Is there a formula for the future value of a series of monthly deposits with interest compounded weekly?

Question

I have searched and not seen (or maybe just didn't recognize it as applying to) a future value formula for the following: Deposit a fixed amount ($100 or whatever) each month into an account/investment. Compound interest is applied weekly (or it could be daily, bi weekly, but less than monthly). Annual interest rate (would be divided by the number of compounding periods per year, I assume). No initial investment (so account/investment would start with $0). Deposit would be at the end of each month.

I've seen a formula (in various places) that is close (or possibly THE formula for this) but can't seem to get it to work with the deposits being monthly and the compounding periods being less than monthly.

If there is a formula somewhere, or if this question has already been answered, could someone please point me to it? I apologize ahead of time if this is the case.

Answer 1

There is no formula that can be applied to most variations of the problem you pose. The reason is that there is no simple, fixed relationship between the two time periods involved: the time interval for successive payments, and the time period for successive interest compounding.

Suppose you have daily compounding and you want to make weekly payments (A case that can be handled). Say the quoted rate is 4.2% per year, compounded daily

Then the rate per day is 4.2/365, or 0.0115068 %

So, in one week, a debt would grow through seven compoundings. A debt of $1 would grow to 1 * (1+.000225068)^7, or 1.000805754

So, the equivalent interest rate for weekly compounding is 0.0805754%

Now you have weekly compounding, and weekly payments, so the standard annuity formulas apply.

The problem lies in that number "7", the number of days in a week. But if you were trying to handle daily / monthly, or weekly / quarterly, what value would you use?

In such cases, the most practical method is to convert any compounding rate to a daily compounding rate, and use a spreadsheet to handle the irregularly spaced payments.

Answer 2

Perhaps there is no single formula that accounts for all the time intervals, but there is a method to get formulas for each compound interest period. You deposit money monthly but there is interest applied weekly. Let's assume the month has 4 weeks.

So you added x in the end of the first month, when the new month starts, you have x money in your account. After one week, you have x + bx money. After the second week, you have x + b(x + bx) and so on. Always taking the previous ammount of money and multiplying it by the interest (b) you have. This gives you for the end of the second month:

This looks complicated, but it's easy for computers. Call it f(0), that is: It is a function that gives you the ammount of money you would obtain by the end of the second month. Do you see that the future money inputs are given with relation to the previous ones? Then we can do the following, for n>1 (notice the x is the end of the formula, it's the deposit of money in the end of the month, I'm assuming it'll pass through the compound interest only in the first week of the next month):

And then write:

There is something in mathematics called recurrence relation in which we can use these two formulas to produce a simplified one for arbitrary b and n. Doing it by hand would be a bit complicated, but fortunately CASes are able to do it easily. I used Wolfram Mathematica commands:

FullSimplify[
 RSolve[{f[0] == 
    x + b x + b (x + b x ) + b (x + b x + b (x + b x )) + 
     b (x + b x + b (x + b x ) + b (x + b x + b (x + b x ))),
   f[n] == b f[n - 1] + b (b f[n - 1]) + 
     b ( b f[n - 1] + b (b f[n - 1])) + 
     b (b f[n - 1] + b (b f[n - 1]) + 
        b ( b f[n - 1] + b (b f[n - 1]))) + x}, f[n], n]
 ]

And it gave me the following formula:

All the work you actually have to do is to figure out what will be f(0) and then write the f(n) for n>0 in terms of f(n-1). Notice that I used the command FullSimplify in my code, Mathematica comes with algorithms for simplyfing formulas so if it didn't find something simpler, you probably won't find it by yourself! If the code looks ugly, it's because of Mathematica clipboard formatting, in the software, it looks like this:

Notice that I wrote the entire formula for f(0), but as it's also a recurrence relation, it can be written as:

That is: f(0)=g(4). This should give you much simpler formulas to apply in this method.