Market Beta for Private Market Valuation

Beta is a measure of how much an investment tends to move when the market moves. For a public stock it’s easy to estimate: prices change every day, so you just run a regression. For private equity, it’s trickier: valuations are reported quarterly, marks get smoothed, and leverage muddies the waters. That means a “naive” beta will often come out too low unless you adjust for these frictions.

This post builds an interactive playground so you can see those frictions at work. You’ll notice three main points:

1. Reported PE returns often lag behind market moves.
2. Appraisal smoothing makes returns look less volatile than they really are.
3. A bottom-up beta built from sectors and leverage gives you a reality check on the regression results.

1) Market and PE returns

The first plot shows three lines of the same story:

• Market: the simulated quarterly market return.
• PE observed: what a fund actually reports (delayed and smoothed).
• PE unsmoothed (est): a simple adjustment that tries to remove some smoothing.

What’s happening. If there’s a sharp shock in the market (gray band), the “PE observed” line often only half-shows it, then dribbles the rest into following quarters. That’s why naive regressions underestimate beta. The dashed unsmoothed line is an attempt to undo some of that dampening.

2) How many lags to include?

If returns are delayed, it helps to regress PE on both this quarter’s market return and a few past quarters too. This chart runs regressions with 0–3 lags and shows:

• Total beta: the sum of contemporaneous and lag betas.
• $R^2$: the fraction of variation explained.

How to read. When you add 1–2 lags, total beta usually rises and $R^2$ improves. That’s evidence of reporting delay. Adding too many lags risks overfitting small samples.

3) What if I unsmooth?

Appraisal smoothing compresses ups and downs. A common fix is “unsmoothing,” which tries to reverse a simple AR(1) smoothing process:

This plot shows how estimated total beta changes as you vary $\rho$.

Takeaway. Unsmoothing usually pushes beta up toward the true latent value. But there’s no single correct $\rho$: treat it as a sensitivity check, not a magic dial.

4) Bottom-up beta (step by step)

So far we’ve estimated beta by looking at returns. But there’s another way: start from the businesses you actually own. This is called a bottom-up beta. It’s a “build it from the parts” approach.

Here’s the process in plain steps:

1. Business risk (unlevered beta). Every sector has a typical unlevered beta — how risky the business is compared to the market, before you add debt. Example: Software companies might have $\beta_U \approx 0.9$, Industrials $\beta_U \approx 0.7$.

2. Portfolio mix. Weight those unlevered betas by how much of your portfolio each sector represents. If you’re 40% in software, 25% in industrials, etc., then

   $$\beta_U^{portfolio} = \sum_i w_i \cdot \beta_{U,i}.$$

3. Relever. Companies in private equity are almost always levered. Debt makes equity riskier, because shareholders absorb losses with borrowed money on top. The formula to add leverage back is:

   $$\beta_L = \beta_U \cdot \left[1 + (1 - \text{tax}) \cdot \tfrac{D}{E}\right].$$
   • $D/E$ is the average debt-to-equity ratio you’re targeting.
   • $(1 - \text{tax})$ reflects the tax shield from interest.

4. Compare. Now you have a levered portfolio beta built entirely from sector composition and capital structure. Compare it to your regression beta. If they line up, great. If not, the gap tells you something about stale marks, sample size, or mismatched index choice.

Why this matters. The bottom-up view is grounded in fundamentals: what industries you own and how much debt they carry. It doesn’t depend on a noisy short return history.

• If regression beta is much lower than bottom-up, it often means the regression is “fooled” by smoothing or lag.
• If regression beta is much higher, maybe you’re using the wrong index, or portfolio leverage is higher than you thought.

Put differently: the regression is what the data says happened. The bottom-up is what the structure says should happen. The truth for risk management is usually somewhere in between.

Wrapping up

• Lags capture delay in reporting.
• Unsmoothing counters appraisal dampening.
• Bottom-up ties beta to business mix and leverage.

Together, they give a fuller, more credible view of how private equity portfolios move with the market.