The LBO as a dynamical system

by Georg R. Pollak4 min read

The previous post priced a business as a static object: multiple × EBITDA, minus net debt. The actual private-equity trade is dynamic — buy with borrowed money, run the system forward for five years, sell — and the right mental model for someone with a math background is not a spreadsheet but a small discrete dynamical system: one state variable that grows, one that amortizes, and an observation function you only get to evaluate twice.

The state and the two equations you control

Write E_t for EBITDA in year t, M for the EV/EBITDA multiple, D_t for net debt. The whole deal is:

Entry (t = 0):   pay   EV₀ = M₀ · E₀
                 fund  D₀ = L · E₀          (L = leverage, e.g. 4×)
                 check eq₀ = M₀·E₀ − D₀     (your money)

Exit  (t = T):   receive eqT = M_T·E_T − D_T

Everything in between is the evolution of E_t and D_t. In the toy model (and to first order, in real deals):

E_t   = E₀ · (1+g)^t                        the business grows
D_t   = D_{t−1} − max(c·E_t − r·D_{t−1}, 0)  the cash sweep

where g is EBITDA growth, r the interest rate, and c the cash conversion — the share of EBITDA that survives taxes, capex and working capital and is free to repay debt. The debt line is the part newcomers underestimate: it is a recurrence in which the business's own cash flow amortizes the acquisition debt. The company buys itself, slowly, on your behalf.

Returns are then two numbers, and with a single entry and exit cash flow they carry the same information:

MoM = eqT / eq₀          money multiple
IRR = MoM^(1/T) − 1      annualized

That exponent is worth staring at: a 2.0× over four years is a 19% IRR; the same 2.0× over seven years is 10%. Time is not a detail — it is in the exponent. This is why "when to sell" is a real decision, and why the third post in this series hands exactly that decision to a reinforcement learner.

The value-creation bridge, derived

Where did the equity gain come from? Expand the difference:

eqT − eq₀ = M_T·E_T − D_T − (M₀·E₀ − D₀)
          = M₀·(E_T − E₀)      growth:        same multiple, bigger business
          + (M_T − M₀)·E_T     multiple:      the market re-rated it
          + (D₀ − D_T)         deleveraging:  the sweep worked

Check the algebra: the three terms telescope exactly to the equity delta — no residual, no hand-waving. This decomposition is the value-creation bridge, and it is how every exited deal gets explained to investors. The three terms also have very different reputations: growth is earned, multiple expansion is weather (you can position for it; you cannot manufacture it), and deleveraging is arithmetic that works as long as nothing breaks.

One more identity worth internalizing — why leverage amplifies. With eq₀ = (M₀ − L)·E₀, the same enterprise gain is divided by a smaller equity base as L → M₀. At M₀ = 8 and L = 4, a 25% rise in EV doubles your equity. The mirror image is equally exact: a 25% fall in a 6×-levered deal wipes it to two turns of cushion. Leverage has no opinion about direction.

Drag it

The model below is the engine from this post running live — the same code the worked numbers above come from. Suggested experiments:

  • Set growth to 0 and both multiples equal: the IRR that remains is pure deleveraging. This is the "boring base case" every lender underwrites.
  • Push leverage from 2× to 6× at fixed growth: watch IRR climb while the margin for error (exit equity vs entry equity under a multiple drop) shrinks.
  • Buy at 8×, sell at 6×: see how much growth it takes just to stand still. Multiple contraction is the quiet killer of vintage returns.
  • Stretch the hold from 4 to 7 years with everything fixed: same MoM mechanics, melting IRR.

What the toy ignores — knowingly

One debt tranche at one rate (real deals stack senior, mezzanine, sometimes PIK); cash conversion as a constant (real working capital breathes with growth); no taxes on exit, no fees, no management incentive plan; and a multiple you choose rather than a market you meet. None of these change the structure — they decorate the same three-term bridge.

The bigger omission is uncertainty. Here, g and M_T are sliders; in reality they are stochastic processes, and the one genuine decision you keep holding is when to stop the process and sell. Making that precise — EBITDA as a random walk with drift, the multiple mean-reverting through cycles, and an agent learning the exit policy from ten thousand simulated deals — is the next post.

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