Probabilistic Equity Valuation with Monte Carlo Simulation, Robust Multivariate Risk Signals, and Calibrated Sentiment

The Hidden Epistemology

Every valuation model encodes a theory of knowledge. The standard discounted cash flow model encodes a particularly aggressive one: that future cash flows, growth rates, and discount rates are known with sufficient precision to justify a point estimate. The model doesn't announce this assumption. It simply produces a number with several significant figures, and the precision implies the epistemology.

This is backwards. The honest sequence is: first establish what is known and with what confidence, then build mathematics that preserves that uncertainty through to the output. A valuation should be a distribution, not a point—a shape that reflects the actual state of knowledge.

This article specifies such a framework. The mathematics is not decorative; each formal choice corresponds to a substantive claim about the structure of the problem.

I. Cash Flows on Their Natural Support

Cash flows are strictly positive quantities. This isn't a modeling choice—it's a constraint from the phenomenon itself. A company's free cash flow might be small, but the generative process being modeled (the capacity to produce cash from operations) lives on $(0, \infty)$.

The naive approach—modeling $CF_i$ directly as a normal random variable—violates this constraint. Normal distributions have support on $(-\infty, \infty)$. You can truncate, but truncation distorts. You can hope negative draws are rare, but hope is not a method.

The principled solution is to model in log-space:

$$\log CF_i = \log CF_0 + \sum_{k=1}^{i} \ell_k, \qquad \ell_k \sim \mathcal{N}(\mu_\ell, \sigma_\ell^2)$$

Here $CF_0 > 0$ is the observed base cash flow, and each $\ell_k = \log(1 + g_k)$ represents the log-growth shock in period $k$. The $\ell_k$ can be positive or negative—growth or contraction—but upon exponentiation to recover $CF_i$, positivity is guaranteed:

$$CF_i = CF_0 \cdot \exp\left(\sum_{k=1}^{i} \ell_k\right) > 0$$

The parameter $\mu_\ell$ encodes expected growth; $\sigma_\ell^2$ encodes uncertainty about that growth. These are estimable from historical data, adjustable by regime, and—crucially—honest about what is not known.

II. Decomposing the Discount Rate

The discount rate $r$ appears as a single symbol in valuation formulas, but it is not a single thing. It is a composite of distinguishable components, each with its own dynamics and uncertainty:

$$r = r_f + \beta \cdot \text{ERP} + s_{\text{size}} + s_{\text{leverage}} + s_{\text{country}} + \epsilon_r$$

where:
- $r_f$ is the risk-free rate (observable, but time-varying)
- $\beta \cdot \text{ERP}$ is systematic risk compensation (estimated, contentious)
- $s_{\text{size}}, s_{\text{leverage}}, s_{\text{country}}$ are factor premiums (empirically calibrated)
- $\epsilon_r$ is residual uncertainty (acknowledged, not hidden)

This decomposition serves two purposes. First, it makes the model auditable: stakeholders can dispute specific components rather than accepting or rejecting an opaque number. Second, it enables structured uncertainty: each component can have its own distribution, estimated from appropriate data, and the composite $r$ inherits uncertainty from all of them.

III. The Terminal Value Constraint

For a forecast horizon of $n$ years, the terminal value captures all cash flows beyond that horizon. The Gordon growth perpetuity formula is:

$$TV = \frac{CF_{n+1}}{r - g_{\text{term}}}, \qquad CF_{n+1} = CF_n(1 + g_{\text{term}})$$

The denominator $(r - g_{\text{term}})$ introduces a hard constraint: $r > g_{\text{term}}$. This is not a recommendation or a "reasonable assumption"—it is a mathematical requirement for the formula to yield a finite, positive value. When $r \leq g_{\text{term}}$, the perpetuity is either undefined or negative, neither of which corresponds to economic reality.

Most implementations ignore this constraint, implicitly assuming it will be satisfied. But if $r$ and $g_{\text{term}}$ are random variables—as they must be in a probabilistic framework—some draws will violate the constraint. These draws don't represent unlikely scenarios; they represent nonsense. The model must exclude them.

Two approaches are defensible:

Rejection sampling: Draw $(r, g_{\text{term}})$ from their joint distribution; discard and redraw if $r \leq g_{\text{term}}$. This is correct but potentially inefficient if violation probability is high.

Truncated joint distribution: Define the joint distribution explicitly on the constrained region $\{(r, g): r > g\}$. This is efficient but requires careful normalization.

Either approach preserves the integrity of the distribution. Post-hoc "repairs"—clamping $g_{\text{term}}$ to be below $r$, for instance—introduce bias and should be avoided.

IV. Dependence Structure

Growth and discount rates are not independent. In recessions, cash flow growth declines while risk premiums rise—a double penalty. In expansions, both move favorably. Modeling these as independent random variables understates tail risk precisely when it matters most.

A copula separates marginal distributions from dependence structure. Let the driver vector be:

$$\mathbf{Z} = (\ell_1, \ldots, \ell_n, g_{\text{term}}, r)$$

The marginals $F_{\ell}, F_{g_{\text{term}}}, F_r$ are specified as above, then bound through a $t$-copula:

$$(U_1, \ldots, U_d) \sim C_\nu(\cdot; \Sigma), \qquad Z_j = F_j^{-1}(U_j)$$

The $t$-copula with $\nu$ degrees of freedom exhibits tail dependence: extreme values in one variable are associated with extreme values in others more strongly than a Gaussian copula would imply. The correlation matrix $\Sigma$ captures the linear dependence structure; the parameter $\nu$ controls tail thickness.

Estimation of $\Sigma$ from limited data is fragile. Hierarchical pooling across sectors and shrinkage toward structured priors reduce noise. The point is not to estimate dependence perfectly—that's impossible—but to represent it honestly rather than assuming it away.

V. The Valuation Functional

Given cash flows and discount rate, enterprise value is:

$$V = \sum_{i=1}^{n} \frac{CF_i}{(1+r)^i} + \frac{TV}{(1+r)^n}$$

This is a deterministic function of random inputs. The output $V$ is therefore a random variable whose distribution can be characterized through Monte Carlo sampling.

VI. Monte Carlo Implementation

The sampling procedure is:

Algorithm: Copula-Based Monte Carlo Valuation

Input: $CF_0$, horizon $n$, marginals $(F_\ell, F_{g_{\text{term}}}, F_r)$, copula $C_\nu(\cdot; \Sigma)$, sample size $N$

For $j = 1$ to $N$:
1. Draw $(U_1^{(j)}, \ldots, U_d^{(j)}) \sim C_\nu(\cdot; \Sigma)$ using Latin Hypercube Sampling
2. Transform: $\ell_k^{(j)} = F_\ell^{-1}(U_k^{(j)})$, $g_{\text{term}}^{(j)} = F_{g_{\text{term}}}^{-1}(U_{n+1}^{(j)})$, $r^{(j)} = F_r^{-1}(U_{n+2}^{(j)})$
3. If $r^{(j)} \leq g_{\text{term}}^{(j)}$: reject and return to step 1
4. Compute $CF_i^{(j)} = CF_0 \cdot \exp(\sum_{k=1}^i \ell_k^{(j)})$
5. Compute $TV^{(j)} = CF_n^{(j)}(1 + g_{\text{term}}^{(j)}) / (r^{(j)} - g_{\text{term}}^{(j)})$
6. Compute $V^{(j)} = \sum_{i=1}^n CF_i^{(j)}/(1+r^{(j)})^i + TV^{(j)}/(1+r^{(j)})^n$

Output: Empirical distribution $\{V^{(j)}\}_{j=1}^N$

Variance reduction techniques—antithetic variates, stratified sampling—improve convergence. Convergence should be monitored on distributional quantities (quantiles, exceedance probabilities), not just the mean.

VII. Decision Statistics

The distribution over $V$ enables probability statements that point estimates cannot express. Let $V_{\text{mkt}}$ be the market-implied enterprise value. Relevant statistics include:

Exceedance probability:
$$P(V > V_{\text{mkt}}) = \frac{1}{N} \sum_{j=1}^N \mathbb{1}\{V^{(j)} > V_{\text{mkt}}\}$$

Expected relative mispricing:
$$\mathbb{E}\left[\frac{V}{V_{\text{mkt}}} - 1\right] = \frac{1}{N} \sum_{j=1}^N \left(\frac{V^{(j)}}{V_{\text{mkt}}} - 1\right)$$

Downside risk:
$$P\left(\frac{V}{V_{\text{mkt}}} < \tau\right)$$

for a user-specified threshold $\tau$.

These quantities support decisions in a way that a point estimate cannot. "The stock is undervalued" is less useful than "There is probability $p$ the stock is undervalued, with expected upside of $x\%$ and probability $q$ it is more than $y\%$ overvalued."

VIII. Nowcasting: Risk Anomalies

The valuation model operates on fundamental assumptions about cash flows and discount rates. But markets also exhibit transient dislocations—volatility spikes, liquidity crunches, correlation breakdowns—that don't map cleanly onto fundamentals but should inform position sizing and confidence.

A robust anomaly detection layer monitors these conditions.

Robust standardization: For indicator series $X_t$, standardize using robust statistics:

$$Z_X^{\text{rob}}(t) = \frac{X_t - \text{Med}(X)}{1.4826 \cdot \text{MAD}(X)}$$

or exponentially-weighted moments for adaptation to volatility clustering:

$$Z_X^{\text{ewma}}(t) = \frac{X_t - \mu_t}{\sigma_t}$$

Multivariate distance: Stack standardized indicators into $\mathbf{X}_t \in \mathbb{R}^d$ and compute Mahalanobis distance:

$$D_t^2 = (\mathbf{X}_t - \hat{\mu})^\top \hat{\Sigma}_R^{-1} (\mathbf{X}_t - \hat{\mu})$$

with $\hat{\Sigma}_R$ estimated via robust methods (Minimum Covariance Determinant). Large $D_t^2$ indicates market conditions outside the model's training regime—a signal to widen uncertainty bands, not to override the model with ad-hoc judgments.

IX. Sentiment Integration

Textual sentiment—from news, filings, analyst reports—provides information about near-term fundamentals. The challenge is incorporating this signal without duplicating price information (which already reflects public sentiment) and without introducing look-ahead bias.

Aggregation: Let sources $i = 1, \ldots, k$ emit sentiment scores $s_i \in [-1, 1]$ at times $t_i$. Aggregate with credibility weights $c_i$ and temporal decay:

$$S_{\text{agg}}(t) = \frac{\sum_{i=1}^k c_i \cdot r_i(t) \cdot s_i}{\sum_{i=1}^k c_i \cdot r_i(t)}, \qquad r_i(t) = \exp\left(-\frac{t - t_i}{\tau}\right)$$

Calibration: The raw aggregate is unitless. Calibrate against a prediction target (e.g., next-$h$-day excess return) using isotonic regression on strictly out-of-sample folds. The calibrated output is a probability statement with measurable reliability.

Linkage to fundamentals: To avoid price-signal duplication, let sentiment update short-horizon growth priors:

$$\ell_k \mid S_{\text{agg}}(t) \sim \mathcal{N}\left(\mu_\ell + \gamma \cdot S_{\text{agg}}(t), \sigma_\ell^2\right)$$

where $\gamma$ is estimated out-of-sample. Sentiment shifts growth expectations, not prices directly. This preserves interpretability and avoids circularity.

X. Evaluation Protocol

A probabilistic model requires probabilistic evaluation. Point-estimate metrics (RMSE, hit rate) are insufficient.

Walk-forward design: Use rolling-origin evaluation with point-in-time data. Fix the model specification before testing. Never use future information to set parameters or select variants.

Proper scoring rules: For predictive CDF $\hat{F}_t$ and realized outcome $y_t$, the Continuous Ranked Probability Score is:

$$\text{CRPS}(\hat{F}_t, y_t) = \int_{-\infty}^{\infty} \left(\hat{F}_t(v) - \mathbb{1}\{v \geq y_t\}\right)^2 dv$$

CRPS is proper: it is uniquely minimized when $\hat{F}_t$ equals the true distribution. Optimizing it rewards both accuracy and calibration.

Calibration diagnostics: Compute probability integral transform (PIT) values $\hat{F}_t(y_t)$. Under correct specification, these are uniformly distributed. Deviations indicate miscalibration: U-shaped histograms suggest overdispersion; inverse-U suggests underdispersion.

Interval coverage: For nominal $(1-\alpha)$ prediction intervals, report empirical coverage (prediction interval coverage probability, PICP) and mean interval width (MPIW). The goal is coverage near $(1-\alpha)$ with minimal width.

XI. Decision Policy

A distribution enables decision rules that explicitly manage tail risk. Consider a constrained Kelly-style allocation on relative value $\alpha = V/V_{\text{mkt}} - 1$:

$$f^* = \arg\max_{0 \leq f \leq f_{\max}} \mathbb{E}[\log(1 + f \cdot \alpha)]$$

subject to:

$$P(\alpha < -L) \leq \epsilon$$

The constraint requires that the probability of losing more than fraction $L$ be at most $\epsilon$. This cannot be expressed without a distribution. Point estimates support only expected-value optimization; distributional estimates support risk-constrained optimization.

XII. Limitations

Terminal value sensitivity: A large fraction of $V$ comes from $TV$, which is highly sensitive to $(r, g_{\text{term}})$. The Gordon formula amplifies uncertainty in these parameters. This is not a flaw in the model—it is a property of the problem. Long-horizon valuation is inherently uncertain. The model makes this visible rather than hiding it.

Regime instability: Dependence structures estimated from historical data may not persist. Factor loadings shift; correlations break down in crises and reconstitute differently afterward. The model should be re-estimated periodically, and regime-switching extensions are a natural development.

Estimation noise in copulas: With typical sample sizes, copula parameters are imprecisely estimated. Hierarchical pooling across peers and shrinkage toward structured priors are necessary. Overconfidence in dependence estimates is as dangerous as ignoring dependence entirely.

Timestamp alignment: Sentiment and anomaly signals must be aligned to trading calendars with care. A timestamp error of one day can introduce look-ahead bias that invalidates out-of-sample evaluation.

XIII. Conclusion

The framework specified here does not make valuation more accurate in the sense of reducing forecast error. The future remains uncertain, and no model changes that. What the framework does is make uncertainty operational: visible in the output, propagated correctly through the mathematics, and usable in decision rules that respect tail risk.

The core contributions are structural:
- Cash flows modeled on their natural support via log-increments
- Economic constraints ($r > g_{\text{term}}$) enforced in sampling, not assumed
- Dependence between growth and discounting represented via copulas
- Nowcasting signals integrated through calibrated updates to priors
- Evaluation targeting distributional quality, not point accuracy

A point estimate compresses uncertainty into false precision. A distribution preserves it. And from preserved uncertainty, it becomes possible to reason honestly about risk.

Contact: mrussmann@proton.me