The report is one page. Two hundred trades, 58% of them winners, average trade +2.5%. You do the obvious thing and raise 1.025 to the two hundredth power, which suggests the account grew about a hundred and forty fold, and you stop believing the whole document. Then you look at the equity curve printed underneath: it ends a little under five times where it started — a good outcome, and about a twenty-eighth of the number you just computed. Both numbers came out of the same trade list. Nobody has faked anything, and the mistake — which was yours, not the report’s — is the single most common misreading of a return series there is.
Returns compound. They combine by multiplication, not by addition, and the moment they do, the average of the individual returns stops being the return of the sequence. The arithmetic mean answers a real question — what did a typical single trade do — and it is the wrong input for the question everyone actually asks, which is what happened to the money.
A shop revises the price of a ₹1,000 item up by 20%, to ₹1,200. A fortnight later it announces a flat 20% off. Up twenty, down twenty, average zero — so the item should be back at ₹1,000. It is at ₹960. The second twenty per cent was taken off a bigger number than the first was added to, and the two per cents were never the same rupees. Four per cent has quietly gone, out of two moves that average exactly nothing.
That ₹40 is the entire subject of this lesson. It is not a rounding error and it is not a trick: it is what happens whenever the same percentage is applied to a base that the previous percentage changed. A trading account is that base, every trade, in both directions.
The size of the gap, and the formula for it
- Arithmetic mean
- The plain average of the period returns, as a decimal
- σ
- The standard deviation of those same period returns, as a decimal
- σ² ÷ 2
- The Volatility drag — what variability alone removes from the compounded outcome
- ≈
- A small-move approximation. It is very close at returns of a few per cent and loosens as the moves get large
Example: A series that alternates +10% and −10% has an arithmetic mean of exactly zero and a σ of 10%, so the drag is 0.10² ÷ 2 = 0.005, or half a per cent per period. Check it directly: 1.10 × 0.90 = 0.99, which is −1% over two periods, or −0.501% per period. The formula and the multiplication agree, and twenty such periods leave 0.99 to the tenth power — 0.904, a loss of 9.6% from a series whose average return is nil.
Where else the same arithmetic turns up
- The recovery asymmetry. A 50% fall needs a 100% rise to get back; a 75% fall needs 300%. The drawdown lesson in this track presents that as a fact about holes, and it is this same multiplication seen once rather than two hundred times.
- Averaging two stocks’ growth rates. The average of two holdings’ compound annual growth rates is not the growth rate of having held both. Compounded returns do not average linearly, and the only honest way to get a portfolio’s figure is to compound the portfolio’s own period returns.
- Leverage. Doubling exposure doubles the arithmetic mean and quadruples the drag, which is why a leveraged version of a modest edge can compound to less than the unleveraged one while being right just as often. The Kelly lesson reaches the same place from the other direction.
- Any average of percentages taken across time. "Average monthly return", "average annual return", "average trade" — all of them are arithmetic means of multiplicative quantities, and none of them may be raised to a power.
- 1Ask for the compounded figure, or the curve
The compounded outcome of the actual sequence is one number and it is not optional. If a report offers only an average trade and a win rate, it has told you about the trades and nothing about the account.
- 2Ask for the standard deviation next to the mean
Without σ you cannot even estimate the drag, and with it you can do so in one line. A report that gives an average and no dispersion is unreadable, not merely incomplete.
- 3Rank alternatives by the compounded result, never by the average trade
The worked example above is the whole reason: the better average trade can be the worse system, reliably and by a wide margin, and the ranking flips with no arithmetic error anywhere.
- 4State your sizing scheme in the same place you state the return
Fixed rupee stake or fixed fraction of the balance. That single line decides which of the two means describes your account, and it is nearly always missing from both strategy reports and personal journals.
The asymmetry of a fall and its recovery is the same multiplication as the drag. Move the loss and watch the gain required grow faster than the loss did.
A report shows 200 trades with an average of +2.5% each and a standard deviation of trade returns of about 18.5%. Positions were sized as a fixed fraction of the account. What does that tell you about the account’s growth?
₹1,000 ki cheez ka rate dukaan 20% badha ke ₹1,200 kar deti hai. Do hafte baad "flat 20% off" ka board lag jaata hai. Upar bees, neeche bees, average zero — toh cheez wapas ₹1,000 pe aani chahiye. Bill ₹960 ka banta hai. Doosra bees percent bade number pe laga tha, aur dono bees percent ek hi rupaye nahi the: do move ka average theek sifar, aur 4% gaayab. Trading account roz wahi badalta hua base hai. Isiliye report ki line "200 trade, average +2.5% per trade" ko 1.025^200 mein daalna sabse aam galti hai — average trade compound karne ke liye nahi hota. Formula ek line ka hai: geometric ≈ arithmetic − σ² ÷ 2. +10%/−10% waali series: average theek sifar, σ 10%, drag 0.10² ÷ 2 = 0.5% per period. Milaao — 1.10 × 0.90 = 0.99, yaani do period mein −1%, aur bees period baad 0.99^10 = 0.904. Jis series ka average sifar tha, usne 9.6% kha liya. Ab do system, 100-100 trade, size hamesha account ka ek hissa: A ka har trade +1.0% aur σ sifar → 1.01^100 = 2.70 guna. B alternate +21% aur −16%, average +2.5% (A se dhai guna zyada) par σ 18.5% → ek jodi 1.21 × 0.84 = 1.0164, aur pachaas jodi = 2.26 guna. Behtar average trade, chhota account. Aur woh carve-out jo poori baat palat deta hai: agar har trade mein tay ₹10,000 hi lagta hai, balance ke hisaab se nahi, toh return jud te hain, drag sifar hota hai, aur arithmetic average bilkul sahi jawab hai. Pehle sizing dekho, phir tay karo kaunsa average padhna hai.
- Returns compound, so the average of a return series is not the return of the series.
- Geometric return is approximately the arithmetic mean less σ² ÷ 2 — the volatility drag.
- Because the drag goes with the square of σ, halving size halves the edge and quarters the drag.
- A system with a better average trade can compound to less; rank by the compounded result only.
- The carve-out: with a fixed rupee stake per trade returns add, and the arithmetic mean is then exactly right.
- An average without a standard deviation beside it, and without the sizing scheme stated, cannot be read at all.
Module 25 checkpoint
4 questions. Answers are revealed once you submit all of them.
1.A pull-back rule tested on one-minute prints of a stock quoted 41.75 bid / 42.35 offer shows a 74% win rate and an average gain of 1.1% a trade. What is the most likely explanation?
2.Two traders apply the same 1% risk rule and the same "stop at two units of volatility" to the same ₹600 share whose daily close-to-close σ is 1.5%. One arrives at 347 shares, the other at 555. Who has made the error?
3.A stock’s pooled daily σ is 1.76%, made up of calm days with a σ near 0.9% and occasional violent days with a σ near 3.5%. Why does a σ-scaled model badly understate how often it moves 6%?
4.A trading journal shows 200 trades averaging +2.5% each, with a standard deviation of trade returns of 18.5%. Which further fact decides whether the account grew by roughly 1.025 to the 200th power or by considerably less?
Mark it done to track your progress through the curriculum.