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Understanding Time Value of Money: Key Concepts and Calculations

The Time Value of Money (TVM) is fundamental in finance, emphasizing that a dollar today is worth more than a dollar in the future due to its potential earning capacity. This guide covers essential elements such as cash flow diagrams, present value (PV), future value (FV), interest rates, and annuities. It explains vital formulas, including FV = PV(1+i)^n, and offers examples demonstrating how to calculate future values based on varying periods and interest rates, enhancing your grasp of this crucial financial principle.

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Understanding Time Value of Money: Key Concepts and Calculations

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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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