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Time Value of Money

Time Value of Money. 1. Cash Flow Diagram ( up arrow is inflow or plus, down arrow is outflow or minus ) 2. Inflow / Outflow 3. P.V. 4. F.V. 5. Interest or Discount Rate (I) 6. N = number of periods or years 7. a = annuity, payment (pmt). Time Value of Money.

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Time Value of Money

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  1. Time Value of Money • 1. Cash Flow Diagram (up arrow is inflow or plus, down arrow is outflow or minus) • 2. Inflow / Outflow • 3. P.V. • 4. F.V. • 5. Interest or Discount Rate (I) • 6. N = number of periods or years • 7. a = annuity, payment (pmt)

  2. Time Value of Money • $110 = FV • 10% interest • 1 year • $100 = PV • $100 : Present Value = PV = P • $110 : Future Value = FV = F • 10% : Interest rate or discount rate = i • 1 year : number of periods = n

  3. Time Value of Money • 50 50 50 50 • 1 2 3 4 • $50 : annuity or payment = pmt =pymt

  4. Time Value of Money • 1 2 3 4 5 • 100 100 100 100 200 • Is this an annuity? • No, because the payments are not of the • same amount. Payments should be equal in every period for annuity. • (Period 1-3) is annuity • (Period 4-5) is not.

  5. Time Value of Money • Notation • GivenFindEquationBookLeTable • P F Fn = P(1+i)n P[FVIF] P[F/P] A-3 • F P Pn = F[1/(1+i)n ]F[PVIF] F[P/F] A-1 • a[ (1+i)n -1 ] • a F F= i a[FVIFA] a(F/A) A-4 • F[i] 1 • Fa a = (1+i)n -1 F[FVIFA] F(a/F)

  6. Time Value of Money • Notation • GivenFindEquationBookLeTable • a[1-1/(1+i)n] • a P P= i a[PVIFA] a(P/a) A-2 • i 1 • P a a=P[1-1/(1+i)n] P[PVIFA] P(a/P)

  7. Time Value of Money • Example: FV = ? • 10% • N = 1 • PV = 100 • PV = -100 i = 10% N = 1 FV = ? • FV = PV ( 1 + i )1 • = 100 ( 1 + 0.1 ) 1 • = 110

  8. Time Value of Money • Example: n=2 Years • Given P, Find F • F = ? • 10% • 1n = 2 2 • 100 • F = 100 ( 1 + 0.10)2 =P(1+i)2 • = 121

  9. Time Value of Money • Fn = P (1+i)n • (1+i)n = F/P = FVIF • Fn = P[F/P] = P[FVIF]

  10. Time Value of Money • Notation • GivenFindEquationBookLeTable • P F Fn = P(1+i)n P[FVIF] P[F/P] A-3 • F P Pn = F[1/(1+i)n ]F[PVIF] F[P/F] A-1 • a[ (1+i)n -1 ] • a F F= i a[FVIFA] a(F/A) A-4 • F[i] 1 • Fa a = (1+i)n -1 F[FVIFA] F(a/F)

  11. Time Value of Money Given P, find F: INPUTS 2 10% -100 N I/YR PV PMT FV 121 OUTPUT

  12. Time Value of Money • Example: F • n=20 • i=10% • PV=100,000 • PV= -100,000 • n=20 • i=10% • FV=?

  13. Time Value of Money • Example: • F = ? • 8% • n = 18 • 10,000 • F = P (F/P) 8% • = 10,000 (3.996)18 • = 39,960.20

  14. Time Value of Money Given P, find F: INPUTS 18 8% -10,000 N I/YR PV PMT FV 39,960 OUTPUT

  15. Time Value of Money Example : Given P, find F. INPUTS 20 15% -10,000 N I/YR PV PMT FV 163,665.40 OUTPUT

  16. Time Value of Money • Notation • GivenFindEquationBookLeTable • P F Fn = P(1+i)n P[FVIF] P[F/P] A-3 • F P Pn = F[1/(1+i)n ]F[PVIF] F[P/F] A-1 • a[ (1+i)n -1 ] • a F F= i a[FVIFA] a(F/A) A-4 • F[i] 1 • Fa a = (1+i)n -1 F[FVIFA] F(a/F)

  17. Time Value of Money • P = F[1/(1+i)n] = F[P/F] = F [PVIF] • Example: F=1,000,000 • n=30 • i=10% • P = ? F=1,000,000 • n=30 • i=10% • P=?

  18. Time Value of Money Given F, find P: INPUTS 20 15% -1,000,000 N I/YR PV PMT FV 61,100 OUTPUT

  19. TVM • Notation • GivenFindEquationBookLeTable • F[i] • FA A = (1+i)n -1 A/F • A[(1+i)n-1] • A F F = iF/A

  20. Time Value of Money • Given A, Find F: • F = ? • 10% • 1 2 • 100 1 100 2 • F2 = 100 2 + 100 1 ( 1 + 0.10 )1 = 210

  21. Time Value of Money • F = a (F/a) • F2 • 1 2 • i = 10% • a1 a2 • $100$100 • F2 = a2 + a1 (1+i)1 • F2 = a+ a(1+i)1

  22. Time Value of Money • F = a (F/a) • F3 • 1 2 3 • a1 a2 a3 • F3 = a3 + a2 (1+i)1 + a1 (1+i)2 • F3 = a+ a(1+i)1 + a(1+i)2

  23. Time Value of Money • Equation 1 • Fn = a + a(1 + i)1 + a(1 + i)2 + a(1 + i)3 +.. ... + a(1 + i)n-1 ] • Equation 2 • Fn = a [1 + (1 + i)1 + (1 + i)2 + (1 + i)3 + ... • + (1 + i)n-1 ] • Equation 3 • F/a = [1 + (1 + i)1 + (1 + i)2 + (1 + i)3 + ... • + (1 + i)n-1 ]

  24. Time Value of Money • F/a = [1 + (1 + i)1 + (1 + i)2 + (1 + i)3 + ... • + (1 + i)n-1 ] {Equation 3} • Multiply each side by (1 + i) to get:{Equation 4} • F/a (1 + i) = [(1 + i) + (1 + i)2 + (1 + i)3+... • + (1 + i)n-1 + (1 + i)n] • (4) - (3): • F/a (1 + i) - F/a = (1 + i)n - 1 • F/a (1 + i- 1) = (1 + i)n - 1

  25. Time Value of Money • F/a (1 + i- 1) = (1 + i)n - 1 • F/a (i) = (1 + i)n - 1 • (1 + i)n - 1 • F/a = • i • F = a [F/a] = a [((1+i)n - 1)/i] = a[FVIFA]

  26. Time Value of Money • Notation • GivenFindEquationBookLeTable • F[i] • FA A = (1+i)n -1 A/F • A[(1+i)n-1] • A F F = iF/A

  27. Time Value of Money • Example: • F20 • 1 2 _ _ _ _ 20 • 2000 2000 2000 • a = pmt = 2000 • n=20 i=10% • F20 =?

  28. Time Value of Money • Example: • F = ? • 15% • 100 100 100 ...........100 • n = 50

  29. Time Value of Money • Example: Given F, find a. • F = 721,770 • 15% • ? ? ? ........... ? • n = 50

  30. Time Value of Money Given a, find F: INPUTS 50 15% 0 -100 N I/YR PV PMT FV 721,770 OUTPUT

  31. Time Value of Money • Notation • GivenFindEquationBookLeTable • F[i] • FA A = (1+i)n -1 A/F • A[(1+i)n-1] • A F F = iF/A

  32. Time Value of Money • a=F[i/((1+i)n-1)]= F[a/F] = F[1/(FVIFA)] • Example: • i=10% 1,000,000 • _ _ _ _ 20 • a a a a • F = 1,000,000 • i = 10% • n =20 a = pmt = ?

  33. Time Value of Money • Given: FV = 721,770 • i = 15% • n = 50 • PV = 0 • a = F [ i ] • (1+i)n - 1 • = 721,770 [ 15% ] = 100 • (1 + 0.15)50- 1

  34. Time Value of Money • Notation • GivenFindEquationBookLeTable • a[1-1/(1+i)n] • a P P= i a[PVIFA] a(P/a) A-2 • i 1 • P a a=P[1-1/(1+i)n] P[PVIFA] P(a/P)

  35. Time Value of Money • P = a (P/a) • P= ( P ) ( F ) • a F a • P/a=[1/(1+i)n] [((1+i)n -1)/i] • P/a=[(1+i)n -1] / [(1+i)n i] • 1 • P = a [ 1 - (1 + i)n] = PVIFA • i

  36. Time Value of Money • Example: • P=? • 1 2 8% 30 • 10,000 10,000 ........... 10,000 • n = 30 • i=8% • n=30 years • pmt=a=10,000/year P =?

  37. Time Value of Money • Notation • GivenFindEquationBookLeTable • a[1-1/(1+i)n] • a P P= i a[PVIFA] a(P/a) A-2 • i 1 • P a a=P[1-1/(1+i)n] P[PVIFA] P(a/P)

  38. Time Value of Money • a = P[a/P] • 1 • P/a = 1 - (1+i)n = PVIFA • i • i 1 • a/P = [ 1 - 1] = PVIFA • (1 + i)n

  39. Time Value of Money • Example: House Price = $1,000,000 • Down Payment = 20% = $200,000 Loan = $800,000 • 800,000 8% • 1 2 3 30 • i = 8% • n = 30 years • PV = 800,000 a = pmt = ?

  40. Time Value of Money • Given P, find a: • i • a = P [ 1 - ( 1)n] • 1 + i • Example : • PV = 94,269 • i = 12% • n = 30 Pmt = ?

  41. Time Value of Money Given P, find a: INPUTS 30 12% -94,269 N I/YR PV PMT FV 11,702.88 OUTPUT

  42. Time Value of Money • Find monthly payment : • PV = 94,269 • n = 30 (12) = 360 • i = 12% / 12 = 1% • a = ?

  43. Time Value of Money Find monthly payment: INPUTS 360 1% -94,269 N I PV PMT FV 969.66 OUTPUT

  44. Time Value of Money • Example : • FV = 110 • i = ? • PV = 100 n = 1 year

  45. Time Value of Money Find i: INPUTS 1 -100 0 110 N I/YR PV PMT FV 10% OUTPUT

  46. Time Value of Money • Find i : • FV = 598.45 i = ? 1 2 3 4 5 n = 5 • PMT 100 100 100 100 100

  47. Time Value of Money Find i : INPUTS 5 0 -100 598.47 N I/YR PV PMT FV 9% OUTPUT

  48. Time Value of Money • Find i : • FV = 600 i = ? • 1 2 3 • P=200 100 100 100 n = 3

  49. Time Value of Money • Find i: INPUTS 3 -200 -100 600 N I/YR PV PMT FV 10.26 OUTPUT

  50. Confused? • Then study more--you’ll get it

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