We Buy Distributions, Not Deals

Private equity underwriting often drifts toward point worship: one entry multiple, one base case, one IRR. But decisions in illiquid, delayed-feedback markets are bets under uncertainty. The right object to buy (and to present to an investment committee) is not a number—it is a distribution.

This post lays out a practical Bayesian frame for underwriting: how to specify priors, update with evidence, and make decisions with loss functions and constraints. We’ll keep the math simple and the outputs auditably clear.

The one-minute Bayesian primer

Updating belief is multiplication in disguise:

• $p(\theta)$ is your prior (sector base rates, manager experience, cycle view).
• $p(y \mid \theta)$ is the likelihood (how today’s evidence speaks).
• $p(\theta \mid y)$ is the posterior—the distribution you buy.

When we say “we buy distributions,” we mean: price, leverage, and sell/hold are chosen against $p(\cdot \mid y)$, not against a single $\hat\theta$.

A minimal generative model of an LBO (enough to be useful)

Think in three layers:

1. Entry: revenue $R_0$, margin $m_0$, multiple $M_{\text{in}}$.
2. Evolution: drivers $g_R$ (revenue growth), $g_m$ (margin drift), deleveraging $\Delta \ell$.
3. Exit: multiple $M_{\text{out}}$ and timing $T$.

Let EBITDA be $E_t = R_t m_t$, enterprise value $V_t = M_t E_t$, and net debt $D_t$. Equity value at exit is $V_T - D_T$. MOIC is

A simple but serviceable stochastic skeleton:

• $\text{HierNormal}$ denotes a hierarchical Normal prior that pools information across sector/region/vintage.
• Shocks $\varepsilon$ can be fat-tailed to avoid fragile tails.

With priors on $(g_R, g_m, \alpha_{\text{sector}}, \beta, \tau, T)$, posterior draws simulate full cash-flows, DPI, and IRR.

Pricing with a corridor, not a dart

The price you pay sets $M_{\text{in}}$. Under posterior draws $\{\theta^{(s)}\}_{s=1}^S$, induce a posterior for MOIC and IRR as a function of price $P$ (or $M_{\text{in}}$). Choose within a pricing corridor that respects both return targets and risk limits.

Decision rule (illustrative):

with a concave utility $U$ (downside-averse), and ruin and loss constraints $(\epsilon,\delta)$ set by the IC.

Posterior draws across entry multiples translate to the corridor below: the fan displays the 10–90% IRR quantiles, while the horizontal bands enforce the median hurdle and ruin constraint.

For instance, at an entry multiple of 8.5x EBITDA the posterior median IRR is ~17% and the chance of breaching DPI < 1 is around 5%, so the point sits inside the shaded corridor. Push the price to 10.5x and the median collapses below 10% while ruin risk jumps above 15%, ejecting the deal from the admissible band—the chart makes that trade-off immediate.

What it achieves: it replaces “pay 9.0x feels fine” with a transparent risk-return frontier over price, making explicit where the bet becomes dominated.

Leverage as a risk budget, not a macho knob

Leverage raises expected IRR but also the tail risk of impairment. Let $\lambda$ be the initial net-debt-to-EBITDA and $c$ the all-in cash cost. Under posterior draws, compute

Leverage choice (example):

The leverage sweep below pairs expected IRR with the ruin probability on a shared $x$-axis; the shaded region marks where the ruin threshold is violated and the dotted guide pins the policy choice $\lambda^\star$.

Concrete read: at $\lambda=3.25$ the expected IRR is roughly 18.4% while ruin risk stays under 10%, so the vertical guide lands just inside the safe zone. Drift up to $\lambda=4.5$ and the IRR barely improves (~17.2%) yet ruin risk surges past 20%, meaning leverage is simply buying downside without return.

What it achieves: it turns capital structure into a policy-consistent optimization problem, rather than a negotiation afterthought.

Sell/Hold is a stopping problem

At time $t$, the continuation value is the discounted posterior expectation of after-tax proceeds if you hold to the optimal exit window; the sale value is today’s bid minus transaction costs. Sell if

Model exit timing $T$ with a (possibly competing-risk) hazard $h(t \mid x_t)$; the continuation value is an expectation over the posterior of $(M_{\text{out}}, T)$.

To close the loop you only need the posterior on the value gap $\Delta = \text{Hold} - \text{Sell}$. Negative draws mean the bid beats continuation, positives mean you are better off harvesting later. Summarize that gap, apply your loss function, and the “should we sell?” argument becomes a posterior probability—not an opinion.

Concrete read: with the bid centered at $214m$, 13 of 30 draws land to the left of zero, so the posterior places about 43% weight on selling. Let net proceeds slip to $208m$ and only about a third of draws stay in sell territory—the dial would swing back to “hold and harvest.”

What it achieves: a crisp, defensible rule where “we think we can do better next year” becomes a posterior odds statement.

What shows up in the IC deck (replace single numbers with distributions)

1. Posterior for exit multiple $M_{\text{out}}$. $\log M_{\text{out}} \sim \mathcal{N}\!\big(\alpha_{\text{sector}}+\beta^\top X, \tau^2\big).$

   The ridges below stack each comp set's posterior for $M_{\text{out}}$; green ticks mark the pooled (shrunken) sector means and the dotted line is the fund-level anchor. Narrow ridges flag well-observed sectors, while wide, flatter ridges show where the model leans on the pool instead of brittle point picks.

   Concrete read: software comps skew high but still pull back toward 9.5x, while industrial tech stays wide—the pooled tick keeps the underwriting guardrails tight even when a single deal whispers "pay 11x."

2. Posterior predictive for IRR and DPI under your proposed price and leverage. Show median, 10–90% band, and $\Pr(\text{DPI}<1)$.

3. Sensitivity as distributions, not tables. For each driver $d\in\{\text{growth}, \text{margin}, M_{\text{out}}\}$, show $\Delta \text{IRR}_d \equiv \text{IRR}(d+\delta) - \text{IRR}(d)$ as a posterior, not a single “$+100$ bps” column.

4. Calibration tiles. For the last $K$ underwritten deals, report two proper scores comparing predicted to realized:

   • Avg log predictive density: $\frac{1}{K}\sum_{i=1}^K \log p_i(r_i).$
   • Brier for the impairment event $\mathbf{1}\{\text{DPI}<1\}$: $\frac{1}{K}\sum_{i=1}^{K} (p_i - y_i)^2.$

   Effect: proves the distributions are not theater; they are calibrated.

Priors you can defend (and improve)

Entry/exit multiples. Hierarchical Normal on $\log M$ with sector and cycle features $X$. Start weakly-informative:

Growth and margin drift. Use business-sane bounds: $g_R \sim \mathcal{N}(\mu_{g}, \sigma_{g}^2)$ with $\mu_g$ tied to macro nowcasts; $g_m \sim \mathcal{N}(0, \sigma_m^2)$ with $\sigma_m$ reflecting operational plan uncertainty.

Debt service. Cost $c$ with a fat-tailed prior to reflect spread volatility; interest coverage constraints built in as hard filters on draws.

Why this matters. Priors are not opinions stapled to a model; they are encoded base rates that stabilize thin evidence and keep underwriting inside a guardrail.

How this changes day-to-day underwriting

• On pricing calls we stop asking "what's the base case IRR?" and start scanning the whole posterior IRR fan. If the ruin probability clears the guardrail, the price is in play. It changes the tone of the room immediately.
• Scenario reviews no longer mean flipping through three color-coded cases. Instead we throw the posterior fan chart on the screen, talk about skew and tails, and everyone sees the same distribution.
• Leverage committees get calmer: the risk-budget dial shows exactly where $\lambda$ trips the ruin limit, so the recommended leverage is a policy answer, not a bravado pitch.
• When it's time to sell or hold, we lead with the value-gap dial. The memo reads, "Sell probability 43%," and the debate is about loss functions, not who feels luckier about next year.

Connections (why this frame is natural for quants)

• Kelly sizing sneaks in: when we cap ruin probability we are basically doing fractional Kelly with a drawdown guardrail, just in LBO clothing.
• Robust mean-variance thinking pops up too: averaging posteriors across models is Bayesian model averaging, and the pricing corridor is the frontier you'd expect if you hate model fragility.
• Credit brain makes an appearance: DPI < 1 is our "default" event, so running a Brier score on it feels exactly like PD calibration. Suddenly the IC nods along.
• And the sell/hold dial? It's an American option decision dressed up in private equity paperwork; the continuation value is literally the option value.

Closing

Buying distributions is not poetry; it’s discipline. It says: we make decisions on random variables with consequences. In practice that means pricing corridors instead of darts, leverage set by a risk budget, and sell/hold as a stopping problem—each grounded in posteriors that you can audit, backtest, and defend.

Once the culture clicks, something lovely happens: the IC starts asking distributional questions (“what’s the 10th percentile DPI at 9.5×?”), and underwriting turns from performing certainty to managing uncertainty.

That is the philosophy: we buy distributions, not deals.