In financial calculations, the term "rent" refers to cash flows, that is, sequential payments stretched over time. A commonly used type of rent is an annuity, where payments are equal and occur at regular intervals, such as in consumer (car, mortgage) lending, insurance premiums, and bond payments.
Since the value of money changes over time, a simple summation of rental payments does not give an accurate representation of the profitability or present value of the rent. Therefore, special formulas for accretion and discounting are required to take into account the interest earned on payments that have already been received. This assumes that the payee has the ability to reinvest the amounts he receives.
Consider the mathematical apparatus of rental payments.
For simplicity, suppose that payments of size R are made at the end of each year, the rent term is n years, and the recipient of the money (the lessor or lender) put them to the bank at the annual compound interest rate j. The interest is paid to the deposit annually.
Thus, the first payment R obtained at the end of the first year by the end of the rent term will rise to
the second payment at the end of the second year by the end of the rent term will rise to
and so on.
The last payment will be equal to R. If you review these values in the reverse order, you can see that this is a Geometric progression, where the first member equals R, and the denominator is 1+j.
Thus, if the amount of loan servicing for the debtor or the lessee is equal to R multiplied by the number of payments, then for the lender or the lessor, the received or accreted sum will be equal (by geometric progression)
Actually, for crediting P is the sum of credit that the bank is trying to pay off, i.e., by giving you a loan at interest j bank calculates your annuity payment based on the desire to get the sum S as if he had just put the money to another bank by the same percentage for the entire term of your loan.
In real life, formulas become more complex as interest accrual j happens monthly and payments are made every month. If we estimate the amount of interest accruals for a year with m, which is usually 12, and also 1,2 or 4, and a number of rental payments with p, that is also usually 1, 2, 4 or 12, the formulas will look like this:
By the way, it is easy to notice that by equatingp and m to 1, we get the original formula.
In this consideration, rent payments were made at the end of the period - such rent is called ordinary or annuity-immediate. If the payments are made at the beginning of the term, such rent is called express or annuity-due. Formulas getting a little bit more complex. In fact, they can be obtained by repeating all the arguments above, taking into account the fact that for the express rent, the first member of geometric progression will be equal to
Actually, this is an additional factor 1 + j, and it appears in the formulas where needed.
The calculator is designed to be used universally, meaning you can enter the known parameters only, and the unknown parameters will be calculated. For example, if you took a loan for 15000 for 3 years at a 19% interest rate, you can use the calculator to compute the annual rent payment, single payment, rent payments amount, and accreted amount.
You should fill the form like this:
Annual rental payment - empty
Interest rate - 19
Rent term - 3
Number of payments - once a month
Number of interest accrual - monthly
Type of payments - ordinary
Accreted amount - empty
Discounted value of rent - 15000
Then you will get the following results:
Annual rent payment - 6598.084 (your expenses for the year of loan servicing)
Single payment - 549.840
Rent payments amount - 19794.251 (your overall expenses for loan servicing)
Accreted amount - 26405.829 (amount received by the bank for your credited 15000)
For those who don't want to bother themselves scrolling this article each time - the link for the calculator page
Accretion and discounting of limited annuities