To save money from inflation, citizens often place them on deposits with banks. But the principle of calculating interest on deposits is not known to all depositors. The process of moving from the present value of money to its future value is called accretion. The amount of future income depends on the term of the deposit and the interest calculation scheme. In banking, simple and compound interest is used.
Calculation of simple interest
Simple interest is used in lending financial transactions with a duration of up to one year. When using this scheme, interest is accrued once, taking into account the unchanged calculation base. For calculus, the following formula applies:
FV = CFo × (1 + n × r), where FV is the future value of funds, r - interest rate, n - term of accrual.
In the event that the duration of the loan operation is less than a calendar year, then the following formula is used for the calculation:
FV = CFo × (1 + t / T × r), where t is the duration of the operation in days, T is the total number of days in a year
Compound interest calculation
When using a complex rate, the annual income in each period is calculated not from the original amount of the deposit, but from the total accumulated amount, including also the previously accrued interest. Thus, as interest is accrued, interest capitalization occurs.
Suppose a depositor has placed 1,000 rubles on a bank deposit at 6% per annum. Determine how much will be accumulated over two years if interest is calculated according to a complex scheme
Interest income = interest rate × initial investment = 1000 × 0.06 = 60 rubles
Thus, by the end of the 1st year, the amount will be accumulated on the deposit:
FV1 = 1000 + 60 = 1060 rubles = 1000 × (1 + 0.06)
If you do not withdraw money from the account, but leave it until next year, then at the end of the 2nd year the amount will be accumulated on the account:
FV2 = FV1 × (1 + r) = CVo × (1 + r) × (1 + r) = CVo × (1 + r) ^ 2 = 1060 × (1 + 0.06) = 1000 × (1 + 0, 06) × (1 + 0, 06) = 1123.6 rubles
The following formula is used to calculate compound interest:
FVn = CVo × FVIF (r, n) = CVo × (1 + r) ^ n
The compound interest multiplier FVIF (r, n) shows what will be equal to one currency unit in n periods at a certain interest rate r.
In practice, very often, for a preliminary assessment of the effectiveness of the interest rate, the period of time required to double the initial investment is calculated. The number of periods in which the original amount will approximately double is 72 / r. For example, at a rate of 9% per annum, the initial capital will double in approximately 8 years.
Comparison of simple and complex interest calculation schemes
To compare different schemes for calculating interest, it is necessary how the accumulation factors change for different values of the indicator n.
If n = 1, then (1 + n × r) = (1 + r) ^ n.
If n> 1, then (1 + n × r) <(1 + r) ^ n.
If 0 <n (1 + r) ^ n.
Thus, if the loan term is less than 1 year, then it is beneficial for the lender to use a simple interest scheme. If the period for calculating interest is 1 year, then the results for both schemes will coincide.
Special cases of interest accrual
In modern banking practice, sometimes there are contacts that are concluded for a period that differs from a whole number of years. In this case, two options for accrual can be used:
1) according to the compound interest scheme
FVn = CFo × (1 + r) ^ w + f;
2) according to the mixed scheme
FVn = CFo × (1 + r) ^ w × (1 + f × r), where w is an integer number of years, f - fractional part of the year.
Suppose a depositor places 40,000 rubles on a deposit for a period of 2 years 6 months at 10% per annum, interest is calculated annually. How much the depositor will receive if the bank calculates interest on a complex or mixed scheme.
1) Calculation according to a complex accrual scheme:
40,000 × (1 + 0, 1) ^ 2, 5 = 50,762, 3 rubles.
2) Calculation on a mixed accrual scheme:
40,000 × (1 + 0, 1) ^ 2 × (1 + 0, 5 × 0, 1) = 50,820 rubles.
For some deposits, interest is accrued more often than once a year. In such cases, the following formula applies:
FVn = CFo × (1 + r / m) ^ m × n, where m is the number of charges per year.
Determine the future value of 7,000 rubles invested for 3 years, at 7% per annum, if interest is charged quarterly?
FV3 = 7000 × (1 + 0.07/4) ^ 3 × 4 = 8620.1 rub.
Please note that when concluding an agreement on a deposit in a bank, you must remember that most often the documents do not use the terms "simple" or "compound" interest. To indicate a simple accrual scheme, the contract may contain the phrase “interest on the deposit is charged at the end of the term”. And when using a complex scheme, the contract may indicate that interest is charged once a year, quarter or month.
