Present value of annuities and annuity payments

The present value of an annuity is the sum of the discounted value of all future cash flows.

You have the opportunity to invest in several annuities. Which of the following $10-$year annuities has the greatest present value $($PV$)?$ Assume that all

annuities earn the same positive interest rate.

An annuity that pays $$500$ at the end of every six months

An annuity that pays $$500$ at the beginning of every six months

An annuity that pays $$1,000$ at the beginning of each year

An annuity that pays $$1,000$ at the end of each year

An ordinary annuity selling at $$2,514\mathrm{.}15$ today promises to make equal payments at the end of each year for the next if

appropriate interest rate $($I$)$ remains at $8\mathrm{.}00\%$ during this time, the annual annuity payment $($PMT$)$ will be

You just won the lottery. Congratulations! The jackpot is $$10,000,000,$ paid in eight equal annual payments. The first payment on the lottery

will be made today. In present value terms, you really won.

$-$assuming annual interest rate of $8\mathrm{.}00\%.$