# Calculate Future Value with Recurring Deposits

I am familiar with the formula for calculating FV and compound interest of a deposit, but I am wondering if there is a formula that will allow me to calculate how much money I will have after depositing recurring amount of money every month, quarter or year, with a fixed annual interest rate and an optional initial deposit?

Let's say:

Initial/present value: 2500

Annual interest: 4%

Recurring deposit every month: 100

How much will the FV be after 5 years?

• What is the frequency of compounding the interest? Monthly? Quarterly? Annually? Does the deposit occur on the first day of each month and interest is paid on the last day of the month? Or some other arrangement? Aug 18, 2012 at 13:31
• I am more interested in a general formula that will help me find out the FV by varying these options. (e.g. monthly, quarterly, annually compounding). As for the deposit, assume it at the beginning of the month and the interest is paid on the last day of the month. Aug 18, 2012 at 13:40
• good article to understand how recurring deposit work. really very appreciable job. can u give me the formula of calculating rate of interest of recurring deposit. please give me the formula. i want to know how to calculate interest rate of recurring deposit.
– user26631
Mar 25, 2015 at 5:47
• FYI: There's generally very little difference between compounding methods. Formula for continuous compounding with additions.
– Zaz
Nov 12, 2018 at 14:28

Using the following values:

``````p = initial value = 2500
n = compounding periods per year = 12
r = nominal interest rate, compounded n times per year = 4% = 0.04
i = periodic interest rate = r/n = 0.04/12 = 0.00333333
y = number of years = 5
t = number of compounding periods = n*y = 12*5 = 60
d = periodic deposit = 100
``````

The formula for the future value of an annuity due is `d*(((1 + i)^t - 1)/i)*(1 + i)`

(In an annuity due, a deposit is made at the beginning of a period and the interest is received at the end of the period. This is in contrast to an ordinary annuity, where a payment is made at the end of a period.)

The formula is derived, by induction, from the summation of the future values of every deposit.

The initial value, with interest accumulated for all periods, can simply be added.

``````pfv = p*(1 + i)^t = 3052.49

total = pfv + fv = 3052.49 + 6652 = 9704.49
``````

So the overall formula is

Let's break this into two parts, the future value of the initial deposit, and the future value of the payments:

• D: deposit
• i: interest rate
• n: number of periods

D(1 + i)n

For the future value of the payments

• A: amount of payments
• i: interest rate
• n: number of payments/periods

A((1+i)n-1) / i)

Adding those two formulas together will give you the amount of money that should be in your account at the end. Remember to make the appropriate adjustments to interest rate and the number of payments. Divide the interest rate by the number of periods in a year (four for quarterly, twelve for monthly), and multiply the number of periods (p) by the same number. Of course the monthly deposit amount will need to be in the same terms.

• Could you elaborate on what is the connection between the two formulas? Should I just calculate the two formulas separately using the same interest rate (the first using the initial sum of 2500 and the second with the monthly deposits of 100) and then just add the two results? Oct 22, 2012 at 10:03
• You have stated the formula for the future value of an ordinary annuity. The Wikipedia link you quoted has the correct formula: `A*(((1 + i)^n - 1)/d)` where `d = i/(i + 1)`. Mar 25, 2015 at 11:08

I noticed there did not necessarily seem to be a caveat for adjusting contribution frequency. I have included a formula below that would take this into account.

A = P(1+r/n)^(nt) + c[a(1 - r/n)^(nfz)] / [1 - (1 + r/n)^(nf)]

P = Principal r = interest rate n = number of compounds per year t = number of years this is compounding c = the amount of the contributions made each period a = will be one of two things depending on when contributions are made [if made at the end of the period, a = 1. If made at the beginning of the period, a = (1 + r/n)^(n*f)] f = frequency of contributions in years (so if monthly, f = 1/12) z = the number of contributions you would make over the life of the account (typically this would be t/f)

For example, suppose I had \$10,000 in an account compounding daily at 4%. If I make contributions monthly of \$100, then what is the value in 10 years? This would be set up accordingly.

Contributions made at the end of the month: A = 10,000(1 + 0.04/365)^(365 * 10) + 100[1(1 - 0.04/365)^(365 1/12(10/(1/12))] / [1 - (1 + 0.04/365)^(365*1/12)]

Simplifying: A = 10,000(1 + 0.04/365)^(3,650) + 100[1(1 - 0.04/365)^(3,650)] / [1 - (1 + 0.04/365)^(365/12)] A = \$29,647.91

Contributions made at the beginning of the month: A = 10,000(1 + 0.04/365)^(365 * 10) + 100[(1 + 0.04/365)^(365*1/12)(1 - 0.04/365)^(365 1/12(10/(1/12))] / [1 - (1 + 0.04/365)^(365*1/12)]

Simplifying: A = 10,000(1 + 0.04/365)^(3,650) + 100[(1 + 0.04/365)^(365/12)(1 - 0.04/365)^(3,650)] / [1 - (1 + 0.04/365)^(365/12)] A = \$29,697.09