Consider the case of the following annuities, and the need to compute either their expected rate of return or duration.

Daniel needed money for some unexpected expenses, so he borrowed $3,965.25 from a friend and agreed to repay the loan in five equal installments of $1,100 at the end of each year. The agreement is offering an implied interest rate of .

Daniels friend, Ian, has hired a financial planner for advice on retirement. Considering Ians current expenses and expected future lifestyle changes, the financial planner has stated that once Ian crosses a threshold of $9,777,839 in savings, he will have enough money for retirement. Ian has nothing saved for his retirement yet, so he plans to start depositing $100,000 in a retirement fund at a fixed rate of 12.00% at the end of each year. It will take for Ian to reach his retirement goal.