Solving for Time to Reach a Savings Target with Monthly Deposits
Summary
The document derives the time needed for an account with an initial balance and regular monthly deposits to reach a target under compound interest. It models the balance recursively: each month’s opening balance earns interest, then a deposit is added. Repeated substitution gives a geometric series for the accumulated deposits, which can be combined with growth of the initial principal. Taking logarithms then gives an expression for the number of months required to reach a specified total; the result is rounded up to a whole month.
The example uses a stated annual rate converted to a monthly rate, with deposits made at month-end, and reports the resulting time for the given target. The method depends on the deposit timing and compounding convention: beginning-of-month deposits, a different effective monthly rate, or changing deposits require adjustments. It assumes a constant rate and does not account for fees, taxes, inflation, or market variability, so it is a deterministic savings calculation rather than an investment return forecast.
Key ideas
- A fixed monthly deposit and constant monthly interest rate produce a geometric series in the account balance.
- The balance recurrence grows the current amount by interest before adding the month-end deposit.
- The target time can be found by isolating the compounding factor and taking logarithms.
- The calculated number of months should be rounded up when the target must be reached or exceeded.
- Deposit timing, rate conversion, fees, and variable returns can change the result.
Tags
Full text
# Compound interest calculator solving for time with deposits
# Compound interest calculator solving for time with deposits
I am attempting to solve a compound interest calculation for time given
```
Principal = 100
Time(years) = t
Rate(per year) = 8%
Deposit(per month) = 5
Total = 300
```
I can find solving for time without regular deposits. Can anyone help me out with a source or a workup of how to do this calculation with a regular monthly deposit?
http://www.mathportal.org/calculators/financial-calculators/compound-interest-calculator.php?formId=0&val1=4500&val2=7&val3=9&combo1=1&combo2=2
I'm attempting to use this calculator but it does not allow for deposits. That is essentially what i need but with deposits.
## Answer by P.Windridge (score 2, accepted)
https://quant.stackexchange.com/a/25587
Look this is just a geometric sum:
Assume interest is paid monthly at rate $r = 0.08/12$ (you can use the exact monthly equivalent if you want) and let $x_n = $total after $n$ months (including that month's interest and deposit).
So $x_0= 100$ and $x_{n+1} = x_n(1+r) + d$, where $d = 5$ is your deposit amount (added at the end of the month).
Applying the recursion repeatedly you see $$ x_n = x_0(1+r)^n + d(1+r)^{n-1} + \ldots + d = x_0(1+r)^n + d\sum_{j=0}^{n-1}(1+r)^j. $$ After applying the geometric sum formula you get $$ x_n = x_0(1+r)^n + \frac{d}{r}((1+r)^n-1). $$
Thus (assuming my metaphorical back-of-envelope calculation is right): $$ (1+r)^n = \frac{x_n + \gamma}{x_0 + \gamma}, $$ where $\gamma = d/r$. Take log of both sides to get $$ n = \log\left(\frac{x_n + \gamma}{x_0 + \gamma}\right)/\log(1+r). $$ I.e. to get the number of months to reach 300 you would take $x_n =300$ here and round $n$ upto the next integer.
It is 32 months for your example.
You could easily modify this so that your deposits increase over time, interest is paid yearly, etc.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.