Module 9 Quiz: Portfolio
This module filled in a card, one column per lesson: the highest correlation a rule shows against the rest, the heat the book carries and how often all of it arrives, the weight the optimiser gives each rule with the short forbidden, and how far that correlation moves when you measure it on half the record instead of all of it. Six questions, all arithmetic. The last one spends the whole card at once and finds that the column which binds is the one nobody argues about.
Covers: Lessons 71 to 75, and the four-column card the module has been filling in since lesson 71.
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. Three medians, three ceilings
Lesson 71 correlated all 31,878 pairs of lesson 63’s 253 rules over the twenty-nine moves on which every rule in the grid is defined. The median pair came to 0.9464 and 8.93 per cent of the pairs were the same series to the last decimal.
Lesson 74 measured the same 31,878 pairs on each half of that window separately. Over the first fifteen moves the median is 0.8835 and 9.37 per cent of pairs are identical. Over the last fourteen the median is exactly 1.0000 and 72.90 per cent are identical.
Ask. How many pairs are identical in each of the three windows, and what ceiling does each median put on the bets a book of these rules can carry, however many of them you run?
2. The divisor, run to its wall
A trader runs several rules whose average pairwise correlation is 0.8800, which is what lesson 74 measured on the worse of its two fortnights. Lesson 67 needed 589 trades to settle whether an edge of a tenth of an R was real, and lesson 65 fixed the pace at forty trades a month.
The bets a book carries are N over one plus N minus one times r, and a rule run alongside others needs the 589 divided by that number.
Ask. What are the bets, the trades and the months at two rules, at four, at six, and in the limit? What correlation would six rules need in order to carry three bets? And what would six independent rules have cost in months?
3. The day the whole book is red
Six positions, each risking 1.5 per cent of the account, at lesson 71’s measured correlation of 0.9464. The variance of a sum of N positions of equal size is the single-position variance multiplied by N times one plus N minus one times r.
Lesson 72 evaluated the one-factor integral for the day on which every position goes against you: at six positions it is 0.3829 at this correlation and 0.0156 if the six were independent. Take a month as 21 trading days, and take lesson 18’s median worst drawdown of 9R.
Ask. What is the heat and what is the day’s standard deviation? How many all-against days does a month contain under each assumption, and what is the ratio between them? And how many trading days of all-against days does it take to reach 9R?
4. Two rules, and the weights you can hold
Lesson 73 forbade the short and the optimiser put everything into two of the four rules. Here are those two, with the standard deviations and the correlation lesson 71 measured over the same twenty-nine moves: the 2-and-5 rule at 0.6015, the 8-and-30 rule at 0.7082, and a correlation between them of 0.6254.
For two assets the minimum-variance weight on the first is the second’s variance minus the covariance, over the sum of the two variances minus twice the covariance.
Ask. What are the two weights, what standard deviation do they deliver, and what is the cut against equal weights? And above what correlation would the optimiser want to short the noisier of the two?
5. The window decides, and the correction that does not save it
Lesson 74 measured the four rules’ average pairwise correlation three ways: 0.5966 over the first fifteen moves, 0.7695 over all twenty-nine, and 0.8800 over the last fourteen. The instrument’s own standard deviation over the second fortnight is 1.240 times its standard deviation over the first.
The standard correction for that bias divides the measured correlation by the square root of one plus delta times one minus the correlation squared, where delta is the ratio of variances minus one. The book is four positions at two per cent each in every case, and the all-against frequencies are 0.2323 at the lowest correlation and 0.3567 at the highest.
Ask. What does the correction do to the 0.8800? What are the bets and the day’s standard deviation at each of the three correlations? And how much more often does the all-against day arrive on the second fortnight than on the first?
6. Which limit binds, and the card spent
An account runs a four per cent daily-loss limit and risks one per cent a trade. A daily-loss limit means nothing unless the whole heat fits inside it, so the position count is the limit divided by the risk per trade.
Take lesson 74’s worst-window correlation of 0.8800, at which the all-against day arrives on 35.67 per cent of days for four positions. Compare the permitted book against a single position carrying the same four per cent of heat, which loses its whole heat whenever it loses at all.
Ask. How many positions does the limit permit, what is the day’s standard deviation for that book against the single position, and by how much does spreading the risk shrink the day and the full-heat day? Then read the card: which rule does the correlation column delete, and which does the weights column zero as well?
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. Three medians, three ceilings
The counts are one multiplication each. 0.0893 × 31,878 = 2,847 over the full window, 0.0937 × 31,878 = 2,987 over the first fifteen moves, and 0.7290 × 31,878 = 23,239 over the last fourteen.
The ceiling is one division. The bets a book carries are N over one plus N minus one times r, and as N grows without limit that goes to one over r. At 0.9464 the ceiling is 1.0566, at 0.8835 it is 1.1319, and at exactly one it is exactly one.
Read the third window twice. A median of 1.0000 does not mean the rules are similar. It means more than half of the pairs in the grid produced the same fourteen numbers as each other, so the ceiling is not approached from below, it is reached: no number of rules from that family, run over that fortnight, carries more than one bet. And the middle window holds fewer identical pairs than the first even though it contains the fortnight in which almost everything tied, because agreeing on twenty-nine numbers is harder than agreeing on fifteen.
Answer. 2,847, 2,987 and 23,239 identical pairs, against ceilings of 1.0566, 1.1319 and exactly 1.
2. The divisor, run to its wall
Four divisions and a table.
| Rules run in parallel | Bets you are carrying | Trades each still needs | Months |
|---|---|---|---|
| Two | 1.0638 | 553.7 | 13.84 |
| Four | 1.0989 | 536.0 | 13.40 |
| Six | 1.1111 | 530.1 | 13.25 |
| As many as you like | 1.1364 | 518.3 | 12.96 |
The limit is one over 0.8800, which is 1.1364, and the wait it leaves is 0.8800 of the original: 0.88 × 589 is 518.3 trades and 12.96 months. There is no number of rules at this correlation that gets the wait below thirteen months, because the limit does not contain N at all.
The correlation that would buy three bets from six rules is the divisor solved backwards: 6 over one plus five r equals 3 gives 1 + 5r = 2 and r = 0.2000. That is not a correlation this course has measured anywhere, on any window, between any two rules of this family.
And six independent rules would have needed 589 ÷ 6 = 98.2 trades each, which is 2.45 months. The distance between 2.45 and 13.25 is the whole of what the word diversification is usually carrying, and it is bought with a number nobody measures.
Answer. 1.0638, 1.0989, 1.1111 and 1.1364 bets; 554, 536, 530 and 518 trades; 13.84, 13.40, 13.25 and 12.96 months; a correlation of 0.2000 for three bets; and 2.45 months if they were independent.
3. The day the whole book is red
Heat is the sum of the stops: six positions at 1.5 per cent is 9.0 per cent, and that is the maximum rather than a summary.
The summary is the standard deviation. One plus five times 0.9464 is 5.7320, six times that is 34.3920, its square root is 5.8645, and at 1.5 per cent a position that is 8.80 per cent. So the day sits 96.2 per cent of the way from the 3.67 per cent six independent positions would give to the 9.0 per cent that arrives when the six move as one.
The frequencies are one multiplication each. 0.3829 × 21 = 8.04 days a month, against 0.0156 × 21 = 0.33, which is one such day every three months. The ratio is 0.3829 ÷ 0.0156 = 24.5, and it is bought with no change to any position size, any stop or any rule.
And the depth. Six positions hand you 6R on the all-against day, so 9R is 1.5 of those days, and at one day in 2.61 they arrive in 3.9 trading days. Four working days for the ingredient of a career-worst drawdown, on a book any checklist would call diversified.
Answer. 9.0 per cent of heat, a day of 8.80 per cent, 8.04 all-against days a month against 0.33, a ratio of 24.5, and 9R in 3.9 trading days.
4. Two rules, and the weights you can hold
The covariance is 0.6254 × 0.6015 × 0.7082 = 0.26641. The two variances are 0.36180 and 0.50155.
The weight on the 2-and-5 rule is (0.50155 − 0.26641) ÷ (0.36180 + 0.50155 − 2 × 0.26641) = 0.23514 ÷ 0.33053 = 0.7114, so the other weight is 0.2886. Those are the 0.711 and 0.289 on the card, arrived at from two standard deviations and one correlation rather than from a matrix inversion.
The variance they deliver is 0.7114² × 0.36180 + 0.2886² × 0.50155 + 2 × 0.7114 × 0.2886 × 0.26641 = 0.33427, whose square root is 0.57816. Equal weights give 0.59080, so the whole prize on this pair is 2.14 per cent of the day.
The short is a comparison rather than a calculation. The weight on the second rule is its own variance minus the covariance, so it turns negative when the covariance exceeds the first rule’s variance, which is when r exceeds 0.6015 ÷ 0.7082 = 0.8493. Below that the optimiser holds both. Above it, it wants to sell the noisier rule to hedge the quieter one, and the correlations lesson 74 measured on its worse fortnight sit above it.
Answer. 0.7114 and 0.2886, delivering 0.57816 against 0.59080 equally weighted, a cut of 2.14 per cent; and the short arrives above a correlation of 0.8493.
5. The window decides, and the correction that does not save it
Delta is 1.240² − 1 = 0.5376. One minus 0.8800² is 0.2256, so the denominator is the square root of 1 + 0.5376 × 0.2256 = 1.12128, which is 1.05891, and 0.8800 ÷ 1.05891 = 0.8310.
That is a real correction and it does not rescue the measurement: 0.8310 still sits far above the first fortnight’s 0.5966, so the volatility explains part of the move and not most of it.
| Correlation measured on | Average pairwise | Independent bets | Standard deviation of the day |
|---|---|---|---|
| First fifteen moves | 0.5966 | 1.4338 | 6.68% |
| All twenty-nine | 0.7695 | 1.2090 | 7.28% |
| Last fourteen moves | 0.8800 | 1.0989 | 7.63% |
And the frequency is one division: 0.3567 ÷ 0.2323 = 1.54. The same four positions, sized identically, hand you a day on which everything goes against you half again as often, and nothing about the book changed. Only the fortnight the correlation was measured in did.
Answer. The 0.8800 corrects to 0.8310; the bets are 1.4338, 1.2090 and 1.0989 and the days 6.68, 7.28 and 7.63 per cent; and the all-against day arrives 1.54 times more often.
6. Which limit binds, and the card spent
The count is one division: 4 ÷ 1 = 4 positions. Not four rules chosen and then sized around, four because that is what one per cent a trade leaves room for under a four per cent limit.
The single position at four per cent has a day of 4.00 per cent and pays its whole heat whenever it loses, which is half the time. Four positions at one per cent give 1 + 3 × 0.8800 = 3.64, four times that is 14.56, and the square root is 3.8158, so the day is 3.82 per cent and the whole heat arrives 35.67 per cent of the time.
So the day is (4.00 − 3.82) ÷ 4.00 = 4.61 per cent smaller and the full-heat day is (0.5000 − 0.3567) ÷ 0.5000 = 28.7 per cent rarer. That is the entire benefit of running four rules rather than one, after five lessons of measurement, and it costs a quartering of the size per trade to collect. Independent, the same swap would have given a day 50 per cent smaller and a full-heat day 87.5 per cent rarer.
And the card, for the four rules the module has carried since lesson 71:
| Rule | Highest correlation, full window | Highest on any window | Weight, short forbidden |
|---|---|---|---|
| 2-and-5 | 0.7326 | 0.7599 | 0.711 |
| 3-and-10 | 0.8287 | 1.0000 | 0 |
| 5-and-20 | 0.9648 | 1.0000 | 0 |
| 8-and-30 | 0.9648 | 1.0000 | 0.289 |
The first column says delete 5-and-20, because it is a duplicate of 8-and-30 at 0.9648 and carries less that the others do not already have. The second says the first column was the optimistic version: on the fortnight that matters, three of the four rules are the same series. The third says the same thing in a different currency, putting nothing at all in 3-and-10 and 5-and-20. Three columns, two rules left, and a cap that had already decided the count before any of them was computed.
Answer. Four positions, a day of 3.82 per cent against 4.00, which is 4.61 per cent smaller, and a full-heat day 28.7 per cent rarer; and the card deletes 5-and-20 and zeroes 3-and-10 as well.
What this quiz was testing
Whether you can tell a book from a list of positions. Handed a median correlation, you turn it into a ceiling and notice that the ceiling does not contain the number of rules; handed a book, you compute both the heat and how often the whole of it arrives, and act on the second; handed two rules, you solve them for weights and find the correlation at which the answer stops being holdable; handed a record, you split it in half and size on the worse half; and handed a loss limit, you divide it by your risk per trade before you argue about which rules to run, because that division has already fixed the count.
Module 10 leaves the arithmetic and starts on the day, and finds the day is not short of hours but short of decisions: of the 253 rules on the grid, 203 produce exactly one entry over the stretch they are defined on, and the busiest cell in the whole grid works out at seventy entries a year.
Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.