Finding a Lease Payment from a Target IRR: Trial and Error and Bisection
Learn how to find a base monthly lease payment for a target annualized IRR, how bisection narrows the search, and how the model handles vehicle price, residual value, insurance, and vehicle tax. Amounts are shown in U.S. dollars.
How the Target-IRR Back-Solve Works
IRR (internal rate of return) measures investment return using cash inflows and outflows over the contract term. There is no simple closed-form formula for a monthly lease payment at a target IRR, so the calculator uses a numerical search: it calculates the IRR for a candidate payment, compares it with the target, and adjusts the candidate. Modeled cash flows include the vehicle price, monthly customer collections, the actual payment timing of insurance and vehicle tax, and the residual value at the end of the term.
The implied annual rate is recalculated from the resulting base lease payment and residual value. Compare the target IRR with the calculated annualized IRR to verify the result.
Overall approach: trial and error
Assume a lease payment, calculate the IRR for that payment, and compare it with the target. Adjust the payment and repeat until it approaches the target.
Specific algorithm: bisection method
Rather than guessing at random, bisection is a systematic numerical search that cuts the range containing the answer in half at each step.
- Set the payment range from $0 to the vehicle price.
- Use the midpoint as the candidate payment and calculate its annualized IRR.
- If the calculated IRR is above the target, lower the upper bound because the payment is too high.
- If the calculated IRR is below the target, raise the lower bound because the payment is too low.
- Repeat the midpoint calculation 200 times as the range narrows.
In short: Trial and error is the overall iterative approach to finding a target value, while bisection is the specific algorithm that halves the search range. They are complementary concepts, not opposites.
View the target-IRR payment cash-flow explanation (PDF)
The diagram labels are in Korean. If the PDF is not displayed, open it in a new tab.
Java, JavaScript, and Python implementation examples
Open the v0.3 source code for the same IRR-based lease-payment calculation in the language of your choice. Click the code to copy all of it.
Annualized IRR is calculated as monthly IRR × 12, not effective annual compounding.
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Calculation inputs
Estimated results
Why is the monthly bill higher than the base lease payment?
The base lease payment is the amount solved for to meet the target IRR. Insurance and vehicle tax are paid upfront by the lessor to insurers or tax authorities, then collected from the customer in monthly installments, so they are added to the base payment on the bill. Because payment timing differs from collection timing, the difference is reflected in the cash flows and IRR.
Insurance and vehicle tax payment schedules
Monthly cash flows
Periods advance one month at a time from the selected delivery month in the current year. Month 0 is the delivery date; (−) indicates cash paid by the lessor and (+) indicates cash received. “Customer bill” is the base lease payment plus insurance and vehicle-tax collections; each component is shown separately in net cash flow.
For illustration only. Amounts are displayed in U.S. dollars, but the insurance and vehicle-tax timing assumptions are carried over from the Korean version. They are not U.S. GAAP, accounting, or tax rules, and this is not a financing estimate or contractual payment.