DRV-004

Why an 8% flat loan is really a 14.13% loan

A flat rate charges interest on money you have already repaid. Converting it to an honest reducing-balance rate needs a numerical solve, and the common shortcut is too crude to publish.

The setup

Under a flat rate the interest is charged on the original principal for the full term, regardless of how much has been repaid. Under a reducing-balance rate it is charged only on what is still outstanding. The two conventions can describe the same cash flows at very different quoted numbers, which is what makes flat rates a marketing device.

The derivation

Step 1: Total the flat interest

Interest is the principal times the rate times the term, with no reference to repayment at all.

total interest = P * flat_rate * years

Step 2: Derive the instalment

The instalment is simply everything owed divided by the number of periods.

E = P * (1 + flat_rate * years) / n

Step 3: Solve DRV-001 backwards

Take that same instalment and ask what reducing-balance rate would produce it. There is no closed form for i, so the answer comes from bisection or Newton–Raphson on the residual.

solve for i: P * i(1+i)^n / ((1+i)^n - 1) - E = 0

solve numerically for i, then annualise

Checks that prove it is right

A formula you cannot test is a formula you have to trust. These take seconds, and they are what separate a derivation from a formula restatement.

n = 1 → The two conventions agree exactly

With a single instalment there is no repaid principal for the flat rate to keep charging on, so the distinction disappears.

T → 0 → The gap vanishes

The shorter the term, the less principal has been repaid at any point, and the smaller the overstatement. A one-year flat loan at 12% is about 21.5% reducing; a five-year one at 8% is 14.13%.

Worked example

₹5,00,000 at 8% flat over 5 years.

Flat interest is ₹2,00,000, so the instalment is ₹11,667 a month over 60 months. Solving the annuity formula backwards for that instalment gives a monthly rate of 1.1775%, or 14.13% a year. The familiar shortcut 2n/(n+1) × r_f gives 15.74% here. Close enough to make the point in conversation, and far too crude to publish as a result. Compute it; do not approximate it.

Bisection to a tolerance of 1e-12, cross-checked against an independent Python implementation.

What it assumes

The flat rate is applied to the original principal for the full stated term.
Instalments are equal and paid at the end of each period.

Where the formula stops being valid

Stating limits plainly is more useful than pretending there are none, and it is the item competitors most consistently omit.

Fees. Processing charges and mandatory insurance raise the effective cost above even the converted figure; the APR calculator handles those.
Prepayment. A flat-rate loan usually charges the full interest regardless of early repayment, so prepaying saves nothing, which is a larger difference than the rate conversion itself.