Question $10$: Your grandmother, who happens to be a retired Finance Professor, wants to give you an annuity for your birthday. She offers you the following choices: $(1)$ $$500$ per year for $25$ years, starting at t$1$ with a discount rate of $4\%$; $(2)$ $$500$ per year for $25$ years, starting at t$0$ with a discount rate of $4\mathrm{.}5\%$; or $(3)$ $$450$ per year for $25$ years, starting at t$1,$ with a discount rate of $4\%,$ growing at a rate of $1\%.$

Which option has the highest PV$?$ Please SHOW YOUR CALCULATIONS for each option.