The schedule identity
Let B be principal at the beginning of an equal period, i the rate for that period and A the payment at the end. Accrued interest is B × i. Principal repayment is A − B × i, and the new balance is B × (1 + i) − A. If the payment does not cover interest, principal does not decline under this model. A real contract may use a more complex accounting and payment order; do not automatically insert daily accrual, arrears, insurance or an unusual payment date.
For n equal payments at a fixed rate, the annuity formula is A = P × i / (1 − (1 + i)^(−n)), where P is initial debt. It follows from the condition that the balance after the final payment is zero. At zero interest, the payment is P / n. An annual rate may be divided by the number of months only when that conversion belongs to the model. Here the rate is stated directly per educational month, so annual-rate conversion is intentionally excluded.
An annuity with fictional figures
Debt is 120,000 roubles, the rate is 1% per educational month, the term is 12 months and there are no additional costs. The equal payment, without intermediate rounding, is approximately 10,661.85 roubles. In month one, interest is 1,200, principal is about 9,461.85 and the balance is about 110,538.15. In month two, interest is about 1,105.38, principal about 9,556.47 and the balance about 100,981.67. The payment is unchanged, but principal declines faster because the interest base is shrinking.
Total interest over the high-precision schedule is approximately 7,942.26 roubles. This is not the printed two-decimal payment multiplied by the number of months: rounding can create a small difference. A real cash schedule must agree its rounding and final correcting payment. The key checks are that principal repayments restore the initial debt and the ending balance reaches zero within the chosen precision.
The differentiated-payment version
With equal principal repayment, 10,000 roubles is returned each month. Interest is charged on the unpaid balance. The first payment is 11,200, the last is 10,100 and total interest is 7,800. In this stated model, interest is lower than under the annuity because principal is repaid faster at the beginning, but the first payment is higher. This is not a universal conclusion about every loan: comparison requires the same amount, term, rate, dates and all costs, not merely similar scheme names.
How to read an offered schedule
Find separate columns for balance, interest, principal and additional payments. Check actual dates, payment count and rate-change terms. Do not confuse interest on the balance with charging the same rate on the original amount for the whole term. Do not call the whole monthly payment interest expense. A smaller first payment does not prove a lower total cost. Rights, disclosure requirements and individual ability to perform obligations require separate review under the applicable jurisdiction.
Check changed assumptions
If the rate changes or an extra repayment is made, do not continue the old equal payment without solving the new problem. Different educational scenarios are possible: the term changes while payment stays fixed, or payment changes while term stays fixed. This does not assert a right to choose either option in a real contract. For a control table, state the new event, reduce the relevant balance and recheck the ending zero. Interest already paid is not recalculated merely because the future base became smaller; different contract mechanics require their own support.
See [simple and compound interest](/guides/simple-and-compound-interest/) for fixed versus changing bases. For comparing published percentages, establish the meaning of [effective rate](/guides/effective-rate/): this payment is not a regulatory total-cost indicator.