Pivot revenue · monthly resolution

The U-curve of a
pivoting startup

After a pivot revenue first falls — the old product's customers churn faster than the new product wins replacements — then climbs back up along the same arc. The fall and the recovery have the same intensity: the curve is a true symmetric U. The fixed parameters set the old base and the fixed costs; the sliders shape the depth and width of the valley. What drives it is the ratio of churn value to acquired value (ρ): it crosses 1 at the bottom of the U, where losses and gains balance. The startup has reserve cash — the real question is whether it outlasts the valley until revenue is back. The horizontal axis is time (months); the U itself is the monthly revenue line.

Fixed parameters

The givens of the pivot: the legacy base and the cost of operating. These set where the U starts and how deep it can go.

Customers at pivot N₀

Old customer value $40/mo

Monthly burn (fixed cost) $/mo

Play with these

The shape of the pivot. The fall and the rise stay mirror images — move these and watch whether the U turns back up before the reserve runs out.

New customer value $40/mo

Reserve cash $1.5Menter in $K

Initial churn at pivot 14%/mo · sets depth

Months to trough τ = 8· sets width

Acquisition cost (CAC) $90/customer

Investor equity 20%stake

Exit valuation multiple 6×× ARR

The pivot first deepens the valley as the old base bleeds out, then the new product's ramp pulls revenue back up on a mirror-image path. The bottom is the month the churn-value / acquired-value ratio ρ crosses 1. Recovery to the pre-pivot level lands at twice the trough month (2τ).

The investor plays for the exit: their stake is a slice of an m× ARR valuation, made whole once it grows past the reserve.

Symmetry · half-width τ

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Trough depth (ρ = 1)

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Cash low point

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Full recovery (back to baseline)

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Investor exit payback · stake (equity × multiple × ARR) clears the reserve

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Monthly revenue (the U-curve) Cumulative reserve (cash) Cash zero (runs out) Pre-pivot level Bottom of the U (ρ=1) Recovery

Revenue (the U): R(t) = R₀ − D·(1 − ((t−τ)/τ)²) — a parabola, trough at τ, the rise mirrors the fall
Value ratio: ρ(t) = (2τ − t) / t — crosses 1 at the trough · Depth: D = p_old·N₀ · (1 − e^(−c₀·τ/2)) · (p_old / p_new)
Reserve: Cash(t) = R_res + Σ[ R(t) − CAC·acq(t) − burn ] · Investor stake e·m·12·R(t), repaid when ≥ R_res