In a previous article, I sang the praises of the Quintessence Implementation Team. I would now like to take some time to focus on one key capability within this team – the business analysts. The best way to describe this capability is to present their work. Towards this end, I will focus a series of articles on how arithmetic performance attribution was tamed by the Quintessence business analysts. I have chosen this example as it is a familiar problem in the investment management industry. Furthermore, many investment professionals understand the challenges that arise when using one particular methodology over another and will thus understand how our analysts overcame these obstacles.

What is Attribution Analysis?

Attribution analysis encompasses a variety of algorithms and methodologies designed to quantify the value (alpha) of the various investment decisions made in managing a fund. The particular methodology implemented is dependent on the investment process of the management firm under consideration. For example, bottom-up equity stock-pickers are only interested in stock selection effects. Asset allocation and sector selection effects come to the fore when the investment process implements a top-down allocation strategy that constrains the bottom-up company selection process.

There are various flavours of attribution analysis. The underlying mathematics changes drastically when one is dealing with different instrument types. For example, fixed-income attribution breaks the alpha into parts that are due to

  • the yield of the bond instrument,
  • rolling down the yield curve,
  • movements in the reference yield curve and
  • credit shifts.

On the other hand, equity attribution focuses on breaking the alpha into

  • a selection effect and
  • an allocation effect.

Attribution analysis algorithms can also be formulated using a geometric as opposed to an arithmetic model. Geometric models have the trait that geometric alphas for sub-periods “multiply-up” to the total geometric alpha over the whole period. On the other hand, most arithmetic models suffer from the characteristic that arithmetic alphas for sub-periods do not “add-up” to the total arithmetic alpha over the said period.

The graphic below shows an example of the generic inputs, data requirements and consequent output of performance attribution algorithms.

Attribution Analysis Input and Output

Single Tier Arithmetic Attribution Analysis with no Trading Execution Effect

Most arithmetic attribution analysis algorithms are based on the Brinson–Hood–Beebower and Brinson–Fachler models. As mentioned, arithmetic attribution models suffer from the malady that the attribution alphas don’t “add-up” over multiple periods. This is the one challenge that our analysts solved. In this first article, I will therefore consider the special case when

  • no classification structure is imposed on the portfolio and the benchmark and
  • no trading happens during the period under consideration.

In other words, I am dealing with

  • a single tier example where the constituents of the portfolio and the benchmark are not categorised in any particular way, be that in asset classes or sectors. Thus only stock selection and allocation effects are relevant.
  • a portfolio where no trades occur during the period under investigation i.e. the instrument return in the portfolio is equal to the instrument return in the benchmark for the period under consideration. This simplifies the algorithm significantly as only one attribution component remains in the Brinson, Hood and Beebower methodology published in 1986.
DescriptionNotationRelationship
Total Return in day t by Instrument k in the Benchmark\rho_t(k)See Equation 3 in the table below.
Total Return in day t by Instrument k in the Portfolior_t(k)See Equation 3 in the table below.
Trading execution effectThe instrument return in the portfolio is the same as that in the benchmark. This will imply no trading execution effect as shown in the rows below.r_t(k)=\rho_t(k)
Stock Selection Effect According to the Brinson, Hood and Beebower Model published in 1986, this will be zero.
(See Equation 1,2 and 3 in the table below.)
\sum\limits_{k}{B_t(k)\times(r_t(k)-\rho_t(k))=0}
Interaction EffectAccording to the Brinson, Hood and Beebower Model, this will also be zero. (See Equation 1,2 and 3 in the table below.)\sum\limits_{k}{(P_t(k)-B_t(k))\times(r_t(k)-\rho_t(k))=0}

This simplification allows one to easily formulate the problem statement and identify the “additive” issues that arise. The solution can then be extended (in further articles) to include multi-tier cases and trading execution effects, resulting in the arithmetic performance attribution malady being tamed by the Quintessence business analysts .

Single Tier Arithmetic Performance Attribution tamed

Consider a portfolio and a benchmark. The union of the universe of instruments in the portfolio and benchmark are subscripted by k. For a given day t, we define the following variables:

DescriptionNotationRelationshipsEquation #
Portfolio weight P in Instrument k at the start of day tP_t(k)\sum\limits_{k}{P_t(k)=1}1
Benchmark weight B in Instrument k at the start of day tB_t(k)\sum\limits_{k}{B_t(k)=1}2
Total Return \rho (the greek letter rho) in day t by Instrument k
Note that since we assume no trades, we only need one variable \rho_t(k).
In later articles when we incorporate the trading execution effect, we will require r_t(k), the total return of the instrument in the portfolio and \rho_t(k), the total return of the instrument in the benchmark.
\rho_t(k)3
Portfolio Total Return r in day tr_tr_t = \sum\limits_{k}{P_t(k) \times \rho_t(k)4
Index Total Return \rho (greek rho) in day t\rho_t\rho_t = \sum\limits_{i}{B_t(k) \times \rho_t(k)5
Arithmetic Alpha in day t for the portfolio relative to the benchmark.\alpha^A_t\alpha^A_t = r_t - \rho_t6
Geometric Alpha in day t for the portfolio relative to the benchmark\alpha^G_t\alpha^G_t = \frac{1 + r_t}{1 + \rho_t} - 17

Some important points are relevant to the above definitions. Equations 4 and 5 (and consequently equation 9 further down) hold if and only if

  • the weights are taken at the beginning of the day.
  • the holdings remain constant over the day under consideration.

In our definitions, the subscript t refers to a day. This can however be any fixed period. One can think of this subscript as a point in time. The only constraint is that between t and t+1, the holdings remain fixed. Now, the purpose of attribution analysis is to break down the total alpha (Equation 6) for a period into the composite parts that represent the investment decision-making process. For a single day, the formula to calculate the arithmetic contribution to the alpha from the active bet in the underlying instrument is provided in equation 8 below.

DescriptionNotationRelationshipsEquation #
Contribution to arithmetic alpha from Instrument k for day t\varepsilon_t(k)\varepsilon_t(k) = [P_t(k) - B_t(k)] \times [\rho_t(k) - \rho_t]8

Using equations 1,2,4 and 5, one can show that this formula behaves as expected; the contributions from each instrument add up to the total alpha for the day.

DescriptionProofEquation #
The proof that the contributions of each instrument adds up to the arithmetic alpha of the day\begin{aligned} \sum\limits_{k} \varepsilon_t(k) &= \sum\limits_{k}([P_t(k) - B_t(k)] \times [\rho_t(k) - \rho_t]) \\&=  \sum\limits_{k}P_t(k)\rho_t(k) - \rho_t \sum\limits_{k}P_t(k) + \rho_t \sum\limits_{k}B_t(k) - \sum\limits_{k}B_t(k)\rho_t(k) \\&=r_t - \rho_t \end{aligned}9

We can thus conclude equation 10 below:

DescriptionProofEquation #
Contributions of all instruments add up to arithmetic alpha for the day  \sum\limits_{k} \varepsilon_t(k) = \alpha^A_t10

What the above analysis shows is that for a period where the holdings remain constant (in both the portfolio and the benchmark), Equation 8 above will give the correct contribution to arithmetic alpha. The next section will show that this simple algorithm hits problems when one calculates contributions over multiple days. In particular, this is due to the fact that alphas compound – simply adding up alphas does not incorporate this compounding effect.

Geometric versus Arithmetic Alpha

It is common knowledge that the simple analysis above breaks down when one tries to perform the calculation described above over multiple periods. It needs to be emphasized that this statement is true even if the portfolio and benchmark holdings remain constant over the period under analysis. Equation 7 above defined the concept of a geometric alpha for the sole purpose of comparing it to the arithmetic alpha. In particular, the key difference between the geometric and the arithmetic alphas defined in equations 6 and 7 is that daily geometric alphas can be multiplied up to obtain the geometric alpha over the whole period. This is because of the compound nature of returns, as shown in the following derivation:

DescriptionProofEquation
Daily Geometric Alphas multiply up correctly to give one the Geometric Alpha for the whole period. \begin{aligned}\alpha^G_T + 1 &= \frac{1 + r_\tau}{1 + \rho_\tau} \\&= \frac{\Pi_t(1 + r_t)}{\Pi_t(1 + \rho_t)} \\&= \Pi\limits_{t}(\frac{1 + r_t}{1 + \rho_t}) \\&= \Pi\limits_{t}({\alpha^G_t} + 1) \end{aligned}11

It is important to observe that this property is not shared in the case of arithmetic alphas. In other words, arithmetic alphas do not add up to the total arithmetic alpha over the whole period. A simple example will clarify this point. Consider a period of 2 days. To emphasise the point that this has nothing to do with trading or portfolio / benchmark changes, let us assume that the holdings remained constant over this period. Assume the following numbers for the portfolio and benchmark total return:

DescriptionCalculationRef #
Portfolio Total Returns in day 1 and day 2 r_1 = 1\% = 0.01
\\r_2 = -0.5\% = -0.005
12
Benchmark Total Return in day 1 and day 2  \begin{aligned} \rho_1 &= 0\% = 0 \\ \rho_2 &= 0.5\% = 0.005 \end{aligned}
Arithmetic Alphas for day 1 and day 2\begin{aligned} \alpha_1 &= r_1 - \rho_1 = 1\% = 0.01 \\ \alpha_2 &= r_2 - \rho_2 = -1\% = -0.01 \end{aligned}
Portfolio Total Return for whole period T = [1,2]\begin{aligned}  r_T&=(1+r_1 )(1+r_2 )-1 \\&=0.495\% \\&=0.00495 \end{aligned}
Benchmark Total Return for whole period T = [1,2]\rho_T = 0.5\% = 0.005
Arithmetic Alpha for period T = [1,2]\begin{aligned} \alpha^A_T &= r_T-\rho_T \\&=0.495\%-0.5\% \\&=-0.005\% \\&=-0.00005 \\&\ne\alpha^A_1+\alpha^A_2 \end{aligned}

What example 12 above shows is that the daily arithmetic alphas do not add up to the total alpha for the period. This is precisely the arithmetic attribution challenge that needs to be tamed by the Quintessence analyst.

DescriptionRelationshipEquation #
Daily arithmetic alphas do NOT add up to the arithmetic alpha of the period.\alpha^A_T \ne \sum\limits_{t}\alpha^A_t13

The fact that one can’t just add up arithmetic contributions (Equation 13) over multiple periods means one cannot define the total contribution of an instrument over the whole period as just being the sum of its daily contributions:

DescriptionExpressionEquation #
The contribution of an Instrument k over the whole period T\varepsilon^*_T(k)14
Defining the total contribution of an instrument for the whole period as being the sum of the daily contributions will not add up to the correct alpha due to equation 13:\begin{aligned} \sum\limits_{k}(\sum\limits_{t}\varepsilon_t(k)) &=  \sum\limits_{t}(\sum\limits_{k}\varepsilon_t(k)) \\ &=  \sum\limits_{t}\alpha_t \\ &\ne \alpha^A_T  \end{aligned}15
We therefore cannot define the contribution of an instrument over the whole period T to be the simple sum of the daily contributions\varepsilon^*_T(k) \ne \sum\limits_{t}\varepsilon_t(k)16

Arithmetic Alpha with Adjustment

Everything discussed above is common knowledge; nothing new or unexpected. This is why some people choose to only consider geometric attribution methodologies. Some investment managers however, find geometric contributions counter intuitive. They want a methodology whereupon contributions add up to the total alpha for the day. For this reason, our analysts identified a mechanism to make this possible.

Equations 17 and 18 below is the key to the methodology implemented.

Firstly, we define \Delta\alpha_t.

DescriptionRelationshipEquation #
Define \Delta\alpha^A_t as being a value that ensures that the daily arithmetic alphas add up to the period arithmetic alpha. This must be true for all periods 1 ≤ τ ≤ T\begin{aligned} \alpha^A_\tau &= r_\tau - \rho_\tau \\&= \Sigma\limits_{n=1}^{\tau}[\alpha^A_n + \Delta\alpha^A_n] \end{aligned}17

It is important to note that the criteria that the numbers sum correctly for any period completely determines \Delta\alpha_t. In particular, we can conclude that \Delta\alpha_1=0. Further, it can be shown that for 2\le t \le T

DescriptionEquationEquation #
One can prove that \Delta\alpha_t defined here in Equation 18 uniquely satisfies equation 17 above\Delta\alpha^A_t = r_t[\Pi\limits_{n=1}^{t-1}(1 + r_n) - 1] - \rho_t[\Pi\limits_{n=1}^{t-1}(1 + \rho_n) - 1]18

Looking back at example 12, we see how equation 18 ensures that the daily arithmetic alphas now add up correctly.

DescriptionCalculationReference #
Portfolio Total Returns in days 1 and 2 \begin{aligned}r_1 &= 1\% = 0.01 \\r_2 &= -0.5\% = -0.005 \end{aligned}19
Index Total Return in days 1 and 2  \begin{aligned} \rho_1&=0\%=0 \\\rho_2&=0.5\%=0.005 \end{aligned}
\Delta\alpha_t for days 1 and 2 as per equation 17 \begin{aligned} \alpha^A_1 &= r_1-\rho_1 = 0.01 - 0 = 0.01 \\\alpha^A_2 &= r_2-\rho_2 =-0.005 - 0.005 = -0.01 \end{aligned}
\Delta\alpha_t for days 1 and 2 as per equation 18 \begin{aligned} \Delta\alpha^A_1 &= 0 \\\Delta\alpha^A_2 &= -0.005 \times (1.01 - 1) \\&= -0.005 \times (1.00 - 1) \\&= -0.00005 \end{aligned}
Daily arithmetic alphas add up to the correct total alpha \begin{aligned} \alpha^A_T &= r_T - \rho_T \\&=-0.00005 \\&= (\alpha^A_1 + \Delta\alpha^A_1) + (\alpha^A_2 + \Delta\alpha^A_2 ) \end{aligned}

The problem now comes in deciding how to distribute this daily delta over all the instruments for the day. This is best achieved by stating it as an optimisation problem. Equations 20 to 24 define this non-linear quadratic optimisation problem:

DescriptionQuadratic Optimisation StatementEquation #
The variable we want to identify is the adjusted daily
contribution of each instrument to alpha.
\varepsilon^*_t(k)20
We want these adjusted contributions to be as close as possible to the original ones calculated using the simple formulae in equation 8. Mathematically, we therefore want to minimise the distance between the adjusted contribution and the original ones calculated using
Equation 8. We take the standard Euclidean distance as our measure.
||\varepsilon\limits^{\rightarrow}_t - \varepsilon\limits^{\rightarrow_*}_t|| = \sum\limits_{k}(\epsilon_t(k) - \epsilon^*_t(k))^221
The adjusted contributions must add up to the adjusted alphas for the day. Because of equation 17, this will ensure that we can then define the total contribution for an instrument over the whole period to be the sum of the daily contributions.\sum\limits_k\varepsilon^*_t(k) = \alpha^A_t + \Delta\alpha^A_t22
Contribution order must be preserved. The instrument contributions calculated in equation 8 are exact for 1 day. Therefore, if (using equation 8) instrument i contributed more than instrument j on day t then this must remain the same when calculating the adjusted contributions.[\varepsilon^*_t(i) - \varepsilon^*_t(j)] \times [\varepsilon_t(i) - \varepsilon_t(j)] \geq 023
The relative contribution size must also remain identical\frac{\varepsilon^*_t(i) - \varepsilon^*_t(j)}{\varepsilon_t(i) - \varepsilon_t(j)} = 124

The solution to this optimisation problem is shown in equation 26 below:

DescriptionQuadratic Optimisation SolutionEquation #
Define the total number of instruments in the union of the portfolio and benchmark on day tm_t25
Adjusted Contribution to arithmetic alpha from Instrument k for day t\varepsilon^*_t(k) = \varepsilon_t(k) + \frac{\Delta\alpha^A_t}{m_t}26
The Total Contribution from Instrument k for the whole period is defined as the sum over the daily contribution\begin{aligned} \varepsilon^*_T(k) &= \sum\limits_{t}\varepsilon^*_t(k) \\&= \sum\limits_{t}[\varepsilon_t(k) + \frac{\Delta\alpha^A_t}{m_t}] \end{aligned}27
The total contributions add up correctly to the total alpha\begin{aligned} \sum\limits_{k}\varepsilon^*_T(k) &= r_\tau - \rho_\tau \\&= \alpha^A_\tau \end{aligned}28

These 4 expressions summarise how arithmetic attribution was tamed by the Quintessence analysts, ensuring that the subperiod alphas add up to the alphas over the complete period.

DescriptionEquationEquation #
Simple Contribution to arithmetic alpha from Instrument k for day t\epsilon_t(k) = [P_t(k) - B_t(k)] \times [\rho_t(k) - \rho_t]8
Daily Alpha Adjustment to ensure correct period summation\Delta\alpha^A_t = R_t[\Pi\limits_{n=1}^{t-1} (1 + r_n) - 1] - \rho_t[\Pi\limits_{n=1}^{t-1} (1 + \rho_n) - 1]18
Adjusted Contribution to arithmetic alpha from Instrument k for day t\epsilon^*_t(k) = \epsilon_t(k) + \frac{\Delta\alpha^A_t}{m_t}26
Total Contribution to arithmetic alpha from Instrument k for the whole period T\epsilon^*_\tau(k) = \sum\limits_{t}\epsilon^*_t(k)27

Two articles to come will show how these concepts are extended to cater for classification structures and execution costs. Feel free to contact me for more information regarding our domain knowledge and how we can assist you in overcoming your challenges.

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