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Compound Interest definition and its formulas

Compound interest can be defined as that interest that is calculated on the previous principal and also on the accumulated interest over the initial periods of a loan. Compound interest can be thought of as an interest upon another interest and will make a loan grow at a rate that is very fast than the usual simple interest which is that interest that is calculated only on the capital amount

The rate at which compound interest accumulates can be dependent on the frequency of compounding, i.e. the higher the number of compounding periods, the greater the compound interest is

Therefore, the total amount of compound interest that has accumulated on $100 compounded at 10% annually will be far lower than on $100 which is compounded at 5% semi-annually over the same time frame. Compound interest is also known to be compounding

There is a formula that can be used in calculating compound interest, this formula is given below:

  • CI (compound Interest) = The total amount of principal and interest in future – The principal amount at present. In other words; CI = [P (1 + i)n] – P = P [(1 + i)n – 1], where P = Principal, i = the nominal annual interest rate in terms of percentage, and n = number of compounding periods for any compound Interest
  • Therefore, compound Interest equation goes as follows: A = P(1 + r/n)nt, where: A is the accrued amount i.e. principal + interest, where P = is the principal amount, R = the annual nominal interest rate in percent, I = is the interest amount, r = the annual nominal interest rate as a decimal, r = R÷100, t = the amount of time in years, 0.5 years is calculated as 6 months, etc. n = is the number of compounding periods per unit t; at the end of each period
  • Formulas where n equals 1 and compounded once per period or unit t: calculate the accumulated amount for any compound Interest, i.e. principal + interest which is A = P(1 + r)t. Next is to calculate the principal amount by solving for P in the equation: P = A / (1 + r)t then calculate the rate of interest in decimal by solving for r in this equation: r = (A/P)1/t – 1, after, calculate the compound Interest rate in percent: R = r * 100. Finally, calculate the time as you solve for t: t = t = ln(A/P) / ln (1 + r) = [ln(A) - ln(P)] / ln (1 + r)
  • Continuous Compounding Formulas (n → ∞). Determine the compound Interest by calculating the total amount that accumulated, i.e. (principal + interest): A = Pert. Also, calculate principal amount, solve for P: P = A / ert. Calculate rate of interest in decimal form, then solve for r, i.e. r = ln(A/P) / t. Calculate rate of the compound Interest in percent: R = r * 100 then finally calculate time, solve for t: t = ln(A/P) / r

If the number of compounding periods for a compound Interest is more than once a year, interest amount as well as the number of compounding period need to be adjusted accordingly. Then, i must be divided by the number of compounding periods per year, and n stands for the number of compounding periods per year times the loan or deposit’s maturity period in years in order to get its compound Interest. For example:

  • The compound interest on a $10,000 compounded annually at 10% (i = 10%) for 10-year period i.e. n equals 10 would be = $25,937.42 - $10,000 = $15,937.42
  • The amount of compound interest on $10,000 compounded semi-yearly at 5% interest rate i.e. i = 5% for 10-year period i.e. n equals 20 would be = $26,532.98 - $10,000 = $16,532.98
  • The amount of compound interest on $10,000 compounded monthly at 10% rate, i.e. i equals 0.833% for a period of 10 years and n equals 120, we will have= $27,070.41 - $10,000 = $17,070.41

Compound interest can significantly increase investment returns over a long term. While a $100,000 deposit that receives 5% simple interest would earn $50,000 in interest over a period of 10 years, with a 5% compound interest on $10,000 would amount to $62,889.46 over the same number of period. While the secret of compounding has led to the apocryphal story of Albert Einstein supposedly referring to it as among the 8th wonder of the world as well as one of human greatest invention. Compounding can as well be imposed on those consumers who have loans with a high interest rates, such as credit-card debt with a credit-card balance of $20,000 that is carried at an interest rate of 20% if compounded monthly and would result in total compound interest of $4,388 over a period of one year

The compounding effect of compound Interest

If you invested $10,000 for 5 years at 5% per year, with interest paid as soon as the term ends, while you earn $2,500 in simple interest after the period of 5 years, $500 every year. This would sum up to a total of $12,500 after 5 years. But if you invested a total of $10,000 for 5 years at 5%, with interest calculated then added each month, it is possible to earn $2,834 in compound interest after 5-year period, which sums up to a total of $12,834. Returns would be higher because you would earn interest on the compound Interest. Here's how the figures above were calculated:

  • Simple interest on a $10,000 investment at 5% per year, that is paid at the end of the term:

     

      Year 1 ($)

     

    Year 2 ($)

     

    Year 3 ($)

     

    Year 4 ($)

     

    Year 5 ($)

     

    Deposit

     

    10,000

     

    0

     

    0

     

    0

     

    0

     

    Interest

     

    0

     

    0

     

    0

     

    0

     

    2,500

     

    Total

     

    10,000

     

    10,000

     

    10,000

     

    10,000

     

    12,500

     

  • Compound interest on a $10,000 investment at 5% per year, paid monthly:

     

      Year 1 ($)

     

    Year 2 ($)

     

    Year 3 ($)

     

    Year 4 ($)

     

    Year 5 ($)

     

    Deposit

     

    10,000

     

    0

     

    0

     

    0

     

    0

     

    Interest

     

    512

     

    538

     

    565

     

    594

     

    625

     

    Total

     

    $10,512

     

    $11,049

     

    $11,615

     

    $12,209

     

    $12,834

     

Although, there are a lot of financial institutions today but not all of them treat cash investments the same way. Some of these institutions compound interest monthly, while other institutions compound quarterly and even annually. In addition, some institution charge extra fees but some other institutions do not. An effective interest rate takes all of this into account and expresses the rate in simple interest terms

Therefore, the balance at the end of one year would be compared to your balance at the beginning of the year and any increase would be shown as a percentage of the opening balance. The effective compound Interest rate is the interest rate you would have been given to achieve a similar balance, if you did not have the benefit of been charged any fees

Calculate the compound Interest that $2,000 will grow to over 2 years for an investment that grows at 5% per annum compounding yearly:

  • A = $2,000 x (1.05)2
  • A = $2,000 x 1.1025
  • A = $2,205.00

Calculate the compound Interest that $2,000 will increase to over 2 years for an investment that grows at 5% per annum compounding monthly:

  • First divide the yearly interest rate by 12 to give: 0.42%. Then calculate the total number of time periods n in months which gives 24
  • Now, input the above values into the formula: A = $2,000 x (1.0042)24
  • A = $2,000 x 1.11
  • A = $2,211.64
  • Knowing the effective investment rate for an investment allows you to check the difference from the kind of investments that was proposed to you by a different financial institutions

A businessman has an investment account that increased from a sum of $30,000 to a total sum of $33,000 within period of 30 months. His local bank offers savings account with daily compounding, what annual compound interest rate does this businessman need to get from them to match the return he got from his investment account?

With of a calculator, select calculate rate represented by letter R. The equation we need to use will be like this: r = n[(A/P)1/nt - 1] and R = r*100. Then input the values like below:

  • Total P+I (A): $33,000
  • Principal (P): $30,000
  • Compound (n): Daily for a year
  • Time (t): 2.5 years, i.e. 30 months
  • Our answer will be: R = 3.8126%/year
  • This means that the businessman in question will need to deposit a sum of $30,000 into a savings account that will pays him a rate of 3.8126%/year with a daily compound interest so as to get the same return as his investment account
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