Lease Calculator

You pay for the part you use up

A loan finances the whole of something. A lease finances only the part of it you consume.

Take a vehicle worth three million that will be worth six hundred thousand in four years. Over those four years you use up 2.4 million of it. A lease charges you for that consumption, plus interest, and at the end you either hand the thing back or pay the six hundred thousand to keep it.

Which is why lease payments are lower than loan payments on the same asset. You are financing less.

This calculator works out that payment. Five inputs, and the interesting one is the residual value, because it does two opposite things at once: it lowers what you pay each month and raises what the arrangement costs you overall. That is the section worth reading.

Five inputs

1. Product Value. The price of the asset today.
2. Down Payment. As a percentage of the value, not an amount.
3. Residual Value. As a percentage of today's value, this is what it is expected to be worth at the end of the term.
4. Annual Interest Rate. The rate the lessor charges.
5. Lease Term. Two boxes, years and months.

You get back the down payment, the residual value and the lease amount in currency, the monthly payment, the total of payments, the total interest and a total cost to own.

Two of those percentages come from the lessor rather than from you. The residual is their estimate of what the asset will be worth at the end, and it is set by them rather than negotiated. The rate is theirs too, though it is often quoted as a money factor in vehicle leasing rather than as a percentage, in which case multiply the money factor by 2,400 to get the annual rate.

The formula, and the term that makes it a lease

The payment comes from the standard amortising relation with one extra term:

Payment = [L × i × (1 + i)ⁿ - R × i] ÷ [(1 + i)ⁿ - 1]

where L is the lease amount after the down payment, i is the annual rate divided by twelve, n is the term in months, and R is the residual value.

The whole difference between a lease and a loan sits in that - R × i in the numerator.

Read it as what it does. The residual is the part of the asset you are not paying off. It stays outstanding for the entire term, so you are charged interest on it every month, but you never repay the principal. At the end it is either handed back with the asset or settled in one payment.

So each monthly payment covers two things: a slice of the depreciation you are consuming, and interest on the full outstanding balance including the residual you will never pay down.

That second part is the one people miss. You are paying interest on the residual value for the entire term, on money that is never reduced by a single payment.

Three million over four years

Value 3,000,000, down payment 10 percent, residual 20 percent, rate 4 percent, term 4 years.

Follow the money. You put 300,000 down, pay 2,371,969 over four years, and then pay 600,000 to keep the asset. That comes to 3,271,969 for something priced at 3,000,000, so the arrangement cost you 271,969 in interest.

If you hand it back instead, you paid 300,000 plus 2,371,969, which is 2,671,969, for four years of use of an asset that depreciated by 2,400,000. The gap is the same 271,969.

The residual cuts your payment and raises your cost

Here is the finding worth taking away from this page.

Same asset, same term, same rate, with only the residual varied:

The payment falls and the interest rises. Every time.

Going from a 20 percent residual to 40 percent takes 11,547 off the monthly payment and adds 45,723 to the total interest.

The reason is exactly the one from the formula section. A higher residual means more of the value stays outstanding for the whole term, accruing interest and never being repaid. You have made the debt cheaper per month by making more of it last longer.

Which reframes what a high residual means. Marketing presents it as good news, and it genuinely does lower the monthly figure, which is what most people are comparing. It also means you are renting more of the asset and buying less of it.

If you intend to hand the asset back at the end, a high residual is straightforwardly good: low payments and no interest in the total that you did not have to pay for the use you got. If you intend to keep it, a high residual costs you real money, and you should compare against an ordinary loan.

At zero residual it becomes an ordinary loan

Set the residual to zero and the extra term in the numerator disappears. What is left is the standard loan payment formula.

On our example, a zero residual gives a monthly payment of 60,963.45. Run the same 2,700,000 at 4 percent over 48 months through an ordinary loan calculation and you get 60,963.45, identical to the cent.

That is a useful check on the arithmetic and a useful piece of intuition. A lease with no residual is a loan. Everything a lease does differently comes from the residual, and the size of the residual is the size of the difference.

It also gives you a clean comparison. If you plan to keep the asset, run the tool twice, once at the lessor's residual and once at zero, and compare the two totals. On our example the loan-equivalent costs 226,245 of interest against the lease's 271,969, a difference of 45,724 for the privilege of a lower monthly payment.

Whether that is worth paying depends on what the cash flow relief is worth to you, and it is a decision that should be made with the number in front of you rather than on the payment alone.

What total cost to own actually assumes

The last output is labelled total cost to own, and it is built as the product value plus the total interest.

Which means it assumes you buy the asset at the end by paying the residual. That is one of the two things you can do, and it is worth being explicit because the other path produces a different figure.

• You keep it. Down payment plus all monthly payments plus the residual. On our example, 3,271,969. This is what the tool reports.
• You hand it back. Down payment plus all monthly payments only. On our example, 2,671,969, and you own nothing.

Both are legitimate outcomes of the same contract. Decide which you intend before you compare against alternatives, because comparing a hand-back lease against a purchase is comparing renting against owning.

Several things sit outside the calculation entirely. Insurance, which lessors usually require at a specified level. Maintenance, unless the contract includes it. Registration, road tax and any local duties. Excess mileage or usage charges on vehicle leases, which can be substantial. And end-of-term condition charges, where the lessor assesses wear and bills for anything beyond fair use.

That last one deserves attention on any hand-back. The residual value assumed a certain condition, and if the asset comes back worse than that, the shortfall is charged to you.

This is a planning estimate rather than a quote, and nothing here is financial advice.

Questions people ask

Why is a lease payment lower than a loan payment?

Because you only finance the part of the asset you use up. The residual value stays outstanding and is settled at the end rather than repaid monthly.

What is residual value?

The lessor's estimate of what the asset will be worth when the lease ends, entered here as a percentage of today's value. It is set by them rather than negotiated.

Is a higher residual better?

It lowers your monthly payment and raises your total interest, because more of the value stays outstanding for longer. Good if you plan to hand the asset back, expensive if you plan to keep it.

Am I paying interest on money I never borrow down?

Yes. The residual stays outstanding for the whole term and accrues interest every month without being reduced by any payment. That is why total interest rises as the residual rises.

What happens at a zero residual?

It becomes an ordinary loan. The formula reduces exactly to the standard amortising payment, which is a useful way to compare leasing against borrowing on the same asset.

Does total cost to own assume I buy it?

Yes. It includes paying the residual at the end. If you hand the asset back, your cost is the down payment plus the monthly payments only, and you own nothing.

My quote gives a money factor, not a rate.

Multiply the money factor by 2,400 to get the equivalent annual percentage rate. A money factor of 0.00167 is about 4 percent.

What is not included?

Insurance, maintenance, registration and road tax, excess mileage or usage charges, and any end-of-term condition charges. On a hand-back, condition charges can be significant, since the residual assumed the asset would come back in a certain state.

References

The payment relation is the standard amortising annuity formula with a balloon term for the residual, under which the residual balance accrues interest across the term without being amortised, as set out in university financial mathematics materials for the actuarial syllabus. The treatment of an annual rate as a periodic rate multiplied by the number of periods in a year, and the actuarial method under which the unpaid balance is increased each period by the finance charge earned and reduced by the payment made, follow Regulation Z. The tax treatment of lease payments and of assets held under lease, including the distinction between renting and acquiring, follows Internal Revenue Service small business guidance.

1. Miguel A. Arcones, Binghamton University, Manual for SOA Exam FM, Chapter 4: Amortization and Sinking Funds. https://people.math.binghamton.edu/arcones/exam-fm/sect-4-1.pdf
2. J. Robert Buchanan, Millersville University, Loan Repayment, MATH 372 Financial Mathematics I. https://sites.millersville.edu/rbuchanan/math372/LoanRepayment-handout.pdf
3. Consumer Financial Protection Bureau (CFPB), Regulation Z, Appendix J to Part 1026: Annual Percentage Rate Computations for Closed-End Credit Transactions. https://www.consumerfinance.gov/rules-policy/regulations/1026/j/
4. Internal Revenue Service, Publication 334: Tax Guide for Small Business. https://www.irs.gov/publications/p334