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Nilai Waktu dari Uang (The Time Value of Money) Sasaran 1.Dapat menjelaskan mekanisme pemajemukan, yaitu bagaimana nilai uang dapat tumbuh saat dinvestasikan,

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Presentasi berjudul: "Nilai Waktu dari Uang (The Time Value of Money) Sasaran 1.Dapat menjelaskan mekanisme pemajemukan, yaitu bagaimana nilai uang dapat tumbuh saat dinvestasikan,"— Transcript presentasi:

1

2 Nilai Waktu dari Uang (The Time Value of Money)

3 Sasaran 1.Dapat menjelaskan mekanisme pemajemukan, yaitu bagaimana nilai uang dapat tumbuh saat dinvestasikan, Menentukan Nilai Masa Depan (Future Value) 2.Menentukan Nilai masa depan (Future Value) atau nilai sekarang (Present Value) atas sejumlah uang dengan periode bunga majemuk yang non tahunan 3.Mendiskusikan hubungan antara pemajemukan dan membawa kembali nilai sejumlah masa sekarang (Present Value)

4 4.Mendefinisikan anuitas biasa dan menghitung nilai majemuknya atau nilai masa depan 5.Membedakan antara anuitas biasa dengan anuitas jatuh tempo sertamenentukan nilai masa depan dan nilai sekarang dari suatu anuitas jatuh tempo 6.Menghitung annual persentase hasil tahunan atau tingkat suku bunga efektif tahunan dan menjelaskan perbedaannya dengan tingkat suku bunga nominal seperti yang tertera

5 Konsep Dasar 1.Terjadi perubahan Nilai Tukar Uang dari waktu ke waktu 2.Keputusan Manajemen Keuangan melalui lintas waktu

6 Bunga Majemuk & Discounted Compounding and Discounting Single Sums

7 Uang yg kita terima hari ini Rp akan bernilai lebih/ tumbuh dimasa yang akan datang. Ini sering di kenal sebagai opportunity costs. Opportunity cost yang diterima Rp akan menjadi lebih dimasa yang akan datang karena adanya bunga Today Future

8 Opportunity cost ini dapat di hitung

9 Rp hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding).

10 Opportunity cost ini dapat di hitung Rp hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding). Today ? Future

11 Opportunity cost ini dapat di hitung Rp hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding). Rp dimasa YAD = ? Hari ini (discounting). Today ? Future

12 Opportunity cost ini dapat di hitung Rp hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding). Rp dimasa YAD = ? Hari ini (discounting). ? Today Future Today ? Future

13 1. Future Value / Nilai Masa Depan

14 Nilai masa depan investasi diakhir tahun ke n FV dapat dihitung dengan konsep bunga majemuk (bunga berbunga) dengan asumsi bunga atau tingkat keuntungan yang diperoleh dari suatu investasi tidak diambil (dikonsumsi) tetapi diinvestasikan kembali dan suku bunga tidak berubah

15 Future Value – Pembayaran Tunggal Kita menyimpan Rp dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ?

16 0 1 PV = FV =

17 Future Value – Pembayaran Tunggal Kita menyimpan Rp dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ? PV = FV = Calculator Solution: P/Y = 1I = 6 N = 1 PV = FV = Rp

18 \ Future Value – Pembayaran Tunggal Kita menyimpan Rp dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ? Calculator Solution: P/Y = 1 I = 6 N = 1 PV = FV = Rp PV = FV =

19 Future Value – Pembayaran Tunggal Kita menyimpan Rp dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ? Mathematical Solution: FV = PV (FVIF i, n ) FV = (FVIF.06, 1 ) (use FVIF table, or) = (1.06) = Rp FV = PV (1 + i) n FV = (1.06) 1 = Rp PV = FV =

20 FV= Nilai masa depan investasi di akhir tahun ke n i= Interest Rate (Tingkat suku bunga atau diskonto) tahunan PV= Present Value (Nilai sekarang atau jumlah investasi mula-mula diawal tahun) (1+i) n dapat dihitung menggunakan tabel A-3 (tabel FVIF-Future Value Interest Factor) atau Lampiran B (Compoud) FV = PV (1 + i) n atau FV = PV (FVIF i, n )

21 Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun?

22 PV = FV =

23 Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun? Calculator Solution: P/Y = 1I = 6 N = 5 PV = FV = Rp PV = FV =

24 Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun? Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF.06, 5 ) (use FVIF table, or) FV = PV (1 + i) n FV = 100 (1.06) 5 = Rp PV = FV =

25 Compounding / Bunga Majemuk dengan periode Non Tahunan Periode bunga majemuk selain tahunan,pada beberapa transaksi periode pemajemukan bisa harian, 3 bulanan atau tengah tahunan

26 Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun?

27 0 ? 0 ? PV = FV =

28 Calculator Solution: P/Y = 4I = 6 N = 20 PV = FV = Rp Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun? PV = FV = ?

29 Calculator Solution: P/Y = 4I = 6 N = 20 PV = FV = $ Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun? PV = FV =

30 Mathematical Solution: FV = PV (FVIF i, n ) FV = (FVIF.015, 20 ) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = (1.015) 20 = Rp Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun? PV = FV =

31 FV n = PV (1+i/m) mn FVn= nilai masa depan investasi diakhir tahun ke-n PV= nilai sekarang atau jumlah investasi mula-mula diawal tahun pertama n= jumlah tahun pemajemukkan i= tingkat suku bunga (diskonto) tahunan m= jumlah berapa kali pemajemukkan terjadi

32 Future Value - continuous compounding Berapa FV dari Rp dengan bunga 8% setelah 100 tahun?

33 0 ? PV = FV =

34 Mathematical Solution: FV = PV (e in ) FV = 1000 (e.08x100 ) = 1000 (e 8 ) FV = Rp , PV = FV = Future Value - continuous compounding Berapa FV dari Rp dengan bunga 8% setelah 100 tahun?

35 PV = FV = Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding, after 100 years? Mathematical Solution: FV = PV (e in ) FV = 1000 (e.08x100 ) = 1000 (e 8 ) FV = Rp , 99

36 Present Value

37 Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

38 0 ? PV = FV = Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

39 Calculator Solution: P/Y = 1I = N = 1 FV = PV = PV = FV = Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

40 Calculator Solution: P/Y = 1I = N = 1 FV = PV = PV = FV = Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

41 Mathematical Solution: PV = FV (PVIF i, n ) PV = (PVIF.06, 1 )(use PVIF table, or) PV = FV / (1 + i) n PV = / (1.06) 1 = Rp PV = FV = Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

42

43 0 ? PV = FV = Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

44 Calculator Solution: P/Y = 1I = 6 N = 5 FV = 100 PV = PV = FV = 100 Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

45 Calculator Solution: P/Y = 1I = 6 N = 5 FV = 100 PV = Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? PV = FV = 100

46 Mathematical Solution: PV = FV (PVIF i, n ) PV = 100 (PVIF.06, 5 ) (use PVIF table, or) PV = FV / (1 + i) n PV = 100 / (1.06) 5 = $74.73 Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? PV = FV = 100

47 Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%?

48 PV = FV = Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%?

49 Calculator Solution: P/Y = 1I = 7 N = 15 FV = 1,000 PV = Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? PV = FV = 1000

50 Calculator Solution: P/Y = 1I = 7 N = 15 FV = 1,000 PV = Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? PV = FV = 1000

51 Mathematical Solution: PV = FV (PVIF i, n ) PV = 100 (PVIF.07, 15 ) (use PVIF table, or) PV = FV / (1 + i) n PV = 100 / (1.07) 15 = $ Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? PV = FV = 1000

52 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

53 PV = FV = Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

54 Calculator Solution: P/Y = 1N = 5 PV = -5,000 FV = 11,933 I = 19% PV = FV = 11,933 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

55 Mathematical Solution: PV = FV (PVIF i, n ) 5,000 = 11,933 (PVIF ?, 5 ) PV = FV / (1 + i) n 5,000 = 11,933 / (1+ i) = ((1/ (1+i) 5 ) = (1+i) 5 (2.3866) 1/5 = (1+i) i =.19 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

56 Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 PV = FV =

57 Calculator Solution: P/Y = 12FV = 500 I = 9.6PV = -100 N = 202 months Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 ? 0 ? PV = -100 FV = 500

58 Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: PV = FV / (1 + i) n 100 = 500 / (1+.008) N 5 = (1.008) N ln 5 = ln (1.008) N ln 5 = N ln (1.008) = N N = 202 months

59 Hint for single sum problems: In every single sum future value and present value problem, there are 4 variables: FV, PV, i, and n When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable. Keeping this in mind makes “time value” problems much easier!

60 The Time Value of Money Compounding and Discounting Cash Flow Streams

61 Annuities Annuity: a sequence of equal cash flows, occurring at the end of each period.

62 Annuities

63 Examples of Annuities: If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. If you borrow money to buy a house or a car, you will pay a stream of equal payments.

64 If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. If you borrow money to buy a house or a car, you will pay a stream of equal payments. Examples of Annuities:

65 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

66 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

67 Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 FV = $3, Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

68 Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 FV = $3, Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

69 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

70 Mathematical Solution: Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

71 Mathematical Solution: FV = PMT (FVIFA i, n ) Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

72 Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

73 Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) FV = PMT (1 + i) n - 1 i Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

74 Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) FV = PMT (1 + i) n - 1 i FV = 1,000 (1.08) = $ Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

75 Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

76 Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

77 Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 PV = $2, Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

78 Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 PV = $2, Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

79

80 Mathematical Solution: Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

81 Mathematical Solution: PV = PMT (PVIFA i, n ) Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

82 Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

83 Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) 1 PV = PMT 1 - (1 + i) n i Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

84 Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) 1 PV = PMT 1 - (1 + i) n i 1 PV = (1.08 ) 3 = $2, Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

85 Other Cash Flow Patterns 0123 The Time Value of Money

86 Perpetuities Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. You can think of a perpetuity as an annuity that goes on forever.

87 Present Value of a Perpetuity When we find the PV of an annuity, we think of the following relationship:

88 Present Value of a Perpetuity When we find the PV of an annuity, we think of the following relationship: PV = PMT (PVIFA i, n ) PV = PMT (PVIFA i, n )

89 Mathematically,

90 (PVIFA i, n ) =

91 Mathematically, (PVIFA i, n ) = (1 + i) n i

92 Mathematically, (PVIFA i, n ) = We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large? (1 + i) n i

93 When n gets very large,

94 1 - 1 (1 + i) n i

95 When n gets very large, this becomes zero (1 + i) n i

96 When n gets very large, this becomes zero. So we’re left with PVIFA = 1 i (1 + i) n i

97 So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity

98 PMT i PV = So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity

99 What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

100 PMT $10,000 PMT $10,000 i.08 i.08 PV = =

101 What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? PMT $10,000 PMT $10,000 i.08 i.08 = $125,000 PV = =

102 Ordinary Annuity vs. Annuity Due $1000 $1000 $

103 Begin Mode vs. End Mode

104 Begin Mode vs. End Mode year year year 5 6 7

105 Begin Mode vs. End Mode year year year PVinENDMode

106 Begin Mode vs. End Mode year year year PVinENDModeFVinENDMode

107 Begin Mode vs. End Mode year year year 6 7 8

108 Begin Mode vs. End Mode year year year PVinBEGINMode

109 Begin Mode vs. End Mode year year year PVinBEGINModeFVinBEGINMode

110 Earlier, we examined this “ordinary” annuity:

111

112 Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that:

113 Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that: The Future Value (at 3) is $3,

114 Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that: The Future Value (at 3) is $3, The Present Value (at 0) is $2,

115 What about this annuity? Same 3-year time line, Same 3 $1000 cash flows, but The cash flows occur at the beginning of each year, rather than at the end of each year. This is an “annuity due.”

116 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?

117 Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = -1,000 FV = $3, Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?

118 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = -1,000 FV = $3,506.11

119 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

120 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i)

121 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or)

122 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + i) n - 1 i (1 + i)

123 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + i) n - 1 i FV = 1,000 (1.08) = $3, (1 + i) (1.08)

124 Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?

125 Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = 1,000 PV = $2, Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?

126 Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = 1,000 PV = $2, Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?

127 Present Value - annuity due Mathematical Solution:

128 Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

129 Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i)

130 Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or)

131 Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) 1 PV = PMT 1 - (1 + i) n i (1 + i)

132 Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) 1 PV = PMT 1 - (1 + i) n i 1 PV = (1.08 ) 3 = $2, (1 + i) (1.08)

133 Is this an annuity? How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate). Uneven Cash Flows ,000 2,000 4,000 6,000 7,000

134 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately ,000 2,000 4,000 6,000 7,000 Uneven Cash Flows

135 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately ,000 2,000 4,000 6,000 7,000 Uneven Cash Flows

136 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately ,000 2,000 4,000 6,000 7,000 Uneven Cash Flows

137 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately ,000 2,000 4,000 6,000 7,000 Uneven Cash Flows

138 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately ,000 2,000 4,000 6,000 7,000 Uneven Cash Flows

139 period CF PV (CF) 0-10, , ,000 1, ,000 3, ,000 4, ,000 4, PV of Cash Flow Stream: $ 4, ,000 2,000 4,000 6,000 7,000

140 Annual Percentage Yield (APY) Which is the better loan: 8% compounded annually, or 7.85% compounded quarterly? We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year! We need to calculate the APY.

141 Annual Percentage Yield (APY)

142 APY = ( 1 + ) m - 1 quoted rate m

143 Annual Percentage Yield (APY) Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m

144 Annual Percentage Yield (APY) Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + )

145 Annual Percentage Yield (APY) Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) APY =.0808, or 8.08%.07854

146 Annual Percentage Yield (APY) Find the APY for the quarterly loan: The quarterly loan is more expensive than the 8% loan with annual compounding! APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) APY =.0808, or 8.08%.07854

147 Practice Problems

148 Example Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?

149 Example $ Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?

150 This type of cash flow sequence is often called a “deferred annuity.” $

151 How to solve: 1) Discount each cash flow back to time 0 separately $

152 How to solve: 1) Discount each cash flow back to time 0 separately $

153 How to solve: 1) Discount each cash flow back to time 0 separately $

154 How to solve: 1) Discount each cash flow back to time 0 separately $

155 How to solve: 1) Discount each cash flow back to time 0 separately $

156 How to solve: 1) Discount each cash flow back to time 0 separately $

157 How to solve: 1) Discount each cash flow back to time 0 separately. Or, $

158 2) Find the PV of the annuity: PV : End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PV = $119, $

159 2) Find the PV of the annuity: PV 3: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PV 3 = $119, $

160 119,

161 Then discount this single sum back to time 0. PV: End mode; P/YR = 1; I = 20; N = 3; FV = 119,624; Solve: PV = $69, , $

162 69, ,624

163 The PV of the cash flow stream is $69, , $ ,624

164 Retirement Example After graduation, you plan to invest $400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years?

165 Retirement Example After graduation, you plan to invest $400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years?

166

167 Using your calculator, P/YR = 12 N = 360 PMT = -400 I%YR = 12 FV = $1,397,

168 Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30?

169 Mathematical Solution:

170 Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n )

171 Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table)

172 Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = PMT (1 + i) n - 1 i

173 Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = PMT (1 + i) n - 1 i FV = 400 (1.01) = $1,397,

174 If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment? House Payment Example

175 If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?

176 ? ? ? ?

177 Using your calculator, P/YR = 12 N = 360 I%YR = 7 PV = $100,000 PMT = -$ ? ? ? ? ? ? ? ?

178 House Payment Example Mathematical Solution:

179 House Payment Example Mathematical Solution: PV = PMT (PVIFA i, n )

180 House Payment Example Mathematical Solution: PV = PMT (PVIFA i, n ) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table)

181 House Payment Example Mathematical Solution: PV = PMT (PVIFA i, n ) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table) 1 PV = PMT 1 - (1 + i) n i

182 House Payment Example Mathematical Solution: PV = PMT (PVIFA i, n ) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table) 1 PV = PMT 1 - (1 + i) n i 1 100,000 = PMT 1 - ( ) 360 PMT=$

183 Team Assignment Upon retirement, your goal is to spend 5 years traveling around the world. To travel in style will require $250,000 per year at the beginning of each year. If you plan to retire in 30 years, what are the equal monthly payments necessary to achieve this goal? The funds in your retirement account will compound at 10% annually.

184 How much do we need to have by the end of year 30 to finance the trip? PV 30 = PMT (PVIFA.10, 5 ) (1.10) = = 250,000 (3.7908) (1.10) = = $1,042,

185 Using your calculator, Mode = BEGIN PMT = -$250,000 N = 5 I%YR = 10 P/YR = 1 PV = $1,042,

186 Now, assuming 10% annual compounding, what monthly payments will be required for you to have $1,042,466 at the end of year 30? ,042,466

187  Using your calculator, Mode = END N = 360 I%YR = 10 P/YR = 12 FV = $1,042,466 PMT = -$ ,042,466

188 So, you would have to place $ in your retirement account, which earns 10% annually, at the end of each of the next 360 months to finance the 5-year world tour.


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