Dymanic Betas Using Kalman Filter

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TLDR: First Principles Thinking

Markets aren't static. A stock's sensitivity to the market (Beta) changes every day. Traditional OLS is like looking at a blurry long-exposure photo; the Kalman Filter is a high-speed video.

The logic: > 1. Predict: Start with your best guess for today's Beta.

  1. Observe: Look at how the stock actually moved compared to the market.

  2. Adjust: If your guess was off, update it. If the market was crazy volatile, trust your guess more. If the market was stable and you missed, trust the data more.

1. Model Configuration

To track a moving target, we define how it behaves (Transition) and how we see it (Measurement).

Parameter Definition First Principle Role
\(\bar\beta\) Long-run average The "gravity" pulling Beta back to normal.
\(\phi\) Mean reversion speed How fast the stock returns to its typical risk profile.
\(Q\) Process noise Internal "jitter"—how much Beta can jump on its own.
\(R\) Measurement noise Market "static"—stock moves unrelated to the market.

2. The Algorithm

Phase A: Initialization

We begin with our best long-term estimate.

  • State: \(\hat\beta \leftarrow \bar\beta\)

  • Uncertainty: \(P \leftarrow \mathrm{SE}(\bar\beta)^2\)

Phase B: The Recursive Loop

For every trading day \(t\), we execute the following:

Step 1: Predict (The "Prior")

We project where Beta should be before seeing today's returns.

\[\hat\beta^{-} = \phi \hat\beta + (1-\phi)\bar\beta\]

Correction: Your uncertainty prediction \(P^{-} = P + Q\) assumes a random walk. Since we are using a mean-reverting transition (\(\phi\)), the variance propagation must account for the shrinkage:

\[P^{-} = \phi^2 P + Q\]

Step 2: Observe

We calculate the "Innovation"—the gap between reality and our model.

  1. Market Signal: \(H_t = r^{mkt}_t\)

  2. Expected Return: \(\hat y_t = H_t \hat\beta^{-}\)

  3. Surprise (Residual): \(e_t = r^{stock}_t - \hat y_t\)

Step 3: The Learning Rate (Kalman Gain)

This determines how much we react to the "Surprise."

  1. System Variance: \(S_t = H_t^2 P^{-} + R\)

  2. Gain: \(K_t = \frac{P^{-} H_t}{S_t}\)

Logic Check

If \(R\) (market noise) is huge, \(K_t\) becomes small. We ignore the surprise because it's probably just noise. If \(P^{-}\) (our uncertainty) is huge, \(K_t\) becomes large. We trust the data more because we don't trust our own guess.

Step 4: Update (The "Posterior")

We refine our estimate and shrink our uncertainty for tomorrow.

  1. New Beta: \(\hat\beta \leftarrow \hat\beta^{-} + K_t e_t\)

  2. New Uncertainty: \(P \leftarrow (1 - K_t H_t) P^{-}\)

3. Key Outputs

  • \(\{\hat\beta_t\}\): Your time-varying Beta series.

  • \(\{P_t\}\): The confidence interval of your estimate (use \(\pm \sqrt{P_t}\) for bands).

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