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Financial Formulas for Investment Growth Analysis

Learn key formulas like Compound Amount Factors and Capital Recovery to maximize investments and savings, with practical examples and calculations.

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Financial Formulas for Investment Growth Analysis

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  1. Chapter 4More Interest Formulas EGN 3615 ENGINEERING ECONOMICS WITH SOCIAL AND GLOBAL IMPLICATIONS

  2. Chapter Contents • Uniform Series Compound Interest Formulas • Uniform Series Compound Amount Factor • Uniform Series Sinking Fund Factor • Uniform Series Capital Recovery Factor • Uniform Series Present Worth Factor • Arithmetic Gradient • Geometric Gradient • Nominal Effective Interest • Continuous Compounding

  3. Uniform Series Compound Amount Factor F1+F2+F3+F4 =F A A A A 0 1 2 3 4 0 1 2 3 4 A F1 0 1 2 3 4 A F2 0 1 2 3 4 A F3 0 1 2 3 4 A=F4 0 1 2 3 4

  4. Uniform Series Compound Amount Factor That is, for 4 periods, F = F1 + F2 + F3 + F4 = A(1+i)3 + A(1+i)2 + A(1+i) + A = A[(1+i)3 + (1+i)2 + (1+i) + 1]

  5. Uniform Series Compound Amount Factor A A A A F For n periods with interest (per period), F = F1 + F2 + F3 + … + Fn-1 + Fn = A(1+i)n-1 + A(1+i)n-2 + A(1+i)n-3 + … + A(1+i) + A = A[(1+i)n-1+ (1+i)n-1 + (1+i)n-3 + …+ (1+i) + 1] 0 1 2 3 4 A A n-1 n

  6. Uniform Series Compound Amount Factor i= interest rate per period n = total # of periods Notation Uniform Series Compound Amount Factor

  7. Uniform Series Formulas (Compare to slide 25) (1) Uniform series compound amount: Given A, i, & n, find F F = A{[(1+i)n – 1]/i} = A(F/A, i, n) (4-4) (2) Uniform series sinking fund: Given F, i, & n, find A A = F{i/[(1+i)n – 1]} = F(A/F, i, n) (4-5) (3) Given F, A, & i, find n n = log(1+Fi/A)/log(1+i) (4) Given F, A, & n, find i There is no closed form formula to use. But rate(nper, pmt, pv, fv, type, guess)

  8. Uniform Series Compound Amount Factor F= 552.6 • Question: If starting at EOY1, five annual deposits of $100 each are made in the bank account, how much money will be in the account at EOY5, if interest rate is 5% per year? i=0.05 0 1 2 3 4 5 $100 $100 $100 $100 $100

  9. QUESTION CONTINUES (USING INTEREST TABLE) F= 552.6 i=0.05 0 1 2 3 4 5 $100 $100 $100 $100 $100

  10. QUESTION CONTINUES(SPREADSHEET) Go to XL--Chap 4 extended examples-A1 Use function: FV(rate, nper, pmt, pv, type)

  11. Uniform Series Compound Amount Factor F1 F2= 580.2 • Question: Five annual deposits of $100 each are made into an account starting today. If interest rate is 5%, how much money will be in the account at EOY5? i=5% 0 1 2 3 4 5 $100 $100 $100 $100 $100

  12. QUESTION CONTINUES(INTEREST TABLE) F1 F2= 580.2 i=5% 0 1 2 3 4 5 $100 $100 $100 $100 $100

  13. QUESTION CONTINUES(SPREADSHEET)

  14. Uniform Series Compound Amount Factor F= ? • Question: If starting at EOY1, five annual deposits of $100 each are made in the bank account, how much money will be in the account at EOY5, if interest rate is 6.5% per year? i=6.5% 0 1 2 3 4 5 $100 $100 $100 $100 $100

  15. INTERPOLATION 6.0 5.6371 0.5 X 1 6.5 0.1136 7.0 5.7507 Interpolation

  16. Uniform Series Sinking Fund Factor F=Given i=Given n=Given 0 1 2 3 4 5 A=? A= Equal Annual Dollar Payments F= Future Some of Money i = Interest Rate Per Period n= Number of Interest Periods

  17. Uniform Series Sinking Fund Factor Notation Uniform Series Sinking Fund Factor

  18. Uniform Series Sinking Fund Factor • Question: A family wishes to have $12,000 in a bank account by the EOY 5. to accomplish this goal, five annual deposits starting at the EOF year 1 are to be made into a bank account paying 6% interest. what annual deposit must be made to reach the stated goal? i=5% n=5 0 1 2 3 4 5 A =$2172 F=$12,000

  19. QUESTION CONTINUES(INTEREST TABLE) i=5% n=5 0 1 2 3 4 5 A = $2172 F=$12,000

  20. Uniform Series Sinking Fund Factor F=$12,000 • Question: A family wishes to have $12,000 in a bank account by the EOY 5. to accomplish this goal, six annual deposits starting today are to be made into a bank account paying 5% interest. What annual deposit must be made to reach the stated goal? i=5% n=6 0 1 2 3 4 5 A = $1764

  21. $1764 QUESTION CONTINUES(INTEREST TABLE) F=$12,000 i=5% n=6 0 1 2 3 4 5 A = $1764

  22. Uniform Series Sinking Fund Factor F=$9000 • Example: the current balance of a bank account is $2,500. starting EOY 1 six equal annual deposits are to be made into the account. The goal is to have a balance of $9000 by the EOY 6. if interest rate is 6%, what annual deposit must be made to reach the stated goal? i=6% 0 1 2 3 4 5 6 P=$2,500 A = $796.23

  23. Uniform Series Capital Recovery Factor P=Given i=Given n 0 1 2 3 4 5 A=? P= Present Sum of Money A= Equal Annual Dollar Payments i = Interest Rate n= Number of Interest Periods

  24. Uniform Series Capital Recovery Factor Notation Uniform Series Capital Recovery Factor

  25. Uniform Series Formulas (Compare to slide 7) (1) Uniform series present worth: Given A, i, & n, find P P = A{[(1+i)n – 1]/[i(1+i)n]} = A(P/A, i, n)(4-7) (2) Uniform series capital recovery: Given P, i, & n, find A A = P{[i(1+i)n]/[(1+i)n – 1]} = P(A/P, i, n) (4-6) (3) Given P, A, & i, find n n = log[A/(A-Pi)]/log(1+i) (4) Given P, A, & n, find i (interest/period) There is no closed form formula to use. But rate(nper, pmt, pv, fv, type, guess)

  26. Uniform Series Capital Recovery Factor P= $100,000 • Example: A person borrows $100,000 from a commercial bank. The loan is to be repaid with five equal annual payments. If interest rate is 10%, what should the annual payments be? i=10% 0 1 2 3 4 5 A =26,380

  27. Example CONTINUES(INTEREST TABLE) P= $100,000 i=10% 0 1 2 3 4 5 A =26,380

  28. Use function: PMT(rate, nper, pv, fv, type) rate = interest rate/period nper = # of periods pv = present worth fv = balance at end of period n (blank means 0). type = 1 (payment at beginning of each period) or 0 (payment at end of a period)(blank means 0) See spreadsheet Uniform Series Capital Recovery (MS EXCEL)

  29. Uniform Series Capital Recovery Factor • Example: At age 30, a person begins putting $2,500 a year into account paying 10% interest. The last deposit is made on the man’s 54th birthday (25 deposits). Starting at age 55, 15 equal annual withdrawals are made. How much should each withdrawal be? Solution • Step 1: First A will be converted into F. • Step2: F will be considered as P. • Step3: P will be converted into Second A

  30. EXAMPLE CONTINUES F = 245,868 i=10% 0 1 2 3 21 22 23 24 A=$2500 i=10% 0 1 2 3 12 13 14 15 A =$32,332 P= $245,868

  31. Uniform Series Present Worth Factor A=Given n=Given 0 1 2 3 4 5 i=Given P=? A= Equal Annual Dollar Payments P= Present Sum of Money (at Time 0) i = Interest Rate/Period n= Number of Interest Periods

  32. Uniform Series Present Worth Factor Notation Uniform Series Present Worth Factor

  33. Uniform Series Present Worth Factor A=26,380 • Example: A special bank account is to be set up. Each year, starting at EOY 1, a $26,380 withdrawal is to be made. After five withdrawals the account is to be depleted. if interest rate is 10%, how much money should be deposited today? 0 1 2 3 4 5 i=10% P= 100,001

  34. EXAMPLE CONTINUES (USING INTEREST TABLE) A=26,380 0 1 2 3 4 5 i=10% P= 100,001

  35. Use function: PV(rate, nper, pmt, fv, type) rate = interest rate/period nper = total # of periods (payments) pmt = constant payment/period fv = balance at end of period n (blank means 0) type = 1 or 0 PV(0.1, 5, -26380) = $100,000.95 See spreadsheet Uniform Series Present Worth (Using MS EXCEL)

  36. Uniform Series Present Worth Factor • Example: Eight annual deposits of $500 each are made into a bank account beginning today. Up to EOY 4, the interest rate is 5%. After that, the interest rate is 8%. What is the present worth of these deposits? i=8% i=5% 0 1 2 3 4 5 6 7 A=500

  37. EXAMPLE CONTINUES i=8% i=5% 0 1 2 3 4 5 6 7 A=500

  38. EXAMPLE CONTINUES (Using MS EXCEL) P1 = PV(0.08, 3, -500)(1+0.05)–4 = (1288.55)(0.8227) = $1,060.09 P2 = PV(0.05, 4, -500) = $1,772.98

  39. Arithmetic Gradient • Arithmetic Gradient series (G): each annual amount differs from the previous one by a fixed amount G. A+4G A+3G 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 A+2G A+G 4G A 3G A A A A A 2G G + = 0

  40. Arithmetic Gradient Present Worth Factor • Given G, i, & n, find P (4-19) Notation Arithmetic Gradient Present Worth Factor

  41. Arithmetic Gradient Present Worth Factor • Question: You has purchased a new car. the following maintenance costs starting at EOY 2 will occur to pay the maintenance of your car for the 5 years. EOY2 $30, EOY3 $60, EOY4 $90, EOY5 $120. If interest rate is 5%, how much money you should deposit into a bank account today? $120 $90 $60 $30 0 i=5% 0 1 2 3 4 5 G=$30 P= $247.11

  42. QUESTION CONTINUES (INTEREST TABLE) $120 $90 $60 $30 0 i=5% 0 1 2 3 4 5 G=$30 P= $247.11

  43. Arithmetic Gradient Present Worth Factor • Question: If interest rate is 8%, what is the present worth of the following sums? 600 400 400 400 400 550 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 500 450 200 150 400 400 100 50 + = 0

  44. QUESTION CONTINUES 400 400 400 400 400 0 0 1 1 2 2 3 3 4 4 5 5 200 150 100 50 0

  45. Arithmetic Gradient Uniform Series Factor Convert an arithmetic gradient series into a uniform series Given G, i, & n, find A (4-20) Notation Arithmetic Gradient Uniform Series Factor

  46. Arithmetic Gradient Uniform Series Factor • Question: Demand for a new product will decrease as competitors enter the market. What is the equivalent annual amount of the revenue cash flows shown below? (interest 12%) 3000 3000 3000 3000 3000 3000 2500 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2000 1500 1000 = + 0 500 1000 1500 2000

  47. Geometric Series Present Worth Factor ? • Geometric series: Each annual amount is a fixed percentage different from the last. In this case, the change is 10%. • We will look at this problem in a few slides. g=10% ? $133 ? $121 $110 ? $100 ? ? 0 1 2 3 4 5 6 7 8 9 10 i=5% P=?

  48. Geometric Gradient • Unlike the Arithmetic Gradient where the amount of period-by-period change is a constant, for the Geometric Gradient, the period-by-period change is a uniform growth rate (g) or percentage rate. First year maintenance cost Uniform growth rate (g)

  49. Geometric Series Present Worth Factor Given A1, g, i, & n, find P (4-29) & (4-30) Geometric Series Present Worth Factor When Interest rate equals the growth rate,

  50. Geometric Series Present Worth Factor ? • Question: What is the present value (P) of a geometric series with $100 at EOY1 (A1), 5% interest rate (i), 10% growth rate (g), and 10 interest periods (n)? g=10% ? $133 ? $121 $110 ? $100 ? ? 0 1 2 3 4 5 6 7 8 9 10 i=5% P=?

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