1. Present Value Professor XXXXX Course Name / Number

2. 2 Future Value The Value of a Lump Sum or Stream of Cash Payments at a Future Point in Time FV n = PV x (1+r) n Future Value depends on: – Interest Rate – Number of Periods – Compounding Interval

3. 3 Future Value of $200 (4 Years, 7% Interest ) What if the Interest Rate Goes Up to 8% ? 0 1 2 3 4 PV = $200 End of Year FV 1 = $214 FV 2 = $228.98 FV 3 = $245 FV 4 = $262.16

4. 4 Future Value of $200 (4 Years, 8% Interest ) 0 1 2 3 4 PV = $200 End of Year FV 1 = $216 FV 2 = $233.28 FV 3 = $251.94 FV 4 = $272.10 Compounding – The Process of Earning Interest in Each Successive Year

5. 5 Periods 0% Future Value of One Dollar ($) 1.00 0 2 4 6 8 10 12 14 16 18 20 22 24 10.00 15.00 20.00 25.00 30.00 5.00 10% 5% 15% 20% The Power Of Compound Interest 40.00

6. 6 Present Value Today's Value of a Lump Sum or Stream of Cash Payments Received at a Future Point in Time

7. 7 Present Value of $200 (4 Years, 7% Interest ) What if the Interest Rate Goes Up to 8% ? 0 1 2 3 4 Discounting PV = $200 FV1 = $214 FV2 = $228.98 FV3 = $245 FV4 = $262.16 End of Year

8. 8 Present Value of $200 (4 Years, 8% Interest ) 0 1 2 3 4 Discounting PV = $200 FV1 = $216 FV2 = $233.28 FV3 = $252 FV4 = $272.10 End of Year

9. 9 The Power Of High Discount Rates Periods Present Value of One Dollar ($) 0 2 4 6 8 10 12 14 16 18 20 22 24 0.5 0.75 1.00 0.25 10% 5% 15% 20% 0%

10. 10 FV and PV of Mixed Stream (5 Years, 4% Interest Rate) PV $5,217.7 0 1 2 3 4 5 -$10,000 $3,000 $5,000 $4,000 $3,000 $2,000.0 Discounting End of Year FV $6,413.8 Compounding - $12,166.5 $3,509.6 $5,624.3 $4,326.4 $3,120.0 $4,622.8 $3,556.0 $2,564.4 $1,643.9 $2,884.6

11. 11 Future Value and Present Value of an Ordinary Annuity Present Value 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 $1,000 Discounting End of Year Future Value Compounding

12. 12 Future Value of Ordinary Annuity (End of 5 Years, 5.5% Interest Rate) 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 $1,000 $1,238.82 $1,174.24 $1,113.02 $1,055.00 $1,000.00 End of Year How is Annuity Due Different ?

13. 13 Future Value of Annuity Due (End of 5 Years, 5.5% Interest Rate) 0 1 2 3 4 5 End of Year FV 5 = $5,888.04 $1,306.96 $1,238.82 $1,174.24 $1,113.02 $1,055.00 $1,000 $1,000 $1,000 $1,000 $1,000 Annuity Due - Payments Occur at the Beginning of Each Period

14. 14 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $947.87 $898.45 $851.61 $807.22 0 1 2 3 4 5 $765.13 Present Value of Ordinary Annuity (5 Years, 5.5% Interest Rate)

15. 15 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $947.87 $898.45 $851.61 $807.22 0 1 2 3 4 5 Present Value of Annuity Due (5 Years, 5.5% Interest Rate) $1000.00

16. 16 Present Value Of Perpetuity ($1,000 Payment, 7% Interest Rate) Stream of Equal Annual Cash Flows That Lasts “Forever” What if the Payments Grow at 2% Per Year?

17. 17 Present Value Of Growing Perpetuity Growing Perpetuity CF 1 = $1,000 r = 7% per year g = 2% per year $1,000 $1,020 $1,040.4 $1,061.2 $1,082.4 … 0 1 2 3 4 5

18. 18 Compounding Intervals m compounding periods The More Frequent The Compounding Period, The Larger The FV!

19. 19 Compounding More Frequently Than Annually – For Quarterly Compounding, m Equals 4: – For Semiannual Compounding, m Equals 2: FV at End of 2 Years of $125,000 Deposited at 5.13% Interest

20. 20 Continuous Compounding In Extreme Case, Interest - Compounded Continuously FV n = PV x (e r x n ) FV at End of 2 Years of $125,000 at 5.13 % Annual Interest, Compounded Continuously FV n = $138,506.01

21. 21 The Stated Rate Versus The Effective Rate Effective Annual Rate (EAR) – The Annual Rate Actually Paid or Earned Stated Rate – The Contractual Annual Rate Charged by Lender or Promised by Borrower

22. 22 FV of $100 at End of 1 Year, Invested at 5% Stated Annual Interest, Compounded: –Annually: FV = $100 (1.05) 1 = $105 –Semiannually: FV = $100 (1.025) 2 = $105.06 –Quarterly: FV = $100 (1.0125) 4 = $105.09 The Stated Rate Versus The Effective Rate Stated Rate of 5% Does Not Change. What About the Effective Rate?

23. 23 Effective Rates - Always Greater Than Or Equal To Stated Rates For Annual Compounding, Effective = Stated For Semiannual Compounding For Quarterly Compounding

24. 24 Deposits Needed To Accumulate A Future Sum Often need to find annual deposit needed to accumulate a fixed sum of money in n years Closely related to the process of finding the future value of an ordinary annuity Find annual deposit needed to accumulate FV n dollars, at interest rate, r, over n years, by solving this equation for PMT:

25. 25 Calculating Deposits Needed To Accumulate A Future Sum You wish to accumulate $35,000 in five years to make a home down payment. Can invest at 4% annual interest. Find the annual deposit required to accumulate FV 5 ($35,000), at r=4%, and n=5years

26. 26 Calculating Amortized Loan Payments Amounts Very common application of TV: Finding loan payment amounts Amortized Loans are loans repaid in equal periodic (annual, monthly) payments Borrow $6,000 for 4 years at 10%. Find annual payment. Divide PV by PVFA 4,10% =3.1700

27. 27 A Loan Amortization Table Loan Amortization Schedule ($6,000 Principal, 10% Interest 4 Year Repayment Period Payments 1$1,892.74$6,000$600$1,292.74$4,707.26 21,892.744,707.26470.731,422.013,285.25 31,892.743,285.25328.531,564.211,721.04 41,892.741,721.04172.101,720.64-a-a End of year a Due to rounding, a slight difference ($.40) exists between beginning-of-year 4 principal (in column 2) and the year-4 principal payment (in column 4) Loan Payment (1) Beginning- of-year principal (2) Interest [.10 x (2)] (3) Principal [(1) – (3)] (4) End-of-year principal [(2) – (4)] (5)

28. Much Of Finance Involves Finding Future And (Especially) Present Values r Central To All Financial Valuation Techniques r Techniques Used By Investors & Firms Alike