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

James needed money for some unexpected expenses, so he borrowed $$4,140\mathrm{.}16$ from a friend and agreed to repay the loan in four equal installments of $$1,250$ at the end of each year. The agreement is offering an implied interest rate of

James's friend, Brandon, has hired a financial planner for advice on retirement. Considering Brandon's current expenses and expected future lifestyle changes, the financial planner has stated that once Brandon crosses a threshold of $$1,386,953$ in savings, he will have enough money for retirement. Brandon has nothing saved for his retirement yet, so he plans to start depositing $$55,000$ in a retirement fund at a fixed rate of $8\mathrm{.}00\%$ at the end of each year. It will take for Brandon to reach his retirement goal.