Your sister just deposited $$13,500$ into an investment account. She believes that she will earn an annual return of $10\mathrm{.}4$ percent for the next $10$ years. You believe that you will only be able to earn an annual return of $9\mathrm{.}6$ percent over the same period. How much more must you deposit today in order to have the same amount as your sister in $10$ years?