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Time Value of Money (CH 4)

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  1. Time Value of Money (CH 4) TIP If you do not understand something, ask me! Future value Present value Annuities Interest rates

  2. Last week • Objective of the firm • Business forms • Agency conflicts • Capital budgeting decision and capital structure decision

  3. The plan of the lecture • Time value of money concepts • present value (PV) • discount rate/interest rate (r) • Formulae for calculating PV of • perpetuity • annuity • Interest compounding • How to use a financial calculator

  4. Financial choices • Which would you rather receive today? • TRL 1,000,000,000 ( one billion Turkish lira ) • USD 652.72 ( U.S. dollars ) • Both payments are absolutely guaranteed. • What do we do?

  5. Financial choices • We need to compare “apples to apples” - this means we need to get the TRL:USD exchange rate • From www.bloomberg.com we can see: • USD 1 = TRL 1,637,600 • Therefore TRL 1bn = USD 610.64

  6. Financial choices with time • Which would you rather receive? • $1000 today • $1200 in one year • Both payments have no risk, that is, • there is 100% probability that you will be paid

  7. Financial choices with time • Why is it hard to compare ? • $1000 today • $1200 in one year • This is not an “apples to apples” comparison. They have different units • $1000 today is different from $1000 in one year • Why? • A cash flow is time-dated money • It has a money unit such as USD or TRL • It has a date indicating when to receive money

  8. Present value • To have an “apple to apple” comparison, we • convert future payments to the present values • or convert present payments to the future values • This is like converting money in TRL to money in USD

  9. Some terms • Finding the present value of some future cash flows is called discounting. • Finding the future value of some current cash flows is called compounding.

  10. 0 1 2 3 10% 100 FV = ? What is the future value (FV) of an initial $100 after 3 years, if i = 10%? • Finding the FV of a cash flow or series of cash flows is called compounding. • FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.

  11. Solving for FV:The arithmetic method • After 1 year: • FV1 = c ( 1 + i ) = $100 (1.10) = $110.00 • After 2 years: • FV2 = c (1+i)(1+i)= $100 (1.10)2 =$121.00 • After 3 years: • FV3 = c ( 1 + i )3 = $100 (1.10)3 =$133.10 • After n years (general case): • FVn = C ( 1 + i )n

  12. Set up the Texas instrument • 2nd, “FORMAT”, set “DEC=9”, ENTER • 2nd, “FORMAT”, move “↓” several times, make sure you see “AOS”, not “Chn”. • 2nd, “P/Y”, set to “P/Y=1” • 2nd, “BGN”, set to “END” • P/Y=periods per year, • END=cashflow happens end of periods

  13. Solving for FV:The calculator method • Solves the general FV equation. • Requires 4 inputs into calculator, and it will solve for the fifth. 3 10 -100 0 INPUTS N I/YR PV PMT FV OUTPUT 133.10

  14. What is the present value (PV) of $100 received in 3 years, if i = 10%? • Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding). • The PV shows the value of cash flows in terms of today’s worth. 0 1 2 3 10% PV = ? 100

  15. Solving for PV:The arithmetic method • i: interest rate, or discount rate • PV = C / ( 1 + i )n • PV = C / ( 1 + i )3 = $100 / ( 1.10 )3 = $75.13

  16. Solving for PV:The calculator method • Exactly like solving for FV, except we have different input information and are solving for a different variable. 3 10 0 100 INPUTS N I/YR PV PMT FV OUTPUT -75.13

  17. Solving for N:If your investment earns interest of 20% per year, how long before your investments double? 20 -1 0 2 INPUTS N I/YR PV PMT FV OUTPUT 3.8

  18. Solving for i:What interest rate would cause $100 to grow to $125.97 in 3 years? 3 -100 0 125.97 INPUTS N I/YR PV PMT FV OUTPUT 8

  19. Now let’s study some interesting patterns of cash flows… • Perpetuity • Annuity

  20. Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT ordinary annuity and annuity due

  21. Value an ordinary annuity • Here C is each cash payment • n is number of payments • If you’d like to know how to get the formula below, see me after class.

  22. Example • you win the $1million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won?

  23. Using the formula

  24. Solving for FV:3-year ordinary annuity of $100 at 10% • $100 payments occur at the end of each period. Note that PV is set to 0 when you try to get FV. 3 10 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 331

  25. Solving for PV:3-year ordinary annuity of $100 at 10% • $100 payments still occur at the end of each period. FV is now set to 0. 3 10 100 0 INPUTS N I/YR PV PMT FV OUTPUT -248.69

  26. Solving for FV:3-year annuity due of $100 at 10% • $100 payments occur at the beginning of each period. • FVAdue= FVAord(1+i) = $331(1.10) = $364.10. • Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity: BEGIN 3 10 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 364.10

  27. Solving for PV:3-year annuity due of $100 at 10% • $100 payments occur at the beginning of each period. • PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55. • Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity: BEGIN 3 10 100 0 INPUTS N I/YR PV PMT FV OUTPUT -273.55

  28. What is the present value of a 5-year $100 ordinary annuity at 10%? • Be sure your financial calculator is set back to END mode and solve for PV: • N = 5, I/YR = 10, PMT = 100, FV = 0. • PV = $379.08

  29. What if it were a 10-year annuity? A 25-year annuity? A perpetuity? • 10-year annuity • N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $614.46. • 25-year annuity • N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $907.70. • Perpetuity (N=infinite) • PV = PMT / i = $100/0.1 = $1,000.

  30. What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%? $297.22 $323.97 $100 $100 $100 $100 0 1 2 3 4 5

  31. 4 0 1 2 3 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV What is the PV of this uneven cash flow stream?

  32. Solving for PV:Uneven cash flow stream • Input cash flows in the calculator’s “CF” register: • CF0 = 0 • CF1 = 100 • CF2 = 300 • CF3 = 300 • CF4 = -50 • Enter I/YR = 10, press NPV button to get NPV = $530.09. (Here NPV = PV.)

  33. Detailed steps (Texas Instrument calculator) • To clear historical data: • CF, 2nd ,CE/C • To get PV: • CF ,↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2, Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT • “NPV=530.0867427”

  34. The Power of Compound Interest A 20-year-old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095=$3*365) in an online stock account. The stock account has an expected annual return of 12%. How much money will she have when she is 65 years old?

  35. Solving for FV:Savings problem • If she begins saving today, and sticks to her plan, she will have $1,487,261.89 when she is 65. 45 12 0 -1095 INPUTS N I/YR PV PMT FV OUTPUT 1,487,262

  36. Solving for FV:Savings problem, if you wait until you are 40 years old to start • If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20. • Lesson: It pays to start saving early. 25 12 0 -1095 INPUTS N I/YR PV PMT FV OUTPUT 146,001

  37. 0 1 2 3 10% 100 133.10 0 1 2 3 4 5 6 0 1 2 3 5% 100 134.01 Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated i% constant? • LARGER, as the more frequently compounding occurs, interest is earned on interest more often. Annually: FV3 = $100(1.10)3 = $133.10 Semiannually: FV6 = $100(1.05)6 = $134.01

  38. What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

  39. Classifications of interest rates • 1. Nominal rate (iNOM) – also called the APR,quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly. • 2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly. • iPER = iNOM / m, where m is the number of compounding periods per year. e.g., m = 12 for monthly compounding.

  40. Classifications of interest rates • 3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding. • If the interest rate is compounded m times in a year, the effective annual interest rate is

  41. Example, EAR for 10% semiannual investment • EAR= ( 1 + 0.10 / 2 )2 – 1 = 10.25% • An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.

  42. keys: description: Sets 2 payments per year [↑] [C/Y=] 2 [ENTER] [2nd] [ICONV] Opens interest rate conversion menu [↓][NOM=] 10 [ENTER] Sets 10 APR. [↓] [EFF=] [CPT] 10.25 EAR on a Financial Calculator Texas Instruments BAII Plus

  43. Why is it important to consider effective rates of return? • An investment with monthly payments is different from one with quarterly payments. • Must use EAR for comparisons. • If iNOM=10%, then EAR for different compounding frequency: Annual 10.00% Quarterly 10.38% Monthly 10.47% Daily 10.52%

  44. If interest is compounded more than once a year • EAR (EFF, APY) will be greater than the nominal rate (APR).

  45. 1 2 3 0 1 2 3 4 5 6 5% 100 100 100 What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually? • Payments occur annually, but compounding occurs every 6 months. • Cannot use normal annuity valuation techniques.

  46. 1 2 3 0 1 2 3 4 5 6 5% 100 100 100 110.25 121.55 331.80 Method 1:Compound each cash flow FV3 = $100(1.05)4 + $100(1.05)2 + $100 FV3 = $331.80

  47. Method 2:Financial calculator • Find the EAR and treat as an annuity. • EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%. 3 10.25 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 331.80

  48. When is periodic rate used? • iPER is often useful if cash flows occur several times in a year.