The Deduction That Changes Nothing

Why a rate is invisible and relief is not

Every rate below is a parameter and no rate on this page is any country’s. That is not evasion, it is the only way to write this once for a course read in twelve languages, and it costs nothing, because the finding is about the shape of the arithmetic rather than about any statute.

Start with the case everybody assumes is the bad one. Suppose the rate is applied to every gain and the same rate relieves every loss. Lesson 67’s trade outcome has a mean of a tenth of an R and a standard deviation of one R. Multiply both by the same factor and the mean becomes 0.0700 and the standard deviation 0.7000 at a rate of thirty per cent, which looks like ruin until you notice that the test in lesson 65 is a ratio. The drift per trade in that test is the mean divided by the standard deviation, and a factor that multiplies both cancels. So the verdict arrives at 589 trades at a rate of zero, at twenty per cent, at forty per cent, and at any rate short of confiscation. This is Domar and Musgrave’s result from 1944 and it is not intuitive: a proportional tax with full loss offset takes a share of your money and leaves your statistics untouched.

Now break the symmetry, because every real system does. Relief is capped, or ring-fenced to the same class of income, or deferred to a later year, or denied on a repurchase. Write the share of a loss that actually reduces what you pay as a fraction of the full rate. Full relief is one and no relief is zero, and everything in practice sits between.

Split the trade outcome to see what that does. For a normal outcome with a mean of a tenth of an R and a standard deviation of one, the winning part averages 0.4509 R a trade and the losing part 0.3509 R a trade, and the edge is what is left after subtracting the second from the first, two numbers each several times its own size. A rate takes a slice off the first number. Relief gives a slice back on the second. When the two slices are equal the edge is unchanged, and when the first is larger the edge shrinks by the difference. That is the whole mechanism, and it explains why a tenth of an R is fragile here in a way that a large edge is not: the tax is levied on 0.4509 and the edge is 0.10, so the rate is working on a base four and a half times larger than the thing it is eating.

Set relief to zero and solve. The edge vanishes when the rate multiplied by the winning part equals the edge itself, which is 0.10 divided by 0.4509, or 22.18 per cent. Below that rate the system still makes money after tax and the test still finishes, later. Above it the system loses money after tax, and no change to any rule, any stop or any position size alters that, because nothing about the rule changed.

Between the two extremes the question is how much relief you need, and that has a closed form too.

Read the second column as the real question to ask about a tax system you trade under. It is not what the rate is. It is what share of a loss comes back, because at a rate of forty per cent a system with a genuine tenth of an R is a losing system unless more than half of every loss reduces the bill.

There is a second arrangement worth pricing, and it is the one every article on this subject leads with: a lower rate on a position held longer. Whether your jurisdiction has one is not this page’s business, but whether this course’s rules could reach it is. Across lesson 63’s 253 cells the mean holding run is 12.74 trading days and the longest single holding observed is 30, against 252 in a year, and the observation window is only fifty-nine moves so the 30 is a floor rather than a measurement. The family would have to be roughly an order of magnitude slower than its slowest cell. Every rule this course has built realises inside a year by construction, so the lower rate is not a discipline problem for this book, it is unreachable.

If you did decide to wait, the amount you can afford to give back while waiting has an exact answer: one minus the quantity one minus the short rate over one minus the long rate, measured as a share of the gain rather than of the position. At a short rate of 37 per cent against a long rate of 20 that is 21.25 per cent of the gain, and where there is no holding distinction at all it is zero and the question does not arise.

Almost nobody has computed their own threshold, and it is two numbers from a record they already have: the average of their winners times the share of trades that win, and their edge. The threshold is the second divided by the first, and it tells you the rate above which your own system stops working if your losses go unrelieved.

Module 10’s list of what the day contains that is not the decision gains its third item: the rate, and the share of a loss that comes back.

Thirty per cent, and the relief that decides it

Take one rate and vary only the relief, so that nothing about the rule, the instrument or the size changes across the rows. The rate is thirty per cent on gains. The system is lesson 67’s: a tenth of an R a trade, a standard deviation of one R, and 589 trades to a verdict before tax.

The first row is Domar and Musgrave and it is the row people assume is impossible: a thirty per cent rate, and the test takes exactly as long as it did untaxed. The last two rows are the same system, the same rate and the same trades, arriving at a verdict of dead because a share of the losses stopped counting.

The third row is the one to sit with. At half your losses relieved the system still has an edge, it is still positive, and the test now needs 11,596 trades rather than 589. Lesson 76 measured this book at 113 orders a year, so 11,596 trades is 103 years. At lesson 65’s forty a month it is twenty-four years. The edge did not disappear. It moved out of reach of measurement, which for a person deciding whether to keep running the system is the same thing.

And notice what the fourth column does not do. It does not fall smoothly. Between 100 per cent relief and 75 the test length nearly triples; between 75 and 50 it multiplies by seven; between 50 and 25 it stops existing. The relief fraction enters the answer through a square, because the trade count in lesson 65 goes as one over the drift squared, so a linear worsening of your position produces a quadratic worsening of your patience. Every other input on this page is linear and this one is not.

So compute your own relief fraction before you compute anything else about tax, and treat every trade count in this course as multiplied by the square of what you find.

What this does not settle

That any of these rates is yours. None of them is anybody’s: this page names no jurisdiction and every rate in both tables is a parameter chosen to show the shape. Nothing here is tax advice, this course is not in a position to give any, and the only instruction on the page is to find your own two numbers and put them through the same arithmetic.

That the trade outcome is normal. It is lesson 67’s assumption and this page inherits it, and the split into a winning part of 0.4509 and a losing part of 0.3509 is a property of that assumption rather than of any record. A distribution with a fatter right tail has a larger winning part, so the rate has a larger base to work on and the 22.18 per cent threshold falls. Your own two numbers are the ones that matter and problem one is how to get them.

That relief is a fraction. It is not, it is a set of rules with a calendar attached: carried forward, ring-fenced to one class, netted within a year, denied on a repurchase, allowed against one kind of income and not another. Collapsing all of that into one number is a simplification whose only defence is that it makes the direction visible and the magnitude roughly right.

That a rate and a calendar are the same thing. They are not, and this page priced only the rate. A gain taxed this year and a loss relieved in three years is not full relief even when the fraction is one, because the money is gone in between and lesson 22’s arithmetic runs on what you are still holding. The calendar makes every figure above optimistic.

That the holding threshold is a reason to hold. The formula says what you can afford to give back on a position you would have held anyway. It says nothing about a position you wanted out of, and the sentence “I am holding this for the tax treatment” converts a rule this course spent seventy-seven lessons building into a discretionary decision made under a deadline.

And the concession that costs most: this page has priced what a rate does to the test and said nothing about what any of it does to the person. It has assumed throughout that this activity is the thing you do, that the hours in lesson 76 and the machinery in lesson 77 and the rate here are all costs of a job rather than of a hobby, and it has never asked whether the job exists. Lesson 79 asks that.