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## Time Value of Money

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**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? The Interest Rate**TIME allows you the opportunity to postpone consumption and**earn INTEREST. Why is TIME such an important element in your decision? Why TIME?**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).**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**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**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)**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)**Why Compound Interest?**Future Value (U.S. Dollars)**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**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.**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.**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.**Valuation Using Table I**FVIFi,nis found on Table I at the end of the book**Using Future Value Tables**FV2 = $1,000 (FVIF7%,2) = $1,000 (1.145) = $1,145[Due to Rounding]**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**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]**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**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**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.**Valuation Using Table II**PVIFi,nis found on Table II at the end of the book**Using Present Value Tables**PV2 = $1,000 (PVIF7%,2) = $1,000 (.873) = $873[Due to Rounding]**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**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]**Project Evaluation: Alternative Methods**• Payback Period (PBP) • Internal Rate of Return (IRR) • Net Present Value (NPV) • Profitability Index (PI)**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%.**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**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**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**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**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**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**$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%)**$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%)**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**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**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**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**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**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]**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**Evaluation Summary**Basket Wonders Independent Project