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 $$1,000$ at the beginning of each year

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 end of each year