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# 14.1 Future Value of an Annuity

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1. 14.1 Future Value of an Annuity • Find the future value of: • an annuity using the simple interest formula • an ordinary annuity using a \$1.00 ordinary annuity future value table • an annuity due using the simple interest formula • an annuity due using a \$1.00 ordinary annuity future value table • an annuity using a formula

2. 14.1.1 Future Value of an Annuity • Calculate the value of a growing account subject to periodic investments of payments. • Some examples include: • Retirement funds • College education • Vacation • Company’s future investment in capital expenses

3. Key Terms • Annuity payment: a payment made to an investment fund each period at a fixed interest rate. • Sinking fund payment: a payment made to an investment fund each period at a fixed interest rate to yield a predetermined future value. • Annuity certain: an annuity paid over a guaranteed number of periods.

4. Key Terms • Contingentannuity: an annuity paid over an uncertain number of periods. • Ordinary annuity: an annuity for which payments are made at the end of each period. • Annuity due: an annuity for which payments are made at the beginning of each period.

5. Future value of an annuity using the simple interest formula • Find the end-of-period principal. First end-of-period principal = annuity payment • For each remaining period in turn:End-of-period principal = previous end-of-period principal x (1 + period interest rate) + annuity payment. • Identify the last end-of-period principal as the future value.Future value = last end-of-period principal

6. Look at this example • What is the FV of an annual ordinary annuity of \$1,000 for 3 years at 4% annual interest? • End-of-year 1 = \$1,000 (no interest earned Y1) • End-of-year 2 = \$1,000 + \$1,000 (1.04) = \$2,040 • End of year 3 = \$1,000 + \$ 2,040 (1.04) = \$3,121.60 • The future value is \$3,121.60.

7. Figure 14-2

8. Try this example • Find the future value of an annual ordinary annuity of \$1,500 for four years at 3% annual interest. • \$6,270

9. 14.1.2 Find the FV Using a \$1.00 Ordinary Annuity FV Table • Using Table 14-1 in your text: • Select the periods row corresponding to the number of interest periods. • Select the rate per month column corresponding to the period interest rate. • Locate the value in the cell where the periods row intersects with the rate-per-period column. • Multiply the annuity payment by the table from step 3.

10. FV = annuity payment x table value • Using Table 14-1 to find the FV of a semiannual ordinary annuity of \$6,000 for five years at 6% annual interest, compounded semiannually. • 5 years x 2 periods per year = 10 periods • 6% annual interest rate = 3% period interest rate 2 periods per year • See Table 14-1 for 10 periods at 3% = 11.464 • FV = \$6,000 x 11.464 = \$68,784 • The future value of this annuity is \$68,784.

11. Try this example • Find the future value of a semiannual ordinary annuity of \$ 5,000 for 10 years at 4% annual interest compounded semiannually. • \$121,485

12. 14.1.3 Find the FV of Annuity Due Using the Simple Interest Formula 1. Find the first end-of-month period principal: multiply the annuity payment by the sum of 1 and the period interest rate. 2. For each remaining period in turn, find the next end-of-period principal= previous end of period principal = annuity payment x 1 + period interest rate 3. Identify the last end-of-period principal as the future value.

13. Look at this example • Find the total interest earned on the annuity of \$6,000 we looked at on Slide 11. • Total invested = \$6,000 x 10 (number of payments) = \$60,000 • Total interest = \$68,784 - \$60,000 = \$8,784. • The total interest earned on this annuity is \$8,784

14. Ordinary annuity versus annuity due • The difference between an ordinary annuity and an annuity due is whether you made the first payment immediatelyorat the end of the first period.

15. Find the FV of this annuity due • Find the FV of annuity due of \$1,000 for three years at 4% annual interest. Find the total investment and total interest earned. • End-of-Y 1 value = \$1,000 x 1.04 = \$1,040. • End-of-Y 2 value = \$2,040 x 1.04 = \$2,121.60 • End-of-Y 3 value = \$3,121.60 x 1.04 = \$3,246.46 • The future value of this annuity is \$3,246.46 • The interest earned = \$246.46

16. Try this example • Find the future value of an annual annuity due of \$5,000 for three years at 4%. Find the total investment amount and the total interest earned. • Total investment = \$15,824.32 • Total interest earned = \$824.32

17. 14.1.4 Find the FV of an Annuity Due Using a \$1.00 Ordinary Annuity FV Table Using Table 14-1: • Select the periods row corresponding to the number of interest periods. • Select the rate-per-period column corresponding to the period interest rate. • Locate the value in the cell where the periods row intersects the rate-per-period column. (next slide)

18. Using a \$1.00 ordinary annuity FV table 4. Multiply the annuity payment by the table value from step 3. This is equivalent to an ordinary annuity. 5. Multiply the amount that is equivalent to an ordinary annuity by the sum of 1 and the period interest rate to adjust for the extra interest that is earned on an annuity due. Future value = annuity payment x table value x (1 = period interest rate)

19. Look at this example • Using Table 14-1, find the FV of a quarterly annuity due of \$2,800 for four years at 8% annual interest, compounded quarterly. • 4 years x 4 periods per year = 16 periods • 8% annual interest rate ÷ 4 periods p/year = 2% • Table 14-1 value for 16 periods at 2% = 18.639 • FV = \$2,800 x 18.639 x 1.02 = \$52,232.98 • The future value of this annuity is \$52,232.98

20. Try this example • Using Table 14-1, find the FV of a quarterly annuity due of \$1,800 for three years at 8% annual interest, compounded quarterly. • \$24,624.43

21. 14.1.5 Find the FV of an Ordinary Annuity or Annuity Due Using a Formula • Identify the period rate (R) as a decimal equivalent, the number of periods (N), and the amount of the annuity payment (PMT). • Substitute the values from Step 1 into the appropriate formula. (next slide)

22. 14.1.5 Find the FV of an Ordinary Annuity or Annuity Due Using a Formula (next slide)

23. Try this example Find the future value of an ordinary annuity of \$100 paid monthly at 5.25% for 10 years. • R = .0525/12 = .004375 (Period Int. Rate) The future value of the ordinary annuity is \$15,737.70.

24. Try this example Find the future value of an annuity due of \$50 monthly at 5.75% for 5 years. • R = .0575/12 = .0047916667 (Period Int. Rate) The future value of the ordinary annuity is \$15,737.70.

25. 14.2 Sinking Funds and the Present Value of an Annuity • Find the sinking fund payment using a \$1.00 sinking fund payment table. • Find the present value of an ordinary annuity using a \$1.00 ordinary annuity present value table. • Find the sinking fund payment or the present value of an annuity using a formula.

26. 14.2.1 Find the Sinking Fund Payment • Select the periods row corresponding to the number of interest periods. • Select the rate-per-period column corresponding to the period interest rate. • Locate the value in the cell where the periods row intersects the rate-per-period column. • Multiply the table value from step 3 by the desired future value Sinking fund payment = FV x Table 14.2 value

27. Look at this example • Using Table 14-2, find the annual sinking fund payment required to accumulate \$140,000 in 12 years at 6% annual interest rate. • Table 14-2 indicates that a 12-period value at 6% is equal to 0.0592770 • SFP = \$140,000 x 0.0592770 = \$8,298.78 • A sinking fund payment of \$8,298.78 is required at the end of each year for 12 years at 6% to yield the desired \$140,000.

28. Try this example • Use Table 14-2 for find the annual sinking fund payment required to accumulate \$100,000 in 10 years at 4% annual interest. • Find the number of periods: 10 • Find the table value where 10 periods and 4% intersect: 0.0832909 • Multiply the desired FV by the table value • The annual sinking fund payment required to accumulate \$100,000 in 10 years is \$8,329.09

29. 14.2.2 Find the PV of an Ordinary Annuity Using a \$1.00 Ordinary Annuity PV Table • Use Table 14-3 in your text to locate the given number of periods and the given rate per period. • Multiply the table value times the periodic annuity payment. Present value of an annuity = periodic annuity payment x table value

30. Look at this example • Use Table 14-3 to find the present value of a semiannual ordinary annuity of \$3,000 for seven years at 6% annual interest, compounded semiannually. • 7 years x 2 periods per year = 14 periods • 6% annual interest rate ÷ 2 periods p/year = 3% period interest rate • PV annuity = \$3,000 x 11.296 (table factor)= \$33,888 • By investing \$33,888 now at 6% interest, compounded semiannually, you can receive an annuity payment of \$3,000 twice a year for seven years.

31. Try this example • Roberto Santos wants to know how much he will have to invest now to receive an annuity payment of \$5,000 twice a year for ten years. The money will be invested at 6% annually compounded semiannually. • Number of periods = 20; Interest per period = 3% • Table factor = 14.877 • 5,000 x 14.877 = 74,385 • He must invest \$74,385 now to receive a \$5,000 annuity payment twice a year for 10 years.

32. 14.2.3 Find the Sinking Fund Payment or the Present Value of an Annuity Using a Formula • Identify the period rate (R) as a decimal equivalent, the number of periods (N), and the future value (FV) of the annuity. • Substitute the values from Step 1 into the appropriate formula. (next slide)

33. 14.2.3 Find the Sinking Fund Payment or the Present Value of an Annuity Using a Formula (next slide)

34. Try this example Find the monthly contribution to reach \$100,000 in 20 years with an annuity fund that earns 5.5% annual interest. R = .055/12 = .0045833333; N = 240 The payment required each month into the sinking fund is \$229.56.

35. Try this example How much is needed in a fund that pays 5.5% to receive \$700 per month for 20 years. R = .055/12 = .0045833333; N = 240 \$85,670.56 is needed in the fund to receive \$700 each month for 20 years.

36. Remember!