Methodology

What the model actually computes, in the order it computes it. This is the full description of the mechanism — enough to reimplement it, disagree with it, or find a mistake in it.

The one rule

A dollar you do not hand to a dealer stays invested. So a dollar paid in month t does not cost you a dollar; it costs you the dollar and the growth it never earned between then and the end of the horizon.

cost of a dollar paid at month t = 1 × (1 + r)^{N − t} r = monthly market return N = horizon in months

Everything follows from this. It is why paying cash is penalised, why a low-rate loan is rewarded for leaving your capital invested, and why the timing of a payment matters as much as its size. Maintenance, insurance, registration and fees are outflows too, so they lose their market growth on exactly the same footing as the payment itself.

Income is identical under every option, so it cancels and is never modelled.

What is scored

net cost = Σ_t outflow_t × (1 + r)^{N − t} − what the car is worth at the horizon + anything still owed at the horizon

Lower wins. The three terms are the whole comparison: what you paid and when, what you have left, and what you are still on the hook for. An option that ends with no car is not being punished for that — it simply has nothing to subtract.

On the shipped scenario, over 8 years: cash comes to $101,727 and the loan to $101,913. A $186 gap on a six-figure decision is a tie, and the page says so rather than declaring a winner.

Three different rate conventions

The model carries three rates and they do not mean the same thing. Confusing them is the most common way a car comparison goes wrong.

The market return is geometric. 7% a year means 7% a year after compounding, so the monthly rate is the twelfth root, not a twelfth: r = (1 + 0.07)^1/12 − 1 = 0.565%. Dividing by twelve would quietly pay you 7.23%.
The loan APR is nominal. US auto loans quote a nominal annual rate compounded monthly, so the monthly rate really is APR / 12. 6.9% becomes 0.575% a month, and the effective annual rate is a little over 7.1%.
The lease uses a money factor. Not a rate at all — a multiplier applied to a sum, described below. Multiply it by 2400 for a comparable APR: the shipped 0.00225 is about 5.4%.

Using one convention for all three is worth hundreds of dollars in a close comparison, which is the only kind worth running.

The depreciation curve

Depreciation is the largest number in car ownership, and it is a curve, not a line. The model uses a steep first year and a gentler decline after:

factor(age) = (1 − d₁)^{min(age, 1)} × (1 − d₂)^{max(age − 1, 0)} d₁ = first-year drop (20% shipped) d₂ = annual drop after (13% shipped)

Two adjustments sit on top. Miles above the baseline reduce the value by a fixed amount per mile. And the result is floored at a share of the new price — 8% shipped — because a running car is never worth nothing, and an unfloored exponential would eventually say it was.

Where you get on this curve is the single largest lever in the model. It is larger than the APR and larger than the lease rate, which is why buying used changes the answer more than shopping for finance does.

What the car is worth to you

Two subtleties, both deliberate.

The curve runs from the car's age, not from your purchase date. Buy a three-year-old car and year four of depreciation is what you get, not year one. That is the entire financial case for buying used: somebody else has already paid the steep part.

What you paid does not enter the resale value. The car is worth what a car of that age is worth. Negotiate well and the bargain stays a bargain rather than being depreciated away; overpay and the model does not politely forget.

What you can actually realise is net of the cost of selling:

asset at horizon = value(age) × (1 − selling cost)

On the shipped scenario the car is worth $12,978 at the 8-year horizon, and $12,588 after the 3% cost of selling it.

The five options

Cash. Price, tax and fees, all at delivery.
Loan. Down payment at delivery, then a level payment for the term. Tax and fees are either financed or paid at signing, your choice.
Lease, return, re-lease. A new contract every term for the whole horizon, with the replacement car's price growing at inflation.
Lease, then buy out for cash.
Lease, then finance the buyout.

Cash and loan are the same code path. Set the down payment to the full price and the loan collapses to the cash purchase exactly — not approximately — because the financed amount becomes zero and the whole sum lands in month 0.

Note that a 0% loan is not the cash case: it is strictly better, because the money stays invested while you pay it off. The engine's own tests assert both of these.

The two buyout options need a full lease term to exist. On a horizon shorter than one term the model returns three options rather than inventing a buyout that could not happen.

Lease arithmetic

A lease is rent for the depreciation, plus interest on the money the bank has tied up:

residual = MSRP × residual% adjusted cap = negotiated price + acquisition fee − cash down depreciation fee = (adjusted cap − residual) / term rent charge = (adjusted cap + residual) × money factor payment = depreciation fee + rent charge (+ tax)

The residual is a percentage of MSRP, not of what you negotiated. This is the detail that makes leases behave differently from purchases: a discount off sticker lowers the cap cost but leaves the residual untouched, so every dollar you negotiate comes straight off the depreciation you are renting. A discount helps a lease more than it helps a purchase.

The rent charge is levied on the cap cost plus the residual, not on the balance outstanding — the bank owns the whole car throughout, not just the part you are using.

Due at signing is the cash down, the first month's payment, the acquisition fee if it is not capitalised, and the usual paperwork. On the shipped scenario that is $4,511 against a $611 monthly payment.

Why a lease is carried as a liability

A lease cannot be cancelled. Signing one commits you to every payment in the term on day one. So at any month in between, the model carries the rest of the contract as a debt you already owe, at present value:

liability(m) = Σ_{k > m} payment / (1 + r)^{k − m} + (disposition fee + mileage bill) / (1 + r)^{end − m}

This is why the lease lines on the chart start high and step up again at each new signing, and it is the honest way to compare a commitment against a purchase you could unwind tomorrow.

It also means the months in between are not fair comparison points. The dots on the chart mark the end of a lease term — the only moments where you could stop with nothing further owed. Compare at the dots, or use a horizon that is a whole number of terms. The model says so in its warnings when your horizon is not.

Running costs

Maintenance compounds on the car's age, not the calendar. Re-leasing keeps resetting the car to nearly new, which is where a lease earns back part of its cost, and it is what an old car gives up:

annual = year-1 cost × (1 + growth)^{max(age − 1, 0)}

Under factory warranty the figure is scaled down rather than zeroed, because a warranty covers failures, not tyres and servicing. The curve is also capped at a fraction of what the car is currently worth: left alone an exponential eventually spends five figures a year maintaining a four-figure asset, which is a state no owner would stay in. The cap binds only well past the ages the growth rate is calibrated for, so it changes nothing in ordinary horizons and keeps the far end finite instead of absurd.

Insurance is charged at the leased rate while leased and the owned rate while owned — leases mandate full coverage and low deductibles, so the two genuinely differ. Registration is flat. All three grow with inflation.

Reading the chart

The line is not cumulative spending. At every month it shows what the option has cost you so far, on the same basis as the final answer:

cost to date(m) = future value of everything paid up to m − what the car is worth at m + what you are still committed to at m

The value at the final month is exactly the net cost in the table above it, so the chart and the verdict are the same number — one measured continuously, one at the end.

Equivalent monthly

Net cost is a future value, which is hard to feel. The equivalent monthly figure is the level payment that would accumulate to the same future value over the same horizon:

equivalent monthly = net cost ÷ [ ((1 + r)^N − 1) / r ]

It is a way to read the answer, not a bill anybody sends you. On the shipped scenario the winning option is $801 a month equivalent — well above its $779 loan payment, because the loan payment is not the cost of the car.

The break-even return

The market return is the assumption the whole comparison turns on, and nobody knows it. So rather than asking you to trust one number, the model solves for the one that would change your mind: the return at which the top two options swap places, found by bisection between −2% and 30%.

On the shipped scenario that is 7.1% — against a 6.9% loan APR. That closeness is the intuition check: borrowing at 6.9% to stay invested is worth it precisely when you beat roughly 6.9% in the market. If the break-even sits far from your honest expectation, the answer is robust. If it sits on top of it, the honest answer is that it is too close to call.

What the model checks

Rather than quietly producing a number for a scenario that does not make sense, the model reports the conditions that would mislead you:

A horizon that is not a whole number of lease terms, so the comparison lands mid-contract.
A negotiated price above MSRP, which inverts the lease arithmetic.
A residual far from the car's projected market value on the same day — the thing that decides whether a buyout is a bargain or a mistake.
A down payment exceeding the price, collapsing the loan to a cash purchase.
Buying used, which makes the comparison lease-new-against-buy-used rather than like-for-like.
A maintenance ceiling that is actually binding within your horizon.

What is not modelled

Fuel and tyres, which are near enough identical across the options and would cancel. Mid-lease exits and lease transfers. Gap insurance premiums. Business use and EV tax treatment. Lot-level capital gains tax — investment tax is applied as a flat drag on the return instead. Wear-and-tear charges at lease return, which are a negotiation rather than an arithmetic.

And the things that carry no dollar figure at all: whether you want a new car every three years, whether you will keep this one for fifteen, and what your money is actually for. Those decide more of this than the APR does.