DBF: direction and breadth of the outcome

Executive summary

Direction–Breadth Factor (DBF) is a compact way to describe two properties of a series of periodic outcomes: the direction of the net result and the extent to which that result is distributed across observations. It does not measure total risk, path quality, or future returns.

The recommended output is not a standalone number, but the vector (CAGR, D±, J, DBF±). CAGR preserves economic severity; D± shows direction and cancellation; J shows breadth; DBF± provides a signed summary. This separation avoids asking one metric to answer incompatible questions.

1 The question: what sustains the outcome?

Two strategies may both finish at +20% and reach that destination through very different mechanisms. One may accumulate small gains throughout most of the year; another may earn everything in a single session; a third may alternate large gains and losses before ending at the same point.

Sharpe asks about mean return relative to dispersion; Calmar relates growth to maximum drawdown; the Ulcer Index summarizes drawdown depth and persistence. DBF asks a different question:

Is the net direction supported by many contributions of comparable magnitude, or does it depend on a few observations?

The word breadth is retained deliberately because it does not simply mean “regularity”. It describes the effective fraction of participating magnitudes, not the visual smoothness of the equity curve. The corresponding effective number is N·J=N₂.

2 Compact formulation

Let rₜ be the log return in period t, and let N be the total number of observations in the window. The unit may be a day, week, month, or trade, but it must be declared before calculating the metric.

The expression reduces to one fraction:

For an all-zero series we adopt D±=J=DBF±=0. There is no directional information or activity to distribute. This is an explicit computational convention, not an economic claim.

• D±=1: every nonzero movement is positive.
• D±=−1: every nonzero movement is negative.
• D±=0: positive and negative movement cancel exactly.
• J=1: every magnitude is equal and no period is zero.
• J=1/N: one observation concentrates the full magnitude.

3 Mathematical anatomy and relationship with Sharpe

DBF does not start from components without precedent. D± is the signed Kaufman Efficiency Ratio applied to the same increments. J is Jain's index applied to |rₜ|; moreover, N·J equals the effective number of order 2 —inverse Herfindahl or Hill N₂— for the weights wₜ=|rₜ|/Σ|rₜ|.

With the periodic Sharpe ratio, zero risk-free rate, and population standard deviation,

the following exact identity holds:

This forces a restrained interpretation: holding J fixed, DBF is a bounded monotonic transformation of Sharpe. Compressing Sharpe creates no new information. Any potential incremental content resides in J, and its economic usefulness still requires out-of-sample validation.

The unsigned version previously published as the Coherence Factor satisfies CF=|DBF±|. DBF preserves sign inside direction itself, without adding a penalty branch or a hyperparameter η.

4 Controlled laboratory: same destination, different mechanism

Every curve below finishes at exactly +20%. The number of contributions, cancellation, and order vary. These are not backtests or market samples: they are economic unit tests designed to isolate behavior.

The component view is most informative. A strategy may have high D± and low J —a concentrated directional gain— or high J and low D± —widely distributed activity that cancels. A low DBF does not identify which mechanism dominates on its own.

5 What DBF does not observe: temporal order

D±, J, Sharpe, and DBF depend on the multiset of returns, not their order. “Clustered losses” and “interleaved” contain exactly the same values and therefore obtain the same DBF by construction. Their drawdowns and economic experiences nevertheless differ.

This is not an implementation bug; it is a boundary of the estimand. Path analysis requires path-aware measures such as maximum drawdown, the Ulcer Index, Martin, or direct inspection of the equity curve.

6 What DBF also does not observe: economic severity

Preserving sign solves an important presentation problem: a sustained loss no longer appears as “positive coherence”. DBF nevertheless remains scale-invariant. A smooth −0.05% loss and a smooth −90% loss have the same relative pattern and may both produce DBF±=−1.

With losses, two states with the same CAGR may motivate different investigations: a gradual loss with high breadth shows a loss distributed within the window; a single crash with low breadth concentrates the adverse outcome in one event and may justify reviewing tail exposure. DBF describes that historical difference, but it does not identify the edge, measure tail risk on its own, or automatically decide which strategy should be closed or rescued.

7 Frequency is part of the metric

DBF does not exist independently of an observational unit. Twelve positive PnLs spread across twelve trades produce J=1; the same twelve PnLs placed within 252 sessions —with 240 inactive days— produce J=12/252. Neither answer is “the true one” without first defining the question.

Temporal aggregation also changes contributions: daily movements that cancel within a week disappear when weekly PnL is calculated. A daily DBF and a trade-level DBF therefore must not be compared as if they measured exactly the same object.

8 Comparison with established industry references

There is no winning metric because each one answers a different question. DBF belongs in a strategy factsheet as a complementary descriptor, not as a replacement for the performance-and-risk core.

8.1 Reading DBF alongside Sharpe

• Sharpe and DBF share information algebraically; they must not be marketed as orthogonal.
• DBF exposes J, but it has not yet been shown that J improves an out-of-sample decision.
• Boundedness in [−1,1] aids communication but compresses extreme differences.
• When the decision requires a scalar economic utility, a utility-derived measure such as MPPM is a more appropriate benchmark than an arbitrary penalty.

9 Potential uses in quantitative finance

DBF may be useful when the question is specified before observing the result and comparisons are made within homogeneous universes: same frequency, horizon, costs, exposure, and PnL convention.

9.1 Minimum recommended protocol

1. Fix the observational unit and window before calculation.
2. Always report (CAGR, D±, J, DBF±) together with drawdown and costs.
3. Compare only strategies with compatible exposure and frequency.
4. Investigate low D± and low J separately: they imply different mechanisms.
5. Do not optimize DBF as a standalone objective until an out-of-sample validation tied to a concrete decision exists.

10 Python formulation from an array of PnLs

The public function is simply named dbf. It accepts periodic PnLs and the initial capital to which they belong. Capital is required to reconstruct a strictly positive equity curve, transform monetary PnL into log returns, and calculate a comparable CAGR.

10.1 Minimal example

If the data are already log returns, the three mathematical expressions may be applied directly. With trade-level PnLs, N is the number of trades; with daily PnLs, inactive days must remain as zeros. Those two estimands are not interchangeable.

The complete reference implementation, tests and executable notebook are maintained in the public GitHub repository linked at the end of this study. The web edition preserves the input contract, formulas, numerical example and resulting values without duplicating source code.

11 Strengths, limitations, and validation agenda

11.1 What remains for the paper

This note establishes the formulation, terminology, and a reproducible implementation. Future academic work must study estimator theory, behavior under dependence and heavy tails, statistical power, frequency robustness, and above all whether J adds incremental value to a real out-of-sample decision.

We do not claim that DBF is superior to Sharpe, predicts returns, measures tail risk, is manipulation-proof, or decides on its own which strategy should be retained. Nor do we claim priority over Kaufman, Jain, Hill, effective numbers, Mallart–Fink, or the algebraic identities derived from those components. QINVIA's contribution is this signed synthesis, its terminology, implementation, and joint interpretation.