Write-Off Shortfall Over a Finance Term

How this is worked out

Why there is a hole at all

A written-off vehicle is settled at what it is worth. The finance is settled at what is owed. Those are two different schedules with nothing in common: a balance falls on a payment plan, and a value falls geometrically from the day it is bought.Where the second is below the first, somebody has to pay the difference, and that somebody is the holder. The regulator that defined the product covering this gap describes it in one line — cover for the difference between a vehicle’s purchase price or outstanding finance and its current market value, in the event it is written off before finance has been repaid — and publishes nothing a consumer could size it with.

The rate is yours, and here is what is actually published

A depreciation path is the one thing this calculation needs that nobody can look up for their own car. There is a citable non-commercial basis and it is worth knowing precisely: the US Bureau of Economic Analysis derives depreciation rates from observed used-asset resale prices, and states its own method — geometric patterns are used because the available data suggest they more closely approximate actual profiles of price declines than straight-line patterns.What it publishes, and what it does not, is the whole reason this page has no default.

The arithmetic

Write P for what is financed, N for the term in months,r for the monthly rate, d for a stated annual depreciation rate andm for a month. Two schedules and one subtraction:

balance(m) = P × ( (1+r)^N − (1+r)^m ) ÷ ( (1+r)^N − 1 )        r above zero
balance(m) = P × ( 1 − m ÷ N )                                   r of zero

value(m)   = price × (1 − d)^(m ÷ 12)

gap(m)     = balance(m) − value(m)

The balance is a closed form rather than a schedule stepped one payment at a time, and that is a correctness decision rather than a preference.An iterated balance does not finish a term at zero in ordinary double precision — it finishes a few ten-billionths above it, and this model reads a positive gap as a shortfall. A version that iterated would have reported an exposure opening in the final month of every interest-bearing agreement that had none.

Three quantities, and why one would be useless

The answer is the month the gap opens, the largest it gets, and the month it closes. A single figure at a single date hides that the exposure has a beginning and an end — and on ordinary arrangements it has both.On a 30,000 vehicle financed in full over sixty months at 9.9 per cent, against 19.25 per cent a year, the gap opens in month 1, peaks at 896 in month 12 and closes in month 26. It exists for fourteen months of a five-year agreement and not for the other forty-six, and “am I exposed” is not a question that arrangement has an answer to.

Why two rates and never one

The depreciation rate is the assumption nobody can settle in advance, and it is also the one that decides whether an arrangement is exposed at all. So this page takes two and publishes a profile at each, and there is no mode that takes one. Where the band straddles zero — a hole at the faster rate and none at the slower — that is the most useful shape this calculation produces, and it is exactly the shape a single figure destroys by choosing for you.

What closes it

The deposit, and not the cover. The same vehicle, the same term, the same rate and the same depreciation, with a ten per cent deposit, produces no shortfall at any month of that sixty-month agreement. A deposit shrinks the balance and leaves the value where it was, so it is the whole of the equity an agreement starts with, and raising it never raises the gap at any month. That is a property of the two schedules rather than a result about one example.

The limits this calculation imposes on itself

• The depreciation rates are yours and this page publishes none. The federal basis it names is a cohort profile rather than one vehicle’s, and the agency states that a cohort profile is more accelerated than a single asset’s — so a rate taken from it describes a class rather than a car.
• That basis is a real rate and everything here is nominal. Using it directly assumes used-vehicle prices do not move with inflation over the term. That is a stated assumption rather than a hidden one, and it is the reason the reference is cited rather than adopted.
• The finance is a level-payment loan and nothing else. A balloon payment, a personal contract purchase, an early-settlement rule and the Rule of 78 are four different amortisations, and none of them is this one with a different rate.
• Nothing is rolled in. Negative equity carried forward from a previous agreement adds to the balance from month zero and is the single largest thing that makes a real shortfall bigger than this page’s.
• No insurer is modelled. A settlement is an insurer’s own assessment of market value, less its own deductions, on its own timetable — this page computes a difference between two schedules you described and has no view on what anybody would pay.

What is not modelled

• what an insurer would actually pay — its own valuation, its deductions, its excess and its timetable
• the price of cover, and whether any is worth buying
• a balloon payment, a personal contract purchase, an early-settlement rule or the Rule of 78 — four different amortisations, and none of them is this one with a different rate
• any charge, arrangement fee, option-to-purchase fee or interest rebate
• negative equity rolled in from a previous agreement, which makes a real gap larger than this one from month zero
• inflation in used-vehicle prices, which is assumed to be nothing
• a vehicle that is written off, repaired, stolen and recovered, or declared a total loss on any particular basis
• the identity of any vehicle, insurer, lender or broker
• tax, in any jurisdiction

The distinction the reference basis turns on — that a rate quoted in the absence of inflation and a rate a price actually moves at are different quantities — is taught in the explainer:nominal return is not real purchasing power.