Chapter 3 - Interest and Equivalence Click here for Streaming Audio To Accompany Presentation (optional) Click here for Streaming Audio To Accompany Presentation (optional) EGR 403 Capital Allocation Theory Dr. Phillip R. Rosenkrantz Industrial & Manufacturing Engineering Department Cal Poly Pomona

EGR Cal Poly Pomona - SA52 EGR The Big Picture Framework: Accounting & Breakeven Analysis “Time-value of money” concepts - Ch. 3, 4 Analysis methods –Ch. 5 - Present Worth –Ch. 6 - Annual Worth –Ch. 7, 8 - Rate of Return (incremental analysis) –Ch. 9 - Benefit Cost Ratio & other techniques Refining the analysis –Ch. 10, 11 - Depreciation & Taxes –Ch Replacement Analysis

EGR Cal Poly Pomona - SA53 Economic Decision Components Where economic decisions are immediate we need to consider: –amount of expenditure –taxes Where economic decisions occur over a considerable period of time we need to also consider the consequences of: –interest –inflation

EGR Cal Poly Pomona - SA54 Computing Cash Flows Cash flows have: –Costs (disbursements) a negative number –Benefits (receipts) a positive number Example 3-1

EGR Cal Poly Pomona - SA55 Time Value of Money Money has value –Money can be leased or rented –The payment is called interest –If you put $100 in a bank at 9% interest for one time period you will receive back your original $100 plus $9 Original amount to be returned = $100 Interest to be returned = $100 x.09 = $9

EGR Cal Poly Pomona - SA56 Simple Interest Interest that is computed only on the original sum or principal Total interest earned = I = P i n, where: –P = present sum of money, or “principal” (example: $1000) –i = interest rate (10% interest is a.10 interest rate) –n = number of periods (years) (example: n = 2 years) I = $1000 x.10/period x 2 periods = $200

EGR Cal Poly Pomona - SA57 Future Value of a Loan With Simple Interest Amount of money due at the end of a loan –F = P + P i n or F = P (1 + i n ) –Where, F = future value F = $1000 ( x 2) = $1200 Simple interest is not used today

EGR Cal Poly Pomona - SA58 Compound Interest Compound Interest is used and is computed on the original unpaid debt and the unpaid interest. Year 1 interest = $1000 (.10) = $100 –Year 2 principal is, therefore: $ $100 = $1100 Year 2 interest = $1100 (.10) = $110 Total interest earned is: $100 + $110 = $210 This is $10 more than with “simple” interest

EGR Cal Poly Pomona - SA59 Compound Interest (Cont’d) Future Value (F) = P + Pi + (P + Pi)i = P (1 + i + i + i 2 ) = P (1+i) 2 = 1000 (1 +.10) 2 = 1210 In general, for any value of n: –Future Value (F) = P (1+i) n –Total interest earned = I n = P (1+i) n - P –Where, P – present sum of money i – interest rate per period n – number of periods

EGR Cal Poly Pomona - SA510 Compound Interest Over Time If you loaned a friend money for short period of time the difference between simple and compound interest is negligible. If you loaned a friend money for a long period of time the difference between simple and compound interest may amount to a considerable difference.

EGR Cal Poly Pomona - SA511 Nominal and Effective Interest Interest rates are normally given on an annual basis with agreement on how often compounding will occur (e.g., monthly, quarterly, annually, continuous). Nominal interest rate /year ( r ) – the annual interest rate w/o considering the effect of any compounding (e.g., r = 12%). Interest rate /period ( i ) – the nominal interest rate /year divided by the number of interest compounding periods (e.g., monthly compounding: i = 12% / 12 months/year = 1%). Effective interest rate /year ( i eff or APR ) – the annual interest rate taking into account the effect of the multiple compounding periods in the year. (e.g., as shown later, r = 12% compounded monthly is equivalent to 12.68% year compounded yearly.

EGR Cal Poly Pomona - SA512 Interest Rates (cont’d) We use “ i ” for the periodic interest rate Nominal interest rate = r (an annual rate) Number of compounding periods/year = m –r = i * m, and i = r / m –Let r =.12 (or 12%) Interest Periodm = interest periods / year i = interest rate / period Annual1.12 Quarter4.03 Month12.01

EGR Cal Poly Pomona - SA513 Effective Interest If there are more than one compounding periods during the year, then the compounding makes the true interest rate slightly higher. This higher rate is called the “effective interest rate” or Annual Percentage Rate (APR) i eff = (1 + i) m – 1 or i eff = (1 + r/m) m – 1 Example: r = 12, m = 12 i eff = (1 +.12/12) 12 – 1 = (1.01) 12 – 1 =.1268 or 12.68%

EGR Cal Poly Pomona - SA514 Consider Four Ways to Repay a Debt Compound and pay at end of loan Interest on unpaid balance Repay Interest Declines at increasing rate Equal installments3 Compounds at increasing rate until end of loan End of loan4 ConstantEnd of loan2 DeclinesEqual installments 1 Interest EarnedRepay Principal Plan

EGR Cal Poly Pomona - SA515 Plan 1 – Equal annual principal payments YearBalancePiPayment

EGR Cal Poly Pomona - SA516 Plan 2 –Annual interest + balloon payment of principal YearBalancePiPayment

EGR Cal Poly Pomona - SA517 Plan 3 – Equal annual payments (installments) YearBalancePiPayment

EGR Cal Poly Pomona - SA518 Plan 4 – Principal & interest at end of the loan YearBalancePiPayment

EGR Cal Poly Pomona - SA519 Which plan would you choose? Total Principal + Interest Paid –Plan 1 = $6500 –Plan 2 = $7500 –Plan 3 = $6595 –Plan 4 = $

EGR Cal Poly Pomona - SA520 Equivalence When an organization is indifferent as to whether it has a present sum of money now or, with interest the assurance of some other sum of money in the future, or a series of future sums of money, we say that the present sum of money is equivalent to the future sum or series of future sums. Each of the four repayment plans are “equivalent” because each repays $5000 at the same 10% interest rate.

EGR Cal Poly Pomona - SA521 To further illustrate this concept, given the choice of these two plans which would you choose? $7000$6200Total $400$14001 Plan 2Plan 1Year To make a choice the cash flows must be altered so a comparison may be made.

EGR Cal Poly Pomona - SA522 Technique of Equivalence Determine a single equivalent value at a point in time for plan 1. Determine a single equivalent value at a point in time for plan 2. Both at the same interest rate Judge the relative attractiveness of the two alternatives from the comparable equivalent values. You will learn a number of methods for finding comparable equivalent values.

EGR Cal Poly Pomona - SA523 Analysis Methods that Compare Equivalent Values Present Worth Analysis (Ch. 5) - Find the equivalent value of cash flows at time 0. Annual Worth Analysis (Ch. 6) - Find the equivalent annual worth of all cash flows. Rate of Return Analysis (Ch. 7, 8) - Compare the interest rate (ROR) of each alternative’s cash flows to a minimum value you will accept. Future Worth Analysis (Ch. 9) - Find the equivalent value of cash flows at time in the future. Benefit/Cost Ratio (Ch. 9) - Use equivalent values of cash flows to form ratios that can be easily analyzed.

EGR Cal Poly Pomona - SA524 Interest Formulas To understand equivalence the underlying interest formulas must be analyzed. We will start with “Single Payment” interest formulas. Notation: i = Interest rate per interest period. n = Number of interest periods. P = Present sum of money (Present worth, PV). F = Future sum of money (Future worth, FV). If you know any three of the above variables you can find the fourth one.

EGR Cal Poly Pomona - SA525 For example, given F, P, and n, find the interest rate (i) or “ROR” Principal outstanding over time (P) Amount repaid (F) at n future periods from now We know F, P, and n and want to find the interest rate that makes them equivalent: If F = P (1 + i) n Then i = (F/P) 1/n - 1 This value of i is the Rate Of Return or ROR for investing the amount P to earn the future sum F

EGR Cal Poly Pomona - SA526 Functional Notation Give P, n, and i, we can solve for F several ways: –Use the formula and a calculator –Use the factors and functional notation in the tables in the back of the text –Use the financial functions (f x ) in EXCEL –Use the financial functions available in many calculators In this course we will use the factors or EXCEL spreadsheet functions unless otherwise instructed

EGR Cal Poly Pomona - SA527 Cash Flow Diagrams We use cash flow diagrams to help organize the data for each alternative. –Down arrow - disbursement cash flow –Up arrow - Income cash flow –n = number of compounding periods in the problem –i = interest rate/period

EGR Cal Poly Pomona - SA528 Notation for Calculating a Future Value Formula: F=P(1+i) n is the single payment compound amount factor. Functional notation: F=P(F/P, i, n) F = 5000(F/P, 6%, 10) F =P(F/P) which is dimensionally correct. Find the factor values in the tables in the back of the text.

EGR Cal Poly Pomona - SA529 Using the Functional Notation and Tables to Find the Factor Values F = 5000(F/P, 6%, 10) To use the tables: –Step 1: Find the page with the 6% table –Step 2: Find the F/P column –Step 3: Go down the F/P column to n = 10 The Factor shown is 1.791, therefore: F = 5000 (1.791) = $8955

EGR Cal Poly Pomona - SA530 Using EXCEL Spreadsheet Functions On the menu bar select the f x icon Select the Financial Function menu Select the FV function to find the Future Value of a present sum (or series of payments): FV(rate, nper, pmt, PV, type) where: –rate = i –nper = n –pmt = 0 –PV = P –type = 0

EGR Cal Poly Pomona - SA531 Notation for Calculating a Present Value P=F(1/1+i) n =F(1+i) -n is the single payment present worth factor Functional notation: P=F(P/F, i, n) P=5000(P/F, 6%, 10)

EGR Cal Poly Pomona - SA532 Example: P=F(P/F, i, n) F = $1000, i = 0.10, n = 5, P = ? Using notation: P = F(P/F, 10%, 5) = $1000(.6209)= $620.90