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First question on here so please go easy if it's a stupid one!

I have created a PHP based FV calculator for use on a client's website. This is the scenario:

They offer returns of 3% per month based on a variable investment amount (for argument's sake, let's say $10,000). Return is calculated monthly so my formula is:

$$ FV = PV (1 + i)^n $$

This works fine. The problem I have is that they charge fees both annually (let's say $1,000) and monthly (say, $60).

I have tried just adding the fees up and taking them from the FV figure but this is obviously wrong as the fees would affect the amount calculated each month and year.

How would I calculate the amount of fees paid and the net profit?

Trevor Wilson
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Wildcard27
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  • It rather depends on whether the fees are charged at the start of end of the month or year and what happens to fees if the investment is for less than a year $(n\lt 12)$. – Henry May 13 '14 at 11:21
  • Apologies, the fees are charged at the beginning of each year and annual fees aren't added to the 1st year – Wildcard27 May 13 '14 at 11:23

1 Answers1

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I guess it depends on when the fees are taken. For example, when the fees are taken at the beginning of the month and year, you could have, in one year,

$$FV = (\cdots (((PV-m)(1+i)-m)(1+i)-m)(1+i) \cdots -m)(1+i) - y (1+i)^{12} $$

where $m$ and $y$ are the monthly and yearly fees, respectively. Note that I am using the future value of the annual fee, as you stated that it is charged at the beginning of the year. Expanding out the expression, you get

$$FV = (PV-y) (1+i)^{12} - m \sum_{n=1}^{12} (1+i)^n = (PV-y) (1+i)^{12} - m (1+i) \frac{(1+i)^{12}-1}{i} $$

Using your numbers, I get $FV = \$ 11,954.80$.

You can then use this future value as a present value one year later to continue the calculation.

I should add that, to compute the fees after one year, simply take the original future value of the initial deposit of $ \$ 10000 $, which is $ PV (1+i)^{12} = \$14,257.60$, then subtract your $FV$ above to get that you paid, in this one year, about $ \$ 2302.83$ in fees, for a net profit of $ \$ 1954.80$.

On the other hand, if you do not pay the annual fee until after the first year, then

$$FV=PV (1+i)^{12} - m (1+i) \frac{(1+i)^{12}-1}{i} -y \approx \$ 12,380.50$$

which implies $\$1877.07$ in fees and a profit of $\$2380.50$.

Ron Gordon
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  • Ok, so apparently I've found myself knee deep into an intellectually superior group! My maths knowledge is nowhere near this caliber. Is it possible to explain the equations in a more simple context? Sorry to be a burden! – Wildcard27 May 13 '14 at 11:30
  • OK, let me see what I can do. In the first line, We start with the initial amount, $PV$, and immediately subtract a monthly fee (as banks are won't to do). At the end of the month, we accrue interest, so multiply by $1+i$. Then next month, subtract a monthly fee, then accrue interest. Repeat. Also subtract the annual fee, which really should have been subtracted at the beginning, so in this line, I merely multiplied by $12$ months of interest. The second line is a bunch of algebra for consolidation, and I used the formula for a geometric series. – Ron Gordon May 13 '14 at 11:34
  • So, I finally got around to understanding your formula! Had to give myself a math lesson in which to do so but I got there! I also got around to programming the equation into a PHP algorithm! So thank you for your help, as far as I can tell, they figures are correct. Suppose I will find out for sure once we get a mathematician as a client! Thanks again! – Wildcard27 May 27 '14 at 02:09