# Accretion and discounting of annuity payments

This online calculator calculates the annuity payment parameters, including accreted amount and discounted value of rent.

In financial calculations, the term **rent** is used to denote cash flows, that is, sequential payments stretched over time. A particular case, most often used in real life and the most developed, is an annuity - a sequence of payments, where all payments are equal to each other and occur at equal intervals of time one after another. This type of rent is quite often used in consumer (car, mortgage) lending, insurance premiums, and bond payments.

Due to the fact that the value of money depends on time (one hundred dollars now is not the same as one hundred dollars in a year), a simple summation of rental payments does not give an idea of the profitability, present value of the rent, etc. The use of special formulas for accretion and discounting is required. 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 getting more complex. Usually, interest accrual *j* happens monthly and the 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 equating*p* 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 in the right places.

The calculator on which this article is written calculates rent parameters. It's made universally - which means you don't have to fill all the fields - you can enter the known parameters only, the unknown parameters will be calculated.

Here is an example - you've taken a loan for 15000 for 3 years at a 19% interest rate.

If you enter the following data:

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 can have the following information:

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

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