$1$a$.$Assume that the current time period is period zero, what is the future value of $$2,000$ invested today if the investment is cashed out exactly $20$ years in the future if interest is compounded annually $($paid one a year$)$ and the annual interest rate is $7\%?$

$1$b$.$ Assume that the current time period is period zero, what is the future value of $$2,000$ invested today if the investment is cashed out in exactly $20$ years in the future if interest is compounded quarterly $($paid four times a year$)$ and the annual interest rate is $7\%?$ Explain the different future value relative to problem $1$a

$1$c$.$ Assume that the current time period is period zero, what is the future value of $$2,000$ invested today if the investment is cashed out in exactly $20$ years in the future if interest is compounded continuously $($paid continuously$)$ and the annual interest rate is $7\%?$ Explain the different future value relative to problems $1$a and $1$b$.$

$2$a$.$ Assume that the current time period is period zero, what is the present value of a $$30,000$ payment paid to you in exactly $20$ years, if interest is compounded annually $($paid once a year$)$ and the annual interest rate is $7\%?$

$2$b$.$ Assume that the current time period is period zero, what is the present value of a $$30,000$ payment paid to you in exactly $20$ years, if interest is compounded quarterly $($paid four times a year$)$ and the annual interest rate is $7\%?$ Explain the different present value relative to problem $2$a$.$

$2$c$.$ Assume that the current time period is period zero, what is the present value of a $$30,000$ payment paid to you in exactly $20$ years, if interest is compounded continuously $($paid continuously throughout the year$)$ and the annual interest rate is $7\%?$ Explain the different present value relative to the present values reported in problems $2$a and $2$b$.$

$3$a$.$ Assume that the current time period is period zero, what is the present value of a perpetuity that pays $$3,000$ annually $($each year$)$ beginning in exactly one year if interest is compounded annually $($paid once a year$)$ and the annual interest rate is $6\mathrm{.}5\%?$

$3$b$.$ Assume that the current time period is period zero, what is the present value of a perpetuity that pays $$3,000$ annually $($each year$)$ beginning in exactly one year if interest is compounded continuously $($continuously paid throughout the year$)$ and the annual interest rate is $6\mathrm{.}5\%?$ Explain the different present value relative to problem $3$a$.$

$3$c$.$ Assume that the current time period is period zero, what is the present value of a perpetuity that pays $$3,000$ annually $($each year$)$ beginning in exactly $15$ years, if interest is compounded annually $($paid once a year$)$ and the annual interest rate is $6\mathrm{.}5\%?$ Explain the difference in the present value you calculated for this problem relative to the values you calculated in problem $3$a$.$

$3$d$.$ Assume that the current time period is period zero, what is the present value of a perpetuity that pays $$3,000$ annually $($each year$)$ beginning in exactly $15$ years, if interest is compounded continuously $($continuously paid throughout the year$)$ and the annual interest rate is $6\mathrm{.}5\%?$ Explain the difference in the present value you calculated for this problem relative to the value you calculated in problem $3$c$.$

$4$a$.$ Assume your financial advisor suggest it is time to take your lifetime retirement savings of $$900,000$ and purchase a retirement annuity that will pay you $$60,000$ per year for twenty years with the first payment beginning in exactly one year. As part of his sales pitch, your advisor informs you the annuity will guarantee you, or your heirs in the event of your death, a total payout of $$1,200,000$ over the $20-$year payout period. Assume the annual interest rate is $4\mathrm{.}5\%$ and interest is compounded annually $($paid once a year$).$ What is the present value of the annuity? Should you purchase the annuity?