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Time Value of Money

Time Value of Money. The Time Value of Money. The Interest Rate Simple Interest Compound Interest Amortizing a Loan. Obviously, $10,000 today . You already recognize that there is TIME VALUE TO MONEY !!. Which would you prefer -- $10,000 today or $10,000 in 5 years ?.

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Time Value of Money

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  1. Time Value of Money

  2. The Time Value of Money • The Interest Rate • Simple Interest • Compound Interest • Amortizing a Loan

  3. Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!! Which would you prefer -- $10,000 today or $10,000 in 5 years? The Interest Rate

  4. TIME allows you the opportunity to postpone consumption and earn INTEREST. Why is TIME such an important element in your decision? Why TIME?

  5. Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Types of Interest • Simple Interest • Interest paid (earned) on only the original amount, or principal borrowed (lent).

  6. Simple Interest Formula FormulaSI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods

  7. SI = P0(i)(n)= $1,000(.07)(2) = $140 Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? Simple Interest Example

  8. FV = P0 + SI = $1,000+ $140 =$1,140 Future Valueis the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. What is the Future Value (FV) of the deposit? Simple Interest (FV)

  9. The Present Value is simply the $1,000 you originally deposited. That is the value today! Present Valueis the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. What is the Present Value (PV) of the previous problem? Simple Interest (PV)

  10. Why Compound Interest? Future Value (U.S. Dollars)

  11. Future Value Single Deposit (Graphic) Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. 0 12 7% $1,000 FV2

  12. Future Value Single Deposit (Formula) FV1 = P0 (1+i)1 = $1,000(1.07) = $1,070 Compound Interest You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest.

  13. Future Value Single Deposit (Formula) FV1 = P0(1+i)1 = $1,000 (1.07) = $1,070 FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = $1,000(1.07)(1.07) = P0(1+i)2 = $1,000(1.07)2 = $1,144.90 You earned an EXTRA$4.90 in Year 2 with compound over simple interest.

  14. General Future Value Formula FV1 = P0(1+i)1 FV2 = P0(1+i)2 General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVIFi,n) -- See Table I etc.

  15. Valuation Using Table I FVIFi,nis found on Table I at the end of the book

  16. Using Future Value Tables FV2 = $1,000 (FVIF7%,2) = $1,000 (1.145) = $1,145[Due to Rounding]

  17. Story Problem Example Julie Miller wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years. 0 1 2 3 4 5 10% $10,000 FV5

  18. Story Problem Solution • Calculation based on general formula:FVn = P0 (1+i)nFV5= $10,000 (1+ 0.10)5 = $16,105.10 • Calculation based on Table I: FV5= $10,000(FVIF10%, 5)= $10,000(1.611) = $16,110 [Due to Rounding]

  19. Present Value Single Deposit (Graphic) Assume that you need $1,000in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 12 7% $1,000 PV0 PV1

  20. Present Value Single Deposit (Formula) PV0 = FV2 / (1+i)2 = $1,000/ (1.07)2 = FV2 / (1+i)2 = $873.44 0 12 7% $1,000 PV0

  21. General Present Value Formula PV0= FV1 / (1+i)1 PV0 = FV2 / (1+i)2 General Present Value Formula: PV0 = FVn / (1+i)n or PV0 = FVn (PVIFi,n) -- See Table II etc.

  22. Valuation Using Table II PVIFi,nis found on Table II at the end of the book

  23. Using Present Value Tables PV2 = $1,000 (PVIF7%,2) = $1,000 (.873) = $873[Due to Rounding]

  24. Story Problem Example Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000in 5 years at a discount rate of 10%. 0 1 2 3 4 5 10% $10,000 PV0

  25. Story Problem Solution • Calculation based on general formula: PV0 = FVn / (1+i)nPV0= $10,000/ (1+ 0.10)5 = $6,209.21 • Calculation based on Table I: PV0= $10,000(PVIF10%, 5)= $10,000(.621) = $6,210.00 [Due to Rounding]

  26. Project Evaluation: Alternative Methods • Payback Period (PBP) • Internal Rate of Return (IRR) • Net Present Value (NPV) • Profitability Index (PI)

  27. Proposed Project Data Julie Miller is evaluating a new project for her firm, Basket Wonders (BW). She has determined that the after-tax cash flows for the project will be $10,000; $12,000; $15,000; $10,000; and $7,000, respectively, for each of the Years 1 through 5. The initial cash outlay will be $40,000. The discount rate is 13%.

  28. Payback Period (PBP) PBPis the period of time required for the cumulative expected cash flows from an investment project to equal the initial cash outflow. 0 1 2 3 4 5 -40 K 10 K 12 K 15 K 10 K 7 K

  29. Payback Solution (#1) PBP = a + ( b- c ) / d = 3 + (40 - 37) / 10 = 3 + (3) / 10 = 3.3 Years (a) 0 1 2 3 4 5 (-b) (d) -40 K 10 K 12 K 15 K 10 K 7 K (c) 10 K 22 K 37 K 47 K 54 K Cumulative Inflows

  30. Payback Solution (#2) PBP = 3 + ( 3K ) / 10K = 3.3 Years Note: Take absolute value of last negative cumulative cash flow value. 0 1 2 3 4 5 -40 K 10 K 12 K 15 K10 K 7 K -40 K -30 K -18 K -3 K 7 K 14 K Cumulative Cash Flows

  31. Yes! The firm will receive back the initial cash outlay in less than 3.5 years. [3.3 Years < 3.5 Year Max.] The management of Basket Wonders has set a maximum PBP of 3.5 years for projects of this type. Should this project be accepted? PBP Acceptance Criterion

  32. IRR is the discount rate that equates the present value of the future net cash flows from an investment project with the project’s initial cash outflow. Internal Rate of Return (IRR) CF1 CF2 CFn ICO = + + . . . + • (1+IRR)1 (1+IRR)2 (1+IRR)n

  33. Find the interest rate (IRR) that causes the discounted cash flows to equal $40,000. IRR Solution $10,000 $12,000 $40,000 = + + (1+IRR)1(1+IRR)2 $15,000 $10,000 $7,000 + + (1+IRR)3(1+IRR)4(1+IRR)5

  34. $40,000 = $10,000(PVIF10%,1) + $12,000(PVIF10%,2) + $15,000(PVIF10%,3) + $10,000(PVIF10%,4) + $ 7,000(PVIF10%,5) $40,000 = $10,000(.909) + $12,000(.826) + $15,000(.751) + $10,000(.683) + $ 7,000(.621) $40,000 = $9,090 + $9,912 + $11,265 + $6,830 + $4,347 = $41,444 [Rate is too low!!] IRR Solution (Try 10%)

  35. $40,000 = $10,000(PVIF15%,1) + $12,000(PVIF15%,2) + $15,000(PVIF15%,3) + $10,000(PVIF15%,4) + $ 7,000(PVIF15%,5) $40,000 = $10,000(.870) + $12,000(.756) + $15,000(.658) + $10,000(.572) + $ 7,000(.497) $40,000 = $8,700 + $9,072 + $9,870 + $5,720 + $3,479 = $36,841 [Rate is too high!!] IRR Solution (Try 15%)

  36. No! The firm will receive 11.57% for each dollar invested in this project at a cost of 13%. [ IRR< Hurdle Rate ] The management of Basket Wonders has determined that the hurdle rate is 13% for projects of this type. Should this project be accepted? IRR Acceptance Criterion

  37. NPV is the present value of an investment project’s net cash flows minus the project’s initial cash outflow. Net Present Value (NPV) CF1CF2CFn -ICO NPV = + + . . . + • (1+k)1 (1+k)2 (1+k)n

  38. Basket Wonders has determined that the appropriate discount rate (k) for this project is 13%. NPV Solution $10,000$12,000$15,000 NPV= + + + (1.13)1(1.13)2(1.13)3 $10,000$7,000 + - $40,000 (1.13)4(1.13)5

  39. NPV Solution NPV = $10,000(PVIF13%,1) + $12,000(PVIF13%,2) + $15,000(PVIF13%,3) + $10,000(PVIF13%,4) + $ 7,000(PVIF13%,5) - $40,000 NPV = $10,000(.885) + $12,000(.783) + $15,000(.693) + $10,000(.613) + $ 7,000(.543) - $40,000 NPV = $8,850 + $9,396 + $10,395 + $6,130 + $3,801 - $40,000 = - $1,428

  40. No! The NPV is negative. This means that the project is reducing shareholder wealth. [Reject as NPV< 0 ] The management of Basket Wonders has determined that the required rate is 13% for projects of this type. Should this project be accepted? NPV Acceptance Criterion

  41. PI is the ratio of the present value of a project’s future net cash flows to the project’s initial cash outflow. Profitability Index (PI) CF1CF2CFn PI = ICO + + . . . + • (1+k)1 (1+k)2 (1+k)n << OR >> PI = 1 + [ NPV / ICO]

  42. No! The PI is less than 1.00. This means that the project is not profitable. [Reject as PI< 1.00 ] PI = $38,572 / $40,000 = .9643 (Method #1, 13-33) Should this project be accepted? PI Acceptance Criterion

  43. Evaluation Summary Basket Wonders Independent Project

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