Compute total performance attribution by Brinson model

## Syntax

``[AggregateTotalAttribution,PeriodicTotalAttribution] = totalAttribution(brinsonAttributionObj)``

## Description

example

````[AggregateTotalAttribution,PeriodicTotalAttribution] = totalAttribution(brinsonAttributionObj)` computes the total performance attribution of the portfolio, summed up for all categories, using the Brinson model.```

## Examples

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This example shows how to create a `brinsonAttribution` object and then use `totalAttribution` to compute the total performance attribution of the portfolio summed up for all categories (sectors).

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 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 ```

Use the brinsonAttribution object with `totalAttribution` to compute the total performance attribution of the portfolio summed up for all categories.

`[AggregateTotalAttribution, PeriodicTotalAttribution] = totalAttribution(BrinsonPAobj)`
```AggregateTotalAttribution=1×4 table Allocation Selection Interaction ActiveReturn __________ _________ ___________ ____________ -0.014353 0.030316 -0.010105 0.0058578 ```
```PeriodicTotalAttribution=3×6 table Period LinkingCoefficient Allocation Selection Interaction ActiveReturn ______ __________________ __________ _________ ___________ ____________ 1 0.97032 -0.019306 0.072391 -0.02413 0.028955 2 1.0838 0.011036 -0.046391 0.015464 -0.019891 3 1.0391 -0.0072954 0.0099623 -0.0033208 -0.00065389 ```

## Input Arguments

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Brinson attribution model, specified as a `brinsonAttribution` object.

Data Types: `object`

## Output Arguments

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Total attribution aggregated over all periods, returned as a table with the following columns:

• `Allocation` — Category allocation effect for the category

• `Selection` — Effect of selecting individual assets within the category

• `Interaction` — Allocation-selection interaction effect

• `ActiveReturn` — Active return compared with benchmark

Total attribution for each period, returned as a table with the following columns:

• `Period` — Time period numbers (1 for the first period, 2 for the second period, and so on)

• `LinkingCoefficient` — Linking coefficients used to aggregate the attribution results over all time periods

• `Allocation` — Category allocation effect for the category

• `Selection` — Effect of selecting individual assets within the category

• `Interaction` — Allocation-selection interaction effect

• `ActiveReturn` — Active return compared with benchmark

## 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