# Accretion and discounting of annuity payments

Some theory first.
In the financial calculations for cash flow indication, i.e. consistent, extended in time payments the term rent is used.
Every single rental payment is called rent member. A special case, most commonly used in real life and the most developed one is annuity - a sequence of payments where all members are equal to each other and the payments take place at regular intervals between each other.
This type of annuity is often used in the consumer (automotive, mortgage) loans, insurance payments, disbursement against obligations.
Due to the fact that the value of money depends on the time (a hundred rubles is not the same as a hundred rubles year later), a simple summation of rental payments do not give any idea of profitability, present rent value, etc. It's necessary to use the special formulas Accretion and discounting. It is assumed that the payee has the option to reinvest the amount he received.

Let's review the mathematical apparatus of rent payments.
For simplicity, assume that the R size payments made at the end of each year, rent term is n years and the payee (the lessor or the creditor) pays the deposit 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 will rise to
$R_1=R(1+j)^{n-1}$, the second
$R_2=R(1+j)^{n-2}$ and so on.
Last payment will be equal to R. If you review these values in the reverse order that you can see that this is a Geometric progression, where the first member equals to R and the denominator is 1+j.
Thus, if the amount of loan servicing for the debtor or the lessee 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)
$S=\frac{R(1+j)^{n}-R}{j}$
Под текущей, или приведенной, или дисконтированной стоимостью ренты P понимается сумма, которая, будучи помещена в банк под те же проценты в начале рентного периода, даст такую же наращенную сумму в конце этого периода.
Для нашего примера с начислением процентов раз в год это будет формула
$S=P(1+j)^{n}$
Приравняв оба уравнения и выразив P, получим
$P=\frac{R}{j}(1-\frac{1}{(1+j)^{n}})$

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:
$S=R\frac{(1+j/m)^{mn}-1}{p*((1+j/m)^{m/p}-1)}$$P=R\frac{1-(1+j/m)^{-mn}}{p*((1+j/m)^{m/p}-1)}$

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 frist member of geometric progression will be equal to
$R(1+j)$
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 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

то из результатов можно почерпнуть следующую информацию
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)

Calculator:

#### Accretion and discounting of limited annuities

Rent member, rental payment
compound interest rate, used for accretion and discounting of payments that make up the rent
total time for which the rent is paid
Present (curent) rent value
Digits after the decimal point: 3
Annual rent payment

Single payment

Rent payments amount

Accreted amount

Present value

Rent term (years)

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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PLANETCALC, Accretion and discounting of annuity payments