Nilai Waktu dari Uang (The Time Value of Money)

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

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

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

4 Konsep Dasar Terjadi perubahan Nilai Tukar Uang dari waktu ke waktu
Keputusan Manajemen Keuangan melalui lintas waktu

5 Bunga Majemuk & Discounted Compounding and Discounting Single Sums

6 Uang yg kita terima hari ini Rp. 100
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

7 Opportunity cost ini dapat di hitung

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

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

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

12 1. Future Value / Nilai Masa Depan

13 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

14 Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100
Future Value – Pembayaran Tunggal Kita menyimpan Rp dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ?

15 Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100
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 =

16 PV = -100.000 FV = Calculator Solution: P/Y = 1 I = 6
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 = 1 I = 6 N = PV = FV = Rp

17 PV = -100.000 FV = Calculator Solution: P/Y = 1 I = 6
\ 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 = I = 6 N = PV = FV = Rp

18 Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100
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 = 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

19 FV = PV (1 + i)n atau FV = PV (FVIF i, n )
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)

20 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun?

21 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun? PV = FV =

22 PV = 100.000 FV = Calculator Solution: P/Y = 1 I = 6
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun? PV = FV = Calculator Solution: P/Y = 1 I = 6 N = PV = FV = Rp

23 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun? PV = FV = 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

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

25 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun?

26 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun? ? PV = FV =

27 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun? PV = FV = ? Calculator Solution: P/Y = 4 I = 6 N = PV = FV = Rp

28 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun? PV = FV = Calculator Solution: P/Y = 4 I = 6 N = PV = FV = $

29 Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100
Future Value – Pembayaran Tunggal Jika anda menabung Rp hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa tabungan anda setelah 5 tahun? PV = FV = 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

30 FVn = 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

31 Future Value - continuous compounding Berapa FV dari Rp. 1
Future Value - continuous compounding Berapa FV dari Rp dengan bunga 8% setelah 100 tahun?

32 Future Value - continuous compounding Berapa FV dari Rp. 1
Future Value - continuous compounding Berapa FV dari Rp dengan bunga 8% setelah 100 tahun? PV = FV = ?

33 Mathematical Solution: FV = PV (e in)
Future Value - continuous compounding Berapa FV dari Rp dengan bunga 8% setelah 100 tahun? PV = FV = Mathematical Solution: FV = PV (e in) FV = (e .08x100) = (e 8) FV = Rp ,99

34 Mathematical Solution: FV = PV (e in)
Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding, after 100 years? PV = FV = 2.980 Mathematical Solution: FV = PV (e in) FV = (e .08x100) = (e 8) FV = Rp ,99

35 Present Value

36 Present Value - single sums Jika anda menerima Rp. 100
Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

37 Present Value - single sums Jika anda menerima Rp. 100
Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6% PV = FV = ?

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

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

40 Present Value - single sums Jika anda menerima Rp. 100
Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6% PV = FV = 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

41 Present Value - single sums Jika anda menerima Rp. 100
Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%

42 Present Value - single sums Jika anda menerima Rp. 100
Present Value - single sums Jika anda menerima Rp tahun yang akan datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6% PV = FV = ?

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

44 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 = Calculator Solution: P/Y = 1 I = 6 N = FV = 100 PV =

45 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 = 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

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

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%? PV = FV =

48 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 = Calculator Solution: P/Y = 1 I = 7 N = FV = 1,000 PV =

49 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 = Calculator Solution: P/Y = 1 I = 7 N = FV = 1,000 PV =

50 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 = 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 = $362.45

51 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?

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? PV = FV =

53 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? PV = FV = 11,933 Calculator Solution: P/Y = 1 N = 5 PV = -5, FV = 11,933 I = 19%

54 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? Mathematical Solution: PV = FV (PVIF i, n ) 5,000 = 11,933 (PVIF ?, 5 ) PV = FV / (1 + i)n 5,000 = 11,933 / (1+ i)5 = ((1/ (1+i)5) = (1+i)5 (2.3866)1/5 = (1+i) i = .19

55 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? PV = FV =

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? ? PV = FV = Calculator Solution: P/Y = 12 FV = 500 I = PV = -100 N = months

57 Mathematical Solution:
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 / ( )N 5 = (1.008)N ln 5 = ln (1.008)N ln 5 = N ln (1.008) = N N = 202 months

58 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!

59 Compounding and Discounting
The Time Value of Money Compounding and Discounting Cash Flow Streams 1 2 3 4

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

61 Annuities Annuity: a sequence of equal cash flows, occurring at the end of each period. 1 2 3 4

62 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.

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

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?
Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = -1,000 FV = $3,246.40

67 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = -1,000 FV = $3,246.40

68 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?
Mathematical Solution:

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

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

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

73 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
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) = $ .08

74 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%?

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%? Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = -1,000 PV = $2,577.10

77 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%? Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = -1,000 PV = $2,577.10

78 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 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%? Mathematical Solution:

80 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%? Mathematical Solution: PV = PMT (PVIFA i, n )

81 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%? Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)

82 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%? Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or) 1 PV = PMT (1 + i)n i

83 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%? Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or) 1 PV = PMT (1 + i)n i PV = (1.08 )3 = $2,577.10 .08

84 Other Cash Flow Patterns
The Time Value of Money 1 2 3 Other Cash Flow Patterns

85 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.

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

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

88 Mathematically,

89 Mathematically, (PVIFA i, n ) =

90 Mathematically, (PVIFA i, n ) = 1 - 1 (1 + i) n i

91 1 - i Mathematically, 1 (PVIFA i, n ) = (1 + i)
We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large? 1 - 1 (1 + i) n i

92 When n gets very large,

93 When n gets very large, 1 - 1 (1 + i) n i

94 When n gets very large, this becomes zero. 1 - 1 (1 + i) n i

95 1 - 1 i i When n gets very large, this becomes zero. 1 (1 + i)
So we’re left with PVIFA = 1 - 1 (1 + i) n i 1 i

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

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

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

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? PMT $10,000 i PV = =

100 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 i = $125,000 PV = =

101 Ordinary Annuity vs. Annuity Due
$ $ $1000

102 Begin Mode vs. End Mode

103 Begin Mode vs. End Mode 4 5 6 7 8 ordinary annuity 1000 1000 1000
year year year ordinary annuity

104 PV Begin Mode vs. End Mode 4 5 6 7 8 ordinary annuity in END Mode
year year year PV in END Mode ordinary annuity

105 PV FV Begin Mode vs. End Mode 4 5 6 7 8 in END Mode in END Mode
year year year PV in END Mode FV in END Mode ordinary annuity

106 Begin Mode vs. End Mode 4 5 6 7 8 annuity due 1000 1000 1000
year year year annuity due

107 PV Begin Mode vs. End Mode 4 5 6 7 8 annuity due in BEGIN Mode
year year year PV in BEGIN Mode annuity due

108 PV FV Begin Mode vs. End Mode 4 5 6 7 8 in BEGIN Mode in BEGIN Mode
year year year PV in BEGIN Mode FV in BEGIN Mode annuity due

109 Earlier, we examined this “ordinary” annuity:

110 Earlier, we examined this “ordinary” annuity:

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

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

113 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,

114 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.”

115 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?

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? Calculator Solution: Mode = BEGIN P/Y = 1 I = 8 N = PMT = -1,000 FV = $3,506.11

117 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 = 1 I = 8 N = PMT = -1,000 FV = $3,506.11

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? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

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: FV = PMT (FVIFA i, n ) (1 + i)

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) FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or)

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) FV = PMT (1 + i)n - 1 i (1 + i)

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 FV = 1,000 (1.08) = $3,506.11 .08 (1 + i) (1.08)

123 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%?

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%? Calculator Solution: Mode = BEGIN P/Y = 1 I = 8 N = PMT = 1,000 PV = $2,783.26

125 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%? Calculator Solution: Mode = BEGIN P/Y = 1 I = 8 N = PMT = 1,000 PV = $2,783.26

126 Present Value - annuity due
Mathematical Solution:

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

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

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) PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or)

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) 1 PV = PMT (1 + i)n i (1 + i)

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 + i)n i PV = (1.08 ) = $2,783.26 .08 (1 + i) (1.08)

132 Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 Is this an annuity?
-10, , , , ,000 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).

133 Uneven Cash Flows -10, , , , ,000 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

134 Uneven Cash Flows -10, , , , ,000 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

135 Uneven Cash Flows -10, , , , ,000 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

136 Uneven Cash Flows -10, , , , ,000 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

137 Uneven Cash Flows -10, , , , ,000 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

138 PV of Cash Flow Stream: $ 4,412.95
-10, , , , ,000 period CF PV (CF) 0 -10, ,000.00 , ,818.18 , ,305.79 , ,507.89 , ,781.09 PV of Cash Flow Stream: $ 4,412.95

139 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.

140 Annual Percentage Yield (APY)

141 Annual Percentage Yield (APY)
APY = ( ) m - 1 quoted rate m

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

143 Annual Percentage Yield (APY)
APY = ( ) m - 1 quoted rate m Find the APY for the quarterly loan: APY = ( ) .0785 4

144 Annual Percentage Yield (APY)
APY = ( ) m - 1 quoted rate m Find the APY for the quarterly loan: APY = ( ) APY = .0808, or % .0785 4

145 Annual Percentage Yield (APY)
APY = ( ) m - 1 quoted rate m Find the APY for the quarterly loan: The quarterly loan is more expensive than the 8% loan with annual compounding! APY = ( ) APY = .0808, or % .0785 4

146 Practice Problems

147 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?

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

150 $ How to solve: 1) Discount each cash flow back to time 0 separately.

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. Or,

157 2) Find the PV of the annuity:
$ 2) Find the PV of the annuity: PV: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PV = $119,624

158 2) Find the PV of the annuity:
$ 2) Find the PV of the annuity: PV3: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PV3= $119,624

159 $ 119,624

160 119,624 Then discount this single sum back to time 0.
$ 119,624 Then discount this single sum back to time 0. PV: End mode; P/YR = 1; I = 20; N = 3; FV = 119,624; Solve: PV = $69,226

161 69,226 $ 119,624

162 The PV of the cash flow stream is $69,226.
$ 119,624 The PV of the cash flow stream is $69,226.

163 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?

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? 1 2 3

165 1 2 3

166 Using your calculator, N = 360 PMT = -400 I%YR = 12 FV = $1,397,985.65
1 2 3 Using your calculator, P/YR = 12 N = 360 PMT = -400 I%YR = 12 FV = $1,397,985.65

167 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?

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? Mathematical Solution:

169 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 )

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 ) FV = 400 (FVIFA .01, 360 ) (can’t use FVIFA table)

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) FV = PMT (1 + i)n - 1 i

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 FV = (1.01) = $1,397,985.65 .01

173 House Payment Example 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?

174 House Payment Example 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?

175 1 2 3 ? ? ? ?

176 P/YR = 12 I%YR = 7 PV = $100,000 PMT = -$665.30
1 2 3 ? ? ? ? Using your calculator, P/YR = 12 N = 360 I%YR = 7 PV = $100,000 PMT = -$665.30

177 House Payment Example Mathematical Solution:

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

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

180 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 + i)n i

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 + i)n i 100,000 = PMT ( ) PMT=$665.30

182 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.

183 How much do we need to have by the end of year 30 to finance the trip?
27 28 29 30 31 32 33 34 35 How much do we need to have by the end of year 30 to finance the trip? PV30 = PMT (PVIFA .10, 5) (1.10) = = 250,000 (3.7908) (1.10) = = $1,042,470

184 27 28 29 30 31 32 33 34 35 Using your calculator, Mode = BEGIN PMT = -$250,000 N = 5 I%YR = 10 P/YR = 1 PV = $1,042,466

185 27 28 29 30 31 32 33 34 35 1,042,466 Now, assuming 10% annual compounding, what monthly payments will be required for you to have $1,042,466 at the end of year 30?

186 27 28 29 30 31 32 33 34 35 1,042,466 Using your calculator, Mode = END N = 360 I%YR = 10 P/YR = 12 FV = $1,042,466 PMT = -$461.17

187 So, you would have to place $461
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.