Suppose that Thor deposited $$7,000$ into an investment account at the beginning of $2018.$ He later withdrew $$6,500$ from that account at the beginning of $2019.$ Upon seeing great historical returns that the S&P $500$ generated in $2019,$ he deposited an additional amount of $$100$ to the account at the beginning of $2020.$

On the other hand, Loki started out his investment with $$50$ at the beginning of $2018.$ Later on$,$ he deposited an additional $$500$ at the beginning $2019.$ Loki did not invest any additional amount at the beginning of $2020.$

Both Thor and Loki liquidated their holdings and cashed out of their investments at the end of $2020$

Following is the historical annual returns from the S&P $500$ index

$2018$ : loss of $4\mathrm{.}38\%$

$2019$: gain of $31\mathrm{.}49\%$

$2020$: gain of $18\mathrm{.}40\%$

Answer the following:

$(1)$ What is the range of returns that you should expect the $3-$year arithmetic average of the S&P $500$ to fall within? And more importantly WHY?

$(2)$ Do you expect the $3-$year geometric average of the S&P $500$ to be higher or lower than the $3-$year arithmetic average? And more importantly WHY?

$(3)$ Accounting both for the $$magic$$ of compounding and for the subsequent cashflows, calculate Thor$$s $3-$year average return and Loki$$s $3-$year average return.