Active share worked stock by stock on one fund, checked by the overlap method, and turned into the fee the client pays for the part that is actually active.
Active share is half the sum of the absolute differences between the fund's weight and the index's weight in every stock held by either, or equivalently 100 per cent minus the overlap. On an illustrative 25-stock fund against its index it comes to 72.5 per cent. Divide the fee by it and the 0.75 per cent the client pays becomes 1.02 per cent on the part of the portfolio that is actually active.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A long-only equity fund holds 25 stocks. Six of them are among the index's ten largest names, fifteen are smaller index constituents and four are not in the index at all. The figures are illustrative and the index is shown in three blocks: its ten largest names, the fifteen smaller constituents the fund owns, and everything else.
| Stock | Index | Fund | |Fund − index| | Overlap (minimum) |
|---|---|---|---|---|
| A | 6.0 | 4.0 | 2.0 | 4.0 |
| B | 5.0 | 0.0 | 5.0 | 0.0 |
| C | 4.5 | 6.0 | 1.5 | 4.5 |
| D | 4.0 | 2.0 | 2.0 | 2.0 |
| E | 3.5 | 5.0 | 1.5 | 3.5 |
| F | 3.0 | 0.0 | 3.0 | 0.0 |
| G | 2.5 | 4.0 | 1.5 | 2.5 |
| H | 2.5 | 0.0 | 2.5 | 0.0 |
| I | 2.0 | 3.0 | 1.0 | 2.0 |
| J | 2.0 | 0.0 | 2.0 | 0.0 |
| 15 smaller index stocks held, 0.6 each in the index | 9.0 | 60.0 | 51.0 | 9.0 |
| 4 stocks outside the index | 0.0 | 16.0 | 16.0 | 0.0 |
| Rest of the index, not held | 56.0 | 0.0 | 56.0 | 0.0 |
| Total | 100.0 | 100.0 | 145.0 | 27.5 |
The last row is the one that matters most and is most often left out. The 56.0 per cent of the index the fund does not own is an active decision, a bet against every one of those stocks, and it belongs in the sum.
Active share = ½ × Σ |wfund,i − windex,i|, over every stock in the fund or the index
= ½ × (22.0 + 51.0 + 16.0 + 56.0) = ½ × 145.0 = 72.5%
Check: active share = 1 − Σ min(wfund,i, windex,i) = 100 − 27.5 = 72.5%
Excel, with index weights in B2:B600 and fund weights in C2:C600 on one list of securities: =SUMPRODUCT(ABS(C2:C600-B2:B600))/2
The halving is there because every overweight is paid for by an underweight of the same size: both portfolios sum to 100, so the absolute differences count each bet twice. The overlap form is the quicker check and the one to use when a holdings file is incomplete: the fund shares 27.5 per cent of its portfolio with the index and the other 72.5 per cent is different.
The ten largest names contribute 22.0 points of the 145.0. Most of the active share comes from what the fund owns in size outside the top of the index (51.0 on the fifteen smaller stocks, 16.0 off-index) and from what it ignores (56.0).
A fund with 72.5 per cent active share can be read as 27.5 per cent of index fund plus 72.5 per cent of genuinely different portfolio. The client could buy the first part for an index fund's fee, illustratively 0.05 per cent. The rest of the 0.75 per cent fee is the price of the active part:
Active fee = (fund fee − (1 − active share) × index fee) / active share
= (0.75 − 0.275 × 0.05) / 0.725 = 0.736 / 0.725 = 1.02%
A second number follows from the same split. To finish level with the index fund after fees, the fund has to beat the index by the fee gap of 0.70 per cent, and only the active 72.5 per cent can do it: the active sleeve must outperform by 0.70 / 0.725 = 0.97 per cent a year. That is the hurdle a manager is really asking the client to believe in.
| Active share | Fund fee | Active fee | Hurdle on the active part |
|---|---|---|---|
| 20% | 0.60% | 2.80% | 2.75% |
| 30% | 0.60% | 1.88% | 1.83% |
| 45% | 0.65% | 1.38% | 1.33% |
| 60% | 0.70% | 1.13% | 1.08% |
| 72.5% (this fund) | 0.75% | 1.02% | 0.97% |
| 90% | 0.90% | 0.99% | 0.94% |
The cheaper-looking fund is often the dearer one. A fund at 30 per cent active share charging 0.60 per cent costs 1.88 per cent on its active part, 1.85 times what this fund charges for the same thing, and needs 1.83 per cent of outperformance from the active part just to tie the index. Cremers and Petajisto, who introduced the measure in 2009, labelled funds between 20 and 60 per cent active share closet indexers. The label matters less than the arithmetic: below 60 per cent, the fee is spread over a small active slice.
Active share measures how different the holdings are, not how much risk the differences carry. A fund with 72.5 per cent active share spread across many small bets can have a lower tracking error than a 40 per cent fund with one large sector tilt. Read it beside tracking error and the information ratio, never instead of it.
Fees, the passive counterfactual and the mandate's own returns are built from one set of inputs in the free workbooks for this book, including the fee at which the active account and the index finish level.
There is no single right figure, but the original 2009 research by Cremers and Petajisto treated funds between 20 and 60 per cent as closet indexers. The more useful test is the fee: at 30 per cent active share a 0.60 per cent fee becomes 1.88 per cent on the active part, while a 72.5 per cent fund charging 0.75 per cent costs 1.02 per cent for the same thing.
Active share measures how different the holdings are; tracking error measures how volatile the return difference is. Many small stock bets give a high active share with a modest tracking error, while one large sector tilt gives the opposite. A fund at 72.5 per cent active share can still track its index closely if its bets are diversified, so the two are read together.
Because both portfolios sum to 100 per cent, every overweight is matched by underweights of the same total, so the absolute differences count each bet twice. In the worked case the differences sum to 145.0 and active share is 72.5 per cent. The equivalent check is 100 minus the overlap, the sum of the smaller of the two weights in each stock: 100 minus 27.5.
This article is one calculation from Asset Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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