Performance attribution

Performance measurement is an integral part of portfolio management. It tells us ‘how’ the portfolio has performed over a temporal difference or simply put, a time period. This can be quantified in numerous ways.

One example of performance measurement is, stock return.

Assuming, a portfolio of stocks, ‘A’, at time t = 0 was valued at K$. Over a time period of say 1 year, the portfolio at time t = 1 is now valued at P$, then the holding period return for that becomes:

or in log-return terms,

This tells us whether the stock has generated positive return or not, whether our investment return objectives have been met or not, or simply, if the required rate of return for a portfolio R(p) has been achieved or not (above which we discuss alpha).

What it does not tell us is, where this return originates from, or in clear terms the ‘source’ of the portfolio return.

This is where performance attribution methodology comes into play. Now, like with any field, be it Physics, Mathematics or Quant finance, what you think might be a novel addition to the field, has already been scoped out and researched upon by someone before you. So is the case with this one.

We have many in-house methodologies as well as industry standard methodologies that help us attribute the source of returns, more specifically, ‘active returns’.

Brinson-Hood-Beebower (BHB)

This model attempts to cleanly segregate the active return of a portfolio into three different terms:

Allocation (Underweight/Overweight decisions)

Where, ‘i’ represents sector, R(bi) is the sector return of benchmark, W(pi) and W(bi) are the sector weights of the portfolio and benchmark respectively.

The idea is simple, keeping the returns the same as the sectoral benchmark has earned, what is the effect of overweighting the sector (as evident with the weight difference terms). This gives us the ‘allocation effect’ of the decision making by a portfolio manager.

Security selection (stock picking)

Under the same notations, we can define the formula for security selection as,

This tells us, keeping the composition of stocks as the same, what effect does the security picking bring out (evident with the returns difference term for a sector).

We notice something similar, we are essentially keeping one or the other effect constant (calculation in vacuum) whilst calculating values for each effect. What if, a portfolio manager (in a concentrated portfolio) overweight the stocks he picked that out-perform or under-perform.

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This calls for a third term,

Interaction effect

This effect is quite controversial, some hold the view that allocation and selection are made by different people or processes (e.g. a top-down asset allocator vs a bottom-up stock picker), so any joint effect that can’t be cleanly attributed to either deserves its own bucket. It’s intellectually honest and says “we don’t know who to credit for this, so we will separate it out.”

Whilst, another school of thought says, the stock picker is the one who generated the excess return within the sector — it doesn’t matter whether the position was large or small, their skill drove it (as we will see, this will lead to another model under a similar setup).

Bidding adieu to the philosophical debate and grounding ourselves to factual mathematics, under the same notations we can define the interaction effect as,

(The controversial interaction term)

Another representation of the same (based on another school of thought) is including the interaction term in the selection effect, which means for the selection effect’s formulation, we replace the sector benchmark weight with portfolio sectoral weight.

The formula, for security selection then looks like,

(Under the second setup for interaction effect)

But, did you spot a flaw in the BHB’s approach to allocation effect?

Here is an example to illustrate the same,

In the mathematical case of where the R(pi) > R(bi), the allocation effect for BHB will always be positive, but what if the sector had under-performed the broad benchmark? The (+) effect from the model then falls apart.

BHB says: you overweighted by 15%, and the sector returned 10% absolutely, so you get credit for 1.50%. But notice — even if Tech had been a below-average sector, say returning 6% when the broader benchmark returned 9%, BHB would still give you positive allocation credit (0.15 × 0.06 = +0.90%) just because the absolute return was positive. That’s the flaw.

Brinson-Fachler (BF)

The formula for ‘Allocation effect’ under Brinson Fachler (which corrects for this) is as follows.

Brinson fachler’s Allocation effect term

It compares the sectoral benchmark return to the broad benchmark to assign an (+) or a (-) score to the effect. This makes the BF more reliable of the two.

Here is an easy way to look at both the models diagrammatically, firstly the BHB under -2- distinct setup.

(Credit to Linxiao Ma)

Then we have the Brinson Fachler,

(Credit to Linxiao Ma)

With that, we end today’s topic of the day — ‘Performance attribution’. Hope you learned something, and please don’t hesitate to reach out to me on Linkedin (for a philosophical or finance debate).