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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
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Abstract
Abstract: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.
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- Source: https://arxiv.org/abs/2609.32746v1
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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 Abstract 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. 1National University of Singapore 2Asian Institute of Digital Finance 3KTH Royal Institute of Technology 4University of Oxford kelvin.koa@u.nus.edu, orestavfilip@gmail.com, shengqionggwu@gmail.com michael.wooldridge@cs.ox.ac.uk, dishkw@nus.edu.sg Introduction Fundamental analysis (FA) aims to estimate the intrinsic value of a company using accounting information such as earnings, cash flows, and assets, and plays a central role in investment decision-making (Bauman 1996). A long-standing goal in this domain is to derive interpretable relationships (expressed as valuation formulas) that connect financial variables to future returns, which provides actionable insights for decision-making (Shmueli 2010). However, constructing such relationships remains a largely manual and iterative process, often relying on domain expertise to propose, test, and refine hypotheses. Symbolic regression (SR) offers a promising direction for automating this process by discovering mathematical expressions directly from data (Cranmer 2023). Recent advances further integrate large language models (LLMs) into SR pipelines, enabling more efficient hypothesis generation and search by leveraging learned priors (Grayeli et al. 2024; Shojaee et al. 2025; Saveliev et al. 2026). These methods have been particularly successful in scientific domains (Romera-Paredes et al. 2024; Novikov et al. 2025; Xie et al. 2026), where the objective is to recover a single, stable governing equation from data. However, unlike the natural sciences, financial valuation constitutes a fundamentally different class of symbolic discovery problems, for which existing SR formulations are structurally misaligned. Specifically, symbolic discovery in the financial domain exhibits three defining characteristics: • Multiple valid perspectives. Unlike physical systems, financial valuation does not admit a single ground-truth equation. Different valuation philosophies (e.g., earnings-based (Buffett and Cunningham 2001), cash flow-based (Damodaran 2012), growth-based (Asness et al. 1997)) can simultaneously provide equally valid but distinct explanations of asset prices. Current agent-based SR methods (Shojaee et al. 2025; Romera-Paredes et al. 2024) that search for a single equation may largely limit expressiveness and display reduced generalizability across different market conditions. • Non-stationary environments. The relationship between financial signals and outcomes varies across market conditions, such as market regimes and company sectors. Because of this, the same actions could lead to different results across learning iterations. The reward pipeline is confounded by the unstable environment, making it difficult to reliably transfer or reuse past experience. • Noisy, continuous feedback. Candidate expressions are evaluated through noisy and continuous performance signals rather than binary correctness signals. Current agent memory systems (Ouyang et al. 2026; Zhang et al. 2026; Allard et al. 2026; Xiong et al. 2026) that distill lessons from binary success and failure signals may misattribute success to spurious patterns arising from noise or distill learnings that are not robust across the noisy dataset. Figure 1: Overview of the challenges in financial SR across different levels of the self-evolving loop. Importantly, these challenges also compound across stages of the agent self-evolving loop (see Figure 1), making the overall symbolic FA progressively harder as a whole. To address these challenges, we propose Multi-Agent Fundamental Analysis with Symbolic Adaptive learning (mufasa), a hierarchical multi-agent framework for symbolic discovery in finance. First, instead of searching for a single universal equation, mufasa performs disentangled symbolic discovery via a set of specialized agents, each operating under a distinct valuation perspective (e.g., cash flow, earnings, etc.) This design encourages diversity in the discovered symbolic relationships while maintaining interpretability within each perspective. Second, to account for non-stationary market conditions, we perform hierarchical-level reasoning by introducing a meta-coordinator agent that conditions on contextual information (e.g., sectors, regimes) to dynamically weigh the outputs of specialist agents. This enables an adaptive combination of perspective-specific knowledge across varying environments. Third, to cope with noisy and continuous feedback, each agent maintains a structured memory of previously explored equations and summaries of their statistical performance. By reasoning over signals such as accuracy, stability, tail risk, etc., agents can identify robust candidates and refine their search strategies despite noisy supervision from individual observations. To demonstrate the effectiveness of mufasa, we perform extensive experiments across companies from multiple countries (e.g., US, UK, China), using their fundamental information obtained from Bloomberg. Our model achieves state-of-the-art value forecasting performance compared to classic financial valuation methods, financial LLMs, and symbolic regression methods, while also producing interpretable equations, which we share with the research community. In addition, we also make publicly available the distilled lessons across evolution iterations and the importance weights of each valuation method across regimes and sectors, which contain insights that could be useful for financial practitioners. In summary, our contributions in this work are: • We identify financial valuation as a distinct class of symbolic regression problems, characterized by multi-perspective validity, non-stationary environments, and noisy, continuous feedback. • We introduce mufasa, a hierarchical framework that decomposes symbolic discovery across specialized agents, performs hierarchical-level reasoning over environmental signals, and distills lessons from statistical feedback rather than binary signals. • We show that mufasa achieves state-of-the-art valuation forecasting performance across multiple global markets, while recovering interpretable and economically grounded symbolic relationships. • We release a rich set of artifacts, including the discovered symbolic equations, distilled lessons across self-evolving iterations, and context-dependent importance of different valuation perspectives, offering insights for future research on financial fundamental analysis. Related Work Financial Fundamental Analysis. Traditional finance literature defines fundamental analysis (FA) as the evaluation of a company’s accounting information (Bauman 1996), such as the information found in financial statements or balance sheets. The aim is to understand the intrinsic value of the company (Graham et al. 1951) and identify stocks that may be undervalued to make a profit. There are multiple approaches to fundamental investing, such as value investing (Graham 2003; Buffett and Cunningham 2001), factor investing (Fama and French 1992; Asness et al. 1997), or investing around macroeconomic trends (Dalio 2018). Each of these involves analyzing different types of financial information and may result in different investment outcomes (Capaul and others 1993; Asness and others 2000; Petkova and Zhang 2005). Unlike traditional deep-learning methods (Xu and Cohen 2018; Hu et al. 2018; Lin et al. 2021), financial approaches emphasize economic interpretability (Shmueli 2010). Recent works have also explored using financial LLMs to perform interpretable analysis over structured accounting data (Qian and others 2025; Liu and others 2025), but these focus on financial QA, and do not deal with forecasting or financial valuation. Symbolic Regression. Symbolic regression (SR) aims to discover mathematical expressions that describe relationships in data (Cranmer 2023). Classical SR approaches (Schmidt and Lipson 2009a; Schmidt and Lipson 2009b) are often based on evolutionary algorithms, such as genetic programming, which iteratively evolve candidate expressions through mutation and recombination. While effective, these methods often suffer from large search spaces and computational inefficiency. The emergence of large language models (LLMs) advanced SR by enabling knowledge-guided hypothesis generation, where models can propose equations informed by prior scientific knowledge (Grayeli et al. 2024; Shojaee et al. 2025). These approaches have demonstrated strong performance in rediscovering known physical laws and uncovering novel relationships in scientific domains (Romera-Paredes et al. 2024; Novikov et al. 2025; Xie et al. 2026). Self-Evolving LLM Agents. Early works have shown that LLMs can function as agents that interact with external tools and environments (Yao and others 2022; Schick and others 2023), which established the foundation for LLM agents, unlocking external capabilities beyond learning model weights. Later works show that LLM agents can self-improve through various techniques: For example, self-reflective frameworks enable agents to improve by learning from feedback signals from the environment (Shinn and others 2023; Zelikman and others 2022). In contrast, agent memory enables agents to accumulate experience over time and refine reasoning and decision-making strategies (Zhang et al. 2026; Cao and others 2025; Long and others 2026). This idea of using LLMs as agents is further extended to self-evolving multi-agent systems, where multiple agents can collectively improve through coordination (Wang and others 2025; Zhai et al. 2025; Xia and others 2025). Within these systems, different agents can take on specialized roles, while the overall system would improve through environment feedback, experiential memory, and interaction across the agents (Qiu and others 2025b; Qiu and others 2025a; Zhang et al. 2025). Figure 2: Overview of the mufasa framework. mufasa uses disentangled multi-agent systems to encourage symbolic expression diversity. A meta-coordinator agent reasons over the non-stationary environment to integrate these expressions. Each agent maintains a memory to identify past statistical patterns. The mufasa Framework In this section, we first study the interpretable fundamental analysis problem and its challenges as compared to the standard symbolic regression task. We then explain the proposed Multi-Agent Fundamental Analysis with Symbolic Adaptive learning (mufasa) framework, shown in Figure 2. Problem Formulation Given a universe of companies and an accounting information database tX_t for time t, our goal is to produce valuations for each company i, which are evaluated against their look-ahead stock prices yi,t+1y_i,t+1. Importantly, we aim to identify an interpretable symbolic relationship, formalized by a symbolic expression f~ f, that explains how company valuations are derived from accounting data. This setting differs from the standard symbolic regression task (Cranmer 2023; Shojaee et al. 2025) in two ways. Firstly, the relevant explanatory variables i,t⊆tx_i,t _t are not explicitly specified a priori, but must instead be identified from the information database during the discovery process. Secondly, the underlying financial relationships are context-dependent, i.e., the symbolic expression f~ f can vary across market conditions. We define a context variable C(⋅)C(·), which can incorporate multiple contextual factors. In this work, we define C(i,t)=(s(i),r(t))C(i,t)= (s(i),r(t) ), representing company sectors s(i)s(i) and market regimes r(t)r(t), respectively. More formally, given tX_t and observed yi,t+1y_i,t+1 for all companies i, our model seeks to discover a set of explanatory variables ^i,t x_i,t and the context-dependent symbolic expressions f~C(i,t) f_C(i,t) such that: f~C(i,t)(^i,t)≈yi,t+1,∀i,t. f_C(i,t)( x_i,t)≈ y_i,t+1, ∀ i,t. (1) The discovered variables and expressions should not only accurately fit the observed targets, but also maintain strong generalizability across unseen out-of-sample data while remaining interpretable. Disentangled Symbolic Discovery Unlike the natural sciences, financial valuation does not admit a single universal functional form. To capture this heterogeneity, we disentangle the symbolic discovery process using a set of perspectives, each associated with a specific valuation philosophy established in finance literature (Buffett and Cunningham 2001; Dalio 2018). Then, each perspective is incorporated into an LLM agent, which seeks to find the best equation based on its beliefs. Each agent operates autonomously: it can propose hypotheses, retrieve relevant financial variables via external tools, and evaluate candidate expressions based on observed outcomes via code. Perspective Decomposition. Formally, we define a set of valuation philosophies a that constrains the hypothesis space. Each philosophy induces inductive biases over candidate equations, including preferred variables and structural patterns, which prevents the collapse of expression diversity. Hypothesis Generation. For each perspective a, we employ an LLM πθ _θ to iteratively generate candidate symbolic expressions f~() f^(a). At each step, a new hypothesis is sampled as: f~()∼πθ(⋅∣()) f^(a) _θ(· ^(a)), where ()p^(a) is the constructed prompt. The prompt encodes the investing philosophy a and accumulated learnings ()Z^(a) derived from historical interactions and statistical feedback. Tool Use: Data Retrieval. Each candidate expression f~() f^(a) implicitly defines a set of variables ^i,t() x^(a)_i,t that the agent may retrieve from the database tX_t using external tools. Specifically, the agent first queries a semantic retrieval mechanism to obtain candidate variable names and descriptions, and subsequently selects and retrieves the corresponding data. If a retrieval attempt fails (e.g., due to unavailability or insufficient coverage), the model receives structured feedback (Schick and others 2023) and adapts its hypothesis accordingly, either by refining the variables used or modifying the proposed expression. Tool Use: Code Evaluation. Each candidate expression is represented as an executable program, enabling direct evaluation. Given the hypothesized expression f~() f^(a) and retrieved variables i,t()x^(a)_i,t, the expression is evaluated to compute predicted values: y^i,t+1()=f~()(i,t()). y^(a)_i,t+1= f^(a) (x^(a)_i,t ). (2) Similarly, if the expression fails to execute (e.g., due to syntax errors, invalid operations, or non-finite outputs), the system returns structured feedback (Gao et al. 2023), which is incorporated into the prompt ()p^(a). The agent then revises its subsequent hypothesis accordingly. Hypothesis Optimization and Assessment. Similar to LLM-SR (Shojaee et al. 2025), we refine candidate expressions through numerical optimization of their coefficients. Given a proposed expression f~() f^(a), we treat any scalar multipliers or parameters as continuous variables and optimize them directly with respect to the training data. This decouples structure discovery from parameter fitting: the LLM defines the functional form of f~() f^(a), while numerical routines adjust its coefficients to improve empirical performance. We use differential evolution (Storn and Price 1997) as our optimization approach. Following optimization, we assess the fitness of each candidate expression by evaluating its predictive performance on observed data. Given the predicted target values y^i,t+1() y^(a)_i,t+1, we define a fitness evaluation score as the prediction error against ground truth: ℒ()=MAPE(y^i,t+1(),yi,t+1).L^(a)=MAPE ( y^(a)_i,t+1,y_i,t+1 ). (3) Here, the output of the philosophy-constrained function is evaluated against the overall look-ahead prices yi,t+1y_i,t+1 to ensure that the most generalizable function for each philosophy is obtained, as detailed in Algorithm 1 of Appendix A. Hierarchical-Level Reasoning Financial market environments are inherently non-stationary. Because of this, the relationship between candidate solutions and their evaluation outcomes is unstable, making it difficult for learning experience to accumulate. To account for this, we introduce hierarchical-level reasoning, where a meta-coordinator agent learns to reason over market conditions to contextualize the different agent outputs. For each context C(i,t)C(i,t), the goal of the meta-coordinator agent πϕ _φ is to produce a weight vector that shows the relative importance of the agents during that context. At each step, the meta-coordinator constructs a prompt metap^meta which includes the current context, the agent-level performances ℒ()L^(a) and the accumulated learnings metaZ^meta. It then selects a subset of agents, and computes their weight vector ~C(i,t) w_C(i,t) via SLSQP-based (Kraft 1988) constrained optimization. Given the agent outputs and the weight vector, we define the aggregated valuation in each context as: y^i,t+1=~C(i,t)⊤⋅^i,t+1, y_i,t+1= w_C(i,t) · y_i,t+1, (4) where ^i,t+1 y_i,t+1 denotes the vector collecting y^i,t+1() y^(a)_i,t+1 for the set of philosophy-disentangled agents a. Similarly, we assess the quality of each candidate weight vector by evaluating its predictive error: ℒmeta=MAPE(y^i,t+1,yi,t+1).L^meta=MAPE ( y_i,t+1,y_i,t+1 ). (5) This hierarchical design enables continual adaptation of agent weights under non-stationary feedback. Statistical Performance Memory Unlike the natural sciences, which seek a single correct expression, or agent-based tasks with binary outcomes, financial expressions are evaluated through noisy and continuous performance measures. This complicates the ability of current agent memory (Ouyang et al. 2026; Zhang et al. 2026) to effectively learn from past experience, as they typically rely on signals derived from successful or failed attempts at completing a task. To cope with this, we augment agent memory with structured statistical information that captures the statistical behavior of previously explored equations. Rather than storing only a scalar loss, memory records each hypothesis together with a statistical summary of its observed behavior, as demonstrated in Table 1. Category What is stored Why Accuracy Mean MAPE, Median MAPE Captures overall predictive accuracy while reducing sensitivity to outliers and skewed observations. Stability Standard Deviation, IQR, Coefficient of Variation Measures whether performance is consistent or highly variable across samples or time periods. Tail Risk p90/p95p90/p95 Error, Worst-Decile Mean Exposes extreme failure modes that are often hidden by average performance metrics. Complexity Variable + Operator Count, Prediction Coverage Reflects interpretability and numerical reliability of the expression in practical use. Bias Mean Signed Percentage Error (MSPE) Detects systematic over- or under-prediction across observations rather than isolated errors. Ranking Spearman Rank Correlation Captures whether the expression preserves cross-sectional ordering beyond absolute scale accuracy. Calibration Signed Pct Error by Quartiles of Actual Price Reveals whether the expression systematically misprices cheap or expensive stocks. Table 1: Statistical information collected by mufasa to enable learning under continuous feedback. From the memory, we distill a set of learnings ()Z^(a). which guide the generation of new hypotheses at the beginning of the loop. These learnings capture recurring statistical regularities that are not apparent from individual binary pass or fail signals, where a single evaluation could be dominated by noise and lead to unreliable scoring of candidate expressions. By aggregating outcomes across repeated evaluations, the system can recover more reliable estimates of relative performance, including properties such as variance and tail behavior that are not observable from isolated observations. Through this process (detailed in Algorithm 2 of Appendix A), mufasa infers heuristically effective strategies that demonstrate consistent performance across its evaluations, improving the overall robustness of the symbolic search. Experiments Dataset. We evaluate mufasa across five equity markets. The primary dataset consists of all S&P 500 constituents from 1990 Q1 to 2023 Q2, with constituent entry and exit dates included to prevent survivorship bias. To demonstrate generalisability across geographies and market structures, we additionally evaluate on the CSI 300 (2002-2021), STOXX 600 (1995-2025), Russell 2000 (1990-2025), and FTSE Asia Pacific ex-Japan (1996-2025), beginning each series at the earliest date with sufficient data coverage. For all markets, data is split chronologically: the first 80% of quarters form the training set and the remaining 20% form the held-out test set. The fundamental data is sourced from Bloomberg and the macroeconomic data primarily from FactSet. We use data from original quarterly filings only, preventing look-ahead bias from subsequent restatements, and define the target variable as the stock price at the end of the fiscal quarter immediately following each filing period. The forecasting resolution is quarterly, as fundamental analysis is typically used for long-term stock investment. (Graham 2003). Market Settings. For regime context, bull and bear markets are defined at the index level following the standard practitioner convention (Pagan and Sossounov 2003): a bull market is declared when the index has risen at least 20% from its most recent trough, and a bear market when it has fallen at least 20% from its most recent peak. Regimes are computed from the index prices sourced from Yahoo Finance. Baselines. We compare mufasa against three categories of closely-related benchmark models. 1. Traditional machine learning methods: Random Forest (Breiman 2001), HistGBM (Ke et al. 2017), and NeuralNet (Rumelhart et al. 1986). 2. Symbolic regression: PySR (Cranmer 2023), FunSearch (Romera-Paredes et al. 2024), and LLM-SR (Shojaee et al. 2025). These models do symbolic discovery for scientific equations using evolutionary algorithms or an iterative single-LLM loop. 3. Financial reasoning LLMs: Fin-R1 (Liu and others 2025) and Fino1 (Qian and others 2025). These pre-trained LLMs were originally finetuned to perform reasoning over financial accounting data, but they do not deal with forecasting. 4. Financial valuation: We compare with some classical financial valuation approaches, including the sector-median P/E, EV/EBIT, EV/EBITDA, and P/B ratios, and also the Dividend Discount Model (DDM), Free Cash Flow to Equity (FCFE), and Free Cash Flow to Firm (FCFF) models (Damodaran 2012). 5. Financial agentic frameworks: FinCon (Yu et al. 2024); FinVision (Fatemi and Hu 2024) Implementation Details. All LLMs are served locally using vLLM on NVIDIA RTX A5000 GPUs, which takes ~10 mins per agent on each processing unit. We use Qwen3-4B-Thinking (Team 2025) for all agents, with a temperature of 0.7 for the agents and 0.3 for the memory components. The model context window is 32,768 tokens; we set a maximum generation length of 16,384 tokens. We evaluate the models on Mean Absolute Percentage Error (MAPE). The lower the results, the better. Results Model S&P 500 Russell 2000 STOXX 600 FTSE APAC CSI 300 All Data Avg Δ Machine Learning Based Methods Random Forest 0.4812 0.6551 0.7490 0.8436 0.5605 0.6579 0.1498 HistGBM 0.5916 0.8121 0.9774 1.3771 0.6041 0.8725 0.3644 NeuralNet 0.6130 1.0302 0.8963 1.5292 0.5242 0.9186 0.4105 Financial LLMs Fino1 (Qian and others 2025) 1.0230 2.1244 1.0958 1.2587 1.3388 1.3681 0.8600 Fin-R1 (Liu and others 2025) 0.5263 0.7324 0.7437 0.8862 0.8095 0.7396 0.2315 Financial Valuation P/E 0.5127 0.8450 0.6209 0.7439 0.6035 0.6652 0.1571 EV/EBIT 0.5580 0.9907 0.6099 0.9150 0.6698 0.7487 0.2406 EV/EBITDA 0.5558 1.1589 0.6808 0.9314 0.6472 0.7948 0.2867 P/B 0.7177 0.8727 2.3048 0.8371 0.8401 1.1145 0.6064 DDM 0.7991 0.9370 1.0324 0.8691 0.6358 0.8547 0.3466 FCFE 0.9865 1.1827 1.3975 0.8700 0.8196 1.0513 0.5432 FCFF 1.2434 1.4313 1.5155 0.9097 0.8412 1.1882 0.6801 Symbolic Regression PySR (Cranmer 2023) 0.6394 0.5892 0.6741 0.7514 0.5215 0.6351 0.1270 FunSearch (Romera-Paredes et al. 2024) 0.7146 1.1524 0.7693 0.7855 0.5735 0.7991 0.2910 LLM-SR (Shojaee et al. 2025) 0.4761 0.7822 0.8269 0.9534 0.5967 0.7271 0.2190 Financial Agentic Frameworks FinCon (Yu et al. 2024) 0.9581 0.9746 1.1822 0.6358 0.5784 0.8658 0.3577 FinVision (Fatemi and Hu 2024) 0.8203 0.5842 1.2521 0.9974 0.9956 0.9299 0.4218 mufasa (Ours) 0.4465 0.5718 0.5064 0.5814 0.4343 0.5081 0.3702 (Avg) Table 2: Performance comparison. The best baselines are underlined, and the best results are bolded. Performance Comparison. Table 2 reports the forecasting performance. We observe the following: • Fin-R1 consistently outperforms Fino1, likely because its emphasis on generalization and transfer better matches the forecasting setting. Nevertheless, both LLMs underperform most specialized baselines. • Traditional valuation ratios (e.g., P/E, EV/EBIT) competitive across most markets. Their simple structure makes them relatively robust to estimation error, although their performance varies considerably across regions, suggesting that markets rely on different fundamental signals. • More complex discounted cash-flow methods, including FCFF and FCFE, perform less well because they depend on noisy estimates such as growth rates and discount factors. We use the single-stage Gordon Growth Model (Gordon 1962), which assumes constant growth; more accurate parameter estimation may improve these methods. • Among the SR baselines, PySR performs strongly, unlike the typical advantage of LLM-based SR. Scientific priors encoded by LLMs may be less useful in finance, where relationships are context-dependent and lack stable governing laws. PySR’s evolutionary search may therefore be better suited to identifying empirically effective expressions. • mufasa achieves the best performance across all datasets, with an average improvement of 19.65%. Existing financial agentic frameworks, FinCon and FinVision, perform competitively in selected markets but generalize inconsistently across regions. In contrast, mufasa outperforms the better agentic baseline in every market, reducing its error by 27.67% on average. This advantage stems from perspective-specific hypothesis generation and statistical memory that retains empirically robust expressions. More details on the statistical significance of the experiments can be found in Appendix B. Model Variant S&P 500 Russell 2000 STOXX 600 FTSE APAC CSI 300 S.P Agent 0.4971 0.8994 0.7608 0.6095 0.5928 M.P (Equal Weights) 0.4652 0.8768 0.5448 0.6442 0.5309 M.P (No Stats Memory) 0.4702 0.7525 0.5398 0.5852 0.4852 M.P (Best Single Agent) 0.4676 1.9173 0.5256 0.5833 0.4739 mufasa (Ours) 0.4465 0.5718 0.5064 0.5814 0.4343 Table 3: Ablation study on perspective variant. S.P and M.P denote the single-/multiple perspective, respectively. Best results are bolded, second best are underlined. Model Equation Theory Equation Theory Fino1 EV/EBITDA⋅(1+gEPS/100)EV/EBITDAmkt⋅EBITDA EV/EBITDA·(1+g_EPS/100)EV/EBITDA_mkt· EBITDA - - Fin-R1 EPSTTM⋅PEEPS_TTM· PE EPS⋅PEEPS· PE Earnings Multiple PySR 1.79⋅|BVPS|⋅|Tax|+Tax⋅|EPS⋅e2.43−eY30+(ePBmkt,med+EPS)⋅4.23| 1.79·|BVPS|· |Tax|+Tax· |EPS· e^2.43-e Y_30+ (e^PB_mkt,med+EPS )· 4.23 | - - FunSearch 0.40⋅EBITDAps+0.30⋅Revenueps+0.20⋅ROE+0.10⋅PE0.40· EBITDA_ps+0.30· Revenue_ps+0.20· ROE+0.10· PE - - LLM-SR 0.01⋅PE−0.03⋅EVEBIT−0.04⋅EVEBITDA+0.58⋅PB+13.99⋅NIps−0.01⋅Revenueps0.01· PE-0.03· EVEBIT-0.04· EVEBITDA+0.58· PB+13.99· NI_ps-0.01· Revenue_ps Ri=α+β1X1,i+β2X2,i+⋯R_i=α+ _1X_1,i+ _2X_2,i+·s Linear Factor mufasa agents Earnings NIps⋅(1+0.01⋅gEPS)⋅PE⋅0.88NI_ps· (1+0.01· g_EPS )· PE· 0.88 EPSr−g EPSr-g Gordon Growth Cashflow FCFps⋅EVEBITDA⋅10088.38FCF_ps· EVEBITDA· 10088.38 ∑t=1∞FCFt(1+r)t _t=1^∞ FCF_t(1+r)^t Discounted Cash Flow Asset BVPS⋅(1+1.01⋅R&DAssets)⋅(1−13829.10⋅R&DAssets)⋅PB−7.24×10−41.34BVPS· (1+1.01· R\&DAssets )· (1-13829.10· R\&DAssets )· PB-7.24× 10^-41.34 BVPS⋅PBBVPS· PB Book Multiple Growth (P/FCF)mkt⋅(FCFps+0.29⋅P/FCF(P/FCF)mkt)⋅(1−14.80⋅(FCFps<1.08))⋅0.01(P/FCF)_mkt· (FCF_ps+0.29· P/FCF(P/FCF)_mkt )· (1-14.80·(FCF_ps<1.08) )· 0.01 Metric⋅MultipleMetric· Multiple Relative Valuation Quality NIps⋅P/FCF⋅(1+log(1+ROE/100)⋅0.87)NI_ps· P/FCF· (1+ (1+ROE/100)· 0.87 ) BV+∑t=1∞(ROEt−r)⋅BVt−1(1+r)tBV+ _t=1^∞ (ROE_t-r)· BV_t-1(1+r)^t Residual Income Table 4: Learnt equations vs. related valuation forms in finance literature, for the S&P500 dataset. Ablation Study. We conduct an ablation study to demonstrate the effectiveness of the model design. • The Single-Perspective Agent uses one LLM agent to search for a unified valuation expression. It is outperformed by the equal-weight multi-perspective variant on four of the five markets, suggesting that representing heterogeneous valuation signals within a single search space limits effective exploration. • Multi-Perspective (Equal Weights) progresses to multi-perspective agents, but without hierarchical reasoning. It improves over the single-perspective variant. Disentangling into specialized agents helps optimization by focusing each agent on a subset of variables and relationships, enabling more diverse exploration and better generalization even when combined using simple averaging. • Multi-Perspective (No Stats Memory) adds hierarchical reasoning with the meta-coordinator agent, but without statistical memory. It outperforms equal weighting on four of the five markets, demonstrating the benefit of hierarchical coordination. Nevertheless, without statistical memory, the system relies primarily on coarse outcome feedback and cannot reliably distinguish consistently effective expressions from those that succeed by chance. • Multi-Perspective (Best Single Agent) retains the specialized agents and statistical memory but uses only the strongest individual agent. It obtains the second-best result on three markets but degrades substantially on Russell 2000, showing that reliance on a single perspective is unstable and requires an agent-selection rule that is not available in advance. • The mufasa framework achieves the best performance across all datasets. We see that integrating multiple perspectives shows improvement over any of the best agents. Overall, combining multi-perspective agents, hierarchical-level reasoning, and statistical memory produces the best results. A more comprehensive ablation study and additional experiments can be found in Appendix F, G. Figure 3: Best error trajectories over the self-evolving iterations for each perspective agent. Self-Evolving Agents. Figure 3 shows the best error trajectories over iterations for each agent. Each agent exhibits a decrease in the lowest error over time, indicating that the symbolic search process iteratively refines candidate equations. Although the general single agent also improves, its final best error remains higher than those achieved by the specialized agents. This result is consistent with the difficulty of searching for a single expression that simultaneously captures heterogeneous valuation mechanisms. By restricting each agent to a specific valuation perspective, the disentangled design enables more focused exploration of the symbolic search space and yields stronger final solutions. Discovered Equations. Table 4 shows the learnt equations on the S&P 500 dataset, together with the most closely related valuation forms that can be found in finance literature (Damodaran 2012; Graham et al. 1951). We observe: • Fin-R1 largely reproduces the standard earnings-multiple formula EPS⋅PEEPS· PE. While interpretable, this fixed form offers limited adaptation to the observed data and represents only one valuation perspective. • PySR discovers a high-performing but financially opaque expression, reflecting its primarily heuristic search. LLM-SR instead combines variables from several perspectives in a linear form, but does not recover the perspective-specific structures observed in classical valuation models. • In contrast, mufasa’s specialized agents recover adapted forms of earnings-multiple, discounted-cash-flow, book-value, relative-valuation, and residual-income models. This suggests that perspective disentanglement promotes economically grounded equation discovery, while allowing the learned forms to adapt to the data. Released Artifacts. The learnt equations for other datasets, the meta-coordinator weights, and the distilled agent learnings ()Z^(a) are in Appendices C, D, and E, respectively. Conclusion In this work, we study symbolic discovery for financial fundamental analysis, which constitutes a distinct class of symbolic regression problems. Unlike the natural sciences, financial valuation admits multiple valid perspectives, operates under non-stationary market conditions, and relies on noisy, continuous performance signals. To address this, we introduce mufasa, a hierarchical multi-agent framework that decomposes the discovery process into specialized agents, integrates them through hierarchical-level context reasoning, and leverages statistical-based memory to guide learning. 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Algorithm 1: Philosophy Agent Training 1: LLM πθ _θ, database tXt, targets yi,t+1y_i,t+1, perspective a, memory ()M^(a), learnings ()Z^(a) 2: // Initialization 3: f~(),i,t()←SeedEquation() f^(a),x^(a)_i,t← SeedEquation(a) 4: ℒ()←MAPE(f~()(i,t()),yi,t+1)L^(a) ( f^(a)(x^(a)_i,t),y_i,t+1 ) 5: for all iterations do 6: // Hypothesis Generation 7: ()←(,f~(),())p^(a)←(a, f^(a),Z^(a)) 8: f~()∼πθ(⋅∣()) f^(a) πθ(· ^(a)) 9: // Tool use: Data Retrieval 10: ^i,t()←Retrieve(t) x^(a)_i,t← Retrieve\! (X_t ) 11: if RetrieveError then 12: ()←()∪ErrorFeedbackp^(a) ^(a)∪ ErrorFeedback 13: end if 14: // Tool use: Code Evaluation 15: y^i,t+1()←Evaluate(f~(),^i,t()) y^(a)_i,t+1← Evaluate\! ( f^(a), x^(a)_i,t ) 16: if EvaluateError then 17: ()←()∪ErrorFeedbackp^(a) ^(a)∪ ErrorFeedback 18: end if 19: // Hypothesis Optimization 20: y^i,t+1()←Optimize(y^i,t+1()) y^(a)_i,t+1← Optimize\! ( y^(a)_i,t+1 ) 21: ℒ()←MAPE(y^i,t+1(),yi,t+1)L^(a) \! ( y^(a)_i,t+1,y_i,t+1 ) 22: // Memory Update 23: ()←()∪(f~(),ℒ())M^(a) ^(a)∪\( f^(a),L^(a))\ 24: ()←UpdateLearnings(())Z^(a)← UpdateLearnings(M^(a)) 25: end for 26: return f~() f^(a) Algorithm 2: Meta-Coordinator Training 1: LLM πϕ _φ, specialist forecasts y^i,t+1() y^(a)_i,t+1, targets yi,t+1y_i,t+1, memory metaM^meta, learnings metaZ^meta 2: for all C(i,t)C(i,t) do 3: ~C(i,t)←InitWeights() w_C(i,t)← InitWeights() 4: for all iterations do 5: // Hypothesis Generation 6: meta←(C(i,t),ℒ(),~C(i,t),meta)p^meta←(C(i,t),L^(a), w_C(i,t),Z^meta) 7: ~C(i,t)∼πϕ(⋅∣meta) w_C(i,t) _φ(· ^meta) 8: // Tool use: Code Evaluation 9: y^i,t+1←~C(i,t)⊤⋅^i,t+1 y_i,t+1← w_C(i,t) · y_i,t+1 10: ℒmeta←MAPE(y^i,t+1,yi,t+1)L^meta \! ( y_i,t+1,y_i,t+1 ) 11: // Memory Update 12: meta←meta∪(~C(i,t),ℒmetaM^meta ^meta∪\( w_C(i,t),L^meta\ 13: meta←UpdateLearnings(meta)Z^meta← UpdateLearnings(M^meta) 14: end for 15: end for 16: return ~C(i,t) w_C(i,t) Appendix B Statistical Significance Analysis To assess the statistical significance of mufasa’s performance results, we apply the Diebold–Mariano (DM) test (Diebold and Mariano 2002) under absolute percentage error loss. We incorporate two corrections. First, we use a Newey–West heteroskedasticity and autocorrelation consistent (HAC) variance estimator (Newey and West 1987) to account for serial correlation in the loss differential series across quarters. Specifically, we employ a Bartlett kernel with bandwidth selected according to Andrews’ rule (Andrews 1991): L=⌊4(T100)2/9⌋,L= 4 ( T100 )^2/9 , (6) where T denotes the number of test quarters. This adjustment is necessary because macroeconomic conditions that make one quarter difficult to forecast often persist, inducing autocorrelation in the loss differentials and leading to underestimated standard errors if uncorrected. Second, we apply the Harvey–Leybourne–Newbold (HLN) small-sample correction (Harvey et al. 1997), which adjusts the DM test statistic and evaluates it against a tT−1t_T-1 reference distribution rather than a standard normal distribution. This accounts for the small number of test quarters available per market. Table 5: Diebold–Mariano (DM) tests comparing mufasa against valuation and learning-based baselines. Entries report mufasa minus baseline in decimal form. Negative values indicate lower error (better performance). Significance: *** p<0.001p<0.001, ** p<0.01p<0.01, * p<0.05p<0.05. Baseline S&P 500 Russell 2000 STOXX 600 FTSE Asia CSI 300 Financial LLMs Fino1 -0.7283*** -0.4515*** -0.3067*** -0.7925*** -0.3408*** Fin-R1 -0.1207*** -0.0375*** -0.1801*** -0.5919*** -0.4835*** Financial Valuation P/E -0.1628*** -0.0371*** -0.0691*** -0.3519*** -0.1158*** EV/EBITDA -0.1174*** -0.1622*** -0.0802*** -0.2941*** -0.0933*** EV/EBIT -0.1235*** -0.0949*** -0.0531*** -0.2965*** -0.0982*** P/B -0.2535*** -0.0967*** -0.1894*** -0.1466*** -0.0618*** DDM -0.2476*** -0.0924*** -0.1514*** -0.1402*** -0.0321** FCFE -0.2890*** -0.2154*** -0.1698*** -0.1001** -0.3787*** FCFF -0.2534*** -0.1774*** -0.1454*** -0.2862*** -0.2745*** Symbolic Regression PySR -0.2695*** -0.0326*** -0.0857*** -0.3287*** -0.0956*** FunSearch -0.3992*** -0.3872*** -0.1696*** -0.4040*** -0.1365*** LLM-SR -0.1101*** -0.1751*** -0.2345*** -0.4354*** -0.1442*** Across all pairwise comparisons, mufasa achieves statistically significant improvements in MAPE, indicating that the observed gains are mostly robust and not driven by random variations. Appendix C Learnt Symbolic Expressions We present the learnt symbolic expressions in Tables 6–9. Overall, we observe that mufasa consistently discovers structured and interpretable expressions that align with well-established financial valuation principles, while adapting them to data-specific and context-dependent variations. A key observation is that the expressions discovered by mufasa closely resemble classical financial valuation formulas, such as discounted cash flow, book multiples and residual income models, as shown in the “Theory” column. Rather than directly reproducing these formulas, mufasa often recovers their core structural components (e.g., scaling by cash flow or book value) and augments them with correction terms that reflect empirical patterns in the data. These deviations from canonical formulas are expected in the financial setting. Unlike theoretical valuation models that rely on simplifying assumptions (e.g., constant growth rates or discount factors), real-world financial data is often noisy, non-stationary, and subject to market-specific distortions. The additional multiplicative or additive adjustments learned by mufasa can thus be interpreted as data-driven refinements that account for such complexities, rather than arbitrary artifacts of overfitting. In contrast, baseline methods such as PySR and FunSearch often produce expressions that are either overly complex or difficult to interpret, lacking clear correspondence to established financial theories. Similarly, LLM-SR capture only coarse additive relationships and fail to represent the non-linear structures commonly used in financial valuation. These comparisons highlight that mufasa strikes a balance between interpretability and flexibility, yielding expressions that are both theoretically grounded and empirically adaptive. Importantly, we observe consistent structural patterns across different datasets (e.g., Russell 2000, STOXX 600, FTSE Asia, and CSI 300), suggesting that the learnt symbolic forms capture generalizable valuation principles rather than dataset-specific artifacts. Model Equation Theory Equation Theory Fino1 EV/EBITDA⋅(1+gEPS/100)EV/EBITDAmkt⋅EBITDA EV/EBITDA·(1+g_EPS/100)EV/EBITDA_mkt· EBITDA - - Fin-R1 EPSTTM⋅PEEPS_TTM· PE EPS⋅PEEPS· PE Earnings Multiple PySR (PBmkt,mean+0.16)⋅(|BVPS|+|e0.98−BVPS|+2.09−BVPS−0.55−2.09)−0.29(PB_mkt,mean+0.16)· ( |BVPS|+|e^0.98-BVPS|+2.09- BVPS-0.55-2.09 )-0.29 - - FunSearch 2⋅BVPS+10⋅NIps2· BVPS+10· NI_ps - - LLM-SR 0.11⋅(PE−Y1)−0.01⋅Y2+0.20⋅NI0.11·(PE-Y_1)-0.01· Y_2+0.20· NI Ri=α+β1X1,i+β2X2,i+⋯R_i=α+ _1X_1,i+ _2X_2,i+·s Linear Factor mufasa agents Earnings NIps⋅PE⋅(1+SustainGrowth100)⋅(1−1.12⋅(0.19−1.35×10−3⋅SustainGrowth))NI_ps· PE· (1+ SustainGrowth100 )· (1-1.12·(0.19-1.35× 10^-3· SustainGrowth) ) EPSr−g EPSr-g Gordon Growth Cashflow FCFps⋅EV/FCF⋅(1+0.23⋅(EV/EBITDA−1.00))FCF_ps· EV/FCF· (1+0.23·(EV/EBITDA-1.00) ) ∑t=1∞FCFt(1+r)t _t=1^∞ FCF_t(1+r)^t Discounted Cash Flow Asset TBVPS⋅PB⋅(1−0.24⋅(PB−1.00))TBVPS· PB· (1-0.24·(PB-1.00) ) BVPS⋅PBBVPS· PB Book Multiple Growth Revenueps⋅(1+gRevenue100)Revenue_ps· (1+ g_Revenue100 ) Revenue0⋅(1+g)Revenue_0·(1+g) Revenue Growth Quality EBITDAps⋅P/FCF⋅(1+EBITDAps−46.35100)EBITDA_ps· P/FCF· (1+ EBITDA_ps-46.35100 ) BV+∑t=1∞(ROEt−r)⋅BVt−1(1+r)tBV+ _t=1^∞ (ROE_t-r)· BV_t-1(1+r)^t Residual Income Table 6: Learnt equations for the Russell 2000 dataset. Model Equation Theory Equation Theory Fino1 EV/EBITDA⋅(1+gEPS/100)EV/EBITDAmkt⋅EBITDA EV/EBITDA·(1+g_EPS/100)EV/EBITDA_mkt· EBITDA - - Fin-R1 EPSTTM⋅PEEPS_TTM· PE EPS⋅PEEPS· PE Earnings Multiple PySR |((−0.99+Div−(EBITDA+R&D+|NI|)−NI)⋅PBmkt,std)+(EBITDA−(0.30⋅Rev+IntInc))Shares|+0.68 | ((-0.99+Div-(EBITDA+R\&D+|NI|)-NI)· PB_mkt,std )+ (EBITDA-(0.30· Rev+IntInc) )Shares |+0.68 - - FunSearch 0.25⋅EBITDAps+0.35⋅OCFps+0.20⋅ROE+0.20⋅BVPSlag40.25· EBITDA_ps+0.35· OCF_ps+0.20· ROE+0.20· BVPS_lag4 - - LLM-SR −0.01⋅PE+0.37⋅PB+0.22⋅(EV/EBIT)+0.36⋅PB-0.01· PE+0.37· PB+0.22·(EV/EBIT)+0.36· PB Ri=α+β1X1,i+β2X2,i+⋯R_i=α+ _1X_1,i+ _2X_2,i+·s Linear Factor mufasa agents Earnings NIps⋅PE⋅(1+gEPS/100)1+EBITDAps/100+(P/FCF/100)⋅1.19 NI_ps· PE·(1+g_EPS/100)1+EBITDA_ps/100+(P/FCF/100)· 1.19 EPSr−g EPSr-g Gordon Growth Cashflow OCFps⋅3.88⋅P/FCF5.27⋅(1+0.02⋅gEBITDA⋅(1−OCFps−6.660.05))OCF_ps· 3.88· P/FCF5.27· (1+0.02· g_EBITDA· (1- OCF_ps-6.660.05 ) ) ∑t=1∞FCFt(1+r)t _t=1^∞ FCF_t(1+r)^t Discounted Cash Flow Asset Assets−LiabilitiesShares⋅PB28435.60+1.42⋅LTInvest+5316.09⋅(LTInvest/Assets−12.78) Assets-LiabilitiesShares· PB28435.60+1.42· LTInvest+5316.09·(LTInvest/Assets-12.78) BVPS⋅PBBVPS· PB Book Multiple Growth Revenueps⋅(1+gRevenue/100)⋅(1+(Revenueps809734.00)34086.20)⋅(1−(1−EV/EBITDA100)⋅0.54)Revenue_ps·(1+g_Revenue/100)· (1+ ( Revenue_ps809734.00 )^34086.20 )· (1- (1- EV/EBITDA100 )· 0.54 ) Revenue0⋅(1+g)Revenue_0·(1+g) Revenue Growth Quality NIps⋅(PE⋅0.86)⋅(1+OCFps,lag1/Assetslag4100)NI_ps·(PE· 0.86)· (1+ OCF_ps,lag1/Assets_lag4100 ) BV+∑t=1∞(ROEt−r)⋅BVt−1(1+r)tBV+ _t=1^∞ (ROE_t-r)· BV_t-1(1+r)^t Residual Income Table 7: Learnt equations for the STOXX 600 dataset. Model Equation Theory Equation Theory Fino1 EV/EBITDA⋅(1+gEPS/100)EV/EBITDAmkt⋅EBITDA EV/EBITDA·(1+g_EPS/100)EV/EBITDA_mkt· EBITDA - - Fin-R1 EPSTTM⋅PEEPS_TTM· PE EPS⋅PEEPS· PE Earnings Multiple PySR 1e−1.27⋅log(|BVPS⋅Debtps−2.91×10−3−Debtps−InterestExpenseR&D+PBmkt,mean⋅1log(PBmkt,mean)|+1.00) 1 e^-1.27· ( | BVPS· Debt_ps-2.91× 10^-3- Debt_ps-InterestExpenseR\&D+PB_mkt,mean· 1 (PB_mkt,mean) |+1.00 ) - - FunSearch 0.20⋅EBITDAps+0.15⋅BVPSlag4+0.10⋅ROElag8+0.10⋅EBITps+0.10⋅EPSlag1+0.05⋅EVEBIT0.20· EBITDA_ps+0.15· BVPS_lag4+0.10· ROE_lag8+0.10· EBIT_ps+0.10· EPS_lag1+0.05· EVEBIT - - LLM-SR 5.86×10−4⋅(EBIT−InterestExpense)−0.21⋅R&D−4.00×10−3⋅GrossProfit+0.76⋅BVPS5.86× 10^-4·(EBIT-InterestExpense)-0.21· R\&D-4.00× 10^-3· GrossProfit+0.76· BVPS Ri=α+β1X1,i+β2X2,i+⋯R_i=α+ _1X_1,i+ _2X_2,i+·s Linear Factor mufasa agents Earnings NIps⋅PE⋅(1+0.001⋅gEPS)⋅(1−1.79⋅σEV/EBIT100)⋅(1+0.03⋅(σEV/EBIT>0.78))NI_ps· PE·(1+0.001· g_EPS)· (1-1.79· _EV/EBIT100 )· (1+0.03·( _EV/EBIT>0.78) ) EPSr−g EPSr-g Gordon Growth Cashflow OCFps⋅(1+0.16⋅gEBIT100)⋅(EV/FCF(EV/FCF)mkt⋅6.87)OCF_ps· (1+ 0.16· g_EBIT100 )· ( EV/FCF(EV/FCF)_mkt· 6.87 ) ∑t=1∞FCFt(1+r)t _t=1^∞ FCF_t(1+r)^t Discounted Cash Flow Asset BVPSlag1⋅PB⋅1.34PB BVPS_lag1· PB· 1.34PB BVPS⋅PBBVPS· PB Book Multiple Growth Revenueps⋅(1+gRevenue/100)⋅(1+EV/EBITDA−(EV/EBITDA)mkt(EV/EBITDA)mkt)Revenue_ps·(1+g_Revenue/100)· (1+ EV/EBITDA-(EV/EBITDA)_mkt(EV/EBITDA)_mkt ) Revenue0⋅(1+g)Revenue_0·(1+g) Revenue Growth Quality NIps⋅PE⋅(1+ROE/100)NI_ps· PE·(1+ROE/100) BV+∑t=1∞(ROEt−r)⋅BVt−1(1+r)tBV+ _t=1^∞ (ROE_t-r)· BV_t-1(1+r)^t Residual Income Table 8: Learnt equations for the FTSE Asia ex Japan dataset. Model Equation Theory Equation Theory Fino1 EV/EBITDA⋅(1+gEPS/100)EV/EBITDAmkt⋅EBITDA EV/EBITDA·(1+g_EPS/100)EV/EBITDA_mkt· EBITDA - - Fin-R1 EPSTTM⋅PEEPS_TTM· PE EPS⋅PEEPS· PE Earnings Multiple PySR |PBmkt,mean⋅0.10⋅BVPSEBITDA−0.03+0.17⋅EPS+0.48+2.16⋅Debtps|+|BVPS0.06−GrossProfit|+EPS | PB_mkt,mean· 0.10· BVPSEBITDA-0.03+0.17· EPS+0.48+2.16· Debt_ps |+ | BVPS0.06-GrossProfit |+EPS - - FunSearch 2⋅BVPS+10⋅NIps2· BVPS+10· NI_ps - - LLM-SR −0.02⋅Debtps+1.53×10−4⋅gRevenue+0.54⋅EV/Sales+0.52⋅Revenueps-0.02· Debt_ps+1.53× 10^-4· g_Revenue+0.54· EV/Sales+0.52· Revenue_ps Ri=α+β1X1,i+β2X2,i+⋯R_i=α+ _1X_1,i+ _2X_2,i+·s Linear Factor mufasa agents Earnings NIps⋅(1+gEPS/100)⋅(PE⋅3.37)⋅0.23NI_ps·(1+g_EPS/100)·(PE· 3.37)· 0.23 EPSr−g EPSr-g Gordon Growth Cashflow 0.09⋅P/OCF+4.99⋅OCFps⋅(1+852.52⋅(OCFps<0))⋅(1+0.63⋅(P/OCF>14.69))0.09· P/OCF+4.99· OCF_ps·(1+852.52·(OCF_ps<0))·(1+0.63·(P/OCF>14.69)) ∑t=1∞FCFt(1+r)t _t=1^∞ FCF_t(1+r)^t Discounted Cash Flow Asset BVPS⋅(PB−0.15⋅PBmkt)⋅1.00BVPS·(PB-0.15· PB_mkt)· 1.00 BVPS⋅PBBVPS· PB Book Multiple Growth Revenueps⋅(1+gRevenue/100)Revenue_ps·(1+g_Revenue/100) Revenue0⋅(1+g)Revenue_0·(1+g) Revenue Growth Quality EBITps⋅(1+ROE/100)⋅P/FCF⋅(1−0.94⋅(RevenuepsEBITps−1))EBIT_ps·(1+ROE/100)· P/FCF· (1-0.94· ( Revenue_psEBIT_ps-1 ) ) BV+∑t=1∞(ROEt−r)⋅BVt−1(1+r)tBV+ _t=1^∞ (ROE_t-r)· BV_t-1(1+r)^t Residual Income Table 9: Learnt equations for the CSI 300 dataset. Appendix D Learnt Context Weighting We illustrate the learnt context-dependent agent weighting over valuation perspectives in Figure 4 across the five datasets. Each heatmap shows how mufasa dynamically allocates importance to the different agent perspectives (e.g., earnings, cash flow, asset, growth, and quality) conditioned on company sector and market regime (bull vs. bear). Several consistent patterns emerge: • First, the model assigns distinct weights across sectors, reflecting domain-specific valuation preferences. For example, asset- or cash flow-based signals are often emphasized in sectors with substantial balance sheet components (e.g., real estate), while earnings-based signals are frequently assigned higher weights in sectors with more stable profitability profiles. • Second, the weights shift across bull and bear regimes, indicating that mufasa adapts its valuation strategy to changing market conditions. For example, it can be noticed that certain perspectives (e.g., growth or earnings) are more emphasized in some bullish settings, while asset- or cash flow-based signals tend to receive relatively higher weights in some bearish contexts. These patterns are not uniform, but reflect a context-dependent reallocation of importance across valuation perspectives. • Importantly, we observe recurring tendencies across geographically distinct markets, including the U.S. (S&P 500, Russell 2000), Europe (STOXX 600), and Asia (FTSE Asia Pacific ex Japan, CSI 300), where certain valuation perspectives (e.g., earnings or cash flow) are frequently assigned higher importance. While the exact weighting patterns vary across sectors and regimes, these recurring trends suggest that the learnt weighting mechanism is able to capture generalizable financial intuitions rather than purely dataset-specific artifacts. Overall, the results demonstrate that the meta-coordinator effectively learns to combine multiple valuation perspectives across diverse market conditions, in a context-aware and interpretable manner. Figure 4: Learnt agent weighting across company sectors and market regimes for all datasets. Appendix E Distilled Agent Learnings Earnings Agent Learnings Summary. Best structure uses earnings * PE with capped growth. The most promising direction is risk-adjusted sector growth caps. Avoid uncapped growth and raw PE multiples due to the persistent Technology sector gap. Key Insights • Earnings-based structures require growth caps to avoid overprediction. • Sector-specific adjustments are necessary for high-PE sectors. • Raw PE-based formulations lead to systematic bias. • Growth signals improve performance only when bounded. Statistical Diagnostics • Technology sector shows persistent overprediction relative to the global baseline. • High-PE sectors exhibit systematic bias in earnings-based models. • Growth variables without caps lead to unstable predictions. • Performance degrades when sector variation is not accounted for. What Works • Using capped growth adjustments in earnings formulations. • Applying sector-specific growth caps. • Combining earnings with sector median PE. • Using bounded multiplicative structures, e.g., growth caps. What Fails • Using IS_EPS_GROWTH_YOY without cap. • Using SECTOR_MEDIAN_PE without growth cap. • Using IS_RD_EXPEND_TTM as a standalone multiplier. • Using uncapped growth-based formulations. Important Variables Variable Observed Role NET_INCOME_TTM_PER_SHARE Core earnings driver. SECTOR_MEDIAN_PE Sector anchoring for valuation. IS_EPS_GROWTH_YOY Growth signal requiring bounding. IS_RD_EXPEND_TTM High-impact but unstable when standalone. Agent Adaptation Behavior • Introduces sector-specific growth caps after detecting overprediction. • Replaces uncapped growth terms with bounded structures. • Combines earnings with sector anchoring variables. • Moves away from standalone multipliers toward structured combinations. Design Principle. Earnings-based models require sector-aware growth caps and bounded multiplicative structures to avoid systematic overprediction. Cash Flow Agent Learnings Summary. Best structure uses cash flow with a sector-adjusted EV/EBITDA multiple. The most promising direction is a Consumer Discretionary discount factor. Avoid static multipliers without growth adjustments due to large sector gaps. Key Insights • Sector-specific adjustments are required to correct extreme underprediction in Consumer Discretionary. • Growth-adjusted factors improve performance compared to static multipliers. • Cash flow quality adjustments are needed for early-stage companies with negative cash flow. • CF_base * (1 + growth_cap) structures outperform pure multiples by incorporating growth dynamics. Statistical Diagnostics • Consumer Discretionary exhibits extreme underprediction driven by growth volatility. • Technology and Financials show systematic deviation due to differences in growth stability and regulatory complexity. • SECTOR_MEDIAN_EV_EBITDA is unstable in asset-light companies due to denominator effects. • Ignoring negative cash flow via max() leads to systematic underestimation in early-stage firms. What Works • Using sector-adjusted EV/EBITDA multiples. • Applying growth-adjusted scaling using SALES_REV_TURN_TTM_LAG4. • Incorporating cash flow quality adjustments for negative cash flow companies. • Using bounded structures, e.g., CF_base * (1 + growth_cap). What Fails • Using EBITDA_YOY as a standalone multiplier. • Using static multipliers without growth adjustments. • Ignoring negative cash flow through truncation via max(). Important Variables Variable Observed Role TRAIL_12M_FREE_CASH_FLOW_PER_SH Core valuation base. SECTOR_MEDIAN_EV_EBITDA Sector-adjusted valuation multiplier. SALES_REV_TURN_TTM_LAG4 Growth-adjustment factor with high sensitivity. NEGATIVE_CASH_FLOW_FACTOR Adjustment for early-stage companies. CONSUMER_DISCRETIONARY_FLAG Sector-specific discount control. Agent Adaptation Behavior • Introduces sector-specific discount factors after detecting extreme underprediction. • Replaces static multipliers with growth-adjusted scaling. • Incorporates adjustments for negative cash flow rather than truncating values. • Proposes new combinations integrating growth, sector, and cash flow quality signals. Design Principle. Cash flow models require sector-aware adjustments, growth scaling, and explicit handling of negative cash flow to avoid systematic underestimation. Asset Agent Learnings Summary. The best structure uses multiplicative adjustment for R&D intensity and sector anchoring, avoiding invalid predictions and leveraging high-impact variables. The most promising direction is a Technology-specific R&D multiplier. Avoid log-based adjustments and standalone BS_CUR_LIAB multipliers due to the persistent Technology sector gap. Key Insights • Sector-specific multipliers are required to correct overprediction in the Technology sector. • R&D intensity relative to assets is a high-impact adjustment factor. • Log-based adjustments introduce invalid predictions and should be avoided. • Liquidity adjustments improve stability in high-variance sectors. • max(BOOK_VAL_PER_SH * x, BOOK_VAL_PER_SH - y) * max(1, z) structures outperform pure multiples. Statistical Diagnostics • Technology sector exhibits extreme overprediction due to overreliance on R&D multipliers. • Log-based transformations result in invalid predictions. • BS_CUR_LIAB as a denominator causes invalid predictions in asset-light companies. • SECTOR_MEDIAN_P_BOOK shows high volatility in Technology, leading to high CV and MAPE. • Consumer Staples shows underprediction due to underweighting of book value. What Works • Using multiplicative adjustments for R&D intensity. • Applying sector-specific adjustments using SECTOR_MEDIAN_P_BOOK. • Using bounded structures, e.g., max(1, …) to avoid invalid predictions. • Incorporating liquidity buffer terms to reduce variance. What Fails • Using BS_CUR_LIAB as a standalone multiplier. • Using log(1 + BOOK_VAL_PER_SH / SECTOR_MEDIAN_P_BOOK). • Using BS_CUR_LIAB as a denominator in asset-light companies. • Using unbounded R&D multipliers in the Technology sector. Important Variables Variable Observed Role BOOK_VAL_PER_SH Core asset value and primary anchor. SECTOR_MEDIAN_P_BOOK Sector anchoring variable. BS_CUR_LIAB Liquidity risk adjustment and variance driver. IS_RD_EXPEND_TTM R&D intensity has a high impact but is prone to overuse. BS_TOT_ASSET Scaling factor for asset normalization. MARKET_MEDIAN_P_BOOK Market context variable with secondary importance. Agent Adaptation Behavior • Introduces Technology-specific multipliers after detecting overprediction. • Replaces log-based adjustments with bounded multiplicative structures. • Adds liquidity buffer terms to reduce high variance. • Explores combinations of R&D intensity and sector anchoring variables. Design Principle. Asset-based models require bounded multiplicative structures, sector-specific adjustments, and controlled use of high-impact variables to avoid instability and invalid predictions. Growth Agent Learnings Summary. The best structure uses MARKET_MEDIAN_P_FCF with growth-capped FCF scaling. The most promising direction is a Consumer Discretionary discount factor. Avoid EBITDA_YOY as a standalone multiplier due to persistent sector misfits. Key Insights • Sector-specific discounts are required to correct extreme bias in Consumer Discretionary. • Growth elasticity improves performance compared to static multiples. • Piecewise growth structures improve flexibility over min/max formulations. • CF_base * (1 + growth_cap) structures improve MAPE by capturing growth dynamics. Statistical Diagnostics • Consumer Discretionary exhibits extreme prediction error driven by FCF volatility. • Technology sector shows systematic overestimation due to growth effects. • TRAIL_12M_FREE_CASH_FLOW_PER_SH as a denominator causes invalid predictions. • SECTOR_MEDIAN_P_FCF / MARKET_MEDIAN_P_FCF shows high volatility in unstable sectors. What Works • Using MARKET_MEDIAN_P_FCF as a core multiplier. • Applying growth-capped scaling on free cash flow. • Using sector-specific scaling for Consumer Discretionary. • Replacing min/max structures with piecewise growth formulations. What Fails • Using EBITDA_YOY as a standalone multiplier. • Using SALES_REV_TURN_YOY in multiplicative models. • Using TRAIL_12M_FREE_CASH_FLOW_PER_SH as a denominator. • Using SECTOR_MEDIAN_P_FCF / MARKET_MEDIAN_P_FCF in unstable sectors. Important Variables Variable Observed Role MARKET_MEDIAN_P_FCF Core multiplier for valuation. TRAIL_12M_FREE_CASH_FLOW_PER_SH Core growth-sensitive signal. SECTOR_MEDIAN_P_FCF Sector adjustment factor. MARKET_MEDIAN_P_SALES Alternative multiplier with instability in low-growth sectors. EBITDA_TTM_PER_SHARE High-sensitivity variable with instability in high-growth sectors. Agent Adaptation Behavior • Introduces sector-specific discount factors after detecting extreme bias. • Replaces min/max structures with piecewise growth formulations. • Shifts from static multiples to growth-velocity models. • Explores combinations of growth scaling and sector adjustment. Design Principle. Growth-based models require sector-specific adjustments, growth elasticity, and bounded scaling to avoid instability and extreme prediction errors. Quality Agent Learnings Summary. The best structure uses income * P_FCF with growth adjustments. The most promising direction is a Technology-specific growth cap. Avoid EBITDA_TTM_PER_SHARE as a standalone multiplier and fixed PE thresholds due to instability in growth regimes. Key Insights • Sector-specific scaling is required to reduce bias in Technology. • Piecewise growth functions improve performance in high-return regimes. • Sector-agnostic growth caps improve stability and reduce overfitting. • CF_base * (1 + growth_cap) structures improve MAPE by balancing growth and stability. • Sector-agnostic growth modifiers improve performance when applied to income * P_FCF structures. Statistical Diagnostics • Technology sector shows bias due to the underestimation of growth. • Utilities show lower error driven by low growth expectations. • RETURN_COM_EQY causes invalid predictions in asset-light companies. • SECTOR_MEDIAN_P_FCF introduces high CV in high-growth sectors. What Works • Using income * P_FCF as a base structure. • Applying piecewise growth functions instead of log growth. • Introducing sector-agnostic growth caps. • Using sector-specific scaling for high-growth sectors. What Fails • Using EBITDA_TTM_PER_SHARE as a standalone multiplier. • Using SECTOR_MEDIAN_PE with fixed thresholds. • Using RETURN_COM_EQY in unstable regimes without constraints. Important Variables Variable Observed Role OPER_CF_TTM_PER_SHARE Strong signal when combined with P_FCF. SECTOR_MEDIAN_P_FCF Core multiplier with sector adjustment role. RETURN_COM_EQY Growth signal with instability in high-return regimes. NET_INCOME_TTM_PER_SHARE Core valuation base. SECTOR_MEDIAN_PE Alternative multiplier with overfitting risk. EBITDA_TTM_PER_SHARE High-volatility variable with unstable performance. Agent Adaptation Behavior • Introduces sector-specific scaling after detecting bias in Technology. • Replaces log growth with piecewise linear growth functions. • Adds sector-agnostic growth caps to reduce overfitting. • Explores combinations of income, growth, and sector multipliers. Design Principle. Quality-based models require growth-aware scaling, bounded transformations, and sector-specific adjustments to balance stability and predictive performance. Analysis of Agent Learnings. We highlight consistent patterns observed across the learnings: • Many candidate formulations fail not due to poor average accuracy, but due to instability or invalid behavior under specific conditions. This is reflected in the statistical diagnostics such as high variance, high CV, or invalid outputs arising from denominator effects (e.g., cash flow or liability terms in asset-light firms). Such failure modes would not be identifiable from scalar loss alone, highlighting the necessity of the statistical memory in guiding the search toward robust expressions. • Instability is consistently conditional on firm type and sector interactions rather than global. For instance, sector-based multiples exhibit high volatility in certain sectors, and variables such as free cash flow or liabilities lead to breakdowns in early-stage or asset-light companies. This explains why a single global expression is insufficient, and also motivates the use of context-dependent aggregation across specialized agents. • A recurring observation is the presence of systematic over- and underestimation across sectors. For example, Technology frequently exhibits persistent overprediction (earnings, asset, growth agents), while Consumer Discretionary shows extreme underprediction or volatility-driven errors (cash flow, growth agents). These directional biases justify the need for sector- and regime-aware weighting. • Variables are rarely discarded outright, but instead require structured use to remain effective. Several high-impact features (e.g., growth signals, R&D intensity, sector multiples) are repeatedly identified as unstable or biased when used in isolation, but become useful when combined with anchoring or adjustment terms. This indicates that the statistical diagnostics not only identify failure modes, but also guide the construction of stable compositions within each valuation perspective. Appendix F Extended Ablation Study Model Variant S&P 500 Russell 2000 STOXX 600 FTSE APAC CSI 300 Binary Memory (Pass/Fail Signals Only) Single-Perspective Agent 0.5699 0.7527 0.7220 0.6095 0.5735 Multi-Agent (Equal Weights) 0.5466 1.0459 0.6454 0.7088 0.5599 Multi-Agent (Best Single Agent) 0.5127 1.5160 0.6212 0.6083 0.5761 Multi-Agent (Meta-Coordinator) 0.4646 0.6445 0.5759 0.6086 0.5070 MAPE-Only Memory (No Additional Statistics) Single-Perspective Agent 0.4983 0.7208 0.5858 0.5947 0.5517 Multi-Agent (Equal Weights) 0.5333 0.9027 0.5881 0.6393 0.5175 Multi-Agent (Best Single Agent) 0.5127 0.6948 0.6212 0.5926 0.5929 Multi-Agent (Meta-Coordinator) 0.4702 0.7525 0.5398 0.5852 0.4852 Full Statistical Memory (Variance, Tail Risk, etc.) Single-Perspective Agent 0.4971 0.8987 0.7608 0.6095 0.5928 Multi-Agent (Equal Weights) 0.4652 0.8831 0.5448 0.6442 0.5309 Multi-Agent (Best Single Agent) 0.4676 1.9170 0.5256 0.5833 0.4739 mufasa (Ours) 0.4465 0.5718 0.5064 0.5814 0.4343 Table 10: Extended ablation study. We compare strategies under different memory settings. We extend the ablation study on the statistical agent memory in Table 10. Single-Perspective Agent uses only a single LLM agent to search for the best expression, Multi-Agent (Equal Weights) use multiple agents without hierarchical-level coordination, Multi-Agent (Best Single Agent) reports the best post-hoc performance out of the multiple agents. Across all memory settings, Multi-Agent (Meta-Coordinator) generally achieves the strongest performance, often outperforming all of the best individual agents. This suggests that the gain does not come simply from finding the single strongest perspective in the agent pool, but from learning how to combine the complementary perspectives. Across the memory variants, Binary Memory refers to storing past equations with binary pass/fail signals only, MAPE-Only Memory stores their MAPE performance, while Full Statistical Memory stores the full suite of statistical information listed in Table 1. We find that using the full statistical memory typically result in the strongest performance across all model variants. Full statistical memory provides information about not only the average error, but also the reliability and risk profile of each agent’s expressions. This allows the agents to distinguish equations that perform well consistently and those that perform well only occasionally, resulting in more robust expressions. Overall, the results show that mufasa benefits from using multi-perspective agents, hierarchical-level coordination and richer statistical feedback, leading to more consistent symbolic discovery across markets. We do a deeper analysis of the statistical memory performance in the following section. Appendix G Additional Statistical Results In Table 1, we show that the full statistical memory stores information such as Spearman Rank Correlation, Percentile Errors, and Mean Signed Percentage Error (MSPE), among others. We perform a deeper analysis of the different statistical performances of the learnt symbolic expressions. Figure 5: Predicted vs. True log prices across Symbolic Regression models on the S&P 500 dataset. Spearman Correlation. Many of the learnt symbolic expressions are non-linear and multiplicative (see Table 4), and the MAPE metric might not be fully indicative of their quality. For example, a mismatched multiplier would result in high MAPE, but could still be correlated with the true symbolic relationship. We additionally analyze the Spearman correlation between the predicted and true prices. Notably, in Figure 5, we find that MAPE and rank correlation capture different aspects of performance: PySR achieved better MAPE than FunSearch in the S&P 500 dataset (see Table 2) but shows lower rank correlation. This might suggest that its heuristic search methodology fits average values without preserving meaningful structural relationships, in contrast to FunSearch, which uses LLM reasoning. We see fewer outlier points going from left to right, given that PySR simply fits average MAPE, while LLM methods like FunSearch search for structural expressions but only evaluate them over the MAPE metric. Our memory structure captures a more holistic set of statistical performances, which includes the rank correlation metric. Overall, mufasa achieved the highest Spearman correlation. Figure 6: Error distribution across sectors. The center lines show median errors, while the top and bottom edges of the box plots show the 25th and 75th percentile errors, respectively. Lower is better. Error Percentiles. In Figure 6, we do a deeper analysis using a breakdown by company sectors. mufasa achieves the strongest performance in sectors such as Utilities, Industrials, Real Estate, and Consumer Staples, where error distributions are both lower and more concentrated. These sectors are typically characterized by their fundamentals, making them well-aligned with the accounting information used by the model. The stability of these sectors allows the model to effectively perform symbolic discovery without being disrupted by the high variability of the stock prices used in evaluation. Model 25th pct. (↓ ) Median (↓ ) 75th pct. (↓ ) PySR 0.4103 0.6249 0.8322 FunSearch 0.6136 0.7500 0.8283 LLM-SR 0.2663 0.4475 0.6086 mufasa 0.1725 0.3382 0.5556 Figure 7: Aggregate APE distributions. While mufasa improves performance across most sectors, it shows slightly higher upper- and lower-percentile errors in the Communications and Consumer Discretionary sectors. A possible explanation is that these sectors are less directly driven by fundamental signals and are more influenced by forward-looking expectations, intangible assets, and market sentiment. As a result, it could fail to capture unexpected jumps caused by news events, which were not considered in our work. However, our median error is still lower or close to benchmark performance in these sectors. In general, mufasa achieves the lowest averaged errors across all percentiles of the distribution, including the median, as shown in Table 7. Similarly, our memory structure (Table 1) also captures and learns from tail errors, allowing it to produce more robust expressions compared to baselines. Directional Bias. Figure 8: Mean Signed Percentage Error across valuation quartiles (Q1: Cheap to Q4: Expensive). In Figure 8, we further analyze directional bias across valuation quartiles, using the mean signed percentage error (MSPE) metric. Across all markets, we find that mufasa tends to overestimate value for cheaper companies (Q1) and underestimate value for expensive ones (Q4). However, compared to baselines, mufasa exhibits lower directional bias overall, with errors remaining closer to zero across all quartiles. This indicates that the model not only improves absolute accuracy (as seen in MAPE), but also avoids systematic mispricing across different valuation regimes. Notably, the bias patterns are also more stable across markets for mufasa, while competing methods display greater variability. This might be attributed to the statistical memory, which also tracks bias-related signals such as MSPE, enabling the agent to reduce systematic over- or under-prediction across the different price regimes. These could be seen in the distilled agent learnings (see Statistical Diagnostics in Section E), which often mention observations on over- and under-estimations. Appendix H Limitations of the work The evaluation of classical valuation baselines depends on a set of modeling assumptions that are standard in finance, but not uniquely defined. For example, our baseline methods require specifying how future cash flows or dividends evolve over time (Damodaran 2012). In this work, we adopt commonly used approaches such as the constant-growth assumption (Gordon 1962) for consistency and interpretability. Importantly, evaluating across all possible assumption variants could be intractable in practice. Exploring different or even adaptive methods to select these assumptions could be a possible direction for future work. Some components in our framework are manually designed, such as the number of agent perspectives or market contexts, which might not capture the full range of heterogeneity present in real-world markets. However, our proposed framework is modular and extensible: additional agents or contexts could be incorporated without modifying any of the core architecture. Future work could also explore higher-level coordination mechanisms that adaptively decide the composition of agents and contexts.