You are considering an investment of Tshs $10\mathrm{.}000,000$ in a mutual fund for the purpose of adding to your retirement income. You expect to retire $20$ years from now. You are choosing between an index tracker fund and an actively managed fund. The index tracker fund has no initial charge, an annual management charge of $2\mathrm{.}5\%$ of the value of the fund and average annual share dealing costs of $1\mathrm{.}3\%.$

The actively managed fund has an initial charge of $5\%$ of the investment, an annual management charge of $1\mathrm{.}5\%$ of the value of the fund, and average annual share dealing costs of $1\%.$ You have read that over the long term the average real rate of renum on a balanced portfolio of shares has been $8\%$ p$.$a$.$ You expect that performance to continue for the next $20$ years. On the assumption that the funds match the general market performance:

Required:

$($a$)$ What do you expect the value of the index tracker fund to be after $20$ years? $(4$ Marks$)$

$($b$)$ What do you expect the value of the actively managed fund to be after $20$ years? $(4$ Marks$)$