Time Value of Money

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

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

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

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

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

Simple Interest Example 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? SI = P0(i)(n) = $1,000(.07)(2) = $140

Simple Interest (FV) What is the Future Value (FV) of the deposit? FV = P0 + SI = $1,000 + $140 = $1,140 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.

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

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 1 2 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.

Single Deposit (Formula) 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,n is 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)n FV5 = $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 $1,000 in 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 1 2 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 1 2 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,n is 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 $10,000 in 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)n PV0 = $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) 0 1 2 3 4 5 -40 K 10 K 12 K 15 K 10 K 7 K PBP is the period of time required for the cumulative expected cash flows from an investment project to equal the initial cash outflow.

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

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

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

Internal Rate of Return (IRR) 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. CF1 CF2 CFn ICO = + + . . . + (1+IRR)1 (1+IRR)2 (1+IRR)n

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

IRR Solution (Try 10%) $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 15%) $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 Acceptance Criterion The management of Basket Wonders has determined that the hurdle rate is 13% for projects of this type. Should this project be accepted? No! The firm will receive 11.57% for each dollar invested in this project at a cost of 13%. [ IRR < Hurdle Rate ]

Net Present Value (NPV) NPV is the present value of an investment project’s net cash flows minus the project’s initial cash outflow. CF1 CF2 CFn - ICO NPV = + + . . . + (1+k)1 (1+k)2 (1+k)n

NPV Solution Basket Wonders has determined that the appropriate discount rate (k) for this project is 13%. $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

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

Profitability Index (PI) PI is the ratio of the present value of a project’s future net cash flows to the project’s initial cash outflow. CF1 CF2 CFn PI = ICO + + . . . + (1+k)1 (1+k)2 (1+k)n << OR >> PI = 1 + [ NPV / ICO ]

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

Basket Wonders Independent Project Evaluation Summary Basket Wonders Independent Project