Annuity
The current article describes the annuity loan scheme and exemplifies the calculation of the payment schedule for this scheme.
In the annuity scheme, a loan is repaid in equal monthly installments during a whole loan period. A total amount of a monthly installment (T) comprises a principal amount and interest amount.
Calculating the monthly installment
The monthly installment is calculated by the following formula:
(1)
where
T - total amount of the monthly installment,
A - loan amount,
IR - monthly interest rate (hereinafter “the interest rate”),
NP - number of installments.
It should be noted that the interest rate in formula (1) is taken in hundredth. Therefore, the interest rate expressed as a percentage must be divided by 100. For example, the interest rate of 1% corresponds to IR = 0.01.
Calculating the monthly interest and monthly principal
A repaid amount of the monthly installment covers the principal and interest. To split each monthly installment into these two payments, the system uses a recursive algorithm where each iteration corresponds to a next installment, i.e. a total number of iterations of the recursion equals NP. In each i-th iteration (each iteration corresponds to the i-th installment), the i-th interest amount and the i-th principal amount are calculated sequentially and the current balance is updated. Calculations are performed according to the following formulas:
(2)
(3)
(4)
where
Ii - interest amount of the i-th installment,
Pi - principal amount of the i-th installment,
Bi - total outstanding principal amount (hereinafter “the balance”) at the moment of the i-th installment,
i - integer index that corresponds to an installment number in the payment schedule and takes on values from 1 to NP sequentially.
Formulas (2)-(4) are used sequentially for calculating corresponding values of the i-th installment (i = 1, 2, ..., NP). It should be noted that an initial balance corresponds to a loan amount, i.e. B1 = A.
Thus, each month, different principal and interest amounts are repaid but the total amount of each monthly installment remains constant and equals T.
Example of the payment schedule calculated using the annuity scheme
Assume a loan is taken out in the amount of 1000 USD at the monthly interest rate of 1% for 3 months, i.e. A = 1000, IR = 0.01 and NP = 3.
Calculating the monthly installment
To calculate the monthly installment, formula (1) is used. Substituting corresponding values of the principal (A = 1000), monthly interest rate (IR = 0.01) and number of installments (NP = 3) into formula (1), we obtain the following value of the monthly installment:
If this data is entered into the system, the output value of the monthly installment (T) will equal 340.02 for some months and 340.03 for other months. This difference is caused by the rounding used in the system.
Calculating the interest and principal
Having calculated the total amount of the monthly installment (T = 340.02USD), we use formulas (2)-(4) to calculate the interest ( Ii ), principal (Pi), and monthly balance (Bi). Calculations are performed with the machine precision and output values contain only two last digits after the point, taking into account the rounding used:
The final balance equals zero, i.e. B4 = 0.00 USD.
As a result, the following payment schedule is generated:
Installment Number | Principal | Interest | Total |
1 | 330.02 | 10.00 | 340.02 |
2 | 333.32 | 6.70 | 340.02 |
3 | 336.66 | 3.37 | 340.02 |
Total | 1000.00 | 20.07 | 1020.07 |