Consider the case of the following annuities, and the need to compute either their expected rate of return or duration. Matthew needed money for some unexpected expenses, so he borrowed $3,900.55 from a friend and agreed to repay the loan in four equal installments of $1,100 at the end of each year. The agreement is offering an implied interest rate of Matthew's friend, Gregory, has hired a financial planner for advice on retirement. Considering Gregory's current expenses and expected future lifestyle changes, the financial planner has stated that once Gregory crosses a threshold of $4,136,860 in savings, he will have enough money for retirement. Gregory has nothing saved for his retirement yet, so he plans to start depositing $40,000 in a retirement fund at a fixed rate of 5.00% at the end of each year. It will take for Gregory to reach his retirement goal.