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 beginning of every six months

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

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

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

You bought an annuity selling at $$6,456\mathrm{.}97$ today that promises to make equal payments at the beginning of each year for the next four years $($ N $).$ If the annuity's appropriate interest rate $($I$)$ remains at $11\mathrm{.}00\%$ during this time, then the value of the annual annuity payment $($PMT$)$ is $$

You just won the lottery. Congratulations! The jackpot is $$35,000,000,$ paid in four equal annual payments. The first payment on the lottery jackpot will be made today. In present value terms, you really won $--$assuming annual interest rate of $11\mathrm{.}00\%.$