Bayesian NAV Updating w/ Kalman Filters

Valuation is not a verdict; it is a belief with receipts. Bayesian updating turns scraps of quarter-to-date evidence into a living, probabilistic estimate of NAV. The math is compact; the philosophy is wide: confidence with a dial. Laplace called probability "common sense reduced to calculation." Tetlock reminded us that "beliefs are hypotheses to be tested, not treasures to be guarded."

This post builds a model step by step. Each step adds one idea and shows how it changes the posterior and the decisions you might take. Think "from toy to tool":

• simple normal update,
• many signals,
• sequential updates,
• better priors,
• robust likelihoods,
• a small state-space model,
• cross-fund pooling,
• decisions and diagnostics.

TL;DR

1. Model the quarter-end return $r_q$ as a hidden variable.
2. Combine a prior (manager, macro) with the likelihood of partial signals.
3. Interpret the posterior as a belief to act on, then test and recalibrate.

Ten-second intuition

Bayes updates beliefs:

• The prior $p(r)$ encodes what you believed before you saw this quarter.
• The likelihood $p(y \mid r)$ says how probable the data would be if the true return were $r$.
• The posterior $p(r \mid y)$ blends the two, with the balance set by comparative uncertainty.

A tighter prior (more certain) moves less; a cleaner signal (less noisy) moves you more. This is the same engine behind weather nowcasting, sensor fusion, and Black-Litterman portfolio updates.

Interactive — Prior vs Posterior (one latent). Use the sliders to change the prior mean and standard deviation. Watch how the posterior curve shifts and tightens as the signal gets cleaner. Treat prior sd as your "confidence dial": narrow = dogmatic, wide = humble.

Setup and notation

Let $V_q$ be NAV at quarter end $q$. Define quarterly log return

We use log returns because they add across time: if monthly logs are $r_1, r_2, r_3$, then $r_q = r_1 + r_2 + r_3$. That makes sequential updating algebraic instead of acrobatic.

During the quarter, partial evidence trickles in:

• exact or near-final month 1,
• provisional month 2,
• proxies for month 3 (sector ETFs, macro surprises, peer marks).

Our task: infer $r_q$ continuously and map it into a distribution for $V_q$.

NAV mapping. If $r_q \sim \mathcal{N}(m, s^2)$, then $V_q \mid \text{data}$ is lognormal with

The minimal model: one latent, one signal

Start with one belief, one clue.

• Prior: $r \sim \mathcal{N}(m_0, s_0^2)$
• Signal: $y \mid r \sim \mathcal{N}(r, s_y^2)$

Posterior:

This is a precision-weighted average. Intuition: if the signal is twice as precise (half the variance) as the prior, it gets twice the weight.

Common pitfalls.

• Do not double-count the same piece of information via multiple "signals".
• Keep units straight: work in logs when you add over months, and convert to percents only for display.

Interactive — Signals and posterior (start simple). Set one signal equal to a clean month-1 return, vary its sd, and watch the posterior mean migrate toward $y$ while the band tightens.

Multi-signal extension (conjugate Normal)

Now allow many clues that linearly load on $r$:

Stack in matrix form: $y = A r + b + \epsilon$ with $A = [a_1,\dots,a_K]^\top$ and $\epsilon \sim \mathcal{N}(0, R)$, $R=\mathrm{diag}(s_k^2)$. The posterior remains Normal:

Why this matters. Different proxies carry different units and cleanliness. A sector ETF with $a_k \approx 1$ and small $s_k$ will move beliefs more than a noisy peer print with $a_k \approx 0.4$.

NAV expectation.

Interactive — Multiple signals. Toggle $a_k$, $b_k$, $y_k$, and $s_k$ to see how weights reallocate when you add or remove evidence. The implied NAV band updates in place.

Sequential updating through the quarter

Evidence arrives in order, but conjugate Normal updates commute: updating with $y_1$ then $y_2$ yields the same posterior as updating once with $(y_1,y_2)$ together. That lets you treat the quarter as a loop:

• T1 (after month 1): prior $(m_0,s_0^2)$ updated by $y_1$ gives $(m_1,s_1^2)$
• T2: update by $y_2$ to $(m_2,s_2^2)$
• T3: add proxies for month 3
• T4: final mark collapses uncertainty

Order-invariance proof sketch. With independent Gaussian signals, posteriors multiply and the product of Gaussians is Gaussian; multiplication is commutative, so the update order does not matter.

Interactive — Timeline scrubber. Scrub T1 to T4 and see the band narrow. The badges report posterior mean and sd at each stage.

Constructing the prior: reputation and regime

A good prior is earned, not guessed.

Reputation as pseudo-data. Suppose a manager has track record mean $\bar r$ and variance $s_{\text{track}}^2$ over $n$ quarters. Use an effective sample size $n_{\text{eff}} \le n$ to discount for drift or luck. Then

Regime priors. Add macro features $f$ with a ridge prior on coefficients:

This is just regularized regression; Black-Litterman is a portfolio-size cousin.

Interactive — Prior builder. Feed in a track record and a peer rate, set $n_{\text{eff}}$, and push $(m_0,s_0)$ into the rest of the post.

Robustness: when reality is not Gaussian

Marks can jump, proxies can be wrong, and volatility is not constant.

• Unknown variance. A Normal-Inverse-Gamma prior on $(r,\sigma^2)$ yields a Student-t posterior on $r$. Fatter tails slow overreaction to outliers.
• Outlier-resistant likelihood. Model $y_k \mid r \sim t_\nu(a_k r + b_k, s_k^2)$. Small $\nu$ down-weights wild points.

Interactive — Likelihood chooser. Flip between Gaussian and t, vary $\nu$, and note how the posterior spreads when you admit heavy tails.

State-space step-up: monthly latent returns (Kalman)

We now model the monthly path, not just the quarterly sum. The latent state is

Process (prior). Each month has baseline mean plus macro tilt and correlated uncertainty:

where $C(\rho)$ is a 3x3 Toeplitz correlation with $C_{ij}=\rho^{|i-j|}$. This makes months related but not identical.

Observations.

• exact months: $y_m = r_m + \epsilon_m$, with small but nonzero sd $s_{\text{obs12}}$ (near-final, not perfect),
• month-3 proxy: $y_3 = r_3 + e_3$, $e_3 \sim \mathcal{N}(0, s_{\text{obs}}^2)$,
• optional final sum: $y_{\Sigma} = r_1 + r_2 + r_3 + \eta$, with sd $s_{\Sigma}$.

Kalman update (scalar measurement). For a measurement $y = Hx + \varepsilon$, $\varepsilon \sim \mathcal{N}(0,R)$, the update is

What the gain $K$ does. $K$ increases when $P^-$ is large (you were uncertain) or $R$ is small (measurement is clean). That is, move fast when you were unsure and the new data are good.

Total-quarter uncertainty. The variance of the quarterly return is

Interactive — Monthly state nowcast.

• Move through stages T1 to T4; see how exact months collapse their bands and the sum variance shrinks.
• Increase $\rho_{\text{proc}}$ to couple months more tightly.
• Use nonzero $s_{\text{obs12}}$ so "exact" months are near-final rather than perfect; this prevents degeneracy and is more realistic.

How to read the geometry.

• The correlation heatmap shows structure across months; off-diagonals indicate shared shocks.
• The $H$ bar shows which latent you are observing at this stage (e.g., month 2 or a sum).
• $R$ is the observation noise; larger $R$ yields smaller $K$ and gentler moves.

Cross-fund pooling: hierarchical Bayes

You rarely follow one fund. To calibrate beliefs across many, you share statistical strength.

Model quarterly fund returns $r_{iq}$ with fund-specific means $\mu_i$ and idiosyncratic noise $\sigma_i$:

Given a fund summary with sample mean $\bar r_i$, variance $s_i^2$, sample size $n_i$, the posterior for $\mu_i$ is

with shrinkage weight

Interpretation: data-poor or high-volatility funds (small $n_i/s_i^2$) shrink more. This is James-Stein in fund clothing.

Interactive — Shrinkage in action. Move $\tau$ to tune pooling strength. Long arrows mean heavy shrinkage. Try increasing volatility for a couple of funds to see how their posteriors retreat toward the sector anchor.

From belief to action

A posterior is not an answer; it is a control surface for choices. Here are three operational views:

1. Tail odds. For a loss threshold $L$ in log terms, compute $P(r_q \le L)$ and decide if rebalancing or hedging triggers.
2. Beat-the-benchmark odds. If the benchmark quarterly log return is $r^*$, then $P(r_q > r^*)$ quantifies the chance of outperformance.
3. Risk-adjusted rule. Define risk-adjusted return

with $m$ the posterior mean and $s$ the posterior sd. Choose thresholds $t_{\text{add}}$ and $t_{\text{trim}}$; Add if $\mathrm{RA} \ge t_{\text{add}}$, Trim if $\mathrm{RA} \le t_{\text{trim}}$, else Hold.

Contours. Lines of constant RA satisfy $s = (m - t)/\lambda$; steeper $\lambda$ penalizes uncertainty more.

Value of information (quick approximation). If the next signal reduces variance from $s^2$ to $s^2 - \Delta$, the RA improves by about $(\lambda/2)\,\Delta/s$ for small changes. Expensive information should clear this expected improvement hurdle.

Interactive — Decision map. The heatmap shows Add/Hold/Trim regions in the nonnegative quadrant. The cross marks your current posterior. Tune $\lambda$ and thresholds to match mandate and risk appetite.

Diagnostics and epistemic humility

Beliefs need audit trails. Three checks help keep you honest.

• Log score. The average log predictive density $\frac{1}{N}\sum_i \log p(r_i \mid \mu_i, s_i)$ rewards sharp and calibrated forecasts.
• Brier score. For a binary event like $r<0$, the mean squared error between predicted probabilities and outcomes.
• Calibration curves and PIT. Probability integral transforms $u_i = F_i(r_i)$ should be uniform if calibrated; reliability curves should sit on the diagonal.

Interactive — Calibration dashboard. Simulate forecasts and outcomes. Inject mean bias or noise mis-specification and see how the PIT warps and scores deteriorate. Then fix your priors or likelihoods upstream.

Implementation notes

• Treat each signal’s uncertainty $s_k$ as first-class metadata.
• Prevent leakage: never let final marks influence interim updates.
• Unit test the algebra: conjugate updates, Kalman recursions (including Joseph form), and posterior predictive checks.
• Beware double counting: a peer print and an ETF move may express the same shock; correlate or down-weight accordingly.

The progression — simple average -> multi-signal -> sequential -> dynamic -> pooled — mirrors the evolution of belief itself: from a single guess to a system that learns, checks itself, and earns its confidence.