Grayson Harris wishes to have $$1\mathrm{.}5$ million in a retirement fund $20$ years from now. He can create the retirement fund by making a single lump$-$sum deposit today.

a$.$ If he can earn $6$ percent on his investments, how much would Grayson deposit today to create the retirement fund?

b$.$ If he can earn only $4$ percent on his investments? Compare and discuss the results of your calculations.