Logistic diffusion model · monthly resolution

The S-curve of revenue
and the cash left in the bank

Slow start → rapid expansion → saturation. The fixed parameters set the shape of the trajectory, while the sliders let you play with price, seed, acquisition — and the investor's terms. The investor plays for the exit: their stake is worth a multiple of revenue, and because the S-curve flattens at the top, there's a best month to sell — the big amber button marks where holding longer stops paying off. The horizontal axis is time (months).

Fixed parameters

The givens of the market and operations. These shape the curve's form and height.

Starting subscribers S₀

Estimated market size (ceiling, K) units

Monthly churn rate 2.5%

Monthly burn (fixed cost) $/mo

Play with these

The startup's upfront decisions. Move them and watch whether it survives the "valley" before the S-curve kicks in.

Subscription fee $30/user/mo

Seed investment $500kenter in $K

New subscribers / mo (at launch) 40

Acquisition cost (CAC) $120/user

Investor equity 20%stake

Exit valuation multiple 6×× ARR

The investor's stake is worth equity × an m× revenue multiple. On an S-curve the value climbs fastest at the inflection, then flattens — so the best exit (★) is the month that maximises the annualized return on the cheque, not simply the latest month. A fainter ring marks the earlier payback point, below which selling loses money.

CAC is included among the sliders: without it the seed + burn cash story would be empty, and marketing efficiency is also something a startup has influence over.

Growth rate (r)

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Effective ceiling (K·(1−c/r))

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Inflection (fastest month)

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Cash low point

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Optimal exit · best annualized return on the investor's stake

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Active subscribers (S-curve) Effective ceiling K_eff Cash (cumulative) Monthly revenue (MRR) Optimal exit (best annualized return)

Engine: S(t) = S(t-1) + r·S(t-1)·(1 − S(t-1)/K) − c·S(t-1)
Revenue: R(t) = fee · S(t) · Profit: π(t) = R(t) − CAC·new(t) − burn · Cash: C(t) = seed + Σπ
Investor stake: Vₑ(t) = 12·e·m·fee·S(t) · MOIC: Vₑ(t)/seed · Optimal exit: argmaxₜ MOIC(t)^(12/t) − 1