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Annuities and perpetuities based problems

Question 1: Calculate the following values, assuming a discount rate of 8%:

a. present value of a perpetuity (also called a perpetual annuity) of $50 received each year at the end of each year

b. present value of an annuity of $50 received at the end of each year for 5 years

c. present value of an annuity of $50 received at the end of each year for 10 years, with the first payment to be received at the end of the 6th year

d. present value of a perpetuity of $50, with the first payment received at the end of the 16th year.

Question 2:

a. Show (with a time line, for example) that the perpetuity in a (above question). is exactly the same as the sum of the annuities and perpetuities in b. to d (above question).

b.Show that their present values add up to the same amount.

Question 3:

a. Jane is 20 years old today. Jane is going to put $1,000 into her savings account on her 21st birthday and again on every birthday for 20 payments (i.e., till her 40th birthday). She will earn 5%, paid annually. How much money will be in the account after she collects her interest and makes her 20th payment?

b. Calculate how much money she could take out each year for the 20 years from her 41st birthday till her 60th birthday, assuming she still earns 5% and takes out the same amount each year, leaving exactly $0 in the account after removing her 20th payment.

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