Understand the Compound Interest with Periodic Deductions with the below given solved problems. In the periodic compound interest, the interest rate is applied and generated at intervals. Periods may vary as annual, bi-annual, monthly, and weekly. The interest amount is added to the principal amount. Periodic Compound Interest Formula, Solved Examples, etc. are provided with a detailed explanation in 10th Grade Math Compound Interest articles.
Read More,
- Compound Interest with Growing Principal
- Introduction to Compound Interest
- Compound Interest as Repeated Simple Interest
Periodic Compound Interest Formula & Derivation
Basically, a periodic interest rate is considered the rate which can be charged on a loan over a specific period of time. Usually, lenders quote interest rates on annual basis. The periodic interest rate is defined as the annual interest rate which is divided by a number of compounding periods.
Effective Annual Rate is considered as that actually gets paid. If the interest is compounded in the year, then the EAR is higher than the mentioned rate.
Suppose that an interest rate like 10% is chopped into “n” periods, compounding every time. With the compound interest formula, we can compound “n” periods using
FV = PV (1 + r)n
From the above formula, the interest rate will not be ‘r’ because it is chopped into “n” periods like r/n
Now, the formula is
FV = PV (1 + (r/n)) n
where,
FV = Future Value,
PV = Present Value,
r = annual rate interest
n = number of periods within the year
Compound Interest with Periodic Deductions Examples with Solutions
Compound Interest with Periodic Deductions is used in various sectors like savings accounts, recurring deposits, fixed deposits, retirement funds, reinvested dividend stocks, etc. Check the step-by-step procedure to solve the problems in the following section.
Question 1. Rohan borrows $ 20,000 at a compound interest of 10% per annum. If he repays $ 4000 at the end of every year, find out the sum at the end of the third year?
Solution: For the first year: Principal = $ 20,000,
Rate = 10%,
Time = 1 year.
Therefore, Interest = $ P * T * R / 100
= $ 20,000 * 1 * 10 / 100
= $ 2,00,000 / 100
= $ 2,000
Therefore, the loan amount after 1 year = Principal + Interest
= $ 20,000 + $ 2000
= $ 22,000
Rohan pays back $ 4,000 at the end of first year.
Therefore, the new principal at the second year beginning = $ 22,000 – $ 4,000
= $ 18,000
For the second year:
Principal = $ 18,000
Rate = 10%
Time = 1 year
Therefore, Interest = $ P * T * R / 100
= $ 18,000 * 10 * 1 / 100
= $ 1,80,000 / 100
= $ 1800
Therefore, the amount of loan after 2 years = Principal + Interest
= $ 18,000 + $ 1,800
= $ 19,800
Rohan pays back $ 4,000 at the end of second year.
Therefore, the new principal at the beginning of the third year = $ 19,800 – $ 4,000
= $ 15,800
For the third year,
Principal = $ 15,800
Rate = 10%
Time = 1 year
Therefore, Interest = $ P * R * T / 100
= $ 15,800 * 10 * 1 /100
= $ 1,58,000 / 100
= $ 1,580
Therefore, the amount of loan (outstanding sum) after 3 years = Principal + Interest
= $ 15,800 + $ 1,580
= $ 17,380
Therefore, the answer is $ 17,380
Question 2. Daniel invests $ 15,000 at the beginning of each year in the bank and earns 8% annual interest, compounded at the end of the year. Find his balance in the bank at the end of three years?
Solution: For the first year, Principal = $ 15,000,
Rate = 8%,
Time = 1 year.
Therefore, Interest = $ P * T * R / 100
= $ 15,000 * 1 * 8 / 100
= $ 1,20,000 / 100
= $ 1,200
Therefore, the amount at the end of 1 year = Principal + Interest
= 15,000 + 1,200
= 16,200
Daniel deposits $15,000 at the beginning of the second year.
Therefore, the new principal for the 2nd year = $ 16,200 + $ 15,000
= $ 31,200
For the second year:
Principal = $ 31, 200
Rate = 8%
Time = 1 year
Therefore, Interest = $ P * T * R / 100
= $ 31,200 * 1 * 8/100
= $ 2,49,600 / 100
= 2,496
Therefore, the amount of loan after 2 years = Principal + Interest
= $ 31,200 + $ 2,496
= $ 33,696
Daniel deposits $ 15,000 at the beginning of third year.
Therefore, the new principal at the beginning of the third year = $ 33,696 + $ 15,000
= $ 48,696
For the third year,
Principal = $ 48,696
Rate = 8%
Time = 1 year
Therefore, Interest = $ P * R * T / 100
= $ 48,696 * 8 * 1 /100
= $ 3,89,568 / 100
= $ 3,895.68
Therefore, the amount at the end of 3rd year = Principal + Interest
= $ 48,696 + $ 3,895.68
= $ 52,591.68
Therefore, the balance in the bank at the end of 3 years will be $ 52,591.68.
Question 3. Noel lends $ 4000 to Emily at an interest rate of 10% per annum. It is compounded half-yearly for 2 years period. Calculate the amount Noel gets after the period of 2 years from Emily?
Solution: Given that, Principal Amount = $ 4000,
Rate of Interest = 10%,
Conversion period = ½ year.
Time period = 2 years
For the first half year,
Principal = $ 4,000
Rate = 10%
Time = ½ year
Therefore, Interest = $ P * T * R / 100
= $ 4,000 * 1 * 10 / 100 * 2
= $ 40,000 / 200
= $ 200
Therefore, the amount at the end of 1 year = Principal + Interest
= 4000 + 200
= 4,200
Daniel deposits $4200 at the beginning of the second year.
Therefore, the new principal for the 2nd year = $ 4200
For the second half year:
Principal = $ 4,200
Rate = 10%
Time = ½ year
Therefore, Interest = $ P * T * R / 100
= $ 4,200 * 1 * 10/100 * 2
= $ 42000 / 200
= $ 210
Therefore, the amount of loan after 2 years = Principal + Interest
= $4,200 + $ 210
= $ 4410
Daniel deposits $4410 at the beginning of third year.
Therefore, the new principal at the beginning of the third year = $4410
For the third half year,
Principal = $ 4410
Rate = 10%
Time = 1/2 year
Therefore, Interest = $ P * R * T / 100
= $ 4410 * 10 * 1 /100 * 2
= $ 44100 / 200
= $ 220.5
Therefore, the amount at the end of 3rd year = Principal + Interest
= $4410 + $220.5
= $4630.5
Therefore, the balance in the bank at the end of 3 years will be $4630.5
For the fourth half year,
Principal = $4630.5
Rate = 10%
Time = 1/2 year
Therefore, Interest = $ P * R * T / 100
= $ 4630.5 * 10 * 1 /100 * 2
= $ 46305 / 200
= $ 231.53
Therefore, the amount at the end of 3rd year = Principal + Interest
= $4630.5 + $231.53
= $4862.03
The total interest to be paid over 2 years = $200 + $210 + $220.5 + $231.53 = $862.03
Total amount = P + I = $4000 + $862.03 = $4862.03
Therefore, the total amount is $4862.03.
FAQs on Calculating Compound Interest Periodically
1. Will compounding calculate interest periodically?
Compound interest calculates by multiplying the principal amount by one and the annual interest raised to compound periods number minus one. Interest can be compounded on a frequency schedule from continuous to daily to annually.
2. How to calculate periodic compound interest?
The periodic rate equals the interest rate annually divided by the number of periods. For example, the interest on a home loan is generally calculated monthly, therefore if the annual interest rate is 6 percent, then you divided that by 12 and you get 2 percent. Hence, it is the monthly interest.
3. What is periodic compound interest?
Compound interest defines as when interest is added periodically, rather than once at the end term. Once the interest is added to the balance, then that interest begins accruing additional interest.
4. Where is the compound interest with periodic deductions used in day-to-day life?
Compound interest with periodic deductions works when you borrow. Whenever you borrow the money, you accrue interest on it and if you don’t pay that interest charges within the stated period in loan, they are “capitalized” or added to the initial loan balance amount.
Summary
Compound Interest with periodic deductions is the biggest friend when it comes to investments and deposits. Compounding interest quarterly, monthly, and half-yearly can spike the interest rate higher. You must have complete knowledge of compound interest with periodic deductions. So, go through every problem mentioned in this article and their solutions for better understanding.