4.1 Assume that the correct discount rate for the following cash flows is 8%. What is the present value of the following cash flows?

• $50 at the end of 3 years $50 at the end of 100 years $50 received at the end of each year for 20 years
• $50 received at the beginning of each year, totaling 20 payments

(8 marks)

4.2 Assuming an 8% discount rate, what is the future value of the following cash flows?

• future value in 3 years of $50 received now future value in 100 years of $50 received now future value at the end of 20 years of $50 received each year at the end of the year
• future value at the end of 20 years of $50 received each year at the beginning of the year, again totaling 20 payments

(8 marks)

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

• present value of a perpetuity (also called a perpetual annuity) of $50 received each year at the end of each year
• present value of an annuity of $50 received at the end of each year for 5 years

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 6^th yearpresent value of a perpetuity of $50, with the first payment received at the end of the 16^th year

(8 marks)

4.4 a. Show (with a time line, for example) that the perpetuity in 4.3a. is exactly the same as the sum of the annuities and perpetuities in 4.3b. to 4.3d.

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

(6 marks)

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

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