Data Warehousing, BI and Data Science

9 December 2018

Performance Attribution

Filed under: Data Warehousing — Vincent Rainardi @ 9:37 pm

In asset management, the business is about managing portfolios. Either it is a public fund where individual investors can put their money in, or a portfolio specifically created for one client such as a pension fund or an insurance.

Almost every portfolio has a benchmark*. The objective a portfolio is to beat the benchmark, meaning that the performance return of the portfolio should be higher than the performance of the benchmark. A benchmark is usually an index which operates in the same asset class, region and sector. For example, if the fund is UK equity, the benchmark can be FTSE 100 All Share.
*Note: an absolute return portfolio doesn’t usually have a benchmark as it is compared to zero (or 3 month Libor in some cases)

A portfolios grows over time, say 12% in the last 1 year. The benchmark to which the porfolio is compared to also grows, say 10% in the last 1 year. The art of understanding where this 2% difference is coming from is called performance attribution. Is it from the US companies or European companies? Is it from technology companies or mining companies?

Fortunately there are mathematics which can explain it. Knowing what makes this 2% or where this 2% is from is very important because it gives us insight about which parts of the portfolio are successful and which parts are not.

This 2% is called Relative Performance. Relative between the portfolio and the benchmark. It is also called Excess Performance or Active Performance. Sometimes people call it outperformance (or underperformance if the portfolio return is less than the benchmark). In this article I would like to describe how to allocate the excess performance into different geography regions/countries, different asset classes, different sectors or different currencies.

But before getting into that I would like give a background on what is a portfolio, what is a holding, what is performance return, what is a benchmark, and finally what is performance attribution. I hope with this background/introduction the readers without any background in investment management will be able to understand what performance return is, before getting into performance attribution.

Part 1. Portfolio, Weight, Benchmark and Performance

In a portfolio there are many assets. They are known as holdings. A holding can be a share, a bond or any other type of security. And cash. A global equity portfolio for example, can contain 50 holdings, some of which are shares of US companies, some are shares of European companies, some are companies in Asia, UK, Japan and Latin America.

Let’s say that the total value of this portfolio is 100 million USD. This $100m is called AUM (Asset Under Management). Let’s say that we invested in 10 US companies, and we put $5 million each, so the value of our holdings in these 10 US companies are $50m. That is 50% of the AUM. This 50% is called weight.

Each of the 50 companies in the portfolios has a share price. The share price changes every day. Some increases, some decreases. Let’s say that from the total value of $100m in 31st Oct 2017, within 1 year it became $111.32m in 31st Oct 2018, an increase of 11.32%. This 11.32% is called Performance Return. We say that our portfolio has a performance return of 11.32%.

In daily conversation people don’t say the “Return”, they just say “performance”. We can measure the performance of the fund in the last 1 month, in the last 3 months, in the last 6 months or in the last 1 year. So if today is 3rd of December 2018, the 1 month performance of our fund is measured by comparing the total value of the fund (the AUM) on 31st Oct 2018 to 30th Sep 2018. The 3 months performance is measured by comparing 31st Oct to 31st July 2018. And 1 year performance is comparing 31st Oct 2018 (the last month end) to 31st Oct 2018 (1 year before).

Accumulative and Annualised Return

In addition to 1,3,6 and 12 months, we also have 3 years, 5 years and 10 years performance return. We can present the performance returns greater than 1 year in two different ways. The first one is called accumulative, i.e. for 3 years we compare the value of the portfolio on 31st Oct 2018 (the latest month end) to the value of the portfolio on 31st Oct 2015 (3 years before). Let’s say that the value of the portfolio in 3 years grows from $100m to $133.1m, so it grows by 33.1%. This 33.1% is the accumulative performance return for 3 years.

The second is called annualised return. What is x, so that if the portfolio grows by x every year for 3 years, it grows from 100m to 133.1m? In this case it is 10%. From $100m it becomes $110m, then £121m, then $133.1m. This 10% is called annualised performance return.

Arithmetic and Geometric Annualised Return

When calculating annualised return we have two options. From the 3 years accumulated return of 33.1% we can divide by 3, getting 11.033% per year. This is called arithmetic annualised return.

The other way is to find x% where $100m * (1+x) * (1+x) * (1+x) = $133.1m. And in this case that x is 10%. This is called geometric annualised return.

We can see here that the geometric annualised return (10%) is lower than the arithmetic annualised return (11.033%). This is because of compound growth, i.e. the amount of growth in year 2 is bigger than the growth in year 1, and the amount of growth in year 3 is bigger than year 2 (for the same % growth).

Geometric annualised return is the real thing. It is the right thing to use. Whereas arithmetic annualised return is technically incorrect, so should not be used. It is only there for a quick “back-of-the-envelope” calculations.

Gross and Net Returns

As a fund manager we charge fees, so there are two option: before the fees are deducted (called gross) or after the fees are deducted (called net). If a fund is growing by 10% per year (that’s the gross performance return), and the total of fees and charges are 1%, the net performance return is 9%.

When investors ask “What’s the performance of your fund?”, they usually mean the net return, i.e. after the costs and charges/fees. But a fund manager usually prefer to specify the gross return, because that’s what she/he delivers (the charges are imposed by the company, not them) and because the gross return is higher, so sounds better.

Usage of Performance Return

Investors don’t like performance return numbers which is less than one year. Many pension funds prefer to invest only in funds which is 3 years old or more. They would like to know the 3 years annualised net performance of the fund, to be compared with the 1 year net return to understand the long term and short term view on performance. And to be compared with the volatility (also 1 year and 3 years) to understand the risk, as well as to Alpha, Beta and Sharpe numbers.

Part 2. Performance Attribution

Now that we know what performance return is, let’s talk about performance attribution, which is the topic of this article.

Let’s have a look at this global equity portfolio:

Region Portfolio Weight Benchmark Weight Portfolio Return Benchmark Return Stock Selection Regional Allocation Interaction Factor Total
US 50% 44% 15% 12% 1.32% 0.19% 0.18% 1.69%
UK 10% 15% 10% 8% 0.30% 0.04% -0.10% 0.24%
Europe 18% 14% 5% 3% 0.28% -0.23% 0.08% 0.13%
Asia Pacific 3% 9% 2% 4% -0.18% 0.29% 0.12% 0.23%
Japan 14% 8% 12% 8% 0.32% -0.05% 0.24% 0.51%
Other 3% 10% 6% 9% -0.30% -0.01% 0.21% -0.10%
Cash 2% 0% 0% 0% 0.00% -0.18% 0.00% -0.18%
Total 100% 100% 11.32% 8.80% 1.74% 0.05% 0.73% 2.52%

Table 1: Example of a Global Equity Portfolio

Portfolio and Benchmark Weight and Return

The performance return of this portfolio is 11.32% (let’s say this is 1 year performance), and the performance return of the benchmark is 8.80%. So the relative performance between the portfolio and benchmark is 11.32% -8.80% = 2.52%.

The portfolio contains equities from the US, UK, Europe (excluding UK), Japan, Asia Pacific (excluding Japan) and other region (such as Africa or Latin America). In the second column we can see that the US equity is 50% of the portfolio, UK is 10% of the portfolio, and so on. The portfolio also contains 2% cash.

The benchmark also contains US, UK, Europe, Japan, Apac and other, but the percentages (the “weights”) are different. US occupies 44% of the benchmark, UK 15%, Europe 14% and so on. The benchmark does not contain any cash. We can see this in the third column.

In the fourth column is the portfolio return. We can see that the US equity grew by 15%, UK by 10%, Europe by 5%, and so on. And the total at the bottom is 11.32%. This is the growth of the portfolio as a whole.

In the fifth column we can see the performance return of the benchmark. The US equity grew by 12%, UK equity by 8%, Europe by 3% and so on. And at the bottom we can see that the benchmark as a whole grew by 8.8%.

The job of a fund manager is to invest in the region which will have most growth, i.e. US, UK and Japan.

Performance Attribution

We are now ready to discuss the last 4 columns: the performance attribution columns. On the last column we can see what makes up the 2.52% difference between the portfolio return and the benchmark return, i.e. 1.69% is because of US equity, 0.24% is because of UK equity, 0.13% is because of Europe equity, and so on.

Region Portfolio Weight Benchmark Weight Portfolio Return Benchmark Return Stock Selection Regional Allocation Interaction Factor Total
US 50% 44% 15% 12% 1.32% 0.19% 0.18% 1.69%
UK 10% 15% 10% 8% 0.30% 0.04% -0.10% 0.24%
Europe 18% 14% 5% 3% 0.28% -0.23% 0.08% 0.13%
Asia Pacific 3% 9% 2% 4% -0.18% 0.29% 0.12% 0.23%
Japan 14% 8% 12% 8% 0.32% -0.05% 0.24% 0.51%
Other 3% 10% 6% 9% -0.30% -0.01% 0.21% -0.10%
Cash 2% 0% 0% 0% 0.00% -0.18% 0.00% -0.18%
Total 100% 100% 11.32% 8.80% 1.74% 0.05% 0.73% 2.52%

Table 2: Performance Attribution – Regional Level

But not only that, we can also breakdown this 2.52% outperformance into:

  1. How much of it is because the allocation into regions
  2. How much of it is because of the stocks/shares that were chosen within each region
  3. And how much of it is because of the combination of #1 and #2 above

In the above case, we can see these 3 numbers down at the bottom of the table:

  1. 05% is because of the allocation into regions.
    For example, the PM (portfolio manager) chose to allocate 50% into US, which enjoyed a high growth (15%).
  2. 74% is because of the stocks/shares that were chosen within each region.
    For example, the US stocks raised by 15% whereas the benchmark raised only by 12%. If we use the benchmark weighting (44%) for both the portfolio and benchmark, we get:
    Performance of the US equity in the portfolio = 44% x 15% = 6.6%
    Performance of the US equity in the benchmark = 44% x 12% = 5.28%
    So the portfolio is outperforming the benchmark by 1.32% as we can see on the table above (the “Stock Selection” factor for US)
  3. 73% is because of the combination between stock selection factor and the sector allocation factor.

For a credit portfolio we have one more factor that we can calculate: the yield curve. This is about how much performance is caused by allocating the portfolio into different maturity buckets, e.g. 1 year, 3 years, 5 years, 10 years, 20 years, etc. Generally speaking, in a normal market condition, the longer the maturity the higher the performance.

Last but not least, each of the region attribution (1.69% for US equity for example) can be allocated into stock selection, sector allocation and interaction.

All this is incredably useful for portfolio managers so that they know whether their allocation is correct or not, and whether their stock selection is correct or not. Note that the words “stock selection” is not accurate for a credit portfolio. “Security selection” is a better choice of words.

Sometimes Performance Attribution is also called Performance Contribution, i.e. how much each factor (or region) contributes to the overall performance.

Calculation

Finally we come to the core of this article: the calculation.

Region Portfolio Weight Benchmark Weight Portfolio Return Benchmark Return Stock Selection Regional Allocation Interaction Factor Total Contribution
US PW BW PR BR SS RA IF TA
UK
Europe
Asia Pacific
Japan
Other
Cash
Total 100% 100% Total PR Total BR Total TA

Table 3: Performance Attribution Calculation

The calculations of the Stock Selection (SS), Regional Allocation (RA), Interaction Factor (IF) and Total Contribution (TC) columns are as follows:

  1. SS = BW x (PR – BR)
  2. RA = (PW – BW) x (BR – Total BR)
  3. IF = (PW – BW) x (PR – BR)
  4. TC = SS + RA + IF
  5. Total PR = Sum (PW x PR)
  6. Total BR = Sum (BW x BR)
  7. Total TC = Sum (TC) = Total PR – Total BR

Logic Behind The Calculations

Now let’s try to understand why the formula for the Stock Selection (SS), Regional Allocation (RA), Interaction Factor (IF) and Total Contribution (TC) columns are as above.

1. Stock Selection

Stock Selection is how different the US stocks/securities in the portfolio perform, compared to the US stocks in the benchmark. So to understand that we use the same weighting for both the portfolio and the benchmark.

In the above case, the US stock has a weight of 50% in the portfolio and 44% in the benchmark. It has 15% performance in the portfolio and 12% in the benchmark. So the performance attribution of the US stock in the portfolio is 50% x 15% = 7.50%. And the performance attribution of the US stock in the benchmark is 44% x 12% = 5.28%.

But to understand the effect of just stock selection, we have to use the same allocation. So let us use 44% for both of them. The performance attribution of the US stock in the portfolio is now 44% x 15% = 6.6%. And the performance of the US stock in the benchmark is still the same as above, which is 5.28%.

Because we use the same allocation (44%) for both the portfolio and the benchmark, the effect we are observing now is purely because of stock selection. The difference between 6.6% and 5.28% is purely because of stock selection. 6.6% minus 5.28% = 1.32% – that is the effect of stock selection.

That is why SS = BW x (PR – BR). It is the difference in return, when the weight is kept constant.

2. Regional Allocation

So how do we find out what is the effect of regional allocation? By keeping the return constant.

So something like this:
RA = (PW – BW) x BR

In the above, we kept the return constant (BR) and so we can understand the effect of the weighting.

Putting in the numbers for the US stocks:
RA for US = (50% – 44% ) x 12% = 6% x 12% = 0.72%

But 12% is not quite right. Because 12% is the return of the US stock. What we need is how well the US stocks perform in comparison to the other regions. Well, as a whole, the benchmark returned 8.8%. So the US is returning 3.2% above the average. That is the figure we need. So the formula becomes:

RA = (PW – BW) x (BR – Total BR) = (50% – 44%) x (12% – 8.8%) = 6% x 3.2% = 0.192%.

This is fairer. The effect of allocating more to the US is not 0.72% but only 0.192%, because it is not out of 12% (the return of US stocks) but out of 3.2% (the outperformance of the US stock compared to the other regions).

3. Interaction Factor

To calculate the Interaction Factor we must not kept the weight constant, and we must not kept the return constant either. We should use the actual differences in weight and return.

The differences in weight for the US stocks is 50% – 44% = 6%
The differences in return for the US stocks is 15% – 12% = 3%

If we multiply them we get the effect caused by the differences of both the weight and the return.

Hence the formula for the Interaction Factor is (PW – BW) x (PR – BR).

4. Total Contribution

Total Contribution is the sum of the 3 factors above, i.e.

TC = SS + RA + IF

Country Level

We can use this method to understand the effect of allocation not only at region level, but also at country level. So in the above, how each country in Europe contributed to the total European return.

For example, it’s like this:

Country Port Weight PW in Europe Bench Weight BW in Europe Port Return Bench Return Stock Selection Country Allocation Interaction Factor Total Contribution
France 4% 22.2% 2% 14.3% 4% 3% 0.14% 0.00% 0.08% 0.22%
Germany 2% 11.1% 1% 7.1% 8% 2% 0.43% -0.04% 0.24% 0.63%
Italy 1% 5.6% 2% 14.3% 4% 4% 0.00% -0.09% 0.00% -0.09%
Ireland 1% 5.6% 3% 21.4% 5% 4% 0.21% -0.16% -0.16% -0.10%
Netherland 2% 11.1% 1% 7.1% 4% 2% 0.14% -0.04% 0.08% 0.18%
Norway 5% 27.8% 3% 21.4% 7% 2% 1.07% -0.06% 0.32% 1.33%
Spain 3% 16.7% 2% 14.3% 2% 3% -0.14% 0.00% -0.02% -0.17%
Total Europe 18% 100% 14% 100% 5.00% 3.00% 1.86% -0.39% 0.53% 2.00%

Table 4: Performance Attribution – Country Level

Here we breakdown the 2% outperformance in European tocks (5% in the portfolio vs 3% in the benchmark) into 0.22% is from France, 0.63% is from Germany, -0.09% is from Italy, and so on.

And within 0.22% for France we further break it down to 0.14% is from Stock Selection, 0% is from Regional Allocation and 0.08% is from Interaction Factor.

They key in breaking down the outperformance from Region level to Country level is to compute the Portfolio Weight in Europe and Benchmark Weight in Europe columns. These 2 columns are calculated by dividing the Portfolio Weight (4% for France, 2% for Germany, and so on) by the total Europe weight (18%). So France’s portfolio weight in Europe is 4% / 18% = 22.2%. Germany portfolio weight in Europe is 2% / 18% = 11.1%. Then we do the same thing for the Benchmark Weight in Europe (BW in Europe column).

The question remain as to why in the country level the total of Stock Allocation for Europe is 1.86% where as in the Region level the total of the Stock Allocation for Europe is 0.28% (see table 5 and 6 below).

Region Portfolio Weight Benchmark Weight Portfolio Return Benchmark Return Stock Selection Regional Allocation Interaction Factor Total Contribution
US
UK
Europe 18% 14% 5% 3% 0.28% -0.23% 0.08% 0.13%
Apac
Japan
Other
Cash
Total 100% 100% 11.32% 8.80% 1.74% 0.05% 0.73% 2.52%

Table 5: European Contribution = 0.13%

Country Portfolio Weight Benchmark Weight Portfolio Return Benchmark Return Stock Selection Country Allocation Interaction
Factor
Total
Contrib
ution
France
Germany
Italy
Ireland
Netherland
Norway
Spain
Total Europe 18% 14% 5.00% 3.00% 1.86% -0.39% 0.53% 2.00%

Table 6: European Contribution = 2%

This is because when calculating the 0.28% we are trying to allocate the 2.52% into region level (table 5). In context of the 2.52%, the Europan stocks get 0.13% and the European Stock Selection get 0.28%. Where as when calculating the 1.86% we are trying to allocate the 2% European outperformance into countries. In this context the total of Stock Selection for Europe is 1.86%. So both are right, it’s just that they have different context. One is global, and the other is within Europe.

Sector, Currency, Asset Class

In addition to the performance attribution by region and country, we can also produce the performance attribution by industry sector, currency and asset class, like this:

Industry Sector Portfolio Weight Benchmark Weight Portfolio Return Benchmark Return Stock Selection Sector Allocation Interaction Factor Total Contribution
Energy 3.7% 7.5% 3.2% 4.3% -0.08% 0.11% 0.04% 0.07%
Materials 5.3% 8.9% 4.7% 5.2% -0.04% 0.07% 0.02% 0.04%
Industrials 6.8% 9.3% 5.7% 6.1% -0.04% 0.03% 0.01% 0.00%
Consumer Discret. 8.9% 9.3% 9.4% 7.3% 0.20% 0.00% -0.01% 0.19%
Consumer Staples 14.5% 8.8% 11.3% 9.7% 0.14% 0.15% 0.09% 0.38%
Health Care 11.6% 9.4% 7.1% 7.7% -0.06% 0.01% -0.01% -0.06%
Financials 7.7% 13.6% 4.3% 5.7% -0.19% 0.09% 0.08% -0.02%
IT 8.6% 11.2% 10.6% 8.2% 0.27% -0.03% -0.06% 0.18%
Communication 6.8% 8.1% 9.3% 10.3% -0.08% -0.04% 0.01% -0.11%
Utilities 13.8% 7.2% 8.9% 6.7% 0.16% -0.03% 0.15% 0.27%
Real Estate 10.3% 6.7% 9.4% 7.9% 0.10% 0.03% 0.05% 0.18%
Cash 1.9% 0.0% 0.0% 0.0% 0.00% -0.14% 0.00% -0.14%
Total 100.0% 100.0% 8.13% 7.15% 0.37% 0.24% 0.37% 0.98%

And this:

Currency Portfolio Weight Benchmark Weight Portfolio Return Benchmark Return Stock Selection Currency Allocation Interaction Factor Total Contribution
USD 27.9% 24.3% 6.7% 5.6% 0.27% -0.04% 0.04% 0.26%
GBP 19.5% 15.6% 5.6% 5.3% 0.05% -0.06% 0.01% 0.00%
EUR 24.8% 28.4% 7.8% 6.2% 0.45% 0.02% -0.06% 0.42%
CAD 6.9% 8.8% 9.4% 8.9% 0.04% -0.04% -0.01% -0.01%
JPY 11.3% 13.6% 11.7% 9.4% 0.31% -0.06% -0.05% 0.20%
AUD 7.7% 9.3% 6.7% 8.5% -0.17% -0.03% 0.03% -0.17%
Other 1.9% 0.0% 0.0% 0.0% 0.00% -0.13% 0.00% -0.13%
Total 100.0% 100.0% 7.38% 6.80% 0.96% -0.34% -0.04% 0.58%

Knowing how the currency allocation affect the performance is useful for the people on the desk (PM and the analysts). And so is knowing how the industry sector allocation affect the performance. It helps the desk knowing where their outperformances are from.

And like when regions can be further broken down into countries, the sectors can further be broken down into industries.

The key to be able to calculate all the above is: a) knowing the value of each position at the beginning and at the end of the period, and b) putting the country of risk, industry sector, currency and asset class for every position in the holding and benchmark. That’s what enables us to group each holding into regions/countries, industries/sectors, currency and asset class/sub class, and table out the weights and performances for each section, both portfolio and benchmark. And from there we can apply the above logic/method to calculate the performance attribution.

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