Suppose the rates of $($realized$)$ return were $100\%$ in the $1$st year, $-50\%$ in the $2$nd year, $30\%$ in the $3$rd year.

$($i$)$ If you had invested $$1,000$ at the beginning of $1$st year, invested $$2,000$ at the beginning of the second year encouraged by the performance in the $1$st year, and withdrawn $$1,000$ at the beginning of the $3$rd year discouraged by the performance in the $2$nd year, how much would you have had at the end of the $3$rd year?

What would have been the annual rate of return on your invested money $($i$.$e$.,$ the dollar$-$weighted average rate of return$)$ over the three$-$year span? Please show your equation and calculation.

$($ii$)$ If you had invested $$1,000$ at the beginning of $1$st year, invested $$1,000$ at the beginning of the second year, and invested $$1,000$ at the beginning of the $3$rd year, how much would you have had at the end of the $3$rd year?

What would have been the annual rate of return on your invested money $($i$.$e$.,$ the dollar$-$weighted average rate of return$)$ over the three$-$year span? Please show your equation and calculation.

$($iii$)$ Comparing $($i$)$ and $($ii$)$ above, which investment strategy between the two above $($i$.$e$.,$ marketing timing vs$.$ constant dollar amount$)$ should you adopt going forward?