How to calculate compound returns when you're subtracting a fixed value for each round of return?

Question

I'm trying to calculate compounded returns, where I make an initial investment and then reinvest the capital and earnings.

In this particular case, my return is equivalent to 60% of invested capital, minus a $10 flat fee.

Without the flat fee, I could easily calculate by n * (1.6^r), where n is initial investment and r is the amount of investment rounds.

The problem is I dont know how to implement the $10 fee into a generalized equation that could cover a variable number of rounds.

As it stands I would have to use: ((((n*1.6)-10)*1.6-10)*1.6-10)..., manually adding each round. Unfortunately this isn't suitable for my purposes.

Thanks

Answer 1

One way to think about this is as two separate cash-flows. The first is a straight compounded, one-time investment, which as you pointed out equates to $p_0 \times 1.6^n$.

The second is the stream of payments that you make. This is a simple annuity [more specifically, an ordinary annuity] which has a present value equal to:

$PV_{OA} = C\times \frac{1-(1 + r)^{-n}}{r}$ where $C$ represents the cash flows for each period, $r$ is the interest rate, and $n$ is the number of periods. From this we can also tell that the future value of the annuity is equal to $FV_{OA} = C\times \frac{(1 + r)^n - 1}{r}$

So, for sake of example, if you have $C = -10$ and $n = 5$ with $60\%$ compounding rate, and an initial value of $1000$ then you would have:

$PV = 1000 + (-10)\times\frac{1-(1+0.6)^{-5}}{0.6} = \$984.82$ or

$FV = 1000\times1.6^5 + (-10)\times\frac{(1+0.6)^{5} - 1}{0.6} = \$10,327.66$

Note that if you take the Present Value (PV) and multiply it by the compounding factor ($1.6^5$) you get the Future Value (FV). You can also verify for yourself, using a table, that the above relationship holds!

$$ \begin{array}{c | c | c} Period & Beginning & After Fee \\ \hline 0 & 1000 & - \\ 1 & 1600 & 1590 \\ 2 & 2544 & 2534 \\ 3 & 4054.40 & 4044.40 \\ 4 & 6471.04 & 6461.04 \\ 5 & 10,337.66 & 10,327.66 \end{array} $$

Answer 2

Write out the brackets. For example taking only 3 terms:

$\Big(1.6\big(1.6( 1.6n -10) -10\big)-10\Big) = 1.6^3n - 1.6^2 \times 10 - 1.6 \times 10 - 10.$

General form would look like: $1.6^n - 10 \sum\limits_{i = 0}^{n-1}1.6^i$. You already know how to handle the first term, the second one is known as geometric series.

Applying the formula we'd get (with M the money after n steps and I the investment in beginning):

$$M = 1.6^n I + \frac{10}{0.6} (1 - 1.6^n) = 1.6^n (I - \frac{10}{0.6}) + \frac{10}{0.6}$$