00096619   
      2       Energy                         -0.0487            -0.0487   
      2       Financials                    0.033379            0.02054   
      2       Health Care                   0.014975           0.015929   
      2       Industrials                   0.047732           0.041906   
      ⋮

Compute Category Attribution

Use the brinsonAttribution object and categoryAttribution to compute the performance attribution of the portfolio for each category (sector).

[AggregateCategoryAttribution,PeriodicCategoryAttribution] = categoryAttribution(BrinsonPAobj)
AggregateCategoryAttribution=9×5 table
           Category           Allocation     Selection     Interaction    ActiveReturn
    ______________________    __________    ___________    ___________    ____________

    Communication Services    -0.0025318    -0.00071062      0.0002061     -0.0030363 
    Consumer Discretionary     0.0024505       -0.00678     -0.0011609     -0.0054903 
    Consumer Staples           0.0023692      0.0093618     -0.0025511        0.00918 
    Energy                    0.00038824     1.3872e-19     2.7745e-20     0.00038824 
    Financials                 0.0013464      0.0097268     -0.0022858      0.0087874 
    Health Care               0.00011567     -0.0013031    -0.00028667     -0.0014741 
    Industrials                0.0016701    -0.00035051    -5.7623e-05       0.001262 
    Information Technology     0.0015686      0.0017812     0.00016898      0.0035187 
    Materials                  0.0040725      0.0050687     -0.0034678      0.0056734 


PeriodicCategoryAttribution=108×7 table
    Period    LinkingCoefficient           Category           Allocation      Selection     Interaction    ActiveReturn
    ______    __________________    ______________________    ___________    ___________    ___________    ____________

      1             1.1901          Communication Services    -0.00034021     0.00038766     -3.489e-05     1.2564e-05 
      1             1.1901          Consumer Discretionary      0.0015405     -0.0020403    -0.00036725    -0.00086699 
      1             1.1901          Consumer Staples           0.00065192     -0.0015322     0.00041752    -0.00046274 
      1             1.1901          Energy                     0.00039006              0              0     0.00039006 
      1             1.1901          Financials                  0.0013774      0.0036342    -0.00085403      0.0041575 
      1             1.1901          Health Care                0.00016968     -0.0010173     -0.0002238     -0.0010714 
      1             1.1901          Industrials               -0.00032634     -0.0013981    -0.00039845     -0.0021229 
      1             1.1901          Information Technology    -0.00045756      0.0059728      0.0007914      0.0063066 
      1             1.1901          Materials                   0.0014797      -0.004811      0.0032234    -0.00010793 
      2             1.1935          Communication Services    -0.00058016      -0.000878      7.902e-05     -0.0013791 
      2             1.1935          Consumer Discretionary    -0.00089975      0.0013325     0.00023985     0.00067261 
      2             1.1935          Consumer Staples           0.00045794     0.00024116    -6.5715e-05     0.00063338 
      2             1.1935          Energy                    -0.00041514              0              0    -0.00041514 
      2             1.1935          Financials                -0.00021838      0.0017119     -0.0004023      0.0010912 
      2             1.1935          Health Care                  5.19e-05    -9.5418e-05    -2.0992e-05     -6.451e-05 
      2             1.1935          Industrials                 0.0016152      0.0011651     0.00033205      0.0031123 
      ⋮

Compute Total Attribution

Use the brinsonAttribution object and totalAttribution to compute the total performance attribution of the portfolio summed up for all categories (sectors).

[AggregateTotalAttribution,PeriodicTotalAttribution] = totalAttribution(BrinsonPAobj)
AggregateTotalAttribution=1×4 table
    Allocation    Selection    Interaction    ActiveReturn
    __________    _________    ___________    ____________

     0.011449     0.016794     -0.0094348       0.018809  


PeriodicTotalAttribution=12×6 table
    Period    LinkingCoefficient    Allocation      Selection     Interaction    ActiveReturn
    ______    __________________    ___________    ___________    ___________    ____________

       1            1.1901            0.0044852    -0.00080413      0.0025538      0.0062349 
       2            1.1935           0.00087285      0.0032617     -0.0025973      0.0015372 
       3            1.1904          -6.9025e-05      0.0052394     0.00069386      0.0058643 
       4             1.208            -0.004707     -0.0095748     -0.0046622      -0.018944 
       5            1.1995          -0.00030704      -0.006091    -0.00061525     -0.0070132 
       6            1.1935            0.0012025    -0.00026724     0.00058932      0.0015246 
       7            1.1813            0.0032236       0.017541     -0.0021062       0.018658 
       8             1.196             0.004629      -0.004929     -0.0018038     -0.0021038 
       9            1.1927            0.0013046     0.00036681     0.00099195      0.0026634 
      10            1.1923            0.0032172      0.0023077     -0.0023663      0.0031586 
      11             1.194            -0.001881     0.00044958      0.0022018     0.00077039 
      12            1.1918           -0.0022819      0.0069324     -0.0007375       0.003913

Generate Summary for Performance Attribution

Use the brinsonAttribution object and summary to generate a table that summarizes the final results (aggregated over all time periods and categories) of the performance attribution using the Brinson model.

SummaryTable = summary(BrinsonPAobj)
SummaryTable=11×1 table
                                     Brinson Attribution Summary
                                     ___________________________

    Total Number of Assets                           30         
    Number of Assets in Portfolio                    22         
    Number of Assets in Benchmark                    30         
    Number of Periods                                12         
    Number of Categories                              9         
    Portfolio Return                            0.22345         
    Benchmark Return                            0.20464         
    Active Return                              0.018809         
    Allocation Effect                          0.011449         
    Selection Effect                           0.016794         
    Interaction Effect                       -0.0094348

Generate Horizontal Bar Charts for Returns, Weights, and Performance Attribution

Use the brinsonAttribution object and categoryReturnsChart, categoryWeightsChart, and attributionsChart to generate horizontal bar charts.

categoryReturnsChart(BrinsonPAobj)

Figure contains an axes object. The axes object with title Portfolio and Benchmark Category Returns, xlabel Category Return, ylabel Category contains 2 objects of type bar. These objects represent Portfolio Return, Benchmark Return.

categoryWeightsChart(BrinsonPAobj)

Figure contains an axes object. The axes object with title Portfolio, Benchmark, and Active Category Weights, xlabel Category Weight, ylabel Category contains 3 objects of type bar. These objects represent Portfolio Weight, Benchmark Weight, Active Weight.

attributionsChart(BrinsonPAobj, Style="stacked")

Figure contains an axes object. The axes object with title Return Attributions by Category, xlabel Return Attribution, ylabel Category contains 3 objects of type bar. These objects represent Allocation Effect, Selection Effect, Interaction Effect.

See Also

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