APR True Cost Calculator
Calculate the true APR of a loan when upfront fees are added to the amount financed.
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What this tool does
This calculator computes the true (effective) APR of a loan when upfront fees or charges are deducted from the cash advanced but the borrower still repays the full principal plus interest. It uses iterative present-value solving: the monthly payment is first computed from the advertised APR and nominal principal, then the internal rate of return is solved such that the present value of all payments equals the net cash actually received (principal minus fees). Outputs include true APR, monthly payment, and total amount paid over the term.
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Formula
How APR True Cost Calculator works
When a lender quotes an APR but also charges an upfront arrangement fee, documentation fee, or processing charge, the effective interest rate the borrower pays is higher than the advertised figure. This calculator reveals that true cost by computing the internal rate of return on the cash flow: the net amount the borrower receives (principal minus fees) versus the stream of monthly payments calculated on the full principal at the advertised rate.
The formula
The tool first calculates the monthly payment M using the standard amortisation formula on the nominal amount financed P at the advertised monthly rate radv = (1 + APRadv)1/12 − 1 over n months. It then solves for the true monthly rate rtrue that satisfies:
P – F = M × [(1 – (1 + rtrue)–n) / rtrue]
where F is the sum of all upfront fees. The tool uses binary search over the monthly rate, bounded between the advertised rate and a ceiling, iterating until convergence. The true APR is rtrue × 12 × 100.
Where this method is most accurate
The calculation assumes fixed monthly payments with no payment holidays, balloon payments, or variable rates. It treats all fees as deductions from the amount advanced at origination; fees paid separately in cash or added to the balance after origination are not modeled. The result is mathematically exact for the stated cash flow but does not include taxes, insurance, or jurisdiction-specific charges that may apply outside the loan contract itself.
What this tool does not do
This calculator does not incorporate optional payment protection insurance, early-repayment penalties, or late fees that depend on borrower behaviour. It does not account for compound frequency differences (daily versus monthly) in some credit products. It does not determine affordability, creditworthiness, or whether a given loan complies with consumer-credit regulations in any jurisdiction. The output is a pure present-value equivalence and does not constitute a recommendation to accept or decline financing.
Disclaimer
This tool is provided for educational and informational purposes only. All results derive exclusively from user-supplied inputs and published present-value equations. No output constitutes financial advice, and MotorMath makes no representation regarding the suitability, legality, or availability of any loan product. Users remain responsible for verifying terms with lenders and ensuring compliance with applicable consumer-credit law.
Questions
- Why is the true APR higher than the advertised rate?
- Because the borrower receives less cash (principal minus fees) but repays the full principal plus interest calculated on the gross amount. The effective rate on the net cash received is therefore higher.
- Does this calculator include early-repayment charges?
- No. It assumes all scheduled payments are made as agreed. Early-repayment fees or rebates of interest are not modeled.
- Can I use this for credit cards or revolving credit?
- No. The formula assumes a fixed-term, closed-end installment loan with equal monthly payments. Revolving credit has variable balances and different interest-accrual mechanics.
- What if the lender adds the fee to the loan balance instead of deducting it?
- If the fee is capitalized into the principal, enter the total financed amount (including fee) as the principal and set fees to zero. The advertised APR will then match the true APR.
- Is the result the same as the 'representative APR' in UK advertising?
- The representative APR quoted in UK car-finance advertisements must already include typical fees and be offered to at least 51% of applicants. This calculator shows what the effective rate becomes when a quoted APR excludes certain upfront charges; comparing the two can reveal whether all costs are disclosed in the headline rate.
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Sources & Methodology
The calculator applies the standard present-value-of-annuity formula to solve for the internal rate of return. Monthly payment M is computed via the amortisation equation at the advertised APR; the true monthly rate is then found by binary search such that the present value of n payments of M equals the net advance (principal minus fees). The true APR is then expressed as an effective annual rate, (1 + true monthly rate)^12 − 1, in line with UK and EU practice. This method is consistent with the United Kingdom Consumer Credit Act 1974 (as amended) definition of total charge for credit and the annual percentage rate of charge, and parallels the U.S. Truth in Lending Act APR calculation for closed-end credit.
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