1. 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

2. EGR 403 - Cal Poly Pomona - SA52 EGR 403 - 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. 12 - Replacement Analysis

3. EGR 403 - 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

4. EGR 403 - Cal Poly Pomona - SA54 Computing Cash Flows Cash flows have: –Costs (disbursements) a negative number –Benefits (receipts) a positive number Example 3-1

5. EGR 403 - 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

6. EGR 403 - 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

7. EGR 403 - 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 (1 +.10 x 2) = $1200 Simple interest is not used today

8. EGR 403 - 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: $1000 + $100 = $1100 Year 2 interest = $1100 (.10) = $110 Total interest earned is: $100 + $110 = $210 This is $10 more than with “simple” interest

9. EGR 403 - 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

10. EGR 403 - 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.

11. EGR 403 - 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.

12. EGR 403 - 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

13. EGR 403 - 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%

14. EGR 403 - 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

15. EGR 403 - Cal Poly Pomona - SA515 Plan 1 – Equal annual principal payments YearBalancePiPayment 1500010005001500 2400010004001400 3300010003001300 4200010002001200 51000 1001100 6500

16. EGR 403 - Cal Poly Pomona - SA516 Plan 2 –Annual interest + balloon payment of principal YearBalancePiPayment 15000500 25000500 35000500 45000500 55000 500 7500

17. EGR 403 - Cal Poly Pomona - SA517 Plan 3 – Equal annual payments (installments) YearBalancePiPayment 15000.00819.00500.001319 24181.00900.90418.101319 33280.10990.99328.011319 42289.111090.09228.911319 51199.021199.10119.901319 6595

18. EGR 403 - Cal Poly Pomona - SA518 Plan 4 – Principal & interest at end of the loan YearBalancePiPayment 1500005000 2550005500 3605006050 466550665.500 57320.500732.058052.55

19. EGR 403 - Cal Poly Pomona - SA519 Which plan would you choose? Total Principal + Interest Paid –Plan 1 = $6500 –Plan 2 = $7500 –Plan 3 = $6595 –Plan 4 = $8052.55

20. EGR 403 - 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.

21. EGR 403 - Cal Poly Pomona - SA521 To further illustrate this concept, given the choice of these two plans which would you choose? $7000$6200Total 540010805 40011604 40012403 40013202 $400$14001 Plan 2Plan 1Year To make a choice the cash flows must be altered so a comparison may be made.

22. EGR 403 - 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.

23. EGR 403 - 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.

24. EGR 403 - 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.

25. EGR 403 - 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

26. EGR 403 - 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

27. EGR 403 - 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

28. EGR 403 - 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.

29. EGR 403 - 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

30. EGR 403 - 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

31. EGR 403 - 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)

32. EGR 403 - 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