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Chapter 2 Time Value of Money

Chapter 2 Time Value of Money. Interest: The Cost of Money Economic Equivalence Development of Interest Formulas Unconventional Equivalence Calculations. Decision Dilemma—Take a Lump Sum or Annual Installments. A suburban Chicago couple won the Power-ball.

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Chapter 2 Time Value of Money

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  1. Chapter 2Time Value of Money • Interest: The Cost of Money • Economic Equivalence • Development of Interest Formulas • Unconventional Equivalence Calculations

  2. Decision Dilemma—Take a Lump Sum or Annual Installments • A suburban Chicago couple won the Power-ball. • They had to choose between a single lump sum $104 million, or $198 million paid out over 25 years (or $7.92 million per year). • The winning couple opted for the lump sum. • Did they make the right choice? What basis do we make such an economic comparison?

  3. What Do We Need to Know? • To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different points in time. • To do this, we need to develop a method for reducing a sequence of benefits and costs to a single point in time. Then, we will make our comparisons on that basis.

  4. Time Value of Money • Money has a time value because it can earn more money over time (earning power). • Money has a time value because its purchasing power changes over time (inflation). • Time value of money is measured in terms of interest rate. • Interest is the cost of money—a cost to the borrower and an earning to the lender

  5. Delaying Consumption

  6. Delaying Consumption

  7. Notation for interest calculations • An: A discrete payment or receipt occurring at the end of some interest period • i: Interest rate per period • N: Total number of interest periods • P: A sum of money at a time chosen for purposes of analysis astime zero– present value/worth • F: A future sum of money at the end of the analysis period

  8. Example: Paying back a loan • You get a loan of $20000 from a bank at a 9% annual interest rate. You also pay a $200 loan origination fee when the loan commences (begins). The bank offers two repayment plans. • Plan 1:Equal payments at the end of every year for the next 5 years • Plan 2:Single payment at the end of the loan period (5 years)

  9. Repayment Plans

  10. Repayment Plans

  11. Cash Flow Diagram

  12. Interest Period 0 1 End of interest period Beginning of Interest period 0 1 End-of-Period Convention Important simplifying assumption: All cash flows are placed at the end of an interest period.

  13. Methods of Calculating Interest • Simple interest: the practice of charging an interest rate only to an initial sum (principal amount) – even though you do not withdraw it • Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.

  14. Simple Interest • P = Principal amount • i = Interest rate • N = Number of interest periods • Example: • P = $1,000 • i = 8% • N = 3 years

  15. Compound Interest • P = Principal amount • i = Interest rate • N = Number of interest periods • Example: • P = $1,000 • i = 8% • N = 3 years

  16. Compounding Process $1,080 $1,166.40 0 $1,259.71 1 $1,000 2 3 $1,080 $1,166.40

  17. Compound Interest • End of 1st period: • End of 2nd period: • End of 3rd period: • At the end of N periods: The Fundamental Law of Engineering Economy

  18. Comparing Simple to Compound Interest

  19. Economic equivalence exists between cash flows that have the same economic effect and could therefore be traded for one another. Even though the amounts and timing of the cash flows may differ, the appropriate interest rate makes them equal. Economic equivalence refers to the fact that a cash flow (which can either be a single payment or a series of payments) can be converted to an equivalent cash at any point in time. Economic Equivalence

  20. Economic Equivalence • The compound interest formula expresses the equivalence between some present amount P and future amount F for given i and N. • Equivalent cash flows are equivalent at any common point in time.

  21. F 0 N P Economic Equivalence • If you deposit P dollars today for N periods at i, you will haveF dollars at the end of period N. • F dollars at the end of period N is equal to a single sum P dollars now, if your earning power is measured in terms of interest rate i.

  22. Typical Repayment Plans for a Bank Loan of $20,000

  23. Equivalence Between Two Cash Flows • You are offered the alternative of receiving either $3000 at the end of 5 years or P dollars today. You have no current need for the money, you would deposit the P dollars in an account paying 8% interest. What value of P would make you indifferent to your choice between P dollars today and the promise of $3000 at the end of 5 years?

  24. $2,042 $3,000 0 5 Equivalence Between Two Cash Flows • Step 1: Determine the base period, say, year 5. • Step 2: Identify the interest rate to use. • Step 3: Calculate equivalence value.

  25. Equivalent Cash Flows are Equivalent at Any Common Point In Time

  26. Equivalence Calculations with Multiple Payments

  27. Practice problem Consider the following sequence of deposits and withdrawals over a period of 4 years. If you earn 10% interest, what would be the balance at the end of 4 years? ? $1,210 1 4 0 2 3 $1,500 $1,000 $1,000

  28. Solution ? $1,210 0 1 3 2 4 $1,000 $1,000 $1,500 $1,100 $1,000 $2,981 $1,210 $2,100 $2,310 + $1,500 -$1,210 $1,100 $2,710

  29. Solution

  30. Practice Problem 2P • How many years would it take an investment to double at 10% annual interest? 0 N = ? P

  31. 2P 0 N = ? P Solution

  32. $1,000 $500 A 0 1 2 3 $502 $502 $502 B 0 1 2 3 Practice Problem At what interest rate would you be indifferent between the two cash flows?

  33. Approach $1,000 • Step 1: Select the base period to compute the equivalent value (say, n = 3) • Step 2: Find the net worth of each at n = 3. $500 A 0 1 2 3 $502 $502 $502 B 0 1 2 3

  34. Establish Equivalence at n = 3 • Find the solution by trial and error, say i = 8%

  35. Types of Cash Flows • Single cash flow • Equal (uniform) payment series • Linear gradient series • Geometric gradient series • Irregular payment series

  36. Single Cash Flow Formula F • Single payment present worth factor (discount factor) • Given: • Find: 0 N P

  37. Uneven Payment Series • Wilson Technology wishes to set aside money now to invest over the next 4 years. The company can earn 10% on a lump sum deposited now. The money will be withdrawn in the following increments: • Year 1: $25000 to purchase computer hardware and software • Year 2: $3000 for additional hardware • Year 3: no expenses • Year 4: $5000 for software upgrades How much must be deposited now?

  38. Uneven Payment Series

  39. Equal Payment Series A 0 1 2 3 4 5 N-1 N F P

  40. Equal Payment Series Compound Amount Factor F 0 1 2 3 N A Example 4.13: • Given: A = $3,000, N = 10 years, and i = 7% • Find: F • Solution: F = $3,000(F/A,7%,10) = $41,449.20

  41. Sinking Fund Factor F 0 1 2 3 N A Example 4.15: • Given: F = $5,000, N = 5 years, and i = 7% • Find: A • Solution: A = $5,000(A/F,7%,5) = $869.50

  42. F = ? First deposit occurs at n = 0 i = 6% 0 1 2 3 4 5 $5,000 $5,000 $5,000 $5,000 $5,000 Handling Time Shifts in a Uniform Series

  43. Capital Recovery Factor P 1 2 3 0 N A Example 4.16: • Given: P = $250,000, N = 6 years, and i = 8% • Find: A • Solution: A = $250,000(A/P,8%,6) = $54,075

  44. Equal Payment Series Present Worth Factor P 1 2 3 0 N A Example 2.14:Powerball Lottery • Given: A = $7.92M, N = 25 years, and i = 8% • Find: P • Solution: P = $7.92M(P/A,8%,25) = $84.54M

  45. Example 2.13 Deferred Loan Repayment Plan • You borrowed $21061.82 to finance educational expenses. The loan will be paid with five payments. You want to defer the first payment until the end of year 2, but still desire to make five annual equal installments. With 6% interest, what are the annual payments?

  46. P =$21,061.82 i = 6% 0 1 2 3 4 5 6 Grace period A A A A A P’ = $21,061.82(F/P, 6%, 1) i = 6% 0 1 2 3 4 5 6 A’ A’ A’ A’ A’ Example 2.13 Deferred Loan Repayment Plan

  47. Two-Step Procedure

  48. ? Option 1: Early Savings Plan 0 1 2 3 4 5 6 7 8 9 10 44 $2,000 Example 2.15 Early Savings Plan – 8% interest ? Option 2: Deferred Savings Plan 0 1 2 3 4 5 6 7 8 9 10 11 12 44 $2,000

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