Module 11 Quiz: The Electives

This module kept a list of the degrees of freedom the course did not have: a rule that may be short, a second place to trade the same thing, the person who has to sit through all of it, and the unit itself. Six questions, all arithmetic, and every one of them runs on a number an earlier lesson measured. The last one puts the four freedoms on one scale and finds that the two largest are the two that are free.

Covers: Lessons 82 to 85, and the four-item list of missing freedoms the module has been keeping since lesson 82.

Every question below hands you numbers and asks for a number back. Work all 6 with a calculator before you scroll to the answers; each answer shows the arithmetic, so a wrong result tells you which step to go back to rather than only that you were wrong.

The questions

1. What a second instrument buys, and what a short leg buys

Lesson 71’s divisor takes a count of positions and their average correlation and returns the independent bets they amount to: the count over one plus the count less one times the average. Two rules on one instrument at lesson 74’s 0.88 carry 1.0638 bets.

Put the same two rules on a second instrument. Four positions make six pairs: two of them sit within an instrument and carry the 0.88, and four cross between the instruments and carry whatever the two instruments correlate at. Take that cross-correlation as 0.35.

Then the other arithmetic. Two equal positions each of standard deviation one have a pair standard deviation of the square root of one plus the correlation over two when both are long, and the plus becomes a minus when one is short.

Ask. At 0.35, how many bets do the four positions carry and what is that against the 1.0638 you already had? What cross-correlation would four long positions need to carry a bet and a half? And at 0.35, what are the two pair standard deviations and the factor between them, and how alike would two instruments have to be before turning one leg over made the pair ten times quieter?

2. The part of the bill a second venue cannot reach

Lesson 83 split lesson 63’s 0.1230 round trip into a spread, slippage and a commission, and found that a second venue competes for the first and the third and cannot touch the second, because slippage is lesson 69’s delay and a delay does not care where the order is sent.

Run the same split on a different instrument. A two-cent spread, eight basis points of slippage a side, and one cent of commission a side, on a price near 48. Lesson 69 priced five minutes between the signal and the order at 32.10 basis points.

Ask. What is this round trip in cash and in basis points, what share of it does the slippage carry, and what share can a second venue ever reach? How far apart would two venues have to quote this instrument before you kept half the difference? And how many whole round trips does one five-minute delay cost?

3. A hundred candidates at a base rate the course did not print

Lesson 84’s test over 156 trades has a power of 0.3461 against lesson 67’s tenth of an R, and a dead system passes 5.05 per cent of the time judged once and 24.25 per cent if you judge after every trade from trade 20. Lesson 67’s eight-R line switches off 59.5 per cent of the systems that genuinely work.

Take a hundred candidate rules of which one in twenty-five is genuinely alive.

Ask. How many rules does each of the two procedures accept, what share of each set is real, and what is the ratio? Then, for each procedure, at what base rate does half of what it accepts turn out to be real?

4. Every distance in the course, measured twice

Lesson 85 measured one bar’s dispersion on all sixty closes at 1.5443, on the first thirty moves at 2.0446 and on the last twenty-nine at 0.7543. Every figure after lesson 63 is quoted in the first of those.

Three of the course’s numbers are distances rather than ratios: lesson 67’s eight-R line, lesson 65’s tenth of an R and lesson 64’s 0.317 of an R a trade. Lesson 75 sizes a position by dividing the money at risk by the distance to the stop; take an account of 250,000 risking two per cent.

Ask. What is the eight-R line in cash, and what is it in each half’s own unit? By what single factor does every distance in the course move into each half, and by what factor between the two halves? How many shares does the sizing rule buy at each dispersion? And what happens to lesson 63’s t of 3.65?

5. The same seven trades in three units

Lesson 63’s winner took seven trades on the sixty closes for a gross of 10.70 a share, paid seven round trips at 0.1230 each, and netted 9.84.

Two of the seven ran wholly inside the first thirty moves, four ran wholly inside the last twenty-nine, and one crossed the split. The three dispersions are 1.5443, 2.0446 and 0.7543.

Ask. What does the record average per trade if you price every trade at the single R, at the loud half’s R and at the quiet half’s? What is the bill as a share of the gross in each of those three? And what are the honest mixed figures, where each trade is priced in the unit of the window it lived through?

6. Four freedoms on one scale

This module named four things the course did not have, and priced each of them against something it had already measured: a rule that may be short, a second venue for the same instrument, the person who has to sit through all of it, and the unit itself.

The four numbers are lesson 82’s 0.9695 against 0.2449 at a correlation of 0.88, lesson 83’s edge over holding of 4.16 a share going to 4.30, lesson 84’s 43.2 per cent going to 6.0 at a base rate of one in ten, and lesson 85’s 2.0446 against 0.7543.

Ask. Turn each of the four into a factor on the quantity its own lesson measured, rank them, and then say which of the four cost something to take and which are free.

The answers

Each one is worked in full. Where a figure comes from a lesson rather than from this page, the lesson is named.

1. What a second instrument buys, and what a short leg buys

The first is one average and one division. Six pairs, two at 0.88 and four at 0.35, average (2 × 0.88 + 4 × 0.35) ÷ 6 = 0.5267, and four positions at that average carry 4 ÷ (1 + 3 × 0.5267) = 1.5504 bets. Against the 1.0638 you carried on one instrument that is 1.46 times, or 0.49 of a bet for two more positions, two more spreads and two more lots of impact.

Running the division backwards gives the cross-correlation a target implies. A bet and a half needs an average of (4 ÷ 1.5 − 1) ÷ 3 = 0.5556, and since two of the six pairs are stuck at 0.88, the four crossing pairs have to come in at (6 × 0.5556 − 1.76) ÷ 4 = 0.3933. Two full bets need 0.0600, and the ceiling at a cross-correlation of exactly zero is 2.1277 — so the whole distance between two bets and the best case a second instrument can ever offer is six hundredths of a correlation.

The short leg is the other formula on the same 0.35. Both long is the square root of 1.35 ÷ 2 = 0.8216, one short is the square root of 0.65 ÷ 2 = 0.5701, and the factor is 1.44. Solve the factor for the correlation and ten times quieter needs (1 + r) ÷ (1 − r) = 100, which is r = 0.9802. The two arithmetics point in opposite directions down the same column, and that is the whole of lesson 82: the correlation that ruins a long-only book is the correlation a spread is made of.

Answer. 1.5504 bets, which is 1.46 times the 1.0638 and 0.49 of a bet bought with two more positions; 0.3933 for a bet and a half; 0.8216 against 0.5701, a factor of 1.44; and a correlation of 0.9802 before a short leg is worth ten times.

2. The part of the bill a second venue cannot reach

Three additions and one division a row.

Part of the round trip	A share	Basis points at 48	Share of the bill	A second venue?
Spread, two cents	0.0200	4.17	17.12%	Yes
Slippage, eight basis points a side	0.0768	16.00	65.75%	No
Commission, one cent a side	0.0200	4.17	17.12%	Yes
The whole round trip	0.1168	24.33	100%	A third of it

The reachable part is 0.0400 of 0.1168, which is 34.25 per cent, against the 16.26 per cent a second venue reaches on lesson 83’s own instrument. A wider spread and a fatter commission on a cheaper share make the second account worth twice as much here, and the reason is arithmetic rather than anything about the venues: the part a venue can reach is the part that is quoted, and the part it cannot is the part that is a delay.

Keeping half a difference means the difference is twice the round trip, because you pay to enter on one side and to exit on the other: 2 × 0.1168 = 0.2336 a share, or 48.67 basis points. And the delay is the number to hold this against. Five minutes at 32.10 basis points on a 48 price is 0.1541 a share, which is 0.1541 ÷ 0.1168 = 1.32 whole round trips. One late order costs more than a second venue gives back on a whole round trip here, and lesson 83 found the same ordering on its own instrument, which is why the evening goes on the delay rather than on the paperwork.

Answer. 0.1168 a share and 24.33 basis points, slippage carrying 65.75 per cent, a second venue reaching 34.25 per cent, a difference of 0.2336 or 48.67 basis points before you keep half, and one five-minute delay costing 1.32 whole round trips.

3. A hundred candidates at a base rate the course did not print

Four alive and ninety-six dead. Judged once, the test accepts 4 × 0.3461 = 1.3844 of the live ones and 96 × 0.0505 = 4.8480 of the dead, so 6.23 rules are accepted and 1.3844 ÷ 6.2324 = 22.21 per cent of them are real.

Now peek and abandon. The dead pass at 0.2425 rather than 0.0505, so 96 × 0.2425 = 23.2800 get through; the eight-R line removes 59.5 per cent of the live ones, so 1.3844 becomes 0.5607. You accept 23.84 rules of which 2.35 per cent are real. Nearly four times as many rules, two fifths as many true ones, and a tenth the chance that any one of them is real, on the same candidates and the same market.

The last part is the same equation solved for the base rate. Half real means the live ones passing equal the dead ones passing: p × 0.3461 = (1 − p) × 0.0505 gives p = 12.73 per cent, and p × 0.1402 = (1 − p) × 0.2425 gives p = 63.37 per cent. A disciplined test reaches a coin flip when one candidate in eight is alive. With two ordinary habits it needs nearly two in three, which is a population nobody searching for a rule has ever been in.

Answer. 6.23 accepted rules of which 22.21 per cent are real, against 23.84 of which 2.35 per cent are; a ratio of 9.44; and half real at a base rate of 12.73 per cent judged once and 63.37 per cent with the two habits.

4. Every distance in the course, measured twice

One multiplication converts the lot. Eight times 1.5443 is 12.3544 a share, and dividing by each half’s own dispersion gives 12.3544 ÷ 2.0446 = 6.04 and 12.3544 ÷ 0.7543 = 16.38. The two divisions are the same two divisions for every distance on the course, because the cash value cancels: 1.5443 ÷ 2.0446 = 0.7553 and 1.5443 ÷ 0.7543 = 2.0474, so lesson 65’s tenth of an R is 0.0755 and 0.2047 of one, and lesson 64’s 0.317 is 0.2394 and 0.6490. Between the two halves the factor is 2.0446 ÷ 0.7543 = 2.711.

The sizing rule is the one thing that repairs itself. Two per cent of 250,000 is 5,000, and 5,000 ÷ 2.0446 = 2,446 shares against 5,000 ÷ 0.7543 = 6,629 — the ratio is 2.711 again, and the risk carried is the same in both. It only repairs itself if you re-measure the input, which is the thing the course did once and then spent for twenty-two lessons.

And the t does not move at all. A t is a mean over a standard error and both are in the same units, so a constant divisor cancels out of it exactly: 3.65 in dollars is 3.65 in R and 3.65 in either half’s R. Every ratio the course built survives this page and every distance it set has to be measured again.

Answer. 12.3544 a share, which is 6.04 of the loud half’s R and 16.38 of the quiet half’s; every distance multiplied by 0.7553 and 2.0474 and therefore by 2.711 between the halves; 2,446 shares against 6,629; and the t unchanged at 3.65.

5. The same seven trades in three units

The record nets 9.839 over seven trades, which is 1.4056 a share, and each unit is one division.

Priced in	The unit	Net per trade, in R
The single R	1.5443	0.910
The loud half’s R	2.0446	0.687
The quiet half’s R	0.7543	1.863

The bill does not move, and that is the half of the page most readers miss. Seven round trips at 0.1230 is 0.861 a share against a gross of 10.70, which is 8.05 per cent whatever unit you write it in, because a share of a gross is a ratio and the divisor cancels. Only when you write the bill as 0.557 of an R does it start to move, because that is a distance.

The honest answer prices each trade in the window it lived through, and the two figures move in opposite directions. The record averages 1.101 R a trade rather than 0.910, because four of the seven were earned in the quiet half; and the bill rises from 0.557 to 0.852 of an R, a factor of 1.53, because four of the seven were paid there too. The t goes from 3.65 to 3.68, which is to say nowhere. The verdict is exactly as safe as module 8 said, and every number underneath it was two numbers.

Answer. 0.910, 0.687 and 1.863 R a trade; 8.05 per cent of the gross in all three, because it is a ratio; and a mixed 1.101 R a trade on a bill of 0.852 R, with the t moving from 3.65 to 3.68.

6. Four freedoms on one scale

One division a row, and they land in an order nobody would guess from how much each is written about.

The freedom	What it moves	The factor
Not breaking two habits	43.2% of accepted rules real against 6.0%	7.16
A rule that may be short	a pair’s day of 0.9695 against 0.2449	3.96
Measuring the unit twice	a dispersion of 2.0446 against 0.7543	2.71
A second venue for the same thing	an edge over holding of 4.16 against 4.30	1.03

Now the bill for each. The short leg needs a security borrowed, which costs a fee this course has never priced and can be recalled, and it carries a loss with no ceiling to size against. The second venue needs an account, a second set of paperwork and a measured difference that clears twelve basis points often enough to be worth having. Those are the two that are paid for, and they are the two smallest factors on the list.

The other two cost nothing at all. Writing the horizon down before the first trade, the depth you will sit through, the count of settings you searched and the position limit is a decision made once instead of a decision made every time you look, and it is worth 7.16. Measuring one bar’s standard deviation on each half of your own record is one line of arithmetic, and it is the difference between a distance and two distances wearing one name. The two largest freedoms in the whole module are free, and both of them are things you do before the market opens rather than things you buy.

Answer. 7.16, 3.96, 2.71 and 1.03, and the two largest are the two that cost nothing.

What this quiz was testing

Whether you can price a freedom before you take it. Handed a second instrument, you run the divisor on it and find it buys half a bet rather than two; handed a short leg, you run the variance of a difference on the same correlation and find it buys more on half the positions; handed a round trip, you split it into the part that is quoted and the part that is a delay; handed a habit, you put it through the same test twice and read the share of your accepted rules that survives; and handed a unit, you measure it on each half of your record and divide, because a distance in a unit that moved is two distances wearing one name.

That is the last question in the course. The professional tier priced a book, then the day it takes to run one, then the profession that would have to pay for it, and then the freedoms none of it had, and every figure in all eighty-five lessons came out of a record small enough to hold in one hand. What you take away is not a system. It is the habit of dividing, of asking what a number is a number of, and of measuring the answer twice.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.