Equity Return Attribution: Brinson-Hood-Beebower Model

The Brinson-Hood-Beebower (BHB) model follows from a couple of papers back in the 1980s that proposed a nice way of carving out various return effects.

The idea follows from a scenario like this: | Sector | Portfolio Weight | Benchmark Weight | Portfolio Return | Benchmark Return | |:-----:|:-----:|:-----:|:-----:|:-----:| | Industrials | 0.4 | 0.2 | 9% | 10% | Does it look like the manager should get a positive or negative credit for the higher allocation to industrials?

That's right.

Actually, this is a positive thing.

The benchmark returned 10%, and the manager overweighted this nice, positive return. That's a plus. How would you rate security selection within that sector?

Not really, no.

Yes!

You can just look at the return difference and see that the manager underperformed the benchmark within that sector. Now take a look at the whole portfolio across three sectors: | Sector | Portfolio Weight | Benchmark Weight | Portfolio Return | Benchmark Return | |:-----:|:-----:|:-----:|:-----:|:-----:| | Industrials | 0.4 | 0.2 | 9% | 10% | | Tech | 0.3 | 0.4 | 14% | 12% | | Energy | 0.3 | 0.4 | -6% | -3% | | **Total** | **100%** | **100%** | **6.0%** | **5.6%** |

The individual sector allocation effect for industrials is calculated as the weight difference multiplied by the benchmark return: >$$ (0.4 - 0.2) \times 10 \% = 2 \%$$. Do this for the other two. What's the total portfolio allocation effect?

Excellent!

Not quite; it's actually 1.1%, once you calculate the other two. >$$ \text{Tech: } (0.3 - 0.4) \times 12 \% = -1.2 \% $$ > >$$ \text{Energy: } (0.3 - 0.4) \times -3 \% = 0.3 \% $$

| Sector | Portfolio Weight | Benchmark Weight | Portfolio Return | Benchmark Return | Allocation Effect | |:-----:|:-----:|:-----:|:-----:|:-----:|:-----:| | Industrials | 0.4 | 0.2 | 9% | 10% | 2.0% | | Tech | 0.3 | 0.4 | 14% | 12% | -1.2% | | Energy | 0.3 | 0.4 | -6% | -3% | 0.3% | | **Total** | **100%** | **100%** | **6.0%** | **5.6%** | **1.1%**|

For security selection, you're again comparing the benchmark returns with the portfolio returns, using the benchmark weights. What individual sector selection effect do you find for industrials?

Perfect! Yes, it's calculated as:

Not quite. This is calculated as:

>$$ 0.2 \times (9 \% - 10 \%) = -0.2 \% $$ and the rest of the values can be filled in: | Sector | Portfolio Weight | Benchmark Weight | Portfolio Return | Benchmark Return | Allocation Effect | Selection Effect | |:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:| | Industrials | 0.4 | 0.2 | 9% | 10% | 2.0% | -0.2% | | Tech | 0.3 | 0.4 | 14% | 12% | -1.2% | 0.8% | | Energy | 0.3 | 0.4 | -6% | -3% | 0.3% | -1.2% | | **Total** | **100%** | **100%** | **6.0%** | **5.6%** | **1.1%**| **-0.6%** |

So consider this summary statement: "The manager demonstrated relatively poor stock security selection in this sector, but separately the choice to underweight the sector worked in the portfolio's favor." To which sector does this refer?

No, this was a good sector to *overweight*. And that's just what the manager did.

Not this one; the manager outperformed in terms of tech selection.

Yes. The manager underweighted energy, which was a good choice given its performance. But relative to the benchmark, the manager made some poor selections.

Finally, one thing just doesn't add up. You can see that the portfolio outperformed the benchmark by 0.4% in total. But then you have 1.1% for allocation effect and -0.6% for selection effect. There's an extra -0.1% unaccounted for to make this total add up. This is the interaction effect. It's not an error or rounding difference; it's how these effects work together, in a way. So for industrials, you have the product of the weight differences and the return differences to arrive at its individual interaction effect: >$$ (0.4 - 0.2) \times (9 \% - 10 \%) = -0.2 \%$$. What is the individual interaction effect for tech?

Good work! Energy can be calculated similarly, and the Brinson-Hood-Beebower model is complete:

Not quite. It's: >$$ (0.3 - 0.4) \times (14 \% - 12 \%) = -0.2 \%$$. Energy can be calculated similarly, and the Brinson-Hood-Beebower model is complete:

| Sector | Portfolio Weight | Benchmark Weight | Portfolio Return | Benchmark Return | Allocation Effect | Selection Effect | Interaction Effect |:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:| | Industrials | 0.4 | 0.2 | 9% | 10% | 2.0% | -0.2% | -0.2% | | Tech | 0.3 | 0.4 | 14% | 12% | -1.2% | 0.8% | -0.2% | | Energy | 0.3 | 0.4 | -6% | -3% | 0.3% | -1.2% | 0.3% | | **Total** | **100%** | **100%** | **6.0%** | **5.6%** | **1.1%**| **-0.6%** | **-0.1%** |

To summarize: [[summary]]

Positive

Negative

Positively

Negatively

Other responses

1.1

Other responses

-0.2

Industrials

Tech

Energy

Other responses

-0.2

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