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:

$$ $$500$ per year for $25$ years, starting at t$1$ with a discount rate of $4\%$;

$$ $$500$ per year for $25$ years, starting at t$0$ with a discount rate of $4\mathrm{.}5\%$; or

$$ $$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.