The Method of Proof

The trader enters a world of numbers and too often leaves it with a superstition. A pattern becomes a promise. A winning month becomes a theory. A loss becomes an insult. The account records a result, but the mind supplies a story about what the result must mean.

The geometric method begins elsewhere. It asks what the terms mean, which premises are being granted, and what follows from them. It does not reward a conclusion for being forceful. The trader who writes “therefore” must show the bridge.

This book uses the architecture familiar from Euclid and Spinoza: definitions, common assumptions, propositions, demonstrations, corollaries, and explanatory notes. It borrows a discipline of reasoning, not the subject matter of Euclidean geometry. A market is not a triangle, and an empirical premise does not become self-evident because it has been called an axiom.

We will establish truths about the arithmetic of trading: expectancy, costs, compounding, drawdowns, position sizing, information, and the conditions of long-run growth. These results are conditional on explicit models. We will then construct a particular method and examine its historical record. The distinction between a theorem and a tested hypothesis is the spine of the book.

Three kinds of statement

A definition fixes how a term is used. Defining an edge as positive expected profit does not demonstrate that a trader possesses one. Defining a trend as a price above an average does not demonstrate that the price will continue upward.

A deduction establishes a conclusion from premises. If a position loses half its value, it must double to recover. Nothing about tomorrow's market is needed to prove that statement. If a proposed argument requires tomorrow's distribution, that distribution must appear among its premises.

An empirical claim concerns what occurred or is likely to occur in the world. Whether a trend filter has reduced drawdowns after realistic costs is such a claim. Data, execution conventions, alternative explanations, and uncertainty belong beside it. A historical result must carry its dates.

The error we are trying to remove is the quiet substitution of one category for another. “I can calculate the optimal stake if I know the odds” becomes “I know the optimal stake.” “This rule succeeded in these observations” becomes “this rule works.” “Every event has causes” becomes “I can predict the next event.” Each substitution removes the most difficult premise from view.

The Spinozan connection

Spinoza distinguishes activity from passivity through adequate causation and adequate ideas (Ethics III, definitions 1–3 and proposition 1). He describes conatus as a thing's striving to persevere (III, propositions 6–7). Those concepts suggest a useful orientation: preserve the capacity to act, examine the causes of your conduct, and do not confuse the intensity of an affect with evidence.

They do not establish that prices have a biological desire to persist, that every frightened person makes a bad trade, or that a trading loss proves inadequate understanding. A well-specified strategy can lose on an outcome its model allows. A poor strategy can win by chance. The mathematical distinctions below prevent the philosophy from becoming another instrument of self-deception.

A statistical regularity can be useful before its entire causal mechanism is known. A compelling causal story can be useless if the relevant effect is already reflected in price. The standard is explanatory honesty joined to observable performance, not a requirement that a finite trader comprehend the whole market.

What this edition promises

Twenty-nine propositions establish conditional results. A fully specified equity/cash trend rule supplies a practical construction. A reproducible historical study evaluates that construction, including costs and adverse comparisons. The operating protocol explains how to monitor its execution and when to stop making claims the evidence no longer supports.

The aim is not a method that cannot lose. Proposition 23 explains why history cannot supply that guarantee. The aim is a method clear enough to execute, limited enough to survive mistakes, and explicit enough to be rejected when it fails its declared purpose.

To read geometrically is to ask of each page: What is defined? What is assumed? What has actually been shown? This is the first discipline of the Spinozan trader.

Definitions and Assumptions

Definitions

D1. Price and information. A price is a quoted or executed exchange price at a specified time and venue. Let F_t denote information available to the strategy at decision time t. Price is not defined here as fair value, intention, or a complete equilibrium of human desires.

D2. Trade outcome. X is a trade's net profit or loss in specified monetary units, after the costs included in its model. A probability distribution for X describes a population of possible outcomes, not a verdict on one observed trade.

D3. Expectancy and edge. Expectancy is E[X], where the expectation exists. A positive-expectancy edge means E[X] > 0 under the relevant distribution. An estimate of edge is an estimate; it is not the distribution itself.

D4. Capital and return. V_t is marked portfolio equity after period t. With no external cash flows, its net return is r_t = V_t / V_(t−1) − 1. Positive equity is required wherever ratios or logarithms are used.

D5. Drawdown. Let H_t be the largest equity previously attained, including initial equity. Drawdown is d_t = 1 − V_t/H_t. Maximum drawdown is the largest d_t in a specified observation interval; a historical maximum is not a future loss limit.

D6. Practical ruin. Ruin means crossing a specified inability-to-continue threshold. Zero equity, a margin breach, and a personal minimum-capital threshold are different definitions and must not be interchanged.

D7. Position and exposure. An equity weight w is the fraction of portfolio value exposed to the equity instrument at a specified time. A rule restricting 0 ≤ w ≤ 1 prohibits shorting and portfolio leverage in this model. It does not remove the instrument's own market risk.

D8. Costs. Costs include modeled commissions, spread, slippage, financing, and other charges. A cost convention specifies whether a rate applies to traded notional or total capital. Taxes and inflation are separate unless explicitly included.

D9. Binary bet. A simplified bet wins net b units for each unit staked with probability p, or loses the stake with probability q = 1−p. A stake fraction f produces wealth factors 1+bf or 1−f. This model is not assumed to describe all market trades.

D10. Log growth. The expected log wealth increment of a strategy is E[ln(1+r)], when defined. For a binary bet, g(f) = p ln(1+bf) + q ln(1−f). Long-run compound growth and arithmetic expected profit are different objectives.

D11. Trading method. A method specifies instruments, data, observation times, signal, size, order timing, costs, and failure handling. A label such as trend following or buying strength is not yet a method.

D12. Empirical adequacy. In this book's practical vocabulary, empirical adequacy means that a claim fits stated evidence and survives relevant checks within a specified scope. This is a contemporary research standard, not an assertion that a backtest attains Spinoza's technical adequate idea.

Common assumptions

A1. Arithmetic and probability. Ordinary arithmetic, linearity of expectation, and probability identities apply where their quantities are defined. Results using variances or expected logarithms explicitly require those moments to exist.

A2. Accounting. External deposits and withdrawals are absent during the compounding examples. Dividends, fees, and corporate actions must be consistently included in the definition of portfolio return. Nothing is earned twice by changing notation.

A3. Positive wealth. Logarithmic results apply only while wealth factors remain strictly positive. The binary examples use b > 0 and 0 ≤ f < 1, with 0 < p < 1 unless a boundary is explicitly considered.

A4. Non-anticipation. Decisions may depend on F_t but not information that becomes available afterward. Knowing a final closing value generally means acting after that observation, not retroactively at the same price.

A5. Nonnegative modeled costs. The costs in comparisons are nonnegative, and the comparisons hold gross positions and outcomes fixed unless they say otherwise. Rebates and market-making economics require a different net-cost specification.

A6. Conditional repetition. Independence, identical distributions, and stable parameters are assumed only in propositions that explicitly invoke them. They are not general facts about financial markets.

A7. Admissible uncertainty. In the no-guarantee construction, both an adverse and a favorable next price move are compatible with the information already observed. No known contractual payoff or arbitrage eliminates that uncertainty for the directional position considered.

A8. Observation versus estimation. A historical record is finite. Its estimated probabilities and costs may differ from the future environment. Claims about future distributions require more than the identity of past observations.

How dependencies are read

“Depends on D4, A2” means that the conclusion relies on the definition of return and the absence of outside cash flows. It does not mean that a real account with deposits can be compared without adjustment. A proof is useful precisely because it exposes what changes when a premise changes.

The practical method later adds empirical hypotheses, designated H1–H3, and design choices, designated R1–R9. They are deliberately not added to the axioms as guaranteed properties of markets.

Expectancy and Compounding

Proposition 1. Net profit is gross profit less cost.

Depends on D2, D8, A2.

Proof. Let G be the marked profit before the modeled costs C. By the accounting definition of net profit, X = G−C. Taking expectations where they exist gives E[X] = E[G]−E[C]. Q.E.D.

Scholium. A commission-free label does not establish C = 0. Spread and execution shortfall can remain. Equally, an expense already embedded in a fund return must not be deducted a second time. Define the ledger before discussing performance.

Proposition 2. A two-class trade population has expectancy pW−(1−p)L−C.

Depends on D2–D3, A1; assumes exhaustive winning and losing classes with finite conditional mean payoffs.

Proof. Let p be the probability of a gross winning trade, W its conditional mean gross gain, and L the conditional mean magnitude of a gross losing trade. Assign any zero outcome consistently. The law of total expectation gives E[G] = pW+(1−p)(−L). Subtract expected cost C by Proposition 1. Q.E.D.

Example. At a 40% win rate, a $200 mean win, a $100 mean loss, and $5 expected cost, expectancy is $80−$60−$5 = $15 per trade. Those numbers define a hypothetical population. A trader must still establish whether observations justify using them.

Proposition 3. Positive expectancy requires p > (L+C)/(W+L).

Depends on Proposition 2; assumes W+L > 0 and fixed W, L, C.

Proof. Positive expectancy means pW−L+pL−C > 0. Thus p(W+L) > L+C. Dividing by the positive denominator yields the stated condition. Q.E.D.

Corollary. Costs increase the break-even winning probability when other quantities remain fixed. They cannot be omitted because the strategy wins often.

Proposition 4. Win rate alone does not determine profitability.

Depends on Proposition 2.

Proof by counterexample. Strategy A wins 90% of the time, gaining $1 on a win and losing $20 on a loss, with zero cost. Its expectancy is $0.90−$2 = −$1.10. Strategy B wins 40% of the time, gaining $3 and losing $1, also without cost. Its expectancy is $1.20−$0.60 = $0.60. The higher-win-rate strategy has negative expectancy and the lower-win-rate strategy has positive expectancy. Therefore win rate is insufficient. Q.E.D.

Scholium. An attractive hit rate may be purchased by occasional large losses. Look at payoff magnitude and tail behavior, not merely the frequency of being right.

Proposition 5. Additional nonnegative costs cannot improve the same gross return path.

Depends on A2, A5; assumes comparable positive multiplicative wealth factors.

Proof. Write each gross wealth factor as a_t > 0. Let the two net factors be a_t−c_t and a_t−k_t with 0 ≤ c_t ≤ k_t < a_t. Each first factor is at least the second. Multiplying positive factors preserves the ordering. The wealth path with larger deductions has no larger terminal wealth. Q.E.D.

Limitation. This does not prove that low turnover always beats high turnover. A different trading rule may earn a different gross return. The proposition isolates costs, not strategy quality.

Proposition 6. Capital compounds by multiplication.

Depends on D4, A2.

Proof. From D4, V_t = V_(t−1)(1+r_t). Repeated substitution gives V_n = V_0 × ∏(1+r_t), for t = 1 through n. Q.E.D.

Corollary. An arithmetic average of returns does not by itself specify terminal wealth. The complete sequence of wealth factors matters through their product. With no intervening flows and fixed factors, rearranging their order leaves the product unchanged, although the interim drawdown path changes.

Proposition 7. Recovering a fractional drawdown d requires gain d/(1−d).

Depends on D5, A2; assumes 0 ≤ d < 1.

Proof. Equity after the loss is H(1−d). To recover H, a subsequent gain g must satisfy H(1−d)(1+g) = H. Dividing and rearranging gives g = 1/(1−d)−1 = d/(1−d). Q.E.D.

These are exact accounting relationships before rounding, not forecasts of recovery time.

Proposition 8. Equal positive and negative percentage returns leave a loss.

Depends on Proposition 6; assumes 0 < a < 1.

Proof. Their combined wealth factor is (1+a)(1−a) = 1−a² < 1. A 20% gain and a 20% loss therefore leave 96% of initial capital, in either order. Q.E.D.

Scholium. This is one reason a volatile sequence can disappoint someone focused on average gains. It does not establish that all volatility is avoidable or that minimizing volatility maximizes wealth.

Proposition 9. Expected log growth cannot exceed the log of the expected wealth factor.

Depends on D10, A1, A3; assumes required expectations exist.

Proof. The logarithm is concave on the positive real numbers. Jensen's inequality gives E[ln(1+r)] ≤ ln(E[1+r]). Equality holds for a constant wealth factor, subject to the usual equality condition. Q.E.D.

Corollary. An expectation stated in dollars and an objective stated in compound growth are not interchangeable. A high arithmetic expected return can coexist with poor expected log growth.

Proposition 10. Zero capital is absorbing in a self-financing multiplicative model.

Depends on Proposition 6, A2; assumes finite subsequent wealth factors.

Proof. If V_t = 0, then V_(t+1) = 0 × (1+r_(t+1)) = 0. Repeating the identity preserves zero. Recovery requires an external inflow or a change in the model. Q.E.D.

Scholium. Practical ruin often arrives before zero. A broker's margin requirement, minimum contract size, or personal reserve may end trading earlier. Those thresholds must be specified rather than hidden inside a dramatic use of the word ruin.

Sizing and Dependence

Proposition 11. A stop-defined stake limits the modeled loss only if execution respects the assumed loss bound.

Depends on D7–D9.

Proof. If a position contains Q units and each can lose at most L monetary units, its loss is at most QL. That follows by addition. If the actual loss per unit can exceed L, the premise needed for the bound is absent; the bound no longer follows. Q.E.D.

Scholium. A planned stop distance is a parameter in an order plan. It is not automatically a contractual maximum loss. Gap, liquidity, execution, and instrument structure determine whether the assumed bound is real.

Proposition 12. A sequence of n full-stake losses removes fraction 1−(1−f)^n of capital.

Depends on D9, Proposition 6; assumes fixed fraction f and no extra costs.

Proof. Each loss multiplies remaining equity by 1−f. After n losses, equity is V_0(1−f)^n. Dividing the reduction by V_0 yields 1−(1−f)^n. Q.E.D.

Example. At 32.5% per bet, five losses remove approximately 85.99%. At 8.125%, the same sequence removes approximately 34.54%. Both are severe. The calculation is not a recommendation to use either stake in real trading.

Proposition 13. Even a favorable independent strategy can produce a losing streak.

Depends on D9, A6; assumes independent bets with 0 < p < 1.

Proof. The probability that a specified block of n bets all lose is q^n > 0. The probability that at least one loss occurs in n bets is 1−p^n > 0. Neither becomes zero merely because expectancy is positive. Q.E.D.

Limitation. The probability of encountering some losing streak somewhere in a long record is not q^n; overlapping possible blocks must be accounted for. Independence may also fail in real returns.

Proposition 14. The binary-bet log-growth optimum is f* = (pb−q)/b when this lies in the allowed interior.

Depends on D9–D10, A3; assumes known, fixed p and b and the binary model.

Proof. Differentiate g(f):

g′(f) = pb/(1+bf) − q/(1−f).

Setting this to zero gives pb(1−f) = q(1+bf). Since p+q = 1, pb−q = bf, hence f* = (pb−q)/b. If the solution falls outside the feasible set, compare permitted boundaries instead. For a long-only stake with pb−q ≤ 0, the optimum is f = 0. Q.E.D.

Example. With p = 0.55 and b = 2, f* = (1.10−0.45)/2 = 0.325. This fraction is optimal for expected logarithmic wealth under the stated model, not for every investor's constraints or preferences.

Proposition 15. The binary log-growth function is strictly concave.

Depends on D10, A3.

Proof. A second differentiation gives

g″(f) = −pb²/(1+bf)² − q/(1−f)² < 0.

Thus an interior stationary point is the unique maximum. Beyond it, increasing f lowers expected log growth. Q.E.D.

Scholium. A trader can have a genuine positive-expectancy opportunity and still stake so much that compounding is unfavorable. Position size is part of the strategy's mathematics, not an afterthought.

Proposition 16. A fixed fraction of Kelly does not preserve a universal fraction of growth.

Depends on D10, Propositions 14–15.

Proof by counterexample. For p = 0.55 and b = 2, substitute f = 0.325 in g. Full-Kelly expected log growth is approximately 0.098557 per bet. At f/2 it is 0.074977, or 76.07% of the optimum. At f*/4 it is 0.044682, or 45.34%. These differ from an exact 75% and 50%. Thus the proposed universal fractions are false. Q.E.D.

Scholium. The familiar half-Kelly approximation is useful within appropriate approximations. It must not be promoted into a universal identity. Fractional staking can reduce sensitivity to an overestimated edge, but does not create an edge or guarantee an acceptable drawdown.

Proposition 17. Estimation error alone does not imply an overestimated edge.

Depends on A8.

Proof by counterexample. Let an estimator take the true value plus ε, where ε is equally likely to equal +a or −a for a > 0. Its error is real, but it overestimates and underestimates with equal probability. Therefore the mere existence of estimation error cannot imply overestimation. Q.E.D.

Scholium. Selection of only the best-looking strategies, data mining, and nonstationarity may create more serious upward bias. A conservative stake should be justified by those uncertainties and the cost of overbetting, not by an invalid claim about all errors.

Proposition 18. A zero-arithmetic-edge positive wealth factor cannot have positive expected log growth.

Depends on Proposition 9; assumes E[r] = 0 and existing moments.

Proof. Jensen's inequality gives E[ln(1+r)] ≤ ln(1+E[r]) = ln(1) = 0. Q.E.D.

Corollary. Sizing alone does not turn a fair net-return gamble into positive expected logarithmic growth. Changing the payoff distribution, price, information, or costs is a different operation.

Proposition 19. Portfolio variance depends on covariance as well as individual variances.

Depends on A1; assumes fixed weights and finite second moments.

Proof. For portfolio return R = Σw_i R_i, expand E[(R−E[R])²]. Linearity gives Var(R) = Σ_i Σ_j w_i w_j Cov(R_i,R_j). The diagonal terms are individual variances; off-diagonal terms capture joint movement. Q.E.D.

Scholium. Five trades in five symbols can be one economic bet if their losses are driven by the same event. Counting positions is not the same as measuring dependence.

Proposition 20. Portfolio risk is not generally the sum of trade risks multiplied by a correlation coefficient.

Depends on Proposition 19.

Proof by counterexample. Take two independent, equal-weight positions with variance σ² > 0. Correlation is zero, yet portfolio variance is σ²/2, not zero. Multiplying the sum of their risks by correlation would incorrectly give zero. Therefore that formula is not a general risk identity. Q.E.D.

Scholium. Historical correlation itself may change. Scenario losses, concentration, and common funding constraints supplement covariance analysis. None of these is reducible to a single reassuring multiplier.

Information and Evidence

Proposition 21. A stop order does not logically guarantee its trigger price as a fill.

Depends on D1, Proposition 11; assumes execution at available prices rather than a contractual guaranteed fill.

Proof by construction. Let a sell stop trigger at 100. Suppose the next executable price after the trigger is 95. An order executing at the available price realizes 95, not 100. Such a path is compatible with the stated execution rule, so a guaranteed fill of 100 does not follow. Q.E.D.

Scholium. A stop-limit order changes the permitted execution price but may fail to execute. Neither order type abolishes the tradeoff between price and execution certainty. The actual venue and order rules must be read before use.

Proposition 22. A signal computed from a completed closing observation must not be credited with an earlier execution.

Depends on D1, A4.

Proof. Let the signal require an observation not contained in F_t before the close. An order selected using that observation before it became available depends on information outside F_t. This violates non-anticipation. Therefore a valid simulation must use an execution convention consistent with the observation's availability. Q.E.D.

Corollary. The practical method below computes the signal after the final monthly close and changes exposure at the following session's open. Existing holdings bear the overnight move before that fill.

Proposition 23. A finite price history does not entail a profitable next directional trade.

Depends on A7.

Proof by construction. Consider two admissible price paths identical through decision time. On the first, the next move favors a nonzero chosen position. On the second, it moves sufficiently against that position to produce a loss after costs. The decision uses the same history and therefore takes the same position on both paths. Since one loses, the shared history cannot logically entail profit. Q.E.D.

Limitation. This proposition concerns uncertain directional exposure. It is not a theorem that every contractual payoff or correctly specified arbitrage has the same uncertainty structure.

Proposition 24. Defining a trend does not prove persistence.

Depends on D11, Proposition 23.

Proof. Define a trend signal as a current observation above an average of past observations. Both the favorable and adverse next paths in Proposition 23 can share that signal. The definition distinguishes histories but does not exclude the adverse future. Therefore persistence requires an additional empirical premise. Q.E.D.

Scholium. A definition of price as conatus in motion conceals exactly this missing premise. Conatus belongs to Spinoza's philosophical vocabulary. A measured continuation effect belongs to market research.

Proposition 25. A zero equity weight removes direct equity-price exposure in the stated equity/cash model.

Depends on D7; assumes the cash component has no equity-price exposure.

Proof. The equity-price contribution to portfolio return is wR_e. At w = 0 it is zero. Q.E.D.

Limitation. Cash still has its own yield, inflation, currency, custody, and instrument risks. Being out of equities does not mean being outside all risk, and exiting after a decline does not undo the decline already experienced.

Proposition 26. A timing rule's average excess return contains an exposure term and a covariance term.

Depends on A1; considers a single-period, frictionless equity/cash model with defined moments.

Proof. Let X = R_e−R_c and let the period's equity weight be w. Portfolio excess return is wX. By the covariance identity,

E[wX] = E[w] E[X] + Cov(w,X).

Subtract expected costs for a net comparison. Q.E.D.

Scholium. A rule may earn money simply because it holds a rising asset part of the time. The covariance term is the part associated with timing exposure to subsequent excess returns. That term must be measured, not inferred from a story about discipline.

Proposition 27. Lower constant exposure can reduce variance without forecasting skill.

Depends on A1; assumes a constant cash return and 0 < w < 1.

Proof. With risky return X and constant cash return c, portfolio return is wX+(1−w)c. Constants contribute no variance, so portfolio variance is w² Var(X), less than Var(X) when Var(X) > 0. Q.E.D.

Corollary. Compare a trend filter with a static stock/cash allocation, not only with fully invested equities. Otherwise a reduction in risk may be mistaken for evidence of successful timing.

Proposition 28. Selecting the largest estimate can create optimism even when each estimate is unbiased.

Depends on A1; assumes integrable estimates.

Proof. For each j, max_i Z_i ≥ Z_j in every outcome. Taking expectations gives E[max_i Z_i] ≥ E[Z_j] for every j, hence E[max_i Z_i] ≥ max_j E[Z_j]. Random variation can make the inequality strict. Therefore the expected selected maximum can exceed the best underlying expected estimate. Q.E.D.

Scholium. Trying many indicators and reporting only the winner purchases apparent certainty with hidden selection. Freeze the principal rule first. Publish neighboring-parameter checks together, not as auditions for a replacement winner.

Proposition 29. Positive expected log growth implies positive asymptotic compound growth under explicit repetition assumptions.

Depends on D10, Proposition 6, A6; assumes independent identically distributed positive wealth factors and E[|ln(1+r)|] < infinity.

Proof. Taking logarithms of the compounding identity gives

ln(V_n/V_0)/n = (1/n) Σ ln(1+r_t).

The strong law of large numbers makes this average converge almost surely to E[ln(1+r)]. If that value is positive, the limiting geometric growth factor is exp(E[ln(1+r)]) > 1. Q.E.D.

Limitation. This is not a finite-horizon guarantee. Market returns need not be independent or stationary, and the expectation is not supplied by the theorem. A strategy that crosses a practical ruin threshold before the long run cannot benefit from an asymptotic result it does not survive to realize.

What the propositions establish together

The accounting is exact within its definitions. The growth results are conditional on their distributions and assumptions. The no-guarantee construction identifies the missing bridge between past observations and future profit. The empirical task begins where those deductions leave an unknown quantity: the distribution of returns produced by an executable method after costs.

The method that follows is offered at that boundary. It is sufficiently precise to be tested, and no stronger than the evidence permits.

A Practical Construction: Monthly Equity and Cash

The propositions tell us what must be accounted for. They do not select a profitable rule. Selection now becomes an empirical undertaking, and the writing must change accordingly.

The method examined here is a long-only monthly trend filter applied to SPY, the State Street SPDR S&P 500 ETF Trust. It holds either the equity fund or cash. It does not sell short, borrow, trade options, or attempt to predict the next intraday movement. Its intended use in this book is an understandable example of systematic exposure management, not a personalized portfolio prescription.

Empirical hypotheses

H1. Some equity declines unfold slowly enough that a lagging trend filter can reduce exposure before the worst of the decline. This may fail in a gap, sudden crash, or rapid reversal.

H2. Avoiding some prolonged declines can reduce realized drawdown and volatility sufficiently to compensate a particular investor for missed gains and whipsaw losses. “Sufficiently” depends on the objective; lower risk is not automatically superior performance.

H3. The effect may survive modest costs and reasonable neighboring parameter choices. This is a question for measurement, not a consequence of the first two hypotheses.

Possible explanations include gradual adjustment to information, persistent flows, or investors' changing risk tolerance. These explanations are plausible research directions, not proved causes of the results. Historical evidence for other trend strategies motivates the test without validating this particular implementation in advance.

The complete rule

R1. Instrument and account. Use SPY as the equity instrument for the historical illustration, denominated in US dollars. Equity target is either 100% or 0%; remaining value is cash. The historical cash return is a Treasury-bill research proxy, not an actual brokerage offer. An implementation must name its own cash vehicle and record the yield and costs actually obtained.

R2. Data. Use split- and distribution-adjusted daily closes for the signal. Take the final trading-session close of each completed calendar month. Ten completed monthly observations are required; until then, hold cash. Do not mix raw closes in one month with total-return-adjusted closes in another.

R3. Signal. Let T_m be the month-end adjusted close and M_m the arithmetic mean of T_m through T_(m−9). If T_m > M_m, target equity. If T_m ≤ M_m, target cash. Equality is assigned to cash. Ten months is a fixed design choice, not a discovered optimum.

R4. Observation and order. Calculate only after the final monthly close is available. Submit a target-changing order for the following trading session's open, using a supported order type and the broker's stated cutoff. If timely data or an eligible order is unavailable, retain existing holdings and log the skipped change rather than claiming a fill that did not occur.

R5. Exposure before the fill. Existing holdings earn or lose the overnight move from month-end close to the next open. A sell signal cannot retroactively place the account in cash during that interval. Gap risk remains.

R6. Size and turnover. Rebalance only when the equity/cash target changes. Account for fees when determining shares so the purchase is funded without borrowing. An unchanged target creates no signal-driven trade. Whole-share rounding, distribution cash, and vehicle requirements may produce small differences in a real account; document them rather than silently redefining the rule.

R7. Costs. The baseline study deducts 0.05% of traded notional on each side, with a 0.25% sensitivity. A purchase and later sale incur two costs. Fund expenses already reflected in the price series are not deducted again. The cash proxy includes no separate trading charge, which can overstate a real implementation's cash return.

R8. Constraints. Do not add leverage to recover missed gains. Do not widen the universe, shorten the signal, or add an indicator merely because the last trade lost. Any such alteration is a new method requiring its own record and evaluation.

R9. Monitoring. Archive observations, targets, order submissions, executions, fees, and deviations. Reconcile the account with the expected holdings. The simulation marks at daily closes; actual intraday or liquidation losses can be larger than its reported maximum drawdown.

A worked decision

Suppose the ten completed month-end observations sum to 4,900, so the average is 490. The newest adjusted close is 510. The target is equity because 510 > 490. If the account is already in equity, there is no change. If it is in cash, the order is scheduled for the next eligible session.

If the next opening price gaps upward, the account buys at the actual opening execution, not at yesterday's close. If it gaps downward while the account is already invested, the account bears that overnight loss even if the signal has changed to cash. The rule governs action; it does not govern the available price.

Suppose a funded cash account has $10,000 and faces a one-way proportional cost c = 0.0005. Buying $10,000 of stock would leave no cash for the fee. In the continuous-unit model the purchased notional is $10,000/(1+c), approximately $9,995.00; the remainder pays the modeled cost. The research code uses this funded accounting rather than creating a small margin balance.

The purpose, fixed before looking

The principal historical objective is positive net compound growth together with lower daily-return volatility and a smaller daily-close maximum drawdown than equity buy-and-hold, in both the full sample and the combined post-2007 sample. This is a risk-management objective, not a requirement to beat SPY's return.

The comparisons include buy-and-hold, Treasury cash, and a static 50/50 equity/cash portfolio rebalanced monthly. The latter matters because lower stock exposure can reduce risk without successful timing (Proposition 27). Eight- and twelve-month filters, higher costs, zero cash income, and next-session-close execution are published as sensitivities. The ten-month rule remains the principal specification even when another lookback looks better afterward.

What would count against the method

Failure to retain positive growth or reduce the specified risks in the declared windows counts against its stated historical objective. Dependence on implausible fills or negligible costs also counts against it. Underperformance to the static mixed portfolio is evidence that the extra operational burden may not be worthwhile for the relevant period and objective.

Even a passing result supports only a bounded statement about a historical simulation. It does not establish an investable edge for all accounts, superior after-tax wealth, protection against all crashes, or an acceptable personal drawdown. The next chapter reports what the fixed test actually found.

What the Historical Test Found

The method passed its declared historical risk-reduction objective against fully invested SPY. It did not establish that timing was superior to simpler allocations, and it did not earn the highest return. The distinction is the result, not a footnote.

The objective and parameters were recorded before this calculation according to the research log. They were not independently preregistered. The sample and strategy family were selected retrospectively in September 2026; the post-2007 period is post-publication history, not a newly untouched prospective test.

Data, execution, and limits

The sample runs from January 3, 1994 through July 31, 2026. Earlier SPY observations supply the signal warm-up. It uses Yahoo Finance daily open, close, and adjusted-close observations retrieved September 5, 2026. Adjusted open equals raw open multiplied by that day's adjusted-close/raw-close ratio. Existing equity earns the previous adjusted close to the next adjusted open; changed holdings then earn the adjusted open to close. Dividends are not added again.

This is a vendor-adjusted total-return simulation, not an exact share ledger with dividend payment dates. Adjusted data can be revised. SPY's fund operating expenses are embedded; the model additionally charges 0.05% per one-way traded notional. It begins in cash and pays the initial entry cost. It does not charge a final liquidation, model taxes, or adjust for inflation.

Cash uses Kenneth French's monthly RF research returns. Each realized monthly return is allocated geometrically over that month's equity trading sessions, half overnight and half intraday. This retrospective synthetic accrual is not a rate known at the start of every day or an available cash-account return. It is not used to choose the signal. The zero-cash-income sensitivity shows how much the modeled interest contributes, but neither case substitutes for an actual cash vehicle's ledger.

CAGR uses elapsed calendar time. Volatility is the sample standard deviation of daily returns multiplied by the square root of 252. Drawdown is measured at daily closes, so it can miss larger intraday losses. Subperiods retain the holdings from the continuous simulation while resetting the reporting high-water mark. All returns are nominal; 2026 is partial through July.

Full-period comparison

The filter's lower drawdown came with a lower compound return than buy-and-hold. It held equity on roughly 78% of daily closing observations. Holding less equity explains some reduction in risk; the result alone does not establish timing skill. The static 50/50 allocation had lower volatility but a larger full-period maximum drawdown than the filter.

The principal filter made 47 nonzero rebalances including its initial entry. Its worst month was August 1998, about −14.12%; its worst calendar year was 2022, about −21.06%. The longest observed period below its prior high lasted 1,001 calendar days. At the final observation it remained below its latest high. These are substantial costs of the method even though it passed its declared comparison with fully invested equities.

Fixed historical slices

1994–2006

2007–2016

2017–July 2026

Combined post-2007

The 2017–July 2026 comparison is especially important: static 50/50 earned a higher CAGR than the filter while showing lower volatility and a smaller drawdown. That observation prevents a claim that the extra timing decisions consistently improved the tradeoff.

In the full sample, the ten-month filter underperformed buy-and-hold in approximately 57.8% of the 332 overlapping sixty-month windows. In the post-2007 slice it underperformed in approximately 88.1% of such windows. Overlapping windows share data and are descriptive observations, not independent statistical trials. The method can demand patience without rewarding that patience relative to the benchmark.

Sensitivities, not replacement winners

The neighboring lookbacks had favorable full-sample results under the specified risk objective, but some outperformed the principal ten-month choice. We retain the original choice and show the alternatives rather than relabeling the best historical parameter as the method discovered by proof. The larger one-way cost reduced the principal CAGR. Zero cash income also reduced it. Next-session-close execution changed the path and drawdown.

The accompanying research tables retain every declared sensitivity in every fixed period, including higher-cost versions of the neighboring filters and benchmarks. The manuscript table is a summary, not a selection of the only results that were calculated. Cash-excess Sharpe ratios are in the full tables; the zero-cash scenario uses a zero reference and should not be compared as if it used the same RF benchmark.

What “works” means here

The supported claim is specific: in this retrospective, cost-adjusted simulation, the rule retained positive growth and reduced daily volatility and maximum daily-close drawdown relative to SPY buy-and-hold in the full and post-2007 intervals. A trader seeking that particular tradeoff has a concrete candidate to examine.

The unsupported claims would be that this proves positive future expectancy, that it beats buy-and-hold or a static mixed portfolio generally, that it is optimal, or that a monthly average prevents catastrophic losses. There are no confidence intervals, dependence-robust significance tests, or estimates of the timing covariance in Proposition 26 here. These descriptive results do not establish a statistically validated edge.

Published work by Faber supports studying a monthly average; research by Moskowitz, Ooi, Pedersen, and Hurst supports studying trend strategies across other markets. Those are distinct studies with distinct data and assumptions. They do not convert this simulation into a theorem or independently replicate its exact implementation.

The appropriate next step is prospective operational verification and a comparison with the simpler alternative the trader could actually hold. If the objective is consistently superior return at comparable risk, this study has not established it. If the objective is a transparent rule with a demonstrated historical reduction in equity drawdown and volatility, the study supplies evidence with clearly visible limits.

Reproduction

The research companion includes the protocol, Python standard-library calculation, focused accounting and timing tests, source URLs, input hashes, and complete numerical tables. Run python3 books/research/backtest.py in the source project to reproduce the archived-input calculation. The public companion explains how to retrieve inputs separately; later vendor downloads may differ from this edition's hashes.

Research companion: https://kuttystauder.com/book-assets/trader-research.zip.

The Discipline of Operation

The method has no opinion about your last trade. It is an instruction conditioned on observations. Your task is to make the observations and execution reliable enough that its actual performance can be compared with its stated performance.

Before considering capital

Run the rule in a prospective paper ledger. This means recording signals as they become available, not reconstructing them after the market has moved. Verify the trading calendar, the month's final observation, the order cutoff, the next execution price, and the handling of distributions and cash. A paper profit does not prove an edge, but a broken paper ledger is sufficient reason to repair operations before exposing capital.

Keep money needed for near-term obligations outside the experiment. The monthly filter provides no guaranteed maximum loss. A person who cannot tolerate the demonstrated drawdowns or long benchmark shortfalls has learned something useful from the study; there is no need to reinterpret discomfort as a philosophical failure.

The monthly record

Record the date and provider of the ten observations. Record their average, the final comparison, the old target, the new target, and whether a trade was required. Record the time the data became available and the time the order was submitted. Finally record the fill, fee, actual cash return, and any reason the account differs from the modeled holdings.

If the signal was equity but an outage prevented the purchase, record a missed execution. Do not rewrite the ledger as though the strategy had been in equity. If an order filled poorly, include that cost. A record that excludes operational failures describes an imaginary account.

Three different failures

An execution failure occurs when the intended rule was not carried out: wrong data, late order, incorrect share size, or an unrecorded fee. Correct the process and retain the cost in the historical record.

A model failure occurs when the assumptions or performance claims no longer describe what the method produces. It cannot be diagnosed from every isolated loss. But neither can it be dismissed indefinitely by claiming that an unseen edge will eventually reveal itself. Compare realized behavior with the declared objective and alternatives at scheduled reviews.

A fit failure occurs when the rule's demonstrated behavior is incompatible with the user's needs. The method can behave exactly as tested and still be unsuitable. Someone requiring reliable short-term cash withdrawals faces a different problem from someone able to accept years of benchmark underperformance.

These failures require different responses. A missed order is not repaired by inventing a new indicator. A vanished return effect is not repaired by greater devotion. An unsuitable objective is not repaired by calling the investor weak.

A review calendar

Check operations after each scheduled decision. Review economic performance at a fixed annual date rather than changing the rules after every uncomfortable month. Include all observations, costs, departures from the rule, and the same benchmarks used at the beginning. If the account's cash return differs materially from the research proxy, recompute the comparison with the actual rate.

A review may conclude that the rule should remain unchanged, be suspended, or be replaced by a simpler allocation. Suspend immediately when data or order integrity is in doubt; there is no requirement to continue an operationally invalid experiment until the annual date. Resume only after the cause is understood and the account reconciled.

If redesign is needed, keep the prior record intact. Specify the new hypothesis, freeze its rules, and label the next experiment separately. Do not splice the best periods of successive strategies into a composite that no one could have followed at the time.

The place of affect

Fear may signal a real risk, a mismatch between size and resources, or a misleading response to noise. Confidence may reflect understanding or merely a recent winning streak. Neither affect carries its own certificate of truth.

The Spinozan exercise is to investigate what the affect is doing to the decision. Is the urge to increase size a response to stronger evidence, or a desire to erase a loss? Is the urge to abandon the rule based on a broken premise, or on seeing another strategy's recent gain? Write the distinction before acting.

This does not make a person emotionless. Spinoza recognizes active joy and desire as well as passive affects (Ethics III, propositions 58–59). There can be satisfaction in accurate accounting, restrained exposure, and a conclusion honestly revised. The account need not rise every month for those activities to have value.

The truth of trading

The truth established by the proofs is conditional and demanding. Outcomes must be measured net of costs. Capital compounds. Loss and recovery are asymmetric. Estimated probabilities are not known probabilities. Information must precede action. A finite history does not command its own continuation.

The empirical truth is narrower: a particular historical simulation produced a particular set of gains, losses, and tradeoffs. It is enough to study that result seriously without turning it into a promise. A useful method is one whose rules can be executed and whose claims remain proportionate to its evidence.

The trader's freedom is not exemption from uncertainty. It is the ability to act within uncertainty without falsifying it. Keep the premises visible. Keep the ledger intact. Let the result correct the story.

Primary references and research record

1. Euclid, Elements, Book I: definitions, postulates, common notions, and propositions. The proof architecture is the inspiration, not a claim that markets satisfy Euclidean postulates. https://mathcs.clarku.edu/~djoyce/elements/bookI/bookI.html
2. Spinoza, Ethics, especially III definitions 1–3, propositions 1, 6–7, 58–59. Contemporary trading applications are this author's interpretations. https://www.gutenberg.org/ebooks/3800
3. Kelly, J. L. (1956), “A New Interpretation of Information Rate.” Expected logarithmic growth and the distinction from expected wealth. https://doi.org/10.1002/j.1538-7305.1956.tb03809.x
4. Faber, M. (2007; updated 2013), “A Quantitative Approach to Tactical Asset Allocation.” The author-hosted version specifies a ten-month moving-average model and discusses historical limitations. Its original execution and cost assumptions differ from this edition's test. https://mebfaber.com/wp-content/uploads/2016/05/SSRN-id962461.pdf
5. Moskowitz, T., Ooi, Y., and Pedersen, L. (2012), “Time Series Momentum.” Evidence across futures and forwards; not a direct validation of the single-ETF rule here. https://www.aqr.com/Insights/Research/Journal-Article/Time-Series-Momentum
6. Hurst, B., Ooi, Y., and Pedersen, L. (2017), “A Century of Evidence on Trend-Following Investing.” Reconstructed diversified trend strategies differ from this book's implementation. https://www.aqr.com/-/media/AQR/Documents/Insights/Journal-Article/AQR-JPM-Fall-2017.pdf
7. State Street, SPY fund information. Instrument identity and inception; this fund tracks large-cap US equities, not the entire global opportunity set. https://www.ssga.com/us/en/intermediary/etfs/state-street-spdr-sp-500-etf-trust-spy
8. Yahoo Finance, historical price adjustments. Vendor-adjusted data are revisable and do not supply an exact dividend-payment-date ledger. https://help.yahoo.com/kb/SLN28256.html
9. Kenneth French data library, factor definitions and RF series. Research Treasury-bill returns are used as a cash proxy with an explicitly synthetic daily accrual convention. https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_factors.html
10. This edition's research protocol, code, tests, and numerical tables: https://kuttystauder.com/book-assets/trader-research.zip. Local archived input checksums identify the run; source data can change on later retrieval.