brinsonAttribution

Create brinsonAttribution object to analyze performance attribution

Since R2022b

Description

Create a brinsonAttribution object for performance attribution using the Brinson model.

Use this workflow to develop and analyze a Brinson model for performance attribution:

1. Prepare data for the AssetTable input.

2. Use brinsonAttribution to create a brinsonAttribution object.

3. Use the following functions with the brinsonAttribution object:

   • categoryAttribution

   • categoryReturns

   • categoryWeights

   • totalAttribution

   • summary

   • categoryReturnsChart

   • categoryWeightsChart

   • attributionsChart

For more detailed information on this workflow, see Analyze Performance Attribution Using Brinson Model.

Creation

Syntax

brinsonAttributionObj = brinsonAttribution(AssetTable)

Description

brinsonAttributionObj = brinsonAttribution(AssetTable) creates a brinsonAttribution object and sets the properties. Use the brinsonAttribution object with the categoryAttribution, categoryReturns, categoryWeights, totalAttribution, and summary functions.

Input Arguments

AssetTable — Information about individual asset returns in the portfolio and benchmark

table

Information about individual asset returns in the portfolio and benchmark, specified as a table with the number of rows equal to NumAssets-times-NumPeriods rows. The column variables are in the following order from left to right:

• Period — Column vector of positive whole numbers containing the time Period numbers. For each one of the NumAssets asset names, the Period numbers range from 1 to NumPeriods with increments of 1, so that there are a total of NumAssets-times-NumPeriods rows. Row 1 is for the first period, row 2 is for the second period, and is repeated for each asset. If the Period column is missing from AssetTable, all Period numbers are internally set to 1 and all asset returns in AssetTable are assumed to be for the same single period.

• Name — String column vector containing the individual asset names for the associated returns. There is a total of NumAssets unique names, and the names are repeated for each Period.

• Return — Numeric column vector containing the asset returns in decimals.

• Category — Categorical vector containing the asset categories (sectors) for the associated asset returns. There is a total of NumCategories unique categories.

• Portfolio Weight — Numeric vector containing the asset portfolio weights in decimals. The weights are internally normalized so that they sum to 1 for each Period.

  Note

  The values for Portfolio Weight must sum to 1 for the Brinson model. If the weights do not sum to 1, the Portfolio Weight values are internally normalized so that they sum to 1 and brinsonAttribution displays a warning. If there is a cash position, you must account for it as an asset with its own weight so that the weights sum to 1.

• Benchmark Weight — Numeric vector containing the asset benchmark weights in decimals. The weights are internally normalized so that they sum to 1 for each Period.

  Note

  The values for the Benchmark Weight must sum to 1 for the Brinson model. If the weights do not sum to 1, the Benchmark Weight values are internally normalized so that they sum to 1 and brinsonAttribution displays a warning. If there is a cash position, you must account for it as an asset with its own weight so that the weights sum to 1.

Note

AssetTable must be a table. Instead of specific dates and times, the Period column of AssetTable must have period numbers that start with 1 and have increments of one. The multiperiod Brinson model assumes that all returns are for time periods of equal intervals (for example, monthly, quarterly, and so on).

Each individual asset must belong to one and only one category. You can apply Brinson attribution to any portfolio containing assets with measured returns if each of the individual asset belongs to one and only one category. Therefore, you can include a bond or cash holding in the portfolio after assigning it to its own category (for example, "Bond" or "Cash") and you enter its measured returns and weights. If the portfolio's benchmark does not have the bond or cash holding, then the benchmark weight for that asset is 0. As for an ETF, you can decompose the ETF into its individual holdings (for example, about 100 stocks in QQQ) and then enter these stocks into the portfolio with the associated measured returns, weights (relative to the overall portfolio, not within the ETF) and associate each stock in the ETF with a category.

Data Types: table

Properties

NumAssets — Total number of assets

numeric

Total number of assets, specified by AssetTable.

Data Types: double

NumPortfolioAssets — Number of assets in portfolio

numeric

Number of assets in the portfolio, specified by AssetTable.

Data Types: double

NumBenchmarkAssets — Number of assets in the benchmark

numeric

Number of assets in the benchmark, specified by AssetTable.

Data Types: double

NumPeriods — Number of time periods

numeric

Number of time periods, specified by AssetTable.

Data Types: double

NumCategories — Number of asset categories

numeric

Number of asset categories, specified by the AssetTable.

Data Types: double

AssetName — Individual asset names

vector

Individual asset names, returned as a NumAssets-by-1 vector.

Data Types: string

AssetReturn — Asset returns

matrix

Asset returns, returned as a NumAssets-by-NumPeriods matrix in decimals.

Data Types: double

AssetCategory — Asset categories

matrix

Asset categories (sectors), returned as a NumAssets-by-NumPeriods matrix.

Data Types: string

PortfolioAssetWeight — Asset portfolio weights

numeric

Asset portfolio weights, returned as a NumAssets-by-NumPeriods matrix in decimals.

Data Types: double

BenchmarkAssetWeight — Asset benchmark weights

numeric

Asset benchmark weights, returned as a NumAssets-by-NumPeriods matrix in decimals.

Data Types: double

PortflioCategoryReturn — Portfolio category returns

numeric

Portfolio category returns, returned as a NumCategories-by-NumPeriods matrix in decimals.

Data Types: double

BenchmarkCategoryReturn — Benchmark category returns

numeric

Benchmark category returns, returned as a NumCategories-by-NumPeriods matrix in decimals.

Data Types: double

PortfolioCategoryWeight — Portfolio category weights

numeric

Portfolio category weights, returned as a NumCategories-by-NumPeriods matrix in decimals.

Data Types: double

BenchmarkCategoryWeight — Benchmark category weights

numeric

Benchmark category weights, returned as a NumCategories-by-NumPeriods matrix in decimals.

Data Types: double

PortfolioReturn — Total portfolio return

numeric

Total portfolio return, returned as a scalar decimal.

Data Types: double

BenchmarkReturn — Total benchmark return

numeric

Total benchmark return, returned as a scalar decimal.

Data Types: double

ActiveReturn — Total active return

numeric

Total active return, returned as a scalar decimal.

Data Types: double

Object Functions

categoryAttribution	Compute performance attribution for portfolio of each category
categoryReturns	Compute aggregate and periodic category returns
categoryWeights	Compute average and periodic category weights
totalAttribution	Compute total performance attribution by Brinson model
summary	Summarize performance attribution by Brinson model
categoryReturnsChart	Create horizontal bar chart of category returns
categoryWeightsChart	Create a horizontal bar chart for category weights
attributionsChart	Create horizontal bar chart of performance attribution

Examples

Create brinsonAttribution Object

This example shows how to create a brinsonAttribution object that you can then use with the categoryAttribution, categoryReturns, categoryWeights, totalAttribution, and summary functions. Also, you can generate plots for the results, using categoryReturnsChart, categoryWeightsChart, and attributionsChart.

Prepare Data

Create a table for the monthly prices for four assets.

 GM =[17.82;22.68;19.37;20.28];
 HD = [39.79;39.12;40.67;40.96];
 KO = [38.98;39.44;40.00;40.20];
 PG = [56.38;57.08;57.76;55.54];
 MonthlyPrices  = table(GM,HD,KO,PG);

Use tick2ret to define the monthly returns.

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

Define the periods.

Period = ones(NumAssets*NumPeriods,1);
  for k = 1:NumPeriods
      Period(k*NumAssets+1:end,1) = Period(k*NumAssets,1) + 1;
  end

Define the categories (sectors) for the four assets.

Name = repmat(string(MonthlyPrices.Properties.VariableNames(:)),NumPeriods,1);
  Categories = repmat(categorical([ ...
      "Consumer Discretionary"; ...
      "Consumer Discretionary"; ...
      "Consumer Staples"; ...
      "Consumer Staples"]),NumPeriods,1);

Define benchmark and portfolio weights.

BenchmarkWeight = repmat(1./NumAssets.*ones(NumAssets, 1),NumPeriods,1);
  PortfolioWeight = repmat([1;0;1;1]./3,NumPeriods,1);

Create AssetTable Input

Create AssetTable as the input for the brinsonAttribution object.

  AssetTable = table(Period, Name, ...
      MonthlyReturns(:), Categories, PortfolioWeight, BenchmarkWeight, ...
      VariableNames=["Period","Name","Return","Category","PortfolioWeight","BenchmarkWeight"])

AssetTable=12×6 table
    Period    Name     Return             Category           PortfolioWeight    BenchmarkWeight
    ______    ____    _________    ______________________    _______________    _______________

      1       "GM"      0.27273    Consumer Discretionary        0.33333             0.25      
      1       "HD"    -0.016838    Consumer Discretionary              0             0.25      
      1       "KO"     0.011801    Consumer Staples              0.33333             0.25      
      1       "PG"     0.012416    Consumer Staples              0.33333             0.25      
      2       "GM"     -0.14594    Consumer Discretionary        0.33333             0.25      
      2       "HD"     0.039622    Consumer Discretionary              0             0.25      
      2       "KO"     0.014199    Consumer Staples              0.33333             0.25      
      2       "PG"     0.011913    Consumer Staples              0.33333             0.25      
      3       "GM"      0.04698    Consumer Discretionary        0.33333             0.25      
      3       "HD"    0.0071306    Consumer Discretionary              0             0.25      
      3       "KO"        0.005    Consumer Staples              0.33333             0.25      
      3       "PG"    -0.038435    Consumer Staples              0.33333             0.25      

Create brinsonAttribution Object

Use brinsonAttribution to create the brinsonAttribution object.

  BrinsonPAobj = brinsonAttribution(AssetTable)

BrinsonPAobj = 
  brinsonAttribution with properties:

                  NumAssets: 4
         NumPortfolioAssets: 3
         NumBenchmarkAssets: 4
                 NumPeriods: 3
              NumCategories: 2
                  AssetName: [4x1 string]
                AssetReturn: [4x3 double]
              AssetCategory: [4x3 categorical]
       PortfolioAssetWeight: [4x3 double]
       BenchmarkAssetWeight: [4x3 double]
    PortfolioCategoryReturn: [2x3 double]
    BenchmarkCategoryReturn: [2x3 double]
    PortfolioCategoryWeight: [2x3 double]
    BenchmarkCategoryWeight: [2x3 double]
            PortfolioReturn: 0.0598
            BenchmarkReturn: 0.0540
               ActiveReturn: 0.0059

You use brinsonAttribution object with the categoryAttribution, categoryReturns, categoryWeights, totalAttribution, and summary functions for the anlaysis. Also, you can generate plots for the results, using categoryReturnsChart, categoryWeightsChart, and attributionsChart.

More About

Brinson Model

Performance attribution supports methods for single periods over relatively short time spans (monthly) and multiple periods compounded over longer time spans (quarterly or a yearly).

The Brinson single-period sector-based (category-based) performance attribution is represented as:

$$\begin{array}{l}{r}_{P,t}={\displaystyle \sum _{j=1}^N{w}_{Pj,t}{r}_{Pj,t}}\\ r_{B,t}={\displaystyle \sum _{j=1}^N{w}_{Bj,t}{r}_{Bj,t}}\\ r_{\upsilon ,t}=r_{P,t}-r_{B,t}={\displaystyle \sum _{j=1}^N{w}_{Pj,t}{r}_{Pj,t}-}{\displaystyle \sum _{j=1}^N{w}_{Bj,t}{r}_{Bj,t}}\\ ={\displaystyle \sum _{j=1}^N(w_{Pj,t}-w_{Bj,t})(r_{Bj,t}-r_{B,t})+}{\displaystyle \sum _{j=1}^N(w_{Pj,t}-w_{Bj,t})(r_{Pj,t}-r_{Bj,t})}+{\displaystyle \sum _{j=1}^N{w}_{Bj,t}(r_{Pj,t}-r_{Bj,t})}\\ ={\displaystyle \sum _{j=1}^N(S_{j,t}+U_{j,t}+I_{j,t})}\end{array}$$

where

r_P,t — Portfolio return for period t

r_b,t — Benchmark return for period t

r_υ,t — Value-added return for period t

N — Number of sectors

w_Pj,t — Portfolio weight for sector j and period t

r_Pj,t — Portfolio return for sector j and period t

w_Bj,t — Benchmark weight for sector j and period t

r_Bj,t — Benchmark return for sector j and period t

S_j,t — Sector allocation effect for sector j and period t

U_j,t — Allocation and sector interaction effect for sector j and period t

I_j,t — Issue sector effect for sector j and period t

Multiperiod performance attribution, uses two main styles of performance attribution: arithmetic and geometric. In the arithmetic performance attribution, which is the method that the Brinson model uses, the relative performance between the portfolio and benchmark is measured by subtracting the benchmark return from the portfolio return. In the geometric performance attribution, the relative performance between the portfolio and benchmark is measured by a ratio based on the portfolio and benchmark returns.

The optimized linking algorithm by Menchero [3] extends the single-period arithmetic performance attribution to multiple periods by introducing the optimized linking coefficient β_t for each period before adding the single-period attributions. This step overcomes the fact that each single-period attribution does not simply add over multiple periods, because the optimized linking coefficient β_t reconciles the difference that arises between simple arithmetic addition and geometric compounding. As a result, the single-period Brinson model can be extended to multiple periods by using the following optimized linking algorithm:

$$\begin{array}{l}{r}_{\upsilon }=r_p-r_B={\displaystyle \sum _{t=1}^T{\beta }_t(r_{P,t}-r_{B,t})=}{\displaystyle \sum _{j=1}^N\left({\displaystyle \sum _{t=1}^T{\beta }_t{S}_{j,t}+{\displaystyle \sum _{t=1}^T{\beta }_t{U}_{j,t}+{\displaystyle \sum _{t=1}^T{\beta }_t{I}_{j,t}}}}\right)={\displaystyle \sum _{j=1}^N(S_j+U_j+I_j)}}\\ {\beta }_t=A+C(r_{p,t}-r_{B,t})\\ A=\frac{\frac{(r_p-r_B)}{T}}{{(1+r_p)}^{\frac{1}{T}}-{(1+r_B)}^{\frac{1}{T}}}\\ C=\frac{r_p-r_B-A{\displaystyle {\sum }_{t=1}^T(r_{P,t}-r_{B,t})}}{{\displaystyle {\sum }_{t=1}^T{(r_{P,t}-r_{B,t})}^2}}\end{array}$$

where

r_p — Portfolio return over multiple periods

r_B — Benchmark return over multiple periods

r_υ — Value-added return over multiple periods

r_P,t — Portfolio return for period t

r_B,t — Benchmark return for period t

β_t — Optimized linking coefficient for period t

T — Number of periods

N — Number of sectors

S_j,t — Sector allocation effect for sector j and period t

U_j,t — Allocation and selection interaction effect for sector j and period t

I_j,t — Issue selection effect for sector j and period t

S_j,t — Sector allocation effect for sector j over multiple periods

U_j,t — Allocation and sector interaction effect for sector j over multiple periods

I_j,t — Issue sector effect for sector j over multiple periods

In Menchero's optimized linking algorithm shown above, the optimized linking coefficient β_t is computed for each period using r_P, r_B, r_P,t, r_B,t, and T. This algorithm allows extending the Brinson model to multiple periods while achieving the resulting desirable properties that are similar to the geometric multiperiod performance attribution.

References

[1] Brinson, G. P. and Fachler, N. “Measuring Non-US Equity Portfolio Performance.” Journal of Portfolio Management. Spring 1985: 73–76.

[2] Brinson, G. P., Hood, L. R., and Beebower, G. L. “Determinants of Portfolio Performance.” Financial Analysts Journal. Vol. 42, No. 4, 1986: 39–44.

[3] Menchero, J. “Multiperiod Arithmetic Attribution.” Financial Analysts Journal. Vol. 60, No. 4, 2004: 76–91.

[4] Tuttle, D. L., Pinto, J. E., and McLeavey, D. W. Managing Investment Portfolios: A Dynamic Process. Third Edition. CFA Institute, 2007.

Version History

Introduced in R2022b

R2023a: Charting functions added for brinsonAttribution object

Using a brinsonAttribution object, you can use the following charting functions: categoryReturnsChart, categoryWeightsChart, and attributesChart.