1. Ch 4: Introduction to Time Value of Money Dr. Yi

2. 2 Outlines of Ch 4 and 5: Time Value of Money (TVM)  Basics  Simple present / future value problem  Simple vs. Compound Interest  Compounding Frequency  Annuity / Annuity Due / Perpetuity  Uneven cash flow  Amortized loan  Effective annual rate

3. 3 Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 0123 i% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. TodayEnd of second year Start of third year

4. 4 The Time Value of Money and Interest Rates  Timelines –Constructing a Timeline –Identifying Dates on a Timeline Date 0 is today, the beginning of the first year Date 1 is the end of the first year

5. 5 The Time Value of Money and Interest Rates  The Time Value of Money –In general, a dollar today is worth more than a dollar in one year If you have $1 today and you can deposit it in a bank at 10%, you will have $1.10 at the end of one year –The difference in value between money today and money in the future is the time value of money. $1.10 minus $1.00 = 10 cents

6. 6 The Time Value of Money and Interest Rates  The Interest Rate: Converting Cash Across Time –By depositing money, we convert money today into money in the future –By borrowing money, we exchange money today for money in the future –Interest Rate (i) The rate at which money can be borrowed or lent over a given period –Interest Factor It is the rate of exchange between dollars today and dollars in the future It has units of “$ in one year/$ today” It is expressed as (1 + i)

7. 7 The Time Value of Money and Interest Rates  The Interest Rate: Converting Cash Across Time –Discount Factors and Rates Money in the future is worth less today so its price reflects a discount Discount Rate ( i ) –The appropriate rate to discount a cash flow to determine its value at an earlier time Discount Factor –The value today of a dollar received in the future, expressed as: $1 / (1 + i)

8. 8 The Rules of Time Travel  Rule 1: Comparing and Combining Values –It is only possible to compare or combine values at the same point in time  Rule 2: Compounding –To calculate a cash flow’s future value, you must compound it  Rule 3: Discounting –To calculate the value of a future cash flow at an earlier point in time, we must discount it.

9. 9 Compounding vs. Discounting

10. 10 Four variables  PV  FV  Interest rate, i  Period, time length, or investment horizon, n FV 012 Year, n i% PV

11. 11 FV and PV  Future Value  Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.  Present Value  Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

12. 12 What’s the FV of an $100 after 3 years if i = 10%? FV = ? 0123 10% Finding FVs (moving to the right on a time line) is called compounding.  Multiply by (1+i) N 100

13. 13 10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding.  Divide by (1+i) N 100 0123 PV = ?

14. 14 Present Values  How much do I have to invest today to have some amount in the future? –FV = PV(1 + i) n –Rearrange to solve for PV = FV / (1 + i) n  When we talk about discounting, we mean finding the present value of some future amount.  When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

15. 15 Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th. Three Known Variables and One Unknown Variable

16. 16 Simple present / future value problem: Type I (FV Problem)  How much will I get tomorrow? FV = ? 0N=3 i%=10 PV=100 FV = PV * (1 + i%) n = 100 * (1+10%) 3 = 100 * FVIF 10%, 3 = 100 * 1.3310 = $133.10 3  N 10  I 100  PV Hit FV

17. 17 Simple present / future value problem: Type II (PV Problem)  How much should I save today? FV=100 0N=3 i%=10 PV = ? PV = FV / (1 + i%) n = 100 * (1/(1+10%) 3 )= 100 * PVIF 10%, 3 = 100 * 0.7513 = $75.13 3  N 10  I 100  FV Hit PV

18. 18 Simple present / future value problem: Type III (# of Period Problem)  How long it will take to reach my goal? FV=133.10 0N = ? i%=10 PV=-100 10  I -100  PV 133.10  FV Hit N FV = PV * (1 + i%) n FV / PV = (1 + i%) n ln(FV/PV) = ln(1 + i%) n ln(FV/PV) = n*ln(1 + i%) ln(FV/PV) / ln(1+i%) = N

19. 19 Simple present / future value problem: Type IV (Interest Rate Problem)  At what interest rate should I save to reach my goal? FV=133.10 0N=3 i% = ? PV=-100 FV = PV * (1 + i%) n FV / PV = (1 + i%) n (FV / PV) 1/n = 1 + i% (FV / PV) 1/n - 1 = i% 3  N -100  PV 133.10  FV Hit I

20. 20 Using a Financial Calculator: Setting the keys  HP 10BII –Gold → C All (Hold down [C] button) Check P/YR –# → Gold → P/YR Sets Periods per Year to # –Gold → DISP → # Gold and [=] button Sets display to # decimal places

21. 21 Examples  Example 1: Heather is considering the purchase of an apartment building. She estimates the building's current value at $400,000. At an annual increase of 8 percent, how much would the investment be worth at the end of 5 years?  Example 2: Chris is considering the purchase of an apartment building. He estimates the building's future value at $587,731 at 5 years from now. At an annual discount rate of 8%, how much would he pay now?

22. 22 Examples  Example 3: Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

23. 23 Finding an interest rate, discount rate, or rate of return on the investment.  Example 4: The entrance cover charge at a local bar at the end of 2000 was $3 and the cover charge at the end of 2002 was $4. At what annual rate did cover charge grow?

24. 24 Another Example  Your daughter is currently 8 years old who will be going to college in ten years. You would like to have $100,000 at that time. You can deposit $60,000 now at 3% per year. Is this enough? How much will you be short of? How much extra dollars do you need to deposit today to achieve $100,000 goal?  FV of 60,000 = 80,634  Short of 19,365  PV of 19,365 = 14,409  Any short-cut available ?  PV of 100,000 = 74,409

25. 25 Growth of a $100 Investment at Various Compound Interest Rates

26. 26 Present Value of $100 at Various Discount Rates

27. 27 Simple vs. Compound Interest (Power of Compound)  Simple Interest –Simple $100 saving today at 10% –One year from today = $100 + 10 =$110.00 –Two years from today = $100 + 10 + 10 = $120.00 –Three years from today = $100 + 10 + 10 + 10 = $130.00 –….. 50 yrs later = $100 + 50($10) = $600  Compound Interest –Compound $100 saving today at 10% –One year from today = $100 + 10 =$110.00 –Two years from today = $100 + 10 + (10 + 1) = $121.00 –Three years from today = $100 + 10 + (10 + 1) + {10 + 1 + 1 +.1} = $133.10 –….. 50 yrs later = 100(1.1) 50 = $11,739

28. 28 Mechanics  Invest $1 @ 10% per year for 1 year FV = $1 x 1.1 = $1.1000 “Get your money back ($1) plus interest (10 cents)”  Invest $1 @ 10% per year for 2 year FV = $1 x 1.1 x 1.1 = $1 x 1.1 2 = $1.2100 “Get your money back ($1) plus interests for two years (20 cents) plus interest on interest (1 cent) for the second year”  Invest $1 @ 10% per year for 1 year FV = $1 x 1.1 x 1.1 x 1.1 = $1 x 1.1 3 = $1.3310 ……

29. 29 Example: Compound Interest  After 1 st year, $1  ($1) + 10 cents = $1.1000  After 2 nd year, $1 + 10 cents  ($1 + 10 cents) + 10 cents + 1 cent = $1 + 20 cents + 1 cent = $1.2100  After 3 rd year, $1 + 10 cents + 10 cents + 1 cent  ($1 + 10 cents + 10 cents + 1 cent) + 10 cents + 1 cent + 1 cent +.1 cent = $1 + 30 cents + 3 cents +.1 cent = $1.3310

30. 30 FV for $100 investing at 10%/Yr  FV at 1 st year = $100 (1+10%)  FV at 2 nd year = $100(1+10%)(1+10%)  FV at 3 rd year = $100(1+10%)(1+10%)(1+10%) …..  FV at 50 th year = $100(1+10%)(1+10%)……..(1+10%) 50 $100 will be “compounded” 50 times. This is nothing new. However, the first interest ($10) will be compounded 49 times, and the second $10 will be compounded 48 times. And money will keep growing itself exponentially!!!

31. 31 Future Value (U.S. Dollars) Power of Compounding

32. 32 0123 10% 0123 5% 456 134.01 100133.10 123 0 100 Annually: FV 3 = $100(1.10) 3 = $133.10. Semiannually: FV 6 = $100(1.05) 6 = $134.01. Compounding Frequency: Annual vs. Semiannual Compounding for FV of $100 after 3 years

33. 33 Future Values at Various Compounding  AnnualFV=100(1+10%) 1 =110  Semi-AnnualFV=100(1+10%/2) 2 =110.25  QuarterlyFV=100(1+10%/4) 4 =110.38  MonthlyFV=100(1+10%/12) 12 =110.47  DailyFV=100(1+10%/365) 365 =110.52

34. 34 Present Values : Annual Vs. Monthly Comp  What is the PV of $100 one year from now with 12 percent interest? Monthly comp: PV 0 = $100  1/(1 +.12/12) (1  12) = $100  1/(1.126825) = $100  (.887449) = $ 88.74 Annual comp: PV 0 = $100  1/(1 +.12) 1 = $100  1/(1.1200) = $100  (.892857) = $ 89.29

35. 35 A New Formula incorporating compounding frequency TVM FormulaAnnual Compounding m compounded per year FV of a lump sum FV = PV (1+i) N FV = PV (1+i/m) Nm PV of a lump sum The interest rate required for a PV to grow to a FV The length of time required for a PV to grow to a FV

36. 36 An Important Question  Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

37. 37 More Practice Problems  Example 5: Your family plan to go to Paris in three years from now. Your travel agency informs that you will need $4,000. Currently, your stock mutual fund yield 12 percent per year. How much must you invest today to achieve the goal? –with annual compounding –with quarterly compounding

38. 38 More Practice Problems  Example 6: You are saving to buy a new $200,000 Ferrari. You have $100,000 today that can be invested at your bank. The bank pays 5 percent annual interest. How long will it be before you have enough to buy the car? –with annual compounding –with quarterly compounding

39. 39 Credit Card Example: Daily Compounding

40. 40 General Rules  Rule 1 –N could be days, months, quarters, semi-years, or years, depending on various ways to compound interest. –For instance, with one-year period, Enter N = 365, 12, 4, 2, or, 1.  Rule 2 –You always input rates on an annual basis, but you must do this with appropriate pmt per year setting. –For instance, with one-year period at 12% annual rate, Enter I = 12 always but with 365 P_YR, 12 P_YR, 4 P_YR, 2 P_YR, or 1 P_YR, depending on different compounding assumption.

41. 41 “Wordings” are confusing!!!  How much will you have per dollar invested in one year? –If you have an investment that will pay you 1.12 percent per month. –If you have an investment that will pay you an interest rate of 13.44 percent compounded monthly. –If you have an investment that will pay you an interest rate of 13.44 percent but interest accrues monthly. –If you have an investment that will pay you an APR of 13.44 percent compounded monthly.

42. 42 More Practice Problems  P 1: Grandparents offer the following: –Option A: pay you $10,000 at the end of year 2, and $20,000 at the end of year 4, –Option B: pay you $20,000 today –Which option would you prefer? Assume R = 10%

43. 43 P2: College Tuition  Your daughter is currently 8 years old who will be going to college in ten years. You will need to pay –$20,000 when she becomes 18 years old –$23,000 when she becomes sophomore –$25,000 when she becomes junior –$29,000 when she becomes senior  How much do you need to deposit today at 3% per year to be able to pay tuition as estimated above?

44. 44 At-the-end-of-chapter problems See class syllabus Practice! Practice! Practice! Remember “You will never have enough practices to master TVM problems!”