1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 9.3 Saving Money

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Understand present value o Understand savings plans

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Saving Money Interest is not always a bad thing. It can lend a helping hand when we are saving money. Whether it is for a particular short term goal or a long term retirement plan, we can use interest to our benefit.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Present Value (PV) Present value (PV) is the amount of principal needed now in order to reach a future value amount. Present value is calculated with the formula where A is the future value, r is the APR, n is the number of compound intervals per year, and t is time, or term, in years.

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Computing Principal Needed for Future Value State College predicts that in 18 years it will take $150,000 to attend the college for four years. Amber has a substantial amount of cash and wishes to save money for her newborn child’s college fund. How much should Amber put aside in an account with an APR of 4%, compounded monthly, in order to have $150,000 in the account in 18 years?

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Computing Principal Needed for Future Value (cont.) Solution We know that Amber wishes to have a future value of $150,000 for her child’s college fund. Since she has found an investment opportunity that will give her monthly compounding at 4%, we know that A = 150,000, r = 0.04, n = 12, and t = 18. Substituting these values into the present value formula to find P, we have the following.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Computing Principal Needed for Future Value (cont.) This means that if Amber could deposit $73,100.31 into the account now, then after 18 years there will be $150,000 in the account.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 How much should you deposit now in an account with an APR of 7% compounded monthly if you wish for the account to have a balance of $200,000 in 25 years? Answers: $34,931.95

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Annuity An annuity is a sequence of regular payments made into an account, or taken out of an account, over time.

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Annuity Formula for Finding Future Value The future value (FV) of an annuity savings account is calculated with the formula

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Annuity Formula for Finding Future Value Annuity Formula for Finding Future Value (cont.) where PMT is the payment amount that is deposited on a regular basis, r is the APR, n is the number of regular payments made each year, and FV is the future value after t years.

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Future Value Using an Annuity Plan Daigle begins placing $75 per month into an annuity savings plan when he begins receiving his first official paycheck in August 2014. The savings plan earns 2.5% APR, compounded monthly. a.Determine the amount of money that Daigle will personally put in the savings plan over 10 years. b.Calculate the future value of Daigle’s savings in August 2024.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Future Value Using an Annuity Plan (cont.) Solution a.To find the amount of money that Daigle will personally add to the savings plan, we need to multiply the monthly payment amount by the number of payments each year and then by the number of years he plans to continue his payments.

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Future Value Using an Annuity Plan (cont.) This gives us the following. So, over the 10 years, Daigle will contribute $9000 to his savings plan.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Future Value Using an Annuity Plan (cont.) b.In order to find the future value of Daigle’s savings, we need several values. First, we are told that he will contribute $75.00 each month. So, we know that PMT = $75.00 and the number of payments made each year gives us n = 12. The APY given is 2.5%, or r = 0.025. The time period is from August 2014 through August 2024, so t = 10 years. Now, we can substitute these values in the annuity formula to find the future value.

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Future Value Using an Annuity Plan (cont.) Therefore, after 10 years, Daigle’s savings account will contain $10,212.90. Since we know that he contributed $9000, over 10 years Daigle’s savings earned interest in the amount of: $10,212.90  $9000 = $1212.90.

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Find the future value of the annuity plan in Example 2 after 15 years. Answers: $16,359.26

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Calculating Future Value for an Annuity (Savings Plan) Use the annuity formula to calculate the future value after six months of monthly $50 payments for an annuity savings plan having an APR of 12%. Solution We have monthly payments of PMT = $50, an annual interest rate of r = 0.12, n = 12 because the payments are made monthly, and because six months is a half year. Using the savings formula, we can find the future value after six months.

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Calculating Future Value for an Annuity (Savings Plan) (cont.) Replacing the variables in the formula with their values we obtain the following. Therefore, after six months, the savings plan has accrued $307.60.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Calculating Future Value of an IRA Using the Annuity Formula At the age of 25, Angela starts an IRA (individual retirement account) to save for retirement. She deposits $200 into the account each month. Based on past performance of similar accounts, she expects to obtain an annual interest rate of 8%. a.How much money will Angela have saved upon retirement at the age of 65? b.Compare the future value to the total amount Angela deposited over the time period.

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Calculating Future Value of an IRA Using the Annuity Formula (cont.) Solution a.Because Angela starts making payments at age 25 and stops at 65, she is making payments over 40 years. Using the savings plan annuity formula, we need the following: payments of PMT = $200, an interest rate of r = 0.08, and n = 12 for monthly deposits. The balance after t = 40 years is as follows.

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Calculating Future Value of an IRA Using the Annuity Formula (cont.)

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Calculating Future Value of an IRA Using the Annuity Formula (cont.) b.Now we wish to calculate the total deposit amount over the 40 years. Keep in mind that over the 40 years, Angela has made 480 monthly deposits of $200. That means she has deposited a total of 480 · $200 = $96,000. The rest is all interest. Or in other words, even though Angela only deposited $96,000, she made $602,201.57 in interest. Just think if she had started saving that same $200 per month when she was 18!

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #3 Repeat Example 4 with Angela saving $200/month from age 18 to age 65. Answers: $1,242,475.48

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Annuity Formula for Finding Payment Amounts The regular payment amount (PMT) needed to reach a future goal with an annuity savings account is calculated with the formula

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Annuity Formula for Finding Payment Amounts Annuity Formula for Finding Payment Amounts (cont.) where FV is the future value desired, r is the APR, n is the number of regular payments made each year for t years.

27. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding Monthly Payments in Order to Meet a Goal Suppose you’d like to save enough money to pay cash for your next computer. The goal is to save an extra $3000 over the next three years. What amount of monthly payments must you make into an account that earns 1.25% interest in order to reach your goal? Solution We can find the monthly payment required to meet the goal by using the annuity formula for payments. The future value is FV = 3000, r = 0.0125, n = 12, and t = 3 years. Substituting these values into the formula, we have the following.

28. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding Monthly Payments in Order to Meet a Goal (cont.) Therefore, in order to reach $3000 in three years with a 1.25% APR, you need to make a monthly payment of $81.82.

29. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Monthly Payments Amber wishes to deposit a fixed monthly amount into an annuity account for her child’s college fund. She wishes to accumulate a future value of $150,000 in 18 years. a.Assuming an APR of 4%, how much money should Amber deposit monthly in order to reach her goal? b.How much of the $150,000 will Amber ultimately deposit in the account, and how much is interest earned?

30. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Monthly Payments (cont.) Solution a.Amber’s goal is to have accumulated a balance of $150,000 after 18 years. She has an interest rate of 4% and is going to make monthly payments. Therefore, we know that P = 150,000, r = 0.04, n = 12, and t = 18. Now, substituting these into the formula, we can calculate the monthly payment needed.

31. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Monthly Payments (cont.) So, Amber will need to deposit at least $475.30 per month in order to meet her goal of $150,000.

32. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Monthly Payments (cont.) b.Amber will need to make 18 years worth of monthly payments of $473.72. We can find her contribution to the savings plan by multiplying these together. 18 ⋅ 12 ⋅ $475.30 = $102,664.80 The interest earned is then the difference between the goal and the contributed amount from Amber. So, we can subtract and have the following. $150,000 – $102,664.80 = $47,335.20 So, the savings plan earned $47,335.20 in interest over the 18 years.

33. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Monthly Payments for a Retirement Fund Assume you have just graduated from college and have obtained your first job making $44,250 per year. Currently, the national average retirement income for people with a bachelor’s degree is $60,000 per year. Assuming you are 22 years of age at graduation and you will retire at age 67, then you will be working—and investing—for 45 years. At current rates, a mutual fund has an APR of 8%. How much do you need to invest each month in order to live off of $60,000 per year when you retire?

34. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Monthly Payments for a Retirement Fund (cont.) Solution You want to build an account large enough so that you can have an annual annuity payment of $60,000 without reducing the principal on your retirement fund. In other words, you will live on the interest the account earns. We can phrase this another way by asking, “What balance is needed so that it earns $60,000 in annual interest?” Since we are assuming there is an APR of 8%, the $60,000 must be equal to 8% of the total balance.

35. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Monthly Payments for a Retirement Fund (cont.) We can write this as a mathematical equation, letting x equal the principal needed, and then solve for x as shown below.

36. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Monthly Payments for a Retirement Fund (cont.) With an 8% APR, a balance of $750,000 allows you to withdraw $60,000 per year without reducing the principal. Now that we know the total amount needed to generate our $60,000 per year income, we can use the monthly annuity formula to compute the monthly payment amount needed to meet this future value goal.

37. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Monthly Payments for a Retirement Fund (cont.) We have that FV = $750,000 from our calculation, r = 0.08, t = 45 years, and n = 12.

38. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Monthly Payments for a Retirement Fund (cont.) So, if you wish to retire and make $60,000 per year without affecting the principal in your retirement fund, you need to start at age 22 depositing approximately $143 per month. With a fixed APR of 8% over the years of your retirement, you will never outlive your retirement savings.