Example $3\mathrm{.}5$

You are considering an investment of $10,000$ in a mutual fund for the purpose of adding to your retirement income, you expect to retire $40$ years from now. You are choosing between an index tracker fund and an activity managed fund. The index tracker fund has no initial charge, an annual management charge of $0\mathrm{.}5\%$ of the value of the fund, and other costs $($including brokerage fees$)$ amount to $0\mathrm{.}3\%$ p$.$a$.$

The actively managed fund has an initial charge of $5\%$ of the value of the investment, an annual management charge of $1\mathrm{.}5\%$ of the value of the fund, and other costs $($including brokerage fees$)$ amount to $1\mathrm{.}1\%$ p$.$a$.$ of the value of the fund. You have read that over the long term the average real rate of return on a balanced portfolio of shares has been $6\%$ p$.$a$.$

You expect that performance to continue for the next $10$ years, on the assumption that the funds match the general market performance.

Required:

$($a$)$ What is the annual total expense ratio $($including the brokerage fees$)$ in each case?

$($b$)$ What do you expect the value of the index tracker fund to be after $10$ years?

$($c$)$ What are the effects of the expenses on the expected average annual return of the tracker fund?

$($d$)$ What do you expect the value of actively managed fund to be after $10$ years?

$($e$)$ What are the effects of the expenses on the average annual return of the actively managed fund?

$($f$)$ What would be the effects of charges on the expected average annual returns of each fund if the investment horizon were $5$ years rather than $10$ years?