1 Chapter 3 – Important Stuff Mechanics of compounding / discounting PV, FV, PMT – lump sums and annuities Relationships – time, interest rates, etc Calculations: PV’s, FV’s, loan payments, interest rates

2 Time Value of Money (TVM) Time Value of Money – relationship between value at two points in time –Today versus tomorrow; today versus yesterday –Because an invested dollar can earn interest, its future value is greater than today’s value Problem types: monthly loan payments, growth of savings account; time to goal

3 Financial Calculator Keys PV-Present value FV-Future value PMT-Amount of the payment N-Number of periods (years?) I/Y-Interest rate per period

4 TI Calculator Manual Strongly Suggested Readings Getting Started – page 6 and 7 Overview – page 1-4, 1-10 and 1-20 Worksheets – pages 2-14 and 2-15 TVM – 3-1 to 3-9

5 Calculator Tips Decimals and Compounding Periods 2 nd (gray), Format (bottom row), 4, enter, CE/C (lower left) - hit twice Compounding: 2 nd, I/Y, 1, enter, CE/C – extremely important !! Right arrow key fixes “misteaks” One cash flow must be negative or error

6 Concept of Compounding Compound Interest – basically interest paid on interest Takes interest earned on an investment and reinvests it –Earn interest on the principal and reinvested interest

7 Compound 6% YearBeginInterestFuture Val 1$100.00$6.00$

8 Future Value Interest Factor

9 Future Value (FV) Algebraically FV n = PV (1 + i) n Underlies all TVM calculations Keystrokes: 100 +/- PV; 3 N; 0 PMT; 6 I/Y; CPT FV = One cash flow must be negative

10 Future Value Interest Factor

11 FV – Other Keystrokes How long for an investment to grow from $15,444 to $20,000 if earn 9% when compounded annually? Must solve for N /- PV; FV; 0 PMT; I/Y 9; CPT N = 3 years What rate earned if start at $15,444 to reach $20,000 in 3 years? Solve for I/Y /- PV; FV; 0 PMT; 3 N; CPT I/Y = 9%

12 FV Can Be Increased By 1. Increasing the length of time it is compounded 2. Compounding at a higher rate And/or 3. Compounding more frequently

13 FV at Different Rates and Periods

14 Present Value (PV) If I earn 10%, how much must I deposit today to have $100 in three years? $75.10 This is “inverse compounding” Discount rate – interest rate used to bring (discount) future money back to present For lump sums (only) PV and FV are reciprocals

15 Present Value Interest Year Year Year Year

16 Present Value Formula [ 1 ] PV=FV n [ (1 + i) n ] PVIF and FVIF for lump sums only are reciprocals. For 5% over ten years FVIF = 1.629=1 /.614 PVIF =.614=1 / 1.629

17 Keystrokes for ten years For PV 100 FV; 0 PMT; 5 I/Y; 10 N; CPT PV = For I/Y 100 FV; 0 PMT; +/ PV; 10 N; CPT I/Y = 5 For N 100 FV; +/ PV; 0 PMT; 5 I/Y;CPT N = 10 years

18 PV Decreases If 1.Number of compounding periods (time) increases, 2.The discount rate increases, And/or 3. Compounding frequency increases

19 Present Value – Different Rates

20 Annuities Series of equal dollar payments –Usually at the end of the year/period If I deposit $100 in the bank each year starting a year from now, how much will I have at the end of three years if I earn 6%?$ We are solving for the FV of the series by summing FV of each payment.

21 FV of $100 6% End of PMTFVIF $ Year 3$ *$ Year Year $ * The payment at end Year 3 earns nothing

22 Annuity Keystrokes What will I have if deposit $100 per year starting at the end of the year for three years and earn 6%? 0 PV; 100+/- PMT; 3 N; 6 I/Y; CPT FV = PV is zero - nothing in the bank today

23 Present Value of an Annuity Amount we must put in bank today to withdraw $500 at end of next three years with nothing left at the end? Present valuing each of three payments Keystrokes: 500+/- PMT; 0 FV; 3 N; 6 I/Y; CPT PV = 1,336.51

24 Nonannual Compounding Invest for ten years at 12% compounded quarterly. What are we really doing? –Investing for 40 periods (10 * 4) at 3% (12%/4) Make sure 2 nd I/Y is set to 1. Need to adjust rate per period downward which is offset by increase in N

25 Nonannual Compounding FV n =PV ( 1 = i/m) m * n m = number of compounding periods per year so per period rate is i/m And m * n is the number of years times the compounding frequency which adjusts to the rate per period

26 Compounding CompoundingOne Year 10 Years Annually$ $ Semiannually Quarterly Monthly

27 Amortizing Loans Paid off in equal installments –Makes it an annuity Payment pays interest first, remainder goes to principal (which declines) $600 loan at 15% over four years with equal annual payments of $210.16

28 $600 Loan Amortization TotalTo Int To PrinEnd Bal Year Year Year Year

29 Loan Interest & Principal Payment

30 Calculate a Loan Payment $8,000 car loan payable monthly over three years at 12%. What is your payment? How many monthly periods in 3 yrs? 36 N Monthly rate? 12%/12 = 1%/mo = I/Y What is FV? Zero because loan paid out 8000+/- PV; 0 FV; 1.0 I/Y; 36 N; CPT PMT=265.71

31 Perpetuities Equal payments that continue forever –Like Energizer Bunny and preferred stock Present Value = Payment Amount Interest Rate Preferred stock pays $8/yr, int rate- 10% Payment fixed at $8/.10 = $80 market price