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Interest and Equivalence in Capital Allocation Theory

Learn about the concepts of interest and equivalence in capital allocation theory, including simple interest, compound interest, future value, and effective interest rates.

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Interest and Equivalence in Capital Allocation Theory

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  1. Chapter 3 - Interest and EquivalenceClick 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 - 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 EGR 403 - Cal Poly Pomona - SA5

  3. 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 403 - Cal Poly Pomona - SA5

  4. Computing Cash Flows • Cash flows have: • Costs (disbursements) a negative number • Benefits (receipts) a positive number Example 3-1 EGR 403 - Cal Poly Pomona - SA5

  5. Time Value of Money • Money has value • Money can be leased or rented • Thepayment 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 403 - Cal Poly Pomona - SA5

  6. 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 403 - Cal Poly Pomona - SA5

  7. 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 EGR 403 - Cal Poly Pomona - SA5

  8. 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 EGR 403 - Cal Poly Pomona - SA5

  9. 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 = In = P (1+i)n - P • Where, • P – present sum of money • i – interest rate per period • n – number of periods EGR 403 - Cal Poly Pomona - SA5

  10. 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 403 - Cal Poly Pomona - SA5

  11. 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 ( ieff 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 403 - Cal Poly Pomona - SA5

  12. 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%) EGR 403 - Cal Poly Pomona - SA5

  13. 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) • ieff = (1 + i)m – 1 or • ieff = (1 + r/m)m – 1 • Example: r = 12, m = 12 • ieff = (1 + .12/12)12 – 1 = (1.01)12 – 1 = .1268 or 12.68% EGR 403 - Cal Poly Pomona - SA5

  14. Plan Repay Principal Repay Interest Interest Earned 1 Equal installments Interest on unpaid balance Declines 2 End of loan Interest on unpaid balance Constant 3 Equal installments Declines at increasing rate 4 End of loan Compound and pay at end of loan Compounds at increasing rate until end of loan Consider Four Ways to Repay a Debt EGR 403 - Cal Poly Pomona - SA5

  15. Plan 1 – Equal annual principal payments EGR 403 - Cal Poly Pomona - SA5

  16. Plan 2 –Annual interest + balloon payment of principal EGR 403 - Cal Poly Pomona - SA5

  17. Plan 3 – Equal annual payments (installments) EGR 403 - Cal Poly Pomona - SA5

  18. Plan 4 – Principal & interest at end of the loan EGR 403 - Cal Poly Pomona - SA5

  19. Total Principal + Interest Paid Plan 1 = $6500 Plan 2 = $7500 Plan 3 = $6595 Plan 4 = $8052.55 Which plan would you choose? EGR 403 - Cal Poly Pomona - SA5

  20. 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 403 - Cal Poly Pomona - SA5

  21. Year Plan 1 Plan 2 1 $1400 $400 2 1320 400 3 1240 400 4 1160 400 5 1080 5400 Total $6200 $7000 To further illustrate this concept, given the choice of these two plans which would you choose? To make a choice the cash flows must be altered so a comparison may be made. EGR 403 - Cal Poly Pomona - SA5

  22. 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 403 - Cal Poly Pomona - SA5

  23. 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 403 - Cal Poly Pomona - SA5

  24. 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 403 - Cal Poly Pomona - SA5

  25. 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 403 - Cal Poly Pomona - SA5

  26. 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 (fx) 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 403 - Cal Poly Pomona - SA5

  27. 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 403 - Cal Poly Pomona - SA5

  28. Notation forCalculating 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 403 - Cal Poly Pomona - SA5

  29. 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 403 - Cal Poly Pomona - SA5

  30. Using EXCEL Spreadsheet Functions • On the menu bar select the fxicon • 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 403 - Cal Poly Pomona - SA5

  31. Notation forCalculating 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 403 - Cal Poly Pomona - SA5

  32. 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 EGR 403 - Cal Poly Pomona - SA5

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