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Interest Calculator (Savings Growth with Regular Contributions)

Enter the initial deposit, contributions (leave unused fields at 0), annual interest rate and time period. If you do not want to include tax or inflation, leave those at 0.

Enter rates as plain percent numbers (for 3%, enter "3"). This is a math simulation with a fixed rate and does not include fees, rate changes or rounding rules of real products. If you enter a tax rate, the tax is taken out of each month's interest and only the after-tax interest is added to the balance. In the US, tax on interest is usually not taken out of the account; you report the interest on your tax return each year. So the real tax and balance can differ slightly from this result.
Result and graph
Enter the starting amount, contributions, interest rate and time period on the left and press "Calculate". The result and a graph will appear here.

What you can do on this page

  • From a starting amount plus yearly and monthly contributions, see right away how much your money grows with compound interest (the ending balance and the total interest). You can choose to contribute at the beginning of each period (deposit first, then earn interest) or at the end (earn interest first, then deposit)
  • Nine compounding frequencies are supported: annually, semiannually, quarterly, monthly, semimonthly, biweekly, weekly, daily and continuously
  • The total interest is split into "interest on the initial deposit" and "interest on contributions", so you can see in numbers what your regular deposits add
  • Enter your tax rate on interest to get the tax and the after-tax interest. Enter an inflation rate to also get the buying power of the ending balance (its value in today's dollars)
  • Includes a balance graph and a year-by-year table. A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page is a math simulation that assumes the interest rate stays the same for the whole period. Real savings and investments involve fees, changing rates, and different tax and rounding rules, so check the exact amounts with your bank or provider. This page does not recommend any financial product or investment. For how compound interest itself works and how it differs from simple interest, see "Compound Interest Calculator" in the related pages.

What is this calculation used for?

Seeing where your monthly savings will go

For example, if you save $500 a month for 10 years at 4% (compounded monthly, deposits at the beginning), you put in $60,000 and end with about $73,870, so about $13,870 is interest.
Seeing "how much the rate changes things" and "how much I will have in 10 years" in numbers first lets you work backward from a goal, such as a down payment on a home or a new car, to decide your monthly savings.

Planning for college costs

If you save $200 a month from a child's birth until age 18 at 5% (compounded monthly, deposits at the beginning), you put in $43,200 and end with about $70,131.
You can try "how much a month is enough" against the expected cost of college. Real college savings, such as a 529 plan, are usually invested, with fees and the risk of loss, so use this fixed-rate result only as a guide.

Estimating retirement savings (compounding over the long run)

Suppose you could save $500 a month for 30 years at 5% (compounded monthly, deposits at the beginning). You put in $180,000 and end with about $417,863. The interest, about $237,863, is more than the money you put in.
You can feel how compounding (interest earning interest) grows stronger over longer times, and it helps you decide to start saving for retirement early, for example in a 401(k) or an IRA. In real investing, however, returns are not fixed, and you can lose money.

Seeing how inflation shrinks the real value of savings

If $10,000 sits at 0% for 10 years while prices rise 2% a year, its buying power becomes \(10000 \div 1.02^{10} \approx 8203\) dollars, a loss of about 18%. (2% is the long-run inflation target of the US Federal Reserve.)
Checking in numbers that "the balance did not go down, but it buys less" builds the habit of comparing your savings rate with the inflation rate.

Knowing how much interest you keep after tax

In the US, the interest from savings accounts and CDs is taxable income, and you report it on your tax return. For example, if you earn $500 of interest and your tax rate is 22%, you keep \(500 \times (1 - 0.22) = 390\) dollars.
Knowing the difference between the advertised rate (before tax) and what you actually keep helps you read the numbers correctly when you compare accounts and plan your savings.

Setting up automatic savings from each paycheck

Automatic transfers from each paycheck into savings, or payroll contributions to a 401(k), are exactly "adding a fixed amount every month", so this monthly-contribution calculation applies directly.
For example, raising $200 a month to $400 a month at 4% for 10 years takes the ending balance from about $29,548 to about $59,096. Comparing in numbers like this helps you choose an amount you can keep up. (For a tax-advantaged account, set the tax rate to 0.)

Formulas and graphs

Recurrence formula for contributions (beginning of period)
Graph
Standard notation (the usual math form)
\(B_k\) \(=\) \((\) \(B_{k-1}\) \(+\) \(C\) \()\) \(\times\) \((1 + r)\)
In words (symbols replaced with words)
④ \(B_k\): balance at the end of this period \(=\) \((\) ① \(B_{k-1}\): balance at the end of the previous period \(+\) ② \(C\): contribution \()\) \(\times\) ③ 1 + rate for one period
The formula in words
① Take the \(B_{k-1}\): balance at the end of the previous period
② first add the \(C\): contribution
③ then multiply by the growth factor for one period, "1 + rate \(r\)"
④ and you get the \(B_k\): balance at the end of this period
Quick example
If you deposit $10,000 at 5% a year (compounded annually) and add $5,000 at the start of each year, the balance at the end of year 1 is
\(B_1\): balance at the end of year 1 \(=\) \((\) starting balance ($10,000) \(+\) contribution ($5,000) \()\) \(\times\) 1 + annual rate (1.05)
\((10000 + 5000) \times 1.05 = 15750\)
Key idea
With contributions at the beginning of the period, you deposit first and then earn interest, so the money you add starts earning interest right away. Repeat this one-period formula once for each period, and you get the ending balance. The calculator on this page first converts the compounding you chose into an equivalent monthly rate (the monthly rate that gives exactly the same growth over one year). Then it repeats this formula month by month. The annual contribution is added in the first month of each year, and the monthly contribution every month.
Recurrence formula for contributions (end of period)
Graph
Standard notation (the usual math form)
\(B_k\) \(=\) \(B_{k-1}\) \(\times\) \((1 + r)\) \(+\) \(C\)
In words (symbols replaced with words)
④ \(B_k\): balance at the end of this period \(=\) ① \(B_{k-1}\): balance at the end of the previous period \(\times\) ② 1 + rate for one period \(+\) ③ \(C\): contribution
The formula in words
① Take the \(B_{k-1}\): balance at the end of the previous period
② multiply it by the growth factor for one period, "1 + rate \(r\)" to add the interest first
③ then add the \(C\): contribution
④ and you get the \(B_k\): balance at the end of this period
Quick example
With the same numbers ($10,000 at 5%, $5,000 contributions) but contributing at the end of each year, the balance at the end of year 1 is
\(B_1\): balance at the end of year 1 \(=\) starting balance ($10,000) \(\times\) 1 + annual rate (1.05) \(+\) contribution ($5,000)
\(10000 \times 1.05 + 5000 = 15500\)
Key idea
With contributions at the end of the period, you earn interest first and deposit afterward, so the new deposit earns no interest in that period. Compared with the beginning-of-period example ($15,750), the difference is $250, which is exactly one period of interest on the $5,000 contribution at 5%. With the same contributions, depositing earlier (at the beginning) earns more interest.
Breakdown of interest (initial deposit and contributions)
Figure
Standard notation (the usual math form)
\(I\) \(=\) \(A\) \(-\) \((\) \(P + S\) \()\)
\(I_C\) \(=\) \(I\) \(-\) \(I_0\)
In words (symbols replaced with words)
③ \(I\): total interest \(=\) ① \(A\): ending balance \(-\) \((\) ② \(P + S\): total deposits \()\)
⑤ \(I_C\): interest on contributions \(=\) \(I\): total interest \(-\) ④ \(I_0\): interest on the initial deposit
The formula in words
① From the \(A\): ending balance
② subtract the total deposits (initial deposit \(P\) + total contributions \(S\))
③ and you get the \(I\): total interest
④ Then subtract the \(I_0\): interest on the initial deposit (the interest you would get with no contributions) from the total interest
⑤ and you get the \(I_C\): interest on contributions
Quick example
In the example for formula 1 (ending balance $15,750, initial deposit $10,000, contribution $5,000, interest on the initial deposit $500):
\(I\): total interest \(=\) ending balance ($15,750) \(-\) total deposits ($15,000)
\(I = 15750 - (10000 + 5000) = 750\)
\(I_C = 750 - 500 = 250\)
Key idea
The interest on the initial deposit, \(I_0\), is the interest you would have earned with no contributions at all. It comes from running the same calculation with the initial deposit only. The difference from the total interest, \(I_C\), is the extra interest created by your contributions. This breakdown shows in numbers what your regular deposits add.
Buying power after inflation
Graph
Standard notation (the usual math form)
\(R\) \(=\) \(A\) \(\div\) \((1 + i)\) \(t\)
In words (symbols replaced with words)
④ \(R\): buying power \(=\) ③ \(A\): ending balance \(\div\) ① 1 + inflation rate \(i\) ② \(t\): years
The formula in words
① Multiply the yearly price factor "1 + inflation rate \(i\)"
② by itself once for each of the \(t\): years to find how many times higher prices have become
③ divide the \(A\): ending balance by that factor
④ and you get the \(R\): buying power (the value in today's dollars)
Quick example
If your balance grew to $11,000 in one year but prices also rose 10% that year, the buying power is
\(R\): buying power \(=\) ending balance ($11,000) \(\div\) 1 + inflation rate (1.1) years (1)
\(11000 \div 1.1 = 10000\)
Key idea
The number grew to $11,000, but if prices rose 10%, it still buys only what $10,000 bought before. This is buying power (the real value of money). If your savings rate is lower than the inflation rate, your balance goes up on paper but shrinks in real terms. For example, if inflation stays at 3% for 5 years, prices rise to about 1.16 times, and a balance of $10,000 has a buying power of about $8,626.09. The key point is that inflation builds up with the same math as compound interest: the same factor is multiplied again and again.
Compound interest with contributions is calculated by repeating a one-period recurrence formula, "(balance + contribution) × (1 + rate)" or "balance × (1 + rate) + contribution", once for each period. On this page you can check the breakdown of interest (initial deposit and contributions), tax and inflation all in one place.

Symbols and terms

Symbols

\(B_k\) B sub k The balance at the end of period \(k\). In the recurrence formula, it is the balance at the end of this period.
\(B_{k-1}\) B sub k minus 1 The balance at the end of the period before. In the recurrence formula, it is the balance at the end of the previous period.
\(C\) C The contribution for one period. It is added to the balance at the start of the period (beginning) or at the end of the period (end).
\(r\) r The interest rate for one period, written as a decimal. With annual compounding it is the annual rate itself (for 3%, \(r = 0.03\)).
\(A\) A The ending balance. The deposits plus interest at the end of the time period.
\(P\) P The initial deposit (principal). The amount you put in at the start.
\(S\) S The total contributions. All the money added through annual and monthly contributions during the period (not including the initial deposit).
\(I\) I The total interest. The ending balance minus the total deposits (initial deposit + total contributions).
\(I_0\) I sub zero The interest on the initial deposit. The interest you would have earned with no contributions.
\(I_C\) I sub C The interest on contributions. The total interest minus the interest on the initial deposit.
\(i\) lowercase i The inflation rate. How much prices rise in one year, written as a decimal (for 2%, \(i = 0.02\)).
\(t\) t The time period in years. How long you save.

Terms

contribution Money you add again and again at regular intervals (every year, every month, and so on). US savings and retirement calculators use this word for regular deposits.
beginning of the period The start of each period. When you contribute at the beginning, the deposit earns interest in that same period, so you earn more interest than with end-of-period contributions of the same size.
end of the period The end of each period. When you contribute at the end, the deposit earns no interest in that period, so you earn less interest than with beginning-of-period contributions of the same size.
tax-advantaged account An account such as a traditional IRA, a Roth IRA or a 401(k), where you pay no tax on the interest each year. In a traditional IRA or 401(k), tax is paid when you withdraw the money; in a Roth account, qualified withdrawals are tax-free. To simulate one, set the tax rate to 0.
compound interest Interest that is added to the balance, so the next interest is earned on the new, larger total. Interest earns interest, so the longer the time, the faster it grows. See "Compound Interest Calculator" in the related pages for details.
compounding frequency How often interest is added to the balance (annually, semiannually, monthly, and so on). With the same annual rate, more frequent compounding grows a little faster. The calculator on this page turns the chosen frequency into the monthly rate that gives the same growth over one year, and calculates month by month.
APY (annual percentage yield) The yearly rate that already includes the effect of compounding, shown by US banks for savings accounts and CDs. It is higher than the stated rate (APR) when interest compounds more than once a year. For example, 3% APR compounded monthly is about 3.04% APY.
recurrence formula A formula that finds the next value from the previous one. Compound interest with contributions is easier to follow as a one-period formula that is repeated than as a single long formula, and the calculator on this page works the same way.
tax on interest Tax on the interest from savings accounts, CDs and bonds. In the US, interest is generally taxed as ordinary income, at the same rate as your wages, by the federal government and in many states. Banks report interest of $10 or more for the year on Form 1099-INT.
buying power The value of money measured by how much it can actually buy, not by the number itself (the nominal amount). As prices rise, the same amount of money buys less. Also called purchasing power.
inflation A steady rise in prices. When prices go up, the same amount of money buys less, so cash and low-interest savings lose real value.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Percents (Grades 6–7)
  • Being able to write "5%" as the decimal 0.05
  • Knowing that "grows by 5%" is the same as "× 1.05"
Basics of compound interest ("Compound Interest Calculator" on this site)
  • Understanding compound interest, where the interest you earn also earns interest
  • Knowing that with the same annual rate, the compounding frequency (annually, monthly and so on) changes the growth a little
Exponents (Grades 6–8)
  • Knowing that \(1.05^{2}\) is "1.05 multiplied by itself"
Sequences and recurrence formulas (high school, advanced)
  • Understanding the idea of a recurrence formula, which finds the next value from the previous one (if not, the calculator still works through it one period at a time)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table for one period (beginning-of-period contribution)
Previous balance 10000
Contribution 5000
Rate per period (decimal; 5% = 0.05) 0.05
New balance =(B1+B2)*(1+B3)
Table for one period (end-of-period contribution)
Previous balance 10000
Contribution 5000
Rate per period (decimal; 5% = 0.05) 0.05
New balance =B1*(1+B3)+B2
Table for the breakdown of interest
Ending balance 15750
Initial deposit 10000
Total contributions 5000
Interest on initial deposit 500
Total interest =B1-(B2+B3)
Interest on contributions =B5-B4
Table for buying power after inflation
Ending balance 11000
Inflation rate (decimal; 10% = 0.1) 0.1
Years 1
Buying power =B1/(1+B2)^B3
Table for the ending balance with monthly contributions (FV function)
Monthly rate (annual ÷ 12; 3% = 0.0025) 0.0025
Months (10 years = 120) 120
Monthly contribution 500
Initial deposit 10000
Ending balance (beginning of period) =FV(B1,B2,-B3,-B4,1)
After pasting, the upper rows in column B are your inputs and the last row is calculated automatically.
"*" is multiplication, "/" is division and "^" is a power (how many times to multiply).
The first table shows 15,750, the second 15,500, the third a total interest of 750 and interest on contributions of 250, and the fourth 10,000 (all in dollars).
The fifth table uses the FV function, Excel's function for savings with regular deposits, and gives about 83,538.92. The FV arguments are, in order: rate, number of periods, payment each period (entered as negative), initial deposit (entered as negative), and 1 for the beginning of the period (0 for the end). With monthly compounding and monthly contributions, it matches the calculator on this page.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table for one period (beginning-of-period contribution)
Previous balance 10000
Contribution 5000
Rate per period (decimal; 5% = 0.05) 0.05
New balance =(B1+B2)*(1+B3)
Table for one period (end-of-period contribution)
Previous balance 10000
Contribution 5000
Rate per period (decimal; 5% = 0.05) 0.05
New balance =B1*(1+B3)+B2
Table for the breakdown of interest
Ending balance 15750
Initial deposit 10000
Total contributions 5000
Interest on initial deposit 500
Total interest =B1-(B2+B3)
Interest on contributions =B5-B4
Table for buying power after inflation
Ending balance 11000
Inflation rate (decimal; 10% = 0.1) 0.1
Years 1
Buying power =B1/(1+B2)^B3
Table for the ending balance with monthly contributions (FV function)
Monthly rate (annual ÷ 12; 3% = 0.0025) 0.0025
Months (10 years = 120) 120
Monthly contribution 500
Initial deposit 10000
Ending balance (beginning of period) =FV(B1,B2,-B3,-B4,1)
The same formulas and the same FV function as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the inputs in column B with your own numbers.

How to calculate it in Python

principal = 10000         # initial deposit ($)
annual_addition = 0       # annual contribution ($)
monthly_addition = 500    # monthly contribution ($)
is_beginning = True       # True = beginning of period, False = end of period
annual_rate = 0.03        # annual rate (0.03 for 3%)
times_per_year = 1        # compounding frequency (annually=1, semiannually=2, monthly=12, daily=365)
years = 10                # time (years)
months = 0                # time (months)
tax_rate = 0.0            # tax rate on interest (0.22 for 22%)

# Convert the compounding into the monthly rate that gives the same growth over one year
monthly_rate = (1 + annual_rate / times_per_year) ** (times_per_year / 12) - 1

balance = principal
total_contribution = 0
total_interest = 0
for k in range(1, years * 12 + months + 1):
    if is_beginning:
        if k % 12 == 1:  # annual contribution in the first month of each year
            balance += annual_addition
            total_contribution += annual_addition
        balance += monthly_addition
        total_contribution += monthly_addition
    interest = balance * monthly_rate
    total_interest += interest
    balance += interest * (1 - tax_rate)  # add only the after-tax interest
    if not is_beginning:
        balance += monthly_addition
        total_contribution += monthly_addition
        if k % 12 == 0:  # annual contribution in the last month of each year
            balance += annual_addition
            total_contribution += annual_addition

print(f"Ending balance: ${balance:,.2f}")
print(f"Total deposits: ${principal + total_contribution:,.2f}")
print(f"Total interest (before tax): ${total_interest:,.2f}")
Runs with the standard library only. Change the initial deposit, contributions, rate, time and so on at the top, then run it. It uses the same method as the calculator on this page (repeat month by month with an equivalent monthly rate). For continuous compounding, add "import math" and change the monthly_rate line to "math.exp(annual_rate / 12) - 1".

How to write it in LaTeX and other math languages (copy and paste)

Recurrence formula for contributions (beginning of period)
Bₖ = (Bₖ₋₁ + C) × (1 + r)
B_k = (B_{k-1} + C)(1 + r)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>B</mi><mi>k</mi></msub>
    <mo>=</mo>
    <mrow>
      <mo>(</mo>
      <msub><mi>B</mi><mrow><mi>k</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub>
      <mo>+</mo>
      <mi>C</mi>
      <mo>)</mo>
    </mrow>
    <mrow>
      <mo>(</mo>
      <mn>1</mn>
      <mo>+</mo>
      <mi>r</mi>
      <mo>)</mo>
    </mrow>
  </mrow>
</math>
B_k = (B_(k-1) + C)(1 + r)
(bprev + c) (1 + r)
B[k] := (B[k-1] + c)*(1 + r);
B(k) = (B(k-1) + c)*(1 + r);
B_k = (B_(k-1) + C)(1 + r)
Recurrence formula for contributions (end of period)
Bₖ = Bₖ₋₁ × (1 + r) + C
B_k = B_{k-1}(1 + r) + C
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>B</mi><mi>k</mi></msub>
    <mo>=</mo>
    <msub><mi>B</mi><mrow><mi>k</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub>
    <mrow>
      <mo>(</mo>
      <mn>1</mn>
      <mo>+</mo>
      <mi>r</mi>
      <mo>)</mo>
    </mrow>
    <mo>+</mo>
    <mi>C</mi>
  </mrow>
</math>
B_k = B_(k-1)(1 + r) + C
bprev (1 + r) + c
B[k] := B[k-1]*(1 + r) + c;
B(k) = B(k-1)*(1 + r) + c;
B_k = B_(k-1)(1 + r) + C
Breakdown of interest (initial deposit and contributions)
I = A − (P + S),  I_C = I − I₀
I = A - (P + S), \quad I_C = I - I_0
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>I</mi>
    <mo>=</mo>
    <mi>A</mi>
    <mo>&#x2212;</mo>
    <mrow>
      <mo>(</mo>
      <mi>P</mi>
      <mo>+</mo>
      <mi>S</mi>
      <mo>)</mo>
    </mrow>
    <mo>,</mo>
    <msub><mi>I</mi><mi>C</mi></msub>
    <mo>=</mo>
    <mi>I</mi>
    <mo>&#x2212;</mo>
    <msub><mi>I</mi><mn>0</mn></msub>
  </mrow>
</math>
I = A - (P + S), I_C = I - I_0
itotal = a - (p + s); ic = itotal - i0
Itotal := A - (P + S); Ic := Itotal - I0;
Itotal = A - (P + S); Ic = Itotal - I0;
I = A - (P + S),  I_C = I - I_0
Buying power after inflation
R = A ÷ (1 + i)ᵗ
R = \dfrac{A}{(1 + i)^{t}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>R</mi>
    <mo>=</mo>
    <mfrac>
      <mi>A</mi>
      <msup>
        <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mo>)</mo></mrow>
        <mi>t</mi>
      </msup>
    </mfrac>
  </mrow>
</math>
R = A / (1 + i)^t
a/(1 + inflation)^t
R := A/(1 + infl)^t;
R = A/(1 + infl)^t;
R = A/(1 + i)^t

How to have ChatGPT  do the calculation

You are an interest calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

An initial deposit of $10,000 earns 3% a year (compounded annually), and $500 is added at the beginning of every month. The time period is 10 years.
Convert the compounding into the equivalent monthly rate whose yearly growth factor is 1.03, and repeat "(balance + contribution) × (1 + monthly rate)" every month.
Find each of the following:
1. The ending balance after 10 years, and the total deposits (initial deposit + total contributions)
2. The total interest, and its breakdown (interest on the initial deposit and interest on contributions)
3. If the interest is taxed at 22%, the total tax and the ending balance after tax

Show the formulas you used and the numbers from the execution result.

How to Use
  1. 1
    Enter your numbers
    Type the numbers you want to calculate with into the input fields
  2. 2
    Calculate
    Press the "Calculate" button
  3. 3
    Check the result
    The result appears on the spot. The same page also explains the idea behind the calculation and the formula
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