# Which compound interest formula can I use to find the final balance with monthly contributions that increase yearly?

I apologise in advance if this has been asked already - I can't seem to find a similar question. I am looking for a formula to find the final balance when provided the following variables:

• Initial Principle
• Time (Years)
• Interest rate (%)
• Monthly contributions
• Monthly compounding
• Yearly increase in contributions (%), compounds

I have found a few formulas that determine the final balance but they all assume the monthly contributions stay constant. My initial formula came from this website: https://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php

Any direction would be a wonderful help!

• You need to tell us how often interest is credited to the account and at what rate, whether or not the monthly contributions occur on the same day of the momth as the crediting of interest, whether the yearly increase in contributions is compounded or a fixed percentage of the original principal amount etc. – Dilip Sarwate Apr 28 '18 at 19:09
• Hi @DilipSarwate, Thank you for your response and sorry for not providing enough detail. I hope the below answers your points adequately: - Compounded monthly - Monthly contributions occur on the same day as crediting - Yearly increase is compounded. So say 1200 -> 1320 -> 1452 if at 10% – Andy Best Apr 28 '18 at 19:39
• What you are describing is the "future value of a growing annuity." Google it and you will find the formula you want. – farnsy Apr 28 '18 at 20:19

Calculation for a growing annuity-due with deposits increasing annually.

``````r is the monthly or quarterly interest rate
y is the number of years
m is the number of months or quarters per year
p is the initial regular deposit
x is the annual deposit percentage increase

fv = (p (1 + r) (-1 + (1 + r)^m) ((1 + r)^(m y) - (1 + x)^y))/
(r (-1 + (1 + r)^m - x))
``````

Example with quarterly deposits increasing annually.

``````r = 0.1
y = 3
m = 4
p = 500
x = 0.05

fv = 12209.85
``````

Quarter-by-quarter calculation

``````y1q1 = 0 + 500
y1q2 = y1q1 (1 + r) + 500
y1q3 = y1q2 (1 + r) + 500
y1q4 = y1q3 (1 + r) + 500
y1q4 (1 + r) = 2552.55

y2q1 = y1q4 (1 + r) + 500 (1 + x)
y2q2 = y2q1 (1 + r) + 500 (1 + x)
y2q3 = y2q2 (1 + r) + 500 (1 + x)
y2q4 = y2q3 (1 + r) + 500 (1 + x)
y2q4 (1 + r) = 6417.37

y3q1 = y2q4 (1 + r) + 500 (1 + x)^2
y3q2 = y3q1 (1 + r) + 500 (1 + x)^2
y3q3 = y3q2 (1 + r) + 500 (1 + x)^2
y3q4 = y3q3 (1 + r) + 500 (1 + x)^2
y3q4 (1 + r) = 12209.85
``````

Derivation

The formula is derived from the following double summation.

Initial principal?

If you have an initial value `v` this can be added on with its own interest.

``````total = fv + v (1 + r)^(m y)
``````

Similar calculation, but for a loan (ordinary annuity)

• Hi Chris, Thank you so much for your answer, this was very helpful. I want to clarify something: I used the following variables for your formula: ` r = 0.1 / 12 y = 3 m = 12 p = 500 x = 0.05 ` To get 22,064.28. I checked these at thecalculatorsite.com/finance/calculators/… and got 22,738.37. Is this due to the fact that additions are made at the start of each compounding period with that calculation? Thank you again for your answer, you have been brilliant. – Andy Best Apr 28 '18 at 23:03
• Hi Andy, No, on the website if you put `Base Amount = 0` you will also get 22,064.28. The base amount is the initial value `v`. So if `v = 500` then `total = fv + v (1 + r)^(m y) = 22,064.28 + 500 (1 + 0.1/12)^36 = 22,738.37`. Your deposits start with a deposit straight away for an annuity-due. So you are starting with `v + p = 1,000` to arrive at 22,738.37. However, the initial value can be considered separately, as can its interest, as shown. – Chris Degnen Apr 28 '18 at 23:31