I already have the answers for these questions but I don't really understand how I got them I just have the numbers matched. It would be helpful for me to have the formula broken out further...
Let the effective annual rate be 5 percent (i.e., r = .05) for all maturities.
a. Calculate the present value of a perpetuity that makes annual payments of $1,000,000 every year forever, with the next payment being made exactly one year from now.
Calculation: P/r = $1,000,000/.05 = $20,000,000
b. Calculate the present value of a perpetuity that makes annual payments of $1,000,000 every year forever, with the next payment being made exactly ten years from now.
Calculation: unsure
c. Calculate the present value of an annuity that makes annual payments of $1,000,000 every year for 9 years, with the next payment being made exactly one year from now.
Calculation: P*((1-((1+r)^-t))/r = $1,000,000*(1-(1.05^-9))/.05 = $7,107,822
d. Calculate the present value of a perpetuity that makes a payment of $1,000,000 every 6 months, with the next payment being made in exactly 6 months from now. Hint: Use the standard perpetuity formula but let the "period" be six months instead of a year and use the effective 6-month rate implied by the annual effective rate of 5 percent.
Calculation: unsure
e. Calculate the present value of an annuity that makes a payment of $1,000,000 every other year for 10 payments with the first payment being made exactly two years from now. Hint: Use the standard annuity formula, but let the "period" be two years (rather than just one year) and use the effective two-year rate implied by the annual effective rate of 5 percent.
Calculation: P*?/((1+r)^2) unsure
f. Calculate the present value of an annuity that makes a payment of $1,000,000 every other year for 10 payments with the first payment being made exactly one year from now. Hint: How is the present value of this annuity related to the annuity value in e above?