Reducing Balance Loans

A flat rate loan calculates interest on the original principal for the entire loan term, regardless of repayments made.

$$\text{Total Interest} = P \times r_{\text{flat}} \times n$$

Example: $20,000 car loan at 6% flat rate over 5 years.

Total interest $= 20000 \times 0.06 \times 5 = \$6{,}000$. Total repayment $= \$26{,}000$. Monthly $= 26000 \div 60 = \$433.33$.

Flat rates are common in car finance, personal loans, and consumer credit. They are banned or heavily regulated in many countries because they mislead borrowers.

A flat-rate loan charges simple interest on the full original principal for the whole term. True monthly repayment = (P + I)/n. Because you owe less as you pay down the loan but interest stays fixed, the effective rate is much higher.

Pause, copy the flat-rate repayment formula: monthly repayment = (P + I_total) / n, and explain why the effective rate is higher: you still pay interest on the original principal even after repaying most of it into your book.

A flat-rate loan charges simple interest on the full original principal for the entire term, you keep paying as if you owe the whole amount even after repaying half. A reducing balance loan charges interest only on the outstanding balance each period: interest = r × current balance. As you repay principal, the interest component falls each period, so total interest paid over the life of the loan is lower than a flat-rate loan at the same stated rate.

A reducing balance loan charges interest only on the outstanding balance each period. As you repay the principal, your interest bill falls.

Same example at 8% reducing balance (compounded monthly) over 5 years:

$r = 0.08 \div 12 = 0.00\overline{6}$, $n = 60$.

$$M = 20000 \times \frac{0.00\overline{6}}{1 - (1.00\overline{6})^{-60}} = \$405.53 \text{ per month}$$

Total repaid $= 405.53 \times 60 = \$24{,}332$. Total interest $= \$4{,}332$.

A reducing balance loan charges interest only on the outstanding balance each period. As the principal falls, so does the interest component, this is how mortgages and most bank loans actually work.

Pause, copy the reducing balance interest formula: interest for each period = r × opening balance for that period, and note the consequence: as the balance falls, the interest component of each repayment falls, making the loan fairer than a flat-rate loan into your book.

Reducing balance loans charge interest = r × current balance each period, so the true rate equals the stated rate. For flat-rate loans, the stated rate understates the true cost. To find the true effective rate of a flat-rate loan: either use the approximation (true rate ≈ 2 × flat rate), or solve for r by setting the present value of all repayments equal to the original loan amount using the PV annuity formula.

To compare a flat rate loan fairly with a reducing balance loan, convert the flat rate to an approximate reducing balance rate:

$$r_{\text{reducing}} \approx \frac{2 \times n \times r_{\text{flat}}}{n + 1}$$

where $n$ is the number of repayment periods.

Example: 6% flat rate over 5 years (60 monthly periods).

$$r_{\text{reducing}} \approx \frac{2 \times 60 \times 0.06}{61} = \frac{7.2}{61} \approx 11.8\%$$

The 6% flat rate is equivalent to approximately 11.8% reducing balance nearly double the advertised rate.

To find the true annual rate of a flat-rate loan: use the reducing balance formula or table to find the rate r such that the present value of repayments equals the original loan. The true rate is typically close to twice the flat rate.

Pause, copy the true-rate rule of thumb (true effective rate ≈ 2 × stated flat rate) and note when to use the full PV method: when exact precision is required, set PV = M × [1 − (1+r)^(−n)] / r and solve for r into your book.