Present Value Professor XXXXX Course Name / Number

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

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

5 Periods 0% Future Value of One Dollar ($) % 5% 15% 20% The Power Of Compound Interest 40.00

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

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

8 Present Value of $200 (4 Years, 8% Interest ) Discounting PV = $200 FV1 = $216 FV2 = $ FV3 = $252 FV4 = $ End of Year

9 The Power Of High Discount Rates Periods Present Value of One Dollar ($) % 5% 15% 20% 0%

10 FV and PV of Mixed Stream (5 Years, 4% Interest Rate) PV $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 Future Value and Present Value of an Ordinary Annuity Present Value $1,000 $1,000 $1,000 $1,000 $1,000 Discounting End of Year Future Value Compounding

12 Future Value of Ordinary Annuity (End of 5 Years, 5.5% Interest Rate) $1,000 $1,000 $1,000 $1,000 $1,000 $1, $1, $1, $1, $1, End of Year How is Annuity Due Different ?

13 Future Value of Annuity Due (End of 5 Years, 5.5% Interest Rate) End of Year FV 5 = $5, $1, $1, $1, $1, $1, $1,000 $1,000 $1,000 $1,000 $1,000 Annuity Due - Payments Occur at the Beginning of Each Period

14 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $ $ $ $ $ Present Value of Ordinary Annuity (5 Years, 5.5% Interest Rate)

15 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $ $ $ $ Present Value of Annuity Due (5 Years, 5.5% Interest Rate) $

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 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 …

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

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 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 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 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 = $ –Quarterly: FV = $100 (1.0125) 4 = $ The Stated Rate Versus The Effective Rate 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 FV n 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 FV 5 ($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 PVFA 4,10% =3.1700

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$4, , , , , , , , , , , , a-a End of year a Due to rounding, a slight difference ($.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)

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