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# Present Value

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1. Present Value Professor XXXXX Course Name / Number

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

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

4. FV4 = \$272.10 FV3 = \$251.94 FV2 = \$233.28 FV1 = \$216 Future Value of \$200 (4 Years, 8% Interest ) PV = \$200 0 1 2 3 4 End of Year Compounding – The Process of Earning Interest in Each Successive Year

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

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

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

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

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

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

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

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

13. FV5 = \$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 Future Value of Annuity Due(End of 5 Years, 5.5% Interest Rate) 0 1 2 3 4 5 End of Year Annuity Due - Payments Occur at the Beginning of Each Period

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

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

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. Present Value Of Growing Perpetuity 0 1 2 3 4 5 \$1,000 \$1,020 \$1,040.4 \$1,061.2 \$1,082.4 … Growing Perpetuity CF1 = \$1,000 r = 7% per year g = 2% per year

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

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

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

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

22. The Stated Rate Versus The Effective Rate • 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 Stated Rate of 5% Does Not Change.What About the Effective Rate?

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

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 FVn dollars, at interest rate, r, over n years, by solving this equation for PMT:

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 FV5 (\$35,000), at r=4%, and n=5years

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 PVFA4,10%=3.1700

27. A Loan Amortization Table Loan Amortization Schedule (\$6,000 Principal, 10% Interest 4 Year Repayment Period Payments End of year Beginning-of-year principal(2) End-of-year principal[(2) – (4)](5) Interest[.10 x (2)](3) Loan Payment(1) Principal[(1) – (3)](4) aDue 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)

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