1. 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)

2. Time Value of Money 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 Time Value of Money 50505050 1234 $50 : annuity or payment = pmt =pymt

4. Time Value of Money Time Value of Money 12345 100 100 100 100 200 Is this an annuity? No 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 Time Value of Money Notation Given Find Equation Book Le Table P FF n = P(1+i) n P[FVIF] P[F/P] A-3 F P P n = 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 F a a = (1+i) n -1 F[FVIFA] F(a/F)

6. Time Value of Money Time Value of Money Notation Given Find Equation Book Le Table a[1-1/(1+i) n ] aP P= i a[PVIFA] a(P/a) A-2 i 1 Pa a=P[1-1/(1+i) n ] P[PVIFA] P(a/P)

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

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

10. Fin362 LE 10 Time Value of Money Time Value of Money Notation Given Find Equation Book Le Table P FF n = P(1+i) n P[FVIF] P[F/P] A-3 F P P n = 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 F a a = (1+i) n -1 F[FVIFA] F(a/F)

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

12. Time Value of Money 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 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 Time Value of Money N 8% Given P, find F: INPUTS I/YRFVPMTPV OUTPUT 18-10,000 39,960

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

16. Fin362 LE 16 Time Value of Money Time Value of Money Notation Given Find Equation Book Le Table P FF n = P(1+i) n P[FVIF] P[F/P] A-3 F P P n = 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 F a a = (1+i) n -1 F[FVIFA] F(a/F)

17. Time Value of Money 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 Time Value of Money N 15% Given F, find P: INPUTS I/YRFVPMTPV OUTPUT 20-1,000,000 61,100

19. TVM TVM Notation Given Find Equation Book Le Table F [ i ] F A A = (1+i) n -1 A/F A [ (1+i) n -1 ] A F F = i F/A

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

21. Time Value of Money Time Value of Money F = a (F/a) F 2 12 i = 10% a 1 a 2 $100 $100 F 2 = a 2 + a 1 (1+i) 1 F 2 = a + a (1+i) 1

22. Time Value of Money Time Value of Money F = a (F/a) F 3 123 a 1 a 2 a 3 F 3 = a 3 + a 2 (1+i) 1 + a 1 (1+i) 2 F 3 = a + a (1+i) 1 + a (1+i) 2

23. Time Value of Money Time Value of Money Equation 1 F n = a + a(1 + i) 1 + a(1 + i) 2 + a(1 + i) 3 +..... + a(1 + i) n-1 ] Equation 2 F n = 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 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 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 Time Value of Money Notation Given Find Equation Book Le Table F [ i ] F A A = (1+i) n -1 A/F A [ (1+i) n -1 ] A F F = i F/A

27. Time Value of Money Time Value of Money Example: F 20 12 _ _ _ _ 20 20002000 2000 a = pmt = 2000 n=20i=10% F 20 =?

28. Time Value of Money Time Value of Money Example: F = ? 15% 100100 100...........100 n = 50

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

30. Time Value of Money Time Value of Money N 15% Given a, find F: INPUTS I/YRFVPMTPV OUTPUT 500 721,770 -100

31. Fin362 LE 31 Time Value of Money Time Value of Money Notation Given Find Equation Book Le Table F [ i ] F A A = (1+i) n -1 A/F A [ (1+i) n -1 ] A F F = i F/A

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

33. Time Value of Money 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. Fin362 LE 34 Time Value of Money Time Value of Money Notation Given Find Equation Book Le Table a[1-1/(1+i) n ] aP P= i a[PVIFA] a(P/a) A-2 i 1 Pa a=P[1-1/(1+i) n ] P[PVIFA] P(a/P)

35. Time Value of Money 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 Time Value of Money Example: P=? 1 2 8% 30 10,00010,000........... 10,000 n = 30 i=8% n=30 years pmt=a=10,000/yearP =?

37. Time Value of Money Time Value of Money N 10% Given a, find P: INPUTS I/YRFVPMTPV OUTPUT 300 94,269 -10,000

38. Fin362 LE 38 Time Value of Money Time Value of Money Notation Given Find Equation Book Le Table a[1-1/(1+i) n ] aP P= i a[PVIFA] a(P/a) A-2 i 1 Pa a=P[1-1/(1+i) n ] P[PVIFA] P(a/P)

39. Time Value of Money 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

40. Time Value of Money Time Value of Money Example:House Price = $1,000,000 Down Payment = 20% = $200,000 Loan = $800,000 800,0008% 12330 i = 8% n = 30 years PV = 800,000a = pmt = ?

41. Time Value of Money Time Value of Money Table A-2: Pmt = 10,000 i = 10% n = 30 PV = 94,269

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

43. Time Value of Money Time Value of Money N 12% Given P, find a: INPUTS I/YRFVPMTPV OUTPUT 30-94,269 11,702.88

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

45. Time Value of Money Time Value of Money N 1% Find monthly payment: INPUTS IFVPMTPV OUTPUT 360-94,269 969.66

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

47. Time Value of Money Time Value of Money N 110 Find i: INPUTS I/YRFVPMTPV OUTPUT 1-100 10% 0

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

49. Time Value of Money Time Value of Money N Find i : INPUTS I/YRFVPMTPV OUTPUT 50 9% -100598.47

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

51. Time Value of Money Time Value of Money Find i: N INPUTS I/YRFVPMTPV OUTPUT 3-200 10.26 -100600

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