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Self-Evolving Multi-Agent Symbolic Discovery for Financial Fundamental Analysis

Kelvin J. L. Koa, Filip Orestav, Shengqiong Wu, Michael J. Wooldridge, Ke-Wei Huang

arXiv:2609.32746v1cs.MAcs.AIcs.LGq-fin.CP

TL;DR

Financial symbolic discovery must handle multiple valid valuation perspectives, shifting market conditions, and noisy feedback. MUFASA combines specialized valuation agents, context-aware coordination, and statistical memory; it reports best performance across datasets while producing interpretable equations.

  • Problem

    Financial valuation differs from scientific symbolic regression because it admits multiple valid perspectives, varies with market conditions, and yields noisy, continuous performance signals.

    Source: Introduction · p. 1

  • Method

    MUFASA uses specialized agents for distinct valuation perspectives, a context-conditioned meta-coordinator, and statistical-performance memory to guide symbolic discovery.

    Source: Introduction · p. 2

  • Results

    MUFASA achieves the best performance across all datasets, with 19.65% average improvement, and reduces the better agentic baseline’s error by 27.67% on average.

    Source: Results · p. 6

  • Takeaways & Limitations

    Released equations, distilled iteration lessons, and context-dependent perspective weights offer insights for future research on financial fundamental analysis.

    Source: Introduction · p. 2

    The number of agent perspectives and market contexts is manually designed and might not capture the full heterogeneity of real-world markets.

    Source: H Limitations of the work · p. 22

Abstract

from arXiv · show

While symbolic regression (SR) has been successfully used in science to discover new equations, its use in financial valuation is hindered by several limitations. Whereas the natural sciences provide objectively correct relationships, financial valuation constitutes a distinct class of symbolic discovery problems, as it admits multiple valid perspectives, operates under non-stationary market conditions, and involves noisy, continuous performance signals. In this work, we propose Multi-Agent Fundamental Analysis with Symbolic Adaptive learning (MUFASA), a hierarchical multi-agent framework for symbolic discovery in finance. MUFASA introduces (1) disentangled equation discovery via specialized agents representing distinct valuation perspectives, (2) a meta-coordinator that performs hierarchical-level reasoning over market context information, and (3) a memory mechanism that reasons over statistical performance summaries (e.g., accuracy, stability, and tail risk) to guide learning under noisy feedback. Experiments across datasets from multiple countries show that MUFASA achieves state-of-the-art performance on the valuation task compared to classical finance methods, financial large language models, and SR approaches, while simultaneously producing interpretable equations, which we share with the community. We also make publicly available the distilled learnings across evolution iterations and context-dependent strategy weights, which might offer useful insights for future research on financial fundamental analysis.

Introduction

Financial symbolic discovery must accommodate multiple valid valuation perspectives, shifting market conditions, and noisy feedback rather than seek one stable equation. MUFASA addresses these challenges with specialized agents, context-aware coordination, and statistical memory, and reports strong cross-market forecasting performance with interpretable outputs.

Introduction

  • Financial valuation poses three challenges for symbolic discovery: multiple valid perspectives, non-stationary environments, and noisy, continuous feedback.Figure 1 overviews how these challenges span levels of the self-evolving loop.Source: Introduction · p. 1, Figure 1 · p. 2
  • MUFASA decomposes discovery among specialized agents, each representing a distinct valuation perspective, to encourage diverse yet interpretable relationships.Perspectives include cash-flow- and earnings-based approaches.Source: Introduction · p. 2
  • A meta-coordinator uses contextual information such as sectors and market regimes to dynamically weight specialist outputs across environments.This combines perspective-specific knowledge as market conditions vary.Source: Introduction · p. 2
  • Agents maintain structured memories of explored equations and statistical performance, including accuracy, stability, and tail risk, to guide search under noisy feedback.The framework uses these summaries to identify robust candidates and refine search strategies.Source: Introduction · p. 2
  • Across multiple countries, MUFASA reports state-of-the-art valuation forecasting against classic finance methods, financial LLMs, and SR, while producing interpretable equations.The released artifacts include equations, distilled lessons across evolution iterations, and context-dependent valuation-perspective weights.Source: Introduction · p. 2

Related Work

Related work spans fundamental analysis, symbolic regression, and self-evolving LLM agents. The paper distinguishes its focus on forecasting and valuation from financial QA and frames its approach against scientific SR and agent-learning methods.

Related Work

  • Financial fundamental analysis uses accounting information to estimate intrinsic company value, while financial LLM work cited here focuses on structured-data QA rather than forecasting or valuation.Fundamental strategies include value, factor, and macroeconomic approaches, which analyze different information and may yield different investment outcomes.Source: Related Work · p. 2
  • Classical symbolic regression often uses evolutionary algorithms to evolve expressions, while LLM-guided methods use learned knowledge to generate hypotheses and have succeeded in scientific domains.Classical approaches can face large search spaces and computational inefficiency.Source: Related Work · p. 2
  • Self-evolving LLM-agent research includes feedback-based self-reflection, memory that accumulates experience, and multi-agent systems with specialized roles and coordination.These systems can improve through environment feedback, experiential memory, and interaction among agents.Source: Related Work · p. 3

The mufasa Framework

MUFASA frames financial valuation as context-dependent symbolic discovery, using specialized valuation perspectives, a coordinating agent, and statistical memory. Its framework combines agent expressions according to market context and uses accumulated performance patterns to guide later hypotheses.

Problem Formulation

  • MUFASA targets interpretable valuations from accounting data that generalize out of sample despite unspecified explanatory variables and context-dependent relationships.The context variable combines company sector and market regime.Source: Problem Formulation · p. 3

Disentangled Symbolic Discovery

  • Valuation philosophies constrain each agent’s equation space through preferred variables and structures, preserving diversity across candidate expressions.Agents generate hypotheses from philosophy-specific prompts and accumulated learnings, then retrieve variables, execute candidates, and optimize their coefficients.Source: Disentangled Symbolic Discovery · p. 3

The mufasa Framework

  • The framework overview depicts specialized agents producing diverse expressions, with a meta-coordinator integrating them and each agent retaining statistical-pattern memory.Table 1 is identified as the collection of statistical information used for learning under continuous feedback.Source: Figure 2 · p. 4, Table 1 · p. 5

Hierarchical-Level Reasoning

  • A meta-coordinator uses market context and agent performance to select agents and assign context-specific weights to their valuations.This hierarchical design supports continual adaptation of agent weights under non-stationary feedback.Source: Hierarchical-Level Reasoning · p. 4

Statistical Performance Memory

  • MUFASA’s memory records equation-level statistical behavior, and distilled learnings guide later hypotheses beyond isolated pass/fail signals.Repeated evaluations can provide more reliable relative-performance estimates, including variance and tail behavior.Source: Statistical Performance Memory · pp. 4–5

Experiments

The experiments evaluate MUFASA on five equity markets using quarterly fundamental and macroeconomic data, chronological train/test splits, and a range of financial, machine-learning, symbolic-regression, and agentic baselines. The supplied passages describe the evaluation setup but report no comparative outcomes.

Dataset

  • The five-market evaluation uses quarterly filings and next-quarter stock prices, with chronological 80/20 train/test splits and controls for survivorship and look-ahead bias.Markets include the S&P 500, CSI 300, STOXX 600, Russell 2000, and FTSE Asia Pacific ex-Japan.Source: Experiments · p. 5

Market Settings

  • For regime context, the study follows standard practitioner convention: at the index level, a bull market is declared after an index rise of at least 20% from its most recent trough.Source: Experiments · p. 5

Baselines

  • Baselines span traditional machine learning, symbolic regression, financial reasoning LLMs, classical valuation models, and financial agent frameworks.Named examples include Random Forest, PySR, Fin-R1, sector-median multiples, and FinCon.Source: Experiments · p. 5

Implementation Details

  • All agents use Qwen3-4B-Thinking, and model performance is evaluated with MAPE, where lower values are better.The implementation uses local vLLM serving on NVIDIA RTX A5000 GPUs.Source: Experiments · p. 5

Results

MUFASA leads the reported valuation benchmarks across datasets, while its ablations and learned equations support the value of perspective-specific discovery. Its specialized agents also finish with lower errors than a general single agent.

Performance Comparison

  • Traditional valuation ratios are competitive across most markets, while Gordon Growth FCFF and FCFE methods perform less well with noisy growth-rate and discount-factor estimates.Fin-R1 outperforms Fino1, though both financial LLMs underperform most specialized baselines; PySR performs strongly among SR baselines.Source: Results · p. 6
  • MUFASA achieves the best performance across all datasets, averaging 19.65% improvement and reducing error by 27.67% versus the better agentic baseline.The authors attribute this advantage to perspective-specific hypothesis generation and statistical memory that retains empirically robust expressions.Source: Results · pp. 5–6, Table 2 · p. 6

Ablation Study

  • Equal-weight multi-perspective agents outperform a single-perspective agent on four of five markets, and adding hierarchical coordination improves on equal weighting on four markets.Without statistical memory, the system cannot reliably distinguish consistently effective expressions from those succeeding by chance.Source: Results · p. 6, Table 3 · p. 6

Self-Evolving Agents

  • Each agent’s lowest error decreases over iterations, but the general single agent ends with higher error than the specialized agents.The authors relate this to the difficulty of capturing heterogeneous valuation mechanisms with one expression.Source: Results · p. 7, Figure 3 · p. 7

Discovered Equations

  • MUFASA’s specialized agents recover adapted earnings-multiple, discounted-cash-flow, book-value, relative-valuation, and residual-income forms on the S&P 500 dataset.Fin-R1 largely reproduces EPS·PE, while PySR’s high-performing expression is described as financially opaque.Source: Results · p. 7, Table 4 · p. 7

Conclusion

MUFASA addresses symbolic discovery in financial fundamental analysis with specialized agents, hierarchical context reasoning, and statistical memory. Across global markets, it achieves strong forecasting performance while producing interpretable, economically grounded expressions.

Conclusion

  • The paper frames financial fundamental analysis as a distinct symbolic-regression problem and introduces MUFASA, releasing its learned expressions and distilled agent learnings publicly.The framework combines specialized agents, hierarchical-level context reasoning, and statistical-based memory to guide learning.Source: Conclusion · p. 7

A Detailed Methods

The detailed methods describe iterative training for perspective agents and a meta-coordinator. The agents generate and evaluate perspective-specific expressions, while the coordinator learns context-dependent weights from forecast errors.

A Detailed Methods

  • The appendix presents these procedures as detailed algorithms for philosophy-agent and meta-coordinator training.The algorithm captions identify the two training procedures.Source: A Detailed Methods · pp. 10–11

Philosophy Agent Training

  • For each perspective, the agent iteratively generates an expression, retrieves data, evaluates it, optimizes hypotheses, and updates memory and distilled learnings.Retrieval or evaluation errors are fed back into the agent’s prompt.Source: A Detailed Methods · p. 10

Meta-Coordinator Training

  • The meta-coordinator initializes context-specific weights, generates weights from market context and learnings, and evaluates the weighted forecast using MAPE.It stores weights with their MAPE losses and updates its learnings from that memory.Source: A Detailed Methods · p. 11

B Statistical Significance Analysis

The significance analysis uses corrected Diebold–Mariano tests to compare MUFASA with baselines. The reported pairwise comparisons show statistically significant MAPE improvements.

Statistical Significance Analysis

  • The DM tests use a Newey–West HAC variance estimator with a Bartlett kernel and Andrews’ rule bandwidth to account for serial correlation across quarters.Persistent difficult forecasting conditions can autocorrelate loss differentials and otherwise underestimate standard errors.Source: B Statistical Significance Analysis · p. 11
  • A Harvey–Leybourne–Newbold correction adjusts the DM statistic and uses a tT −1 reference distribution to account for few test quarters per market.Table 5 reports MUFASA-minus-baseline differences; negative values indicate lower error, with stars denoting significance thresholds.Source: B Statistical Significance Analysis · p. 11, Table 5 · p. 11
  • Across all pairwise comparisons, MUFASA achieves statistically significant improvements in MAPE.The authors report this indicates that the observed gains are mostly robust and not driven by random variations.Source: B Statistical Significance Analysis · p. 11, Table 5 · p. 11

C Learnt Symbolic Expressions

MUFASA discovers interpretable valuation expressions that retain recognizable financial structures while adapting to empirical patterns. Its meta-coordinator also reallocates valuation-perspective weights across sectors and market regimes.

C Learnt Symbolic Expressions

  • Across datasets, MUFASA’s expressions recover components of classical valuation formulas while adding data-driven adjustments, unlike several baselines’ harder-to-interpret structures.The learned forms show recurring patterns across markets, while adapting to dataset-specific conditions.Source: C Learnt Symbolic Expressions · p. 12, Table 6 · p. 12, Table 7 · p. 12
  • PySR and FunSearch often produce complex or hard-to-interpret expressions, while LLM-SR captures coarse additive relationships rather than common nonlinear valuation structures.MUFASA’s expressions balance interpretability and flexibility, with recurring structures across Russell 2000, STOXX 600, FTSE Asia, and CSI 300.Source: C Learnt Symbolic Expressions · p. 12

D Learnt Context Weighting

  • MUFASA’s meta-coordinator varies the importance of earnings, cash flow, asset, growth, and quality perspectives by sector and bull-or-bear regime.Weighting tendencies recur across geographically distinct markets, although exact patterns vary by sector and regime.Source: D Learnt Context Weighting · p. 13, Figure 4 · p. 14

E Distilled Agent Learnings

Distilled agent learnings favor structured, bounded combinations over isolated or unbounded financial variables. The diagnostics identify sector- and firm-dependent instability, informing adjustments tailored to valuation perspectives.

Earnings-based learnings

  • Earnings formulations favor bounded growth caps and sector anchoring, while uncapped growth and standalone multipliers are associated with systematic overprediction.The distilled summary identifies earnings multiplied by PE with capped growth as the best structure and flags a persistent Technology-sector gap.Source: E Distilled Agent Learnings · p. 15

Cash-flow learnings

  • Cash-flow formulations favor sector-adjusted EV/EBITDA, growth-adjusted scaling, and explicit handling of negative cash flow.Static multipliers and truncating negative cash flow with max() are identified as problematic; bounded growth structures are preferred.Source: E Distilled Agent Learnings · p. 16

Asset-based learnings

  • Asset-based formulations favor bounded multiplicative structures and sector adjustments, with controlled use of high-impact variables such as R&D intensity.The learnings associate unbounded R&D multipliers with Technology overprediction and log-based adjustments with invalid predictions.Source: E Distilled Agent Learnings · p. 17

Growth-based learnings

  • Growth-based formulations favor growth-capped scaling and sector adjustments; the distilled summary identifies income multiplied by P_FCF with growth adjustments as a preferred structure.The learnings also report persistent Technology overprediction and warn against standalone EBITDA_TTM_PER_SHARE multipliers and fixed PE thresholds.Source: E Distilled Agent Learnings · p. 19

Agent adaptation

  • The distilled learnings describe agent adaptation through bounded growth terms, sector-specific scaling, and combinations of valuation, growth, and cash-flow-quality signals.Cash-flow agents also introduce adjustments for negative cash flow rather than truncating it.Source: E Distilled Agent Learnings · pp. 16, 17, 19

Statistical diagnostics

  • Candidate expressions can fail through instability or invalid outputs in specific conditions, which scalar loss alone may not reveal.Diagnostics such as variance, coefficient of variation, and denominator effects help guide the search toward robust expressions.Source: E Distilled Agent Learnings · p. 20

Cross-agent patterns

  • Instability depends on firm type and sector interactions, and recurring sector over- and underestimation supports context-dependent weighting.The learnings cite Technology overprediction and Consumer Discretionary underprediction or volatility-driven errors across valuation agents.Source: E Distilled Agent Learnings · p. 20
  • High-impact variables are not necessarily discarded; diagnostics favor combining them with anchoring or adjustment terms rather than using them alone.Growth signals, R&D intensity, and sector multiples are repeatedly described as unstable or biased in isolation.Source: E Distilled Agent Learnings · p. 20

F Extended Ablation Study

The extended ablation compares agent coordination strategies and memory settings. Its results favor meta-coordination and fuller statistical feedback for more consistent symbolic discovery across markets.

F Extended Ablation Study

  • Across memory settings, the meta-coordinator generally performs strongest, often exceeding the best individual agent and indicating gains from combining complementary perspectives.The comparison includes single-perspective, equal-weight multi-agent, best-single-agent, and meta-coordinator strategies.Source: F Extended Ablation Study · p. 20, Table 10 · p. 20
  • Full Statistical Memory typically performs strongest across model variants, compared with binary pass/fail and MAPE-only memories.It stores information about average error, reliability, and risk profile, helping distinguish consistently performing equations from occasional successes.Source: F Extended Ablation Study · p. 20
  • Overall, MUFASA benefits from multi-perspective agents, hierarchical coordination, and richer statistical feedback for more consistent symbolic discovery across markets.The extended study compares strategies under different memory settings.Source: F Extended Ablation Study · p. 20, Table 10 · p. 20

G Additional Statistical Results

Additional analyses report strong rank correlation, low error distributions in several sectors, and lower directional bias overall, while identifying sector-specific tail-error and design boundaries. The weighting and evaluation conclusions remain qualified by market context and manually chosen model components.

G Additional Statistical Results

  • MUFASA achieves the highest Spearman correlation in the S&P 500 comparison, despite PySR having better MAPE than FunSearch there.The paper notes that MAPE and rank correlation capture different aspects of performance.Source: G Additional Statistical Results · p. 21, Figure 5 · p. 21
  • MUFASA shows lower and more concentrated sector error distributions in Utilities, Industrials, Real Estate, and Consumer Staples.Its upper- and lower-percentile errors are slightly higher in Communications and Consumer Discretionary; unexpected news-driven jumps were not considered.Source: G Additional Statistical Results · pp. 21–22, Figure 6 · p. 21
  • MUFASA has lower directional bias than baselines overall, with errors closer to zero across valuation quartiles, but tends to overestimate Q1 and underestimate Q4 companies.Figure 8 reports mean signed percentage error across the cheap-to-expensive quartiles.Source: G Additional Statistical Results · p. 22, Figure 8 · p. 22

H Limitations of the work

  • Classical valuation baselines use modeling assumptions such as constant growth, and evaluating every possible assumption variant may be intractable.The paper identifies adaptive selection of these assumptions as a possible future direction.Source: H Limitations of the work · p. 22
  • The manually designed number of agent perspectives and market contexts may not capture all real-world market heterogeneity.The framework is modular, so additional agents or contexts could be incorporated without modifying its core architecture.Source: H Limitations of the work · p. 22

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