Beyond Sharpe

Executive summary

This study does not attempt to crown a universal ratio. It proposes three complementary answers to a concrete question: how can we distinguish two strategies with similar returns but radically different economic experiences?

1. Return, risk and path quality

A return figure tells us how much was gained or lost. It does not tell us whether the result was stable, depended on a single session, required enduring a prolonged drawdown, or whether an apparently clean path concealed continuous cancellation between gains and losses.

The following metrics describe a historical sample. Strategy selection must also consider costs, capacity, tails, dependence, statistical uncertainty and out-of-sample validation.

2. What established measures contribute

None of these limitations invalidates the measure. Each ratio compresses a different question. The error begins when a partial answer is treated as a universal definition of “quality”.

3. From CWR to a multidimensional reading of the path

CWR —Consistency-Weighted Return— was an initial attempt to answer a question that distribution-based ratios do not solve on their own: if two strategies earn the same return, how should we reward the one that gets there more consistently? Its intuition is to multiply return by the fit of the cumulative path to a linear trend.

3.1 Intuition and construction

R²₀ is obtained by fitting cumulative arithmetic PnL against time with a line forced through the origin. A path close to that line retains more of its return; an irregular path is discounted. The proposal has two important virtues: it is easy to communicate and places the shape of the equity curve at the center of the analysis.

3.2 Why it is not sufficient as a general quality measure

This review does not invalidate the original question. It identifies where one regression compresses economically distinct phenomena:

1. Accumulation and apparent fit. Cumulative series are persistent by construction; a high R² may reflect integration rather than a stable source of alpha.
2. A line forced through the origin. The result depends on the initial level, horizon and timing of the profit. Earning early and earning late can receive very different readings without changing total PnL.
3. Unidentified risks. Maximum drawdown, time under water and concentration in a few sessions are not observable components; if they appear at all, they are mixed into the linear fit.
4. Negative domain. Multiplying a loss by a factor between zero and one moves it toward zero. A strategy that loses heavily can therefore appear less bad precisely when its path is worse.
5. Frequency and sample length. Moving from daily to monthly data changes the geometry of the regression and therefore its reading of consistency.

3.3 The role of CWR in this study

CWR remains a historical reference and conceptual starting point, not an opponent to defeat. TRACE retains the multidimensional ambition, TRACE Lite reformulates regularity through geometric distance, and the Coherence Ratio separates direction and concentration in closed form. The main evolution is to expose the components and define consistently what happens when the outcome is negative.

4. Three proposals

4.1 TRACE

TRACE stands for Trajectory, Risk, Alignment and Concentration Efficiency. The name reflects its three blocks: path regularity, drawdown risk and the alignment/concentration of contributions. Let R = CAGR:

• P_R: proximity to the log-linear path, regularized by variability.
• U = 1/(1+UI/u₀): a drawdown factor based on the Ulcer Index.
• Cdir: distribution of contributions aligned with the sign of the outcome.

The negative branch differs because multiplying a loss by a factor below one would move it toward zero and reward a worse path. This study uses γ=δ=λ=η=1 and u₀=10% by default; these values must be disclosed and sensitivity-tested.

4.2 TRACE Lite

If xₜ = log(Eₜ) and aₜ is the line joining the initial and terminal values:

It is easy to communicate and requires no calibration, although a visually straight cumulative curve may conceal returns that continuously cancel one another.

4.3 Coherence Factor and Coherence Ratio

For log returns ℓₜ:

This expression equals directional efficiency × temporal breadth. CF=1 requires equal movements in one direction; a single impulse produces CF=1/N; cancellation pushes CF toward zero.

CF is dimensionless; the Coherence Ratio retains return units. It is simple and has no hyperparameters, but it is order-invariant: it does not distinguish when returns occurred.

5. Experimental design

The study uses two synthetic laboratories:

• Same outcome: 16 curves finish at 1.25 after 504 sessions. CAGR is identical; only geometry, volatility, concentration, autocorrelation and order change.
• Signed domain: 11 one-year curves span +25%, zero and losses down to −90%, including smooth, volatile and single-jump or single-crash paths.

The curves are not intended to imitate a specific strategy. They are economic unit tests: each isolates one property and reveals what every formula actually decides.

6. Comparative results

Absolute values from different ratios are not directly comparable. The heatmap uses within-column percentiles: it shows which paths each definition favors; it does not imply that 0.8 in one metric equals 0.8 in another.

6.1 What disagreements reveal

• One jump: TRACE and the Coherence Ratio penalize it through concentration; TRACE Lite only sees the distance from the equity curve to a line and is less severe.
• Alternating sawtooth: the cumulative curve may look clean —high LQ— while CF falls because returns continuously cancel one another.
• Increasing and decreasing volatility: CF is identical because the multiset of returns is the same; TRACE can distinguish them through the drawdown path.
• Long time under water: Martin and TRACE penalize it naturally; a purely distributional measure does not know the temporal duration of pain.
• Sporadic strategies: CF assigns 1/N to a single impulse. This is transparent, but it may reject legitimately sparse alpha such as convex or event-driven payoffs.

These differences suggest a better practice than choosing a champion: use TRACE as a broad diagnostic, TRACE Lite as a geometric explanation, and CF/Coherence Ratio as a simple detector of cancellation and temporal concentration.

7. Invariances and limits

A publishable metric should state not only what it calculates, but also which transformations leave it unchanged.

Common limitation: none of the three formulas estimates uncertainty, corrects for multiple selection or establishes persistence. A high historical score is not a probability of future success.

8. Potential uses in quantitative finance

These metrics do not attempt to forecast the next return. Their potential value is to describe, compare and monitor how PnL is generated, which matters when several models display similar aggregate returns.

8.1 A reasonable workflow

1. Report CAGR, volatility, maximum drawdown, Ulcer Index, Sharpe and Calmar first.
2. Add TRACE, TRACE Lite and the Coherence Ratio as a second diagnostic layer.
3. Compare in-sample, out-of-sample and rolling windows; one figure for the entire history conceals regime changes.
4. Repeat the analysis across several frequencies and economically coherent initial-capital assumptions.
5. Test whether the metrics improve a real decision —selection, limits or allocation— relative to a baseline model that does not use them.

8.2 Uses to avoid

They should not become a single objective function for optimizing strategies: doing so would encourage overfitting to the formula itself. Nor do they replace costs, capacity, factor exposure, inter-strategy correlation, tails or statistical uncertainty. A high historical score is a descriptive observation, not a probability of future profitability.

9. Practical comparison with reference ratios

Established ratios and the new proposals answer different questions. The useful comparison is not to declare a winner, but to identify which dimension is visible and which remains outside the formula.

9.1 Strengths and potential weaknesses of our proposals

10. Reproducible implementation

The three public functions share the same inputs: a np.ndarray of periodic PnLs and their initial capital. Capital converts monetary PnL into comparable equity, CAGR and drawdown.

The complete TRACE, TRACE Lite and Coherence Ratio implementation—including input validation, signed treatment, the minimal example and tests—is maintained in the public GitHub repository linked at the end of this study.