Chapter 3 – Important Stuff

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 ManualStrongly 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 • 2nd (gray), Format (bottom row), 4, enter, CE/C (lower left) - hit twice • Compounding: 2nd , 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 Interest @ 6% YearBeginInterestFuture Val 1 $100.00 $6.00 $106.00 2 106.00 6.36 112.36 3 112.36 6.74 119.10

8. Future Value Interest Factor Year@2%@6%@10% • 1.020 1.060 1.100 • 1.040 1.124 1.210 3 1.104 1.191 1.611 10 1.219 1.791 2.594

9. Future Value (FV) Algebraically FVn = PV (1 + i)n Underlies all TVM calculations Keystrokes: 100 +/- PV; 3 N; 0 PMT; 6 I/Y; CPT FV = 119.10 One cash flow must be negative

10. 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. • 15444 +/- PV; 20000 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. • 15444 +/- PV; 20000 FV; 0 PMT; 3 N; CPT I/Y = 9%

11. 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

12. FV at Different Rates and Periods

13. 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

14. Present Value Interest Factor @2%@5%@10% Year 1 .980 .952 .909 Year 2 .961 .907 .826 Year 3 .942 .864 .751 Year 10 .820 .614 .386

15. Present Value Formula [ 1 ] PV = FVn[ (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

16. Keystrokes$100 @5% for ten years • For PV 100 FV; 0 PMT; 5 I/Y; 10 N; CPT PV = 61.39 • For I/Y 100 FV; 0 PMT; +/-61.39 PV; 10 N; CPT I/Y = 5 • For N 100 FV; +/-61.39 PV; 0 PMT; 5 I/Y; CPT N = 10 years

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

18. Present Value – Different Rates

19. 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%? $318.36 • We are solving for the FV of the series by summing FV of each payment.

20. FV of $100 Annuity @ 6% End of PMTFVIF $ Year 3 $100 1.0000 * $100.00 Year 2 100 1.0600 106.00 Year 1 100 1.1236 112.36 $318.36 * The payment at end Year 3 earns nothing

21. 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 = 318.36 PV is zero - nothing in the bank today

22. 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 if we earn 6%? Present valuing each of three payments Keystrokes: 500+/- PMT; 0 FV; 3 N; 6 I/Y; CPT PV = 1,336.51

23. 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 2nd I/Y is set to 1. • Need to adjust rate per period downward which is offset by increase in N

24. Nonannual Compounding • FVn = 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

25. Compounding $100 @10% CompoundingOne Year10 Years Annually $110.00 $259.37 Semiannually 110.25 265.33 Quarterly 110.38 268.51 Monthly 110.47 270.70

26. 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

27. $600 Loan Amortization TotalTo IntTo PrinEnd Bal Year 1 210.16 90.00 120.16 479.84 Year 2 210.16 71.98 138.18 341.66 Year 3 210.16 51.25 158.91 182.75 Year 4 210.16 27.41 182.75 0

28. Loan Interest & Principal Payment

29. 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

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