In quantitative finance, standard regression losses are misaligned with the economics of return prediction. As the conditional mean of financial log-returns is close to zero, symmetric losses such as the mean squared and mean absolute errors make the constant zero forecast a near-optimal solution, penalizing models with genuine but noisy directional skill. This applies both during training, where predictions shrink toward zero, and during evaluation, where trivial forecasters can lead loss-based rankings. Under a Gaussian linear prediction model, we show that all symmetric monotonic losses share a universal breakeven directional accuracy against the zero predictor, which rises sharply and becomes unobtainable as the prediction noise approaches the standard deviation of the returns. We introduce the CZAR (Composite Zero-Agnostic Return) loss function, a piecewise quadratic loss built around five key requirements: convexity in the prediction, asymmetry with true return direction that vanishes at zero, near-linear penalization of undershoots and wrong-direction predictions, divergence for large errors, and an adaptive loss floor for evaluation. CZAR is provably convex in the prediction at fixed true value, has closed-form gradient and Hessian suitable for custom objectives in gradient-boosted libraries, and its four hyperparameters reduce to a single choice through correlated defaults. In idealized tests, the minimum directional accuracy required for a CZAR-evaluated forecaster to outperform the zero predictor under mean log loss remains near the 50% chance level, whereas the corresponding threshold for symmetric losses rises sharply with prediction noise. This advantage persists under heavy-tailed return distributions. In a LightGBM experiment on intraday BTC log-returns, CZAR-trained models reduce the `zero-returns bias' and improve directional accuracy on large-magnitude returns.
Figures & tables
Figure 1: Scatter plots of predicted vs. true returns under the Gaussian linear model y^=ρy+N(0,σN2) for signal coefficient ρ∈{0,0.1} (rows) and noise scale σN∈{0,0.5,1} (columns). The red diagonal marks perfect prediction. Directional accuracy (DA), signal-to-noise ratio, mean absolute error, and mean squared error are annotated in each panel. The top-left panel ( ρ=0,σN=0 ) is the zero-returns predictor with y^≡0 , collapsing all predictions to the horizontal axis. The remaining ρ=0 panels are pure-noise predictors y^=σNε . Despite zero DA, the zero-returns predictor achieves lower MAE (0.795) and MSE (1.007) than the signal-carrying predictor at ρ=0.1,σN=0.5 (MAE 0.826, MSE 1.079), illustrating the zero-returns attractor problem.
Figure 2: Breakeven directional accuracy vs. noise scale σN for asymmetric MAE (top row) and asymmetric MSE (bottom row), evaluated under mean linear loss (left) and mean log loss (right). Black solid lines show the universal symmetric analytical breakeven, orange dashed lines show symmetric MAE or MSE baseline, gray dashed lines show symmetric loss with an additive Gaussian floor ( λ∈{1,3} ), and colored dash-dotted lines show asymmetric loss downweighting overshoots by factor α∈{0,0.25,0.5} (linear panels) or α∈{0.25,0.5} (log panels). In all panels, the relations become vertical when Gaussian linear models cannot achieve lower losses than the zero-returns model. Left panels: For linear losses, asymmetry reduces the breakeven DA while remaining >50% for all σN . A linear undershooting term (MAE, α=0 , upper left panel) achieves far lower breakeven than a quadratic one (MSE, lower left panel), and the α sensitivity is negligible for MSE. Right panels: For log losses, asymmetry pushes the breakeven below 50% at small σN , meaning a model predicting consistently in the wrong direction could achieve a lower mean log loss than the zero-returns predictor, a clearly undesirable property. The Gaussian floor instead keeps the breakeven ≥50% while reducing it at intermediate σN .
Figure 3: LCZAR as a function of standardized prediction z^ for fixed true value z=2 , showing sensitivity to the two primary hyperparameters. The left panel shows varying α at fixed β=1 . Larger α steepens the quadratic penalty in both regions. The right panel shows varying β at fixed α=1 . Larger β suppresses Region B (overshoot, z^>z ), making the asymmetry more pronounced.
Figure 4: Gradient and Newton step of LCZAR for standardized true values z∈{−1,0,1,5} . The left panel shows the gradient σ∂LCZAR/∂y^ vs. standardized prediction z^ . Each curve has a step discontinuity at z^=z of magnitude 1−βeff(∣z∣) . The gradient approaches −sign(z)(1−βeff) from Region A and 0 from Region B, so it always jumps upward as z^ increases through z . For z>0 (e.g. z=1,5 ) Region A lies to the left of the region boundary and the gradient jumps from −(1−βeff) up to 0 . For z<0 (e.g. z=−1 ) Region A lies to the right and the jump runs from 0 up to +(1−βeff) . For large ∣z∣ the step grows as βeff→0 , while for z=0 ( βeff=1 ) the gradient is continuous and linear (MSE-like), with no discontinuity. The right panel shows the Newton step (∂L/∂y^)/(∂2L/∂y^2) vs. z^ . In Region B the factor βeffα is shared by the gradient and Hessian and cancels in the ratio, leaving a Newton step of σ,Δz=y^−y that is independent of α and βeff (i.e. the overshoot damping seen in the loss and gradient does not carry over to the Newton step). In Region A the constant Hessian ( α/σ2 ) gives a Newton step σ,Δz+σ(1−βeff)/α , i.e. the MSE step plus a constant offset from the MAE term that grows as ∣z∣ increases and as α shrinks. In both panels, solid lines show α=1 and dashed lines show α=0.01 . Both use the correlated asymmetry term β=β∗(α) from Equation ( 10 ) and floor level C=C∗(α) from Equation ( 11 ) (with τ=0.5 and μ=0 ).
Parameter
Role
Domain
Default
α
Quadratic regularization strength
α>0
1
β
Asymmetry decay rate
β≥0
β∗(α) , eq. ( 10 )
C
Floor target level
C≥0
C∗(α) , eq. ( 11 )
τ
Hinge smoothing width
τ>0
0.5
Table 1: CZAR hyperparameters. At β=0 , βeff=1 for all z and both regions are purely quadratic (MSE-like). Larger β increases the asymmetry between Region A (MAE+quadratic) and Region B (discounted quadratic) for returns with large ∣z∣ .
Figure 5: LCZAR as a function of standardized prediction z^ for several standardized true values z (default parameters, μ=0 ) and α=1 (solid lines) and 0.01 (dashed lines). Each curve attains its minimum at z^=z . For large ∣z∣ , Region A (undershooting or wrong direction) adopts a near-linear profile while Region B (overshoot) remains quadratic, reflecting the asymmetry encoded by βeff . The loss floor raises the minimum loss for small ∣z∣ , visible as the elevated minimum of the z=0 curve.
Figure 6: Loss in the (z,z^) plane for CZAR (top) and MSE (bottom). Left panels show loss on a linear scale, as seen by a gradient-based optimizer; right panels show log10(loss) , more representative of the dynamic range used when ranking models. The CZAR linear panel uses α=0.01 (where the MAE-like undershoot regime is most visible on a linear scale) while the CZAR log panel uses α=1 (where the dynamic range is best resolved on a log scale); both use the correlated defaults β∗(α) , C∗(α) , and τ=0.5 .
Figure 7: Breakeven directional accuracy vs. noise scale σN under the Gaussian linear prediction model. The left axis shows the minimum DA required to beat the zero-returns predictor; the right axis shows the equivalent breakeven signal-to-noise ρ/σN . The black solid line shows universal analytical breakeven for symmetric monotonic losses, DA=0.5+arctan(ρmin/σN)/π with ρmin=1−1−σN2 ; the gray dashed line shows the lower envelope achievable with the asymmetric MAE loss ( α=0 ) under mean linear loss (reproduced from Figure 2 ); the light blue dashed line shows CZAR under mean linear loss ( α=0.01 ); and the dark blue solid shows CZAR under mean log loss ( α=1 ). Both CZAR curves use the correlated defaults β∗(α) , C∗(α) , and τ=0.5 .
Figure 8: Breakeven directional accuracy vs. noise scale σN under Gaussian (black lines) and heavy-tailed Student- t truth distributions (blue lines for ν=5 and orange lines for ν=3 ), all normalized to unit variance. The left panel shows mean linear loss. The right panel shows mean log loss. In both panels, solid lines show the CZAR loss with the correlated defaults β∗(α) , C∗(α) , τ=0.5 ( α=0.01 for linear loss, α=1 for log loss), and dashed lines show the MSE loss. Under linear loss the breakeven curves are close to distribution-invariant. Under log loss heavier tails raise the breakeven DA for both losses, but CZAR retains its margin over MSE at every σN and tail index.
Figure 9: Sample-averaged log10 -loss vs. prediction-quality metric for the Gaussian linear model, swept over ρ∈[−1.5,1.5] and σN∈[0,1.5] . Each point is one (ρ,σN) pair, with color encoding ρ (blue for positive, red for negative); a small floor ϵ=10−3 is added before taking the log to avoid divergence at zero residuals. The floor is orders of magnitude below the zero-returns baseline of every loss shown and matters only for near-perfect predictors ( ρ≈1 , σN≈0 ), so the position of the dashed line and the orderings discussed in the text are unaffected by its value. The dashed vertical line in each panel marks the log-loss of the zero-returns predictor ( ρ=0 , σN=0 ); points to its left beat the zero-returns baseline. Columns compare MAE, MSE, Huber ( δ=1 ) and CZAR ( α=1 with the correlated defaults β∗(α) , C∗(α) , and τ=0.5 ). Top: directional accuracy (DA) on the vertical axis. Bottom: log aspect ratio log10(σpred/σtrue) on the vertical axis.
Training loss
DA
DA IQR
DA 1σ
log10 AR
RP
IC
Sharpe naive
BTC 15-minute
L1
0.510
0.487
0.473
−0.93
0.019
0.018
−4.97
L2
0.514
0.508
0.499
−1.18
0.016
0.019
0.96
CZAR ( α=0.005 )
0.531
0.523
0.537
−0.46
0.046
0.034
5.62
CZAR ( α=0.01 )
0.522
0.504
0.501
−0.44
0.017
0.033
−0.95
CZAR ( α=0.05 )
0.523
0.525
0.534
−0.59
0.033
0.050
3.95
Table 2: Held-out metrics for LightGBM models trained with each loss on BTC log-returns, at the 15-minute (top) and 1-hour (bottom) horizons. DA 1σ and DA IQR denote directional accuracy restricted to truths with ∣z∣>1σ and to truths outside the interquartile range, respectively. The symmetric losses L1 and L2 shrink their predictions well below the truth ( log10AR≈−0.9 to −1.2 , i.e. roughly an order of magnitude too small) and their directional accuracy on large moves sits at or below the 50% chance line. Every CZAR variant keeps log10AR closer to zero, and often raise DA on large moves above chance, lift RP and IC several-fold over the symmetric baselines, and generally improve the naive Sharpe. Full-sample DA differences are smaller and often within sampling error. Binomial standard errors on the 2000 -candle test window are ≈1.1 percentage points on the full-sample DA, and ≈2 and ≈1.6 points on DA 1σ and DA IQR , which use only the roughly one-third and one-half of truths in the tails. The naive Sharpe ratios exclude costs (i.e. are not comparable to live trading) and are annualized from test windows of ≈21 days (15-minute) and ≈83 days (1-hour).
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure D1 : CZAR breakeven directional accuracy at σN=1 as a function of α and β (both axes log-scale). Color indicates the minimum DA required to beat a zero-returns predictor under linear loss averaging; lower (redder) is better. The black curve marks the optimal β∗(α) and the red dashed curve the two-term power-law fit from Equation ( 10 ). The shallow basin around the optimum shows that the default is robust to moderate departures from β∗ .
Figure D2 : Expected log-loss Ey∼N(0,1)[log10L(y,y^)] as a function of standardized prediction z^ for τ∈[0.1,1.0] (color) at three α values, with C=2 and β=β∗(α) . Solid curves are monotonic in ∣z^∣ and define admissible τ values; dotted curves develop spurious minima near z^≈±2 and are excluded. The smallest admissible τ grows with α ; τ=0.5 is admissible across α≤1 and is adopted as the default.
Figure D3 : Optimization of the loss floor parameter C . The left panel shows log-loss breakeven DA versus σN at the optimized C∗(α) (color), with τ=0.5 and β=β∗(α) . The universal symmetric (MSE) breakeven (thick dashed black) is shown for reference; the dashed gray lines bound the 95% binomial CI for DA=0.5 at N=106 . The right panel shows optimal C∗(α) (black points) and the asymmetric Hill-bell fit of Equation ( 11 ) (red line). C∗ is selected as the value for which a linear fit to the breakeven DA in a small window around σN=0.1 equals 0.5 .
The direction of change --- whether a series will move up or down --- is often as important as its exact value in decisiondriven applications such as risk management and financial forecasting. However, most forecasting losses optimize either point magnitude or shape and frequency structure, and none explicitly targets the direction of change. In this paper, we find that MSE-trained forecasters fail on the direction of small moves. To address this, we propose CosDir, a simple yet effective direction-aware loss that aligns the difference vectors of the prediction and the target via cosine similarity. Being scale-invariant, CosDir keeps a directional gradient on small moves, re-injecting learning signal exactly where MSE neglects it. CosDir is a lightweight, plug-in term that attaches to any backbone without architectural modification. Since the best ratio for mixing the directional and magnitude terms differs across datasets, we further propose CosDir-UW, an extension that makes this ratio adaptive by learning it during training, matching a per-dataset tuned weight with no hyperparameter. We conduct over 100K experiments, demonstrating that our method consistently and significantly improves directional accuracy while preserving magnitude accuracy, and that it outperforms various loss functions. Code is available at: https://github.com/seunghan96/cosdir.
Volatility forecasts play a central role in financial risk management because their overall level and day-to-day movements affect downstream decisions. Most studies compare forecasting models while keeping the training loss fixed. Yet losses emphasise different errors and can target different properties of future volatility, so raw comparisons may combine persistent forecast-level differences with differences in daily forecast movements. This leaves unresolved whether the importance of loss choice comes mainly from the forecast level it targets or from differences that remain after level adjustment. We address this gap through a comparison of seven losses and five models across major cryptocurrencies. Validation-based alignment adjusts the forecast level before the raw and aligned forecasts are evaluated using statistical scores and one-day Value-at-Risk. Before alignment, marginal score variation is greater across losses. After alignment, model choice becomes the larger source of variation in the full five-model comparison, while cross-loss differences in VaR breach rates narrow substantially. Our contribution is a comprehensive evaluation of loss and model choice that shows why losses can appear so influential in raw comparisons and how this interpretation changes when forecast level and downstream risk are considered explicitly.
Financial time series forecasting presents structural challenges absent from standard benchmarks. Log-returns are non-stationary, exhibit exceptionally low signal-to-noise (SNR) ratios, and are governed by regime-dependent temporal dependencies. We identify a key limitation of state-of-the-art (SOTA) time series models in financial settings. A fixed context window is mismatched to the time-varying optimal look-back of non-stationary price processes. We propose the Regime-Aware Variable-context Expert Network (RAVEN), a Mixture-of-Experts framework designed to adaptively determine the temporal context for each input sample. Instead of relying on a fixed look-back horizon, RAVEN constructs a hierarchy of nested contiguous windows whose lengths are determined by the data itself. Specifically, RAVEN scores patches by learned importance in reverse chronological order and applies the Cumulative Importance Thresholding (CIT) mechanism to derive nested prefix windows, each routed to a scale-specialized expert. A Global Compressed Representation (GCR) branch runs in parallel over the full context, preserving global temporal coherence that local experts cannot guarantee. Because the nested routing induces structured overlap among expert inputs, we introduce a Correlation-Aware Weighting (CAW) to align variable-length expert outputs and penalize pairwise cosine similarity prior to aggregation. Experiments on cumulative log-return prediction (HS300, S&P500) and fund sales forecasting demonstrate that RAVEN achieves SOTA performances, improves Pearson correlation by 9.2% on HS300 and 20.2% on S&P500, and reduces MSE by 18.2% on fund sales forecasting, while achieving the best results in 14 of 16 metrics on four PEMS traffic benchmarks.
Cheng He, Zhenyu Guan, Xijie Liang +6
University of Science and Technology of China, China · Shanghai Black Wing Asset Management Co., Ltd., China · College of Sciences, Shanghai University, China +2