The Benchmark You Chose
Lesson 63’s winning rule made 9.84 a share net against 5.68 for buying and holding, so 57.7 per cent of what it returned was the market and 42.3 per cent was the rule. That is the arithmetic every attribution does, and on this record it is wrong. The rule held a position over 30 of the 59 bars, so it was exposed to barely more than half the market’s move, and charging it for all of that move is charging it for a return it could not have collected. Weight the market by the exposure the rule actually carried and its share falls to 30.0 per cent, which nearly doubles what the rule is credited with, from 4.16 a share to 7.75.
Prerequisites: Lesson 63, for the rule being decomposed and the 9.84 against 5.68, lesson 65, for what it took to be allowed to believe the record at all, and lesson 11, for the round-trip cost that is one of the three parts below.
A return has no parts until you name the comparison
Attribution sounds like accounting and it is not. A profit-and-loss figure is observed; the question of what produced it is not, and cannot be, because the alternative you are implicitly measuring against never happened. Every attribution is therefore a subtraction, and the whole content of it sits in what you chose to subtract. Change the comparison and the same record hands you a different answer with no arithmetic error anywhere.
So print the record and then print the comparisons. The rule is lesson 63’s winner, a 2-bar average against a 5-bar one, run over the sixty closes with next-bar execution and 0.1230 a share of round-trip cost. It takes 7 trades and makes 10.70 gross and 9.84 net. Buying at the first close and selling at the last makes 5.80 gross and 5.68 net. All four of those are lesson 63’s figures unchanged.
Here is the one number lesson 63 did not print, and it decides everything on this page. Walk the rule bar by bar and count the bars over which it actually held a position: 30 of the 59. The rule was in the market 50.85 per cent of the time and flat for the rest. It was not a competitor to buying and holding; it was a half-sized position in the same instrument, taken at chosen moments.
That converts the return into three parts that can be computed rather than argued about. The market part is the exposure the rule carried multiplied by the move the market made: 0.5085 times 5.80, which is 2.95. The selection part is whatever is left of the gross after that: 10.70 minus 2.95, which is 7.75, and it is the return that came from being long at those particular times rather than at an average selection of times. The cost part is 7 round trips at 0.1230, which is 0.86, and it is the only one of the three that is certain.
| Component | Dollars a share | Share of the net |
|---|---|---|
| Market: exposure of 0.5085 times a move of 5.80 | +2.95 | 0.300 |
| Selection: being long at those bars rather than at any bars | +7.75 | 0.788 |
| Costs: 7 round trips at 0.1230 | -0.86 | -0.087 |
| Net | +9.84 | 1.000 |
Those three add to the net exactly, which is the only property a decomposition has to have and is worth checking every time, because an attribution whose parts do not sum has a fourth part in it that somebody has not named. Notice also what the shares say against the claim above: the market’s contribution is 30.0 per cent of the net here, and it was 57.7 per cent when the benchmark was a fully invested one. Neither figure is a mistake. They are answers to two different questions, and the sentence “most of the return was the market” is true under one and false under the other.
The sheet this module has been building since lesson 62 gains a fifth column, and it is the exposure share. Almost nobody records it. It takes one pass over the position history, it is the first thing a professional allocator asks for and the last thing a retail platform reports, and without it every comparison to an index on the statement above it is uninterpretable.
The same 9.84, against three benchmarks
Three comparisons are defensible on this record, and it is worth running all three rather than picking one, because the spread between them is the honest measure of how much the answer depends on the question.
The first is buying and holding, fully invested. That charges the rule 5.68 and credits it with 4.16. The second is the market return at the rule’s own exposure. That charges it 2.95 and credits it with 7.75. The third has no closed form and needs a simulation: pick 30 of the 59 bars at random, be long over exactly those, and see what that earns. Do it 4,000 times from seed 20260903.
| Benchmark | What it charges the rule | What the rule is credited with |
|---|---|---|
| Buy and hold, fully invested, net | 5.68 | 4.16 |
| Market return at an exposure of 0.5085 | 2.95 | 7.75 |
| Same exposure, 30 bars chosen at random, median | 2.90 | 7.80 |
The second and third rows agree to within five cents, and that agreement is the load-bearing result on this page. They were computed by completely different routes: the second is one multiplication, and the third is four thousand random selections of thirty bars. Random timing at a given exposure earns the exposure component and nothing else, which is what makes the middle row a decomposition rather than an assumption. The first row, the one every platform and every fund factsheet reports, is the outlier.
A return has no parts until you name what it is being compared against.
The third benchmark also does something the other two cannot, which is carry a probability. Of the 4,000 random selections of thirty bars, 373 matched or beat the rule’s 10.70 gross, which is 9.32 per cent. So the rule’s timing was better than coin-flip timing at the same exposure, at a rate that a reader can compare against the bar lesson 64 built. Nine per cent is not the same claim as “beat the index by 4.16”, and it is a great deal more useful, because the first says how often nothing at all would have done as well and the second says only that something happened.
So compute your exposure share first, before any comparison to anything.
What this does not settle
That exposure can be measured in bars. It cannot, not properly. Counting the bars over which a position was open treats one share held for thirty bars and ten shares held for three bars as the same exposure, and they have nothing in common except a product. A real attribution weights every bar by the capital at risk in it, and this page does not have position sizes because lesson 63’s rule does not specify any. Every figure in the first table would move under a sizing rule, and the direction it moved would depend on whether the rule was larger in the bars that went its way.
That the costs belong to selection. The decomposition charges all 0.86 against the selection component, because the trading is what the rule chose to do and holding would have paid one round trip rather than seven. An allocator would argue that the first round trip is the price of the exposure and only the other six are the price of the timing, which moves 0.12 from selection to market. The choice is defensible either way and the page made it silently until this sentence.
That the random-timing null is the right null. Picking thirty bars at random scatters them, and a real rule holds runs of consecutive bars, so the null is comparing against a portfolio that no rule could actually trade. Consecutive bars share regime and volatility, which makes real exposure lumpier and its outcomes more variable than the null’s, so the 9.32 per cent is too flattering rather than too harsh. A block version that drew runs instead of single bars is the honest fix and this page does not run one, for the same reason lesson 63 did not.
That any ratio settles it instead. The obvious retort to all of this is to skip the benchmark and quote a risk-adjusted number, so here it is on this record: the per-trade Sharpe ratio is 0.910 over 0.659, which is 1.381, and on 7 observations its standard error is 0.528. The 95 per cent interval therefore runs from 0.35 to 2.42, which spans the entire distance from ordinary to exceptional. The ratio has not escaped the sample size; it has only stopped displaying it.
That one rule on one rising series generalises. Long-only on a series that went up is the exact case where the fully invested benchmark is least fair and where exposure weighting flatters the rule most, and the argument would look different on a falling stretch, where a rule that spends half its time flat beats the index by not participating and deserves very little credit for it. The mechanism holds in both directions. The numbers on this page hold in one.
And the concession that costs most: all of this partitions the past and none of it says which part will happen again. The selection component is the interesting one precisely because it is the part attributable to the rule, and lesson 63 already showed that a search of 253 configurations manufactures selection-looking returns from series with no structure in them. Attribution tells you where a number came from. It cannot tell you whether it was earned. Lesson 67 takes the next question, which is the one a reader in a losing month actually has: it finds that a system with a genuine edge of a tenth of an R a trade spends an ordinary 156-trade run inside a worst drawdown of 8.82R, which is deeper than the one that makes most people switch it off.
Problems
- Compute your exposure share. Take your last two hundred bars on one instrument and count the bars over which you held any position at all, divided by two hundred. Ten minutes, and you end holding one number, your exposure share, which is the multiplier that turns the index return on your statement into the part of your return you did not cause.
- Decompose one month three ways. Take a month of your own record and compute the market component at your exposure, the selection component as the remainder, and the costs, and check that the three sum to your actual net. Half an hour, and you end holding one number, the selection component, which is the only part of the month that was yours.
- Run the random-timing null on yourself. Take the bar-to-bar changes of that same month, pick at random as many bars as you were actually exposed for, sum them, and repeat a thousand times from a seed you write down. Count how often random timing at your exposure beat your actual result. An evening, and you end holding one number, that fraction, which is how often your own exposure taken at random moments would have done as well as you did.
Sources. Gary P. Brinson, L. Randolph Hood and Gilbert L. Beebower, “Determinants of Portfolio Performance” (Financial Analysts Journal, 1986), for the decomposition of a return into an exposure component and a selection component, which is the first table on this page reduced to one instrument. Michael C. Jensen, “The Performance of Mutual Funds in the Period 1945–1964” (The Journal of Finance, 1968), for the argument that a manager’s contribution is only what survives after the market exposure is paid for, and for the measured finding that on that basis most of it did not survive. Andrew W. Lo, “The Statistics of Sharpe Ratios” (Financial Analysts Journal, 2002), for the standard error this lesson puts on a Sharpe ratio, which is the formula behind the interval from 0.35 to 2.42. William F. Sharpe, “The Sharpe Ratio” (The Journal of Portfolio Management, 1994), for the author’s own insistence that the ratio is a comparison between two things measured over the same period rather than a score.
Backtesting as Evidence
The rule being decomposed, and the 9.84 against 5.68 this lesson takes apart.
Read Lesson →The Horizon You Fix First
What it took to be allowed to believe the record before attributing it.
Read Lesson →Slippage and Impact at Retail Size
The round-trip cost that is one of the three components.
Read Lesson →Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.
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