3-7 3.7 FUTURE VALUE OF INVESTMENTS Banking 11/27/2018 3-7 3.7 FUTURE VALUE OF INVESTMENTS OBJECTIVES Calculate the future value of a periodic deposit investment. Graph the future value function. Interpret the graph of the future value function. Chapter 1

Key Terms future value of a single deposit investment periodic investment biweekly future value of a periodic deposit investment

How can you effectively plan for the future balance in an account? How can you calculate what the value of a deposit will be after a certain amount of time? If you want your balance to be a specific amount at the end of a period of time, how do you determine how much your initial deposit and subsequent deposits should be? Remember to consider the annual interest rate when considering your answer.

Future value of a periodic deposit investment B = balance at end of investment period P = periodic deposit amount r = annual interest rate expressed as decimal n = number of times interest is compounded annually t = length of investment in years

Example 1 Rich and Laura are both 45 years old. They open an account at the Rhinebeck Savings Bank with the hope that it will gain enough interest by their retirement at the age of 65. They deposit $5,000 each year into an account that pays 4.5% interest, compounded annually. What is the account balance when Rich and Laura retire? Round to the nearest cent. B= P 1+ r n nt −1 r n = 5000 1+ .045 1 1∗20 −1 .045 1 =$𝟏𝟓𝟔,𝟖𝟓𝟕.𝟏𝟏

How much interest will Rich and Laura earn over the 20-year period? Example 2 How much interest will Rich and Laura earn over the 20-year period? Interest = Ending Balance – Amount Rich & Laura Deposited Rich & Laura Deposited: $5000×20=$100,000 $156,857.11−$100,000=$𝟓𝟔,𝟖𝟓𝟕.𝟏𝟏

CHECK YOUR UNDERSTANDING How much more would Rich and Laura have in their account if they decide to hold off retirement for an extra year? 5000 1+ .045 1 1∗21 −1 .045 1 =$𝟏𝟔𝟖,𝟗𝟏𝟓.𝟔𝟖 $𝟏𝟔𝟖,𝟗𝟏𝟓.𝟔𝟖−$𝟏𝟓𝟔,𝟖𝟓𝟕.𝟏𝟏=$𝟏𝟐,𝟎𝟓𝟖.𝟓𝟕

EXAMPLE 3 Linda and Rob open an online savings account that has a 3.6% annual interest rate, compounded monthly. If they deposit $1,200 every month, how much will be in the account after 10 years? B= P 1+ r n nt −1 r n = 1200 1+ .036 12 12×10 −1 .036 12 =$𝟏𝟕𝟑,𝟎𝟐𝟐.𝟖𝟕

EXAMPLE 4 Construct a graph of the future value function that represents Linda and Rob’s account for each month. Use the graph to approximate the balance after 5 years. 1200 1+ .036 12 𝑥 −1 .036 12

CHECK YOUR UNDERSTANDING Construct a graph for Rich and Laura’s situation in Example 1.

More Realistic Investment Invest $100, monthly at 3.6% for 30 years. B= P 1+ r n nt −1 r n = 100 1+ .036 12 12×30 −1 .036 12 ≈ $𝟔𝟒,𝟔𝟔𝟒