This is a tough problem, and requires a lot of diligence in order to solve it.

Consider what this question is asking, in that all distributions are reinvested in the mutual fund. In this case, we can treat the annual investments as earning interest independent of one another:

- Funds compounding at 10% interest for 20 years will be scaled by (1.10)
^{20}.
- Funds compounding at 10% interest for 19 years will be scaled by (1.10)
^{19}.
- Funds compounding at 10% interest for 18 years will be scaled by (1.10)
^{18}.

…

- Funds compounding at 10% interest for 1 year will be scaled by (1.10)
^{1}.

Each of these scaling factors will affect one of the annual investment amounts (The initial investment will compound for 20 years, the next year for only 19, and so on), meaning that we need to sum each of the scale factors multiplied by the annual investment amount.

Depending on the fee incorporated, the formula will need to be adjusted accordingly. For load fees, the amount of the principle that gets invested is lessened, so we adjust the principle. For 12b- 1 fees, the distributions are affected, so you adjust the interest rate. For exit fees, the final fund amount is the only thing affected, so we adjust the amount accordingly.

So, the amount in the accounts at the end of the 20 year investment period is as follows:

a) $ 3,000 * [ (1.10)^{20} + (1.10)^{19} + (1.10)^{18} + … + (1.10)^{1} ] =
**$ 189,007.50**

b) ($ 3,000 *** 0.95**) * [ (1.10)^{20} + (1.10)^{19} + (1.10)^{18} + … + (1.10)^{1} ] =** $ 179,557.12**

c) $ 3,000 * [ (1.10)^{20} *** (0.99)**^{20} + (1.10)^{19}
*** (0.99)**^{19} + … + (1.10)^{1} *** (0.99)**^{1} ] =
**$ 165,276.02**

d) $ 3,000 * [ (1.10)^{20} + (1.10)^{19} + (1.10)^{18} + … + (1.10)^{1} ]
*** 0.95** = **$ 179,557.12**

Notice the differences in where the fees affect the formula! A small fee that is compounded (like the 12b- 1 fee) will make a huge dent in your gains, whereas a larger fee that is not (like the front-end and back-end load fees) will have a smaller effect. Another point to watch in this problem: since the interest gained is independent of the amount invested (which is not entirely accurate for a stock market investment), the front-end and back-end load fees result in the same final amounts for subproblems b) and d).

You can save yourself a lot of trouble if you write a simple computer program (e.g., using a graphing calculator or other computational tool) to compute the sums for you.

For reference, the Wolfram Alpha commands to compute the above sums are here:

[ (1.10)^{20} + (1.10)^{19} + (1.10)^{18} + … + (1.10)^{1} ]

[ (1.10)^{20} * (0.99)^{20} + (1.10)^{19} * (0.99)^{19} + … + (1.10)^{1} * (0.99)^{1} ]

- Sum (1.1*.99)^j, j=1 to 20