# Compound interest calculator | Compound interest formulae

This calculator not only calculates compounding interest but also calculates compounded amount or accumulated amount for single as well as periodic investments.

Steps to use this calculator

there are two sections in this calculator one for single investment and other for recurring investment

Total Time: 1 minute

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### enter period in year

period should be entered in years only of your period is in months then you can convert it to years by using months/12 formulae or you can use the converter below this calculator

### select whether you are also investing on recurring basis or not

if you don’t invest on recurring basis then you don’t have anything else to do but if you invest periodically then follow the next steps

### enter investment timing

whether the investment is made at the start or end of period.

### enter the frequency of your investment

how frequently you invest like monthly, quarterly, semiannually etc.

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### enter period in years

period for which you keep investing this amount.

1.What is Compound interest :

Compound interest is an interest which is not only charged on principal investment but also the accumulated interest earned. It can be understood with the following derivation.

2.Compound interest formula derivation:

So here is how we calculate Compound interest.
Let say we invest \$100 in an account which earn 10%. In year1 the interest = 100 * 0.10 = 10 and balance at the end of year 1 = 100 + 100*0.10 = 100(1+0.10) = 100(1.1)
Interest in year 2 = 100(1.1) * 0.10 and balance at the end of year 2 = 100(1.1) + 100(1.1) * 0.10 = 100(1.1)(1.1) = 100(1.1)^2
Interest in year 3 = 100(1.1)^3 * 0.10  and balance at the end of year 3 = 100(1.1)2 + 100(1.1)^2 * 0.10 = 100(1.1)^2 * (1.1) = 100(1.1)^3
Since the balance at the end of year 3 = 100(1.1)^3, the balance at the end of year n would be = 100(1.1)^n
And if the investment is C and interest rate is r  then
Balance at the end of n year = C(1+r)^n

And interest = balance at the end of period – initial investment = C(1+r)^n  – C = C((1+r)^n  -1)

3.Compound Interest formulae:

3.1For single investment:

Accumulated amount = C(1+r/p)^(pt)

Compound interest = C(1+r/p)^(pt)  – C =  C((1+r/p)^(pt) – 1)
Where
C = amount invested
r = interest rate per annum Compounding P times in a year
For semiannual Compounding, P = 2, for annual, P = 1, For quarterly, P = 4, For monthly, P = 12, For daily, P = 360
t = period of investment in years

3.2For single investment with continuous:

Accumulated amount = Ce^(yt)
Compound interest = C(e^(yt) -1)
Where
C = cashflow
y = interest rate Per annum Compounding continuesly
t = period of investment in years
e = exponential constant whose value = 2.718281828

3.2.1For recurring or periodic investments at the end of period(month, quarter, year etc.):

Accumulated amount = E((1+r/p)^(pt)  – 1)/(r/p)

Compound interest = Accumulated amount- Total of all investments = A – (E*p*t)
Where
E = periodic investment p times a year
r = interest rate per annum Compounding P times in a year (remember that compounding frequency should be equal to payment frequency)
For semiannual Compounding, P = 2, for annual, P = 1, For quarterly, P = 4, For monthly, P = 12, For daily, P = 360
t = period of investment in years

3.2.1For recurring or periodic investments at the beginning of period(month, quarter, year etc.):

Accumulated amount = E((1+r/p)^(pt)  – 1)/(r/p) * (1+r/p)
Compound interest = Accumulated amount- Total of all investments = A – (E*p*t)

Where
E = periodic investment p times a year
r = interest rate per annum Compounding P times in a year (remember that compounding frequency should be equal to payment frequency)
For semiannual Compounding, P = 2, for annual, P = 1, For quarterly, P = 4, For monthly, P = 12, For daily, P = 360
t = period of investment in years

3.2.3 For recurring or periodic investments with continuous cashflows.

Accumulated amount = E(e^(yt)  – 1)/r

Compound interest = A – E*t
Where
y = interest rate compounding continuously
t = time in years

How to convert interest rate for finding accumulated amount:

Interest rate can be converted using following formula:

r(p) = ((1+r(m)/m)^(m/p)   – 1) * p
Where
m = given compounding times
p = required compounding times
r(m) = rate given at p times compounding
per year
r(p) = rate desired at m times  compounding per year.

In case of continuous compounding, conversion formulae are.

From continuous to any frequency.

r(p) = e^(y/m)  – 1

From any frequency to continuous.
y = ln((1+r(p)/p)^p)

Note : Ln is natural log.

Example : suppose you invest \$1000 now for 10 years and you invest  \$100 every month for next 10 years in an account that pays 10% p.a. Compounding quarterly what will be value of investment and Compounding interest.

Accumulated value of initial investment = 1000*(1+.10/4)^(4*10) = 2685
Compound interest = 2685 – 1000 = \$1685

For monthly investments, we need to convert the interest rate from quarterly to monthly compounding as follows.

r(12) = ((1+r(4)/4)^(4/12)  – 1)*12
r(12) = ((1+0.10/4)^(1/3)  -1)*12 = 9.9178%

Now the accumulated value = 100 * ((1+0.099178/12)^(12*10)  – 1)/(.099178/12)= \$20388

Compound interest = \$20388 – (100*12*10) = \$8388

Total accumulated amount = 20388 + 2685 = \$23073
Total Compound interest = 1685 + 8388 = \$10073

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