3. In recent years, the use of mortgage loans by young families has become quite popular. Consider the possibility of obtaining a mortgage with a constant, fixed interest rate. Assume that the repayment of such a loan is made through equal (annuity) payments at the end of each payment period stipulated by the contract. Suppose that, according to the terms of the mortgage agreement, partial early repayment of the loan is possible without penalty for any amount up to the outstanding loan balance on the date of partial early repayment.

Let's assume that a mortgage borrower has additional funds equal to the current monthly payment in the middle of one of the payment periods, and he decides to use these funds to pay off the debt in addition to the annuity payment. Should the borrower immediately (i.e., in the middle of the payment period) make a partial early repayment, or should he wait until the end of the period and make the payment together with the annuity payment? For simplicity, assume that in the second case, the additional funds will remain on the current account with zero yield until the end of the period. Provide an intuitive explanation and support your reasoning analytically.

Hint: To answer the question posed in the assignment, you will need to use additional information about mortgage lending and the allocation of funds for interest and principal repayment in the case of partial early repayment.
Solution: If the borrower immediately directs the funds to a partial prepayment, in this case, part of these funds will go towards paying the accrued interest for half the period, and the rest will go towards repaying the principal. As a result, due to the reduction in the principal amount, less interest will be accrued on the remaining debt for the second half of the period. At the end of the period, the regular annuity payment will be divided into the payment of accrued interest for the second half of the period and the repayment of the principal.

If the borrower waits until the end of the period, then the entire additional amount will go towards repaying the principal. Since the funds will be held in a current account with zero yield, this decision will not result in any additional effects.

Let's write this analytically.

Let \( \mathrm{T} \) be the amount of money the borrower pays to the bank monthly, \( \mathrm{S} \) be the debt amount at the beginning of the period, and \( \mathrm{r} \) be the interest rate for the period.

In the first case, with an additional payment of \( \mathrm{T} \) in the middle of the period, the debt amount will decrease by \( T - 0.5 r S > 0 \), so the interest for the second half of the period will be \( 0.5 r(S - T + 0.5 r S) \), and the debt amount at the end of the period will be \( S - T + 0.5 r S - T + 0.5 r(S - T + 0.5 r S) = S - 2T + r S - 0.5 r T + (0.5 r S)^2 \).

If the additional payment is made at the end of the period, the remaining debt will be \( S - T + r S - T = S - 2T + r S \), which is greater than in the first case.

Thus, all other things being equal, it is more advantageous to pay immediately.

## Criteria:

5 points - intuitive justification;

10 points - intuitive justification + justification with a specific example;

15 points - intuitive justification + justification in general form.