nilai waktu dari uang (the time value of money)
DESCRIPTION
Nilai Waktu dari Uang (The Time Value of Money). Sasaran. Dapat menjelaskan mekanisme pemajemukan, yaitu bagaimana nilai uang dapat tumbuh saat dinvestasikan, Menentukan Nilai Masa Depan (Future Value) - PowerPoint PPT PresentationTRANSCRIPT
Nilai Waktu dari Uang (The Time Value of
Money)
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. 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
Konsep Dasar
1. Terjadi perubahan Nilai Tukar Uang dari waktu ke waktu
2. Keputusan Manajemen Keuangan melalui lintas waktu
Bunga Majemuk & DiscountedCompounding and Discounting Single Sums
Uang yg kita terima hari ini Rp. 100.000 akan bernilai lebih/ tumbuh dimasa yang akan datang . Ini sering di kenal sebagai opportunity costs.
Opportunity cost yang diterima Rp. 100.000 akan menjadi lebih dimasa yang akan datang karena adanya bunga
Today Future
Opportunity cost ini dapat di hitung
Opportunity cost ini dapat di hitung
• Rp. 100.000 hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding).
Opportunity cost ini dapat di hitung
• Rp. 100.000 hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding).
Today
?
Future
Opportunity cost ini dapat di hitung
• Rp. 100.000 hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding).
• Rp. 100.000 dimasa YAD = ? Hari ini (discounting).
Today
?
Future
Opportunity cost ini dapat di hitung
• Rp. 100.000 hari ini menjadi ? Di masa YAD / Future menjadi ? (compounding).
• Rp. 100.000 dimasa YAD = ? Hari ini (discounting).
?
Today Future
Today
?
Future
1. Future Value /Nilai Masa Depan
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
Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100.000 dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ?
Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100.000 dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ?
0 1
PV =PV = FV = FV =
Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100.000 dalam tabungan dengan tingkat suku bunga majemuk 6% per tahun. Berapa nilai uang kita setelah 1 tahun ?
00 1 1
PV = -100.000PV = -100.000 FV = FV =
Calculator Solution:
P/Y = 1 I = 6
N = 1 PV = -100.000
FV = Rp. 106.000
\ Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100.000 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 = -100.000
FV = Rp. 106.000
00 1 1
PV = -100.000PV = -100.000 FV = FV =
Future Value – Pembayaran Tunggal Kita menyimpan Rp. 100.000 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 = 100.000 (FVIF .06, 1 ) (use FVIF table, or) = 100.000 (1.06) = Rp.106.000• FV = PV (1 + i)n
FV = 100.000 (1.06)1 = Rp.106.000
00 1 1
PV = -100.000PV = -100.000 FV = FV = 106.000106.000
FV = Nilai masa depan investasi di akhir tahun ke ni = Interest Rate (Tingkat suku bunga atau diskonto) tahunanPV = 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 )
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun?
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun?
00 5 5
PV =PV = FV = FV =
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, berapa tabungan anda setelah 5 tahun?
Calculator Solution:
P/Y = 1 I = 6
N = 5 PV = -100.000
FV = Rp.133.820
00 5 5
PV = 100.000PV = 100.000 FV = FV =
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 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. 133.820
00 5 5
PV = 100.000PV = 100.000 FV = 133.820 FV = 133.820
Compounding / Bunga Majemuk dengan periode Non Tahunan
• Periode bunga majemuk selain tahunan,pada beberapa transaksi periode pemajemukan bisa harian, 3 bulanan atau tengah tahunan
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa
tabungan anda setelah 5 tahun?
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa
tabungan anda setelah 5 tahun?
00 ? ?
PV = 100.000PV = 100.000 FV = FV =
Calculator Solution:
P/Y = 4 I = 6
N = 20 PV = -100.000
FV = Rp. 134.680
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa
tabungan anda setelah 5 tahun?
00 20 20
PV = 100.000PV = 100.000 FV = ?FV = ?
Calculator Solution:
P/Y = 4 I = 6
N = 20 PV = -100.000
FV = $134.680
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa
tabungan anda setelah 5 tahun?
00 20 20
PV = 100.000PV = 100.000 FV = 134.680FV = 134.680
Mathematical Solution:• FV = PV (FVIF i, n )
FV = 100.000 (FVIF .015, 20 ) (can’t use FVIF table)
• FV = PV (1 + i/m) m x n
FV = 100.000 (1.015)20 = Rp. 134.680
Future Value – Pembayaran Tunggal Jika anda menabung Rp. 100.000 hari ini dengan tingkat bunga 6%, dimajemukan kuartalan berapa
tabungan anda setelah 5 tahun?
00 20 20
PV = 100.000PV = 100.000 FV = 134.680FV = 134.680
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
Future Value - continuous compounding Berapa FV dari Rp. 1.000 dengan bunga 8% setelah
100 tahun?
Future Value - continuous compounding Berapa FV dari Rp. 1.000 dengan bunga 8%
setelah 100 tahun?
0 ?
PV =PV = FV = FV =
Mathematical Solution:
FV = PV (e in)
FV = 1000 (e .08x100) = 1000 (e 8)
FV = Rp. 2.980.957,99
00 100 100
PV = -1000PV = -1000 FV = FV =
Future Value - continuous compounding Berapa FV dari Rp. 1.000 dengan bunga 8%
setelah 100 tahun?
00 100 100
PV = -1000PV = -1000 FV = FV = 2.9802.980
Future Value - continuous compoundingWhat 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. 2.980.957,99
Present Value
Present Value - single sumsJika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
0 ?
PV =PV = FV = FV =
Present Value - single sums Jika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
Calculator Solution:
P/Y = 1 I = 6
N = 1 FV = 100.000100.000
PV = -94.340
00 1 1
PV = PV = FV = FV = 100.000100.000
Present Value - single sums Jika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
Calculator Solution:
P/Y = 1 I = 6
N = 1 FV = 100.000100.000
PV = -94.340
PV = PV = -94.-94.3434 FV = FV = 100.000100.000
00 1 1
Present Value - single sums Jika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
Mathematical Solution:• PV = FV (PVIF i, n )
PV = 100.000 (PVIF .06, 1 )(use PVIF table, or)
• PV = FV / (1 + i)n
PV = 100.000 / (1.06)1 = Rp. 94.340
PV = PV = -94.-94.3434 FV = FV = 100.000100.000
00 1 1
Present Value - single sums Jika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
Present Value - single sums Jika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
0 ?
PV =PV = FV = FV =
Present Value - single sums Jika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
Calculator Solution:
P/Y = 1 I = 6
N = 5 FV = 100
PV = -74.73
00 5 5
PV = PV = FV = 100 FV = 100
Present Value - single sums Jika anda menerima Rp. 100.000 1 tahun yang akan
datang, berapa nilai sekarangnya (PV) jika biaya opportunity 6%
Calculator Solution:
P/Y = 1 I = 6
N = 5 FV = 100
PV = -74.73
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
00 5 5
PV = PV = -74.-74.7373 FV = 100 FV = 100
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 sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
00 5 5
PV = PV = -74.-74.7373 FV = 100 FV = 100
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = FV = FV =
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
Calculator Solution:
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -362.45
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = FV = 1000 FV = 1000
Calculator Solution:
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -362.45
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = -362.-362.4545 FV = 1000 FV = 1000
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
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = -362.-362.4545 FV = 1000 FV = 1000
Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
00 5 5
PV = PV = FV = FV =
Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
Calculator Solution:
P/Y = 1 N = 5
PV = -5,000 FV = 11,933
I = 19%
00 5 5
PV = -5000PV = -5000 FV = 11,933 FV = 11,933
Present Value - single sumsIf 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
.419 = ((1/ (1+i)5)
2.3866 = (1+i)5
(2.3866)1/5 = (1+i) i = .19
Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
Present Value - single sumsSuppose 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?
00
PV = PV = FV = FV =
Calculator Solution:
• P/Y = 12 FV = 500
• I = 9.6 PV = -100
• N = 202 months
Present Value - single sumsSuppose 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?
00 ? ?
PV = -100PV = -100 FV = 500 FV = 500
Present Value - single sumsSuppose 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)
1.60944 = .007968 N N = 202 months
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!
The Time Value of Money
Compounding and Discounting
Cash Flow Streams
0 1 2 3 4
Annuities
• Annuity: a sequence of equal cash flows, occurring at the end of each period.
• Annuity: a sequence of equal cash flows, occurring at the end of each period.
0 1 2 3 4
Annuities
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.
• 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:
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
Future Value - annuityIf 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
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
10001000 10001000 1000 1000
Calculator Solution:
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = $3,246.40
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
10001000 10001000 1000 1000
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Mathematical Solution:
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Mathematical Solution:
FV = PMT (FVIFA i, n )
Future Value - annuityIf 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)
Future Value - annuityIf 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
Future Value - annuityIf 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)3 - 1 = $3246.40
.08
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Present Value - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
0 1 2 3
Present Value - annuityWhat 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
0 1 2 3
10001000 10001000 1000 1000
Present Value - annuityWhat 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
0 1 2 3
10001000 10001000 1000 1000
Present Value - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
Present Value - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
Mathematical Solution:
Present Value - annuityWhat 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 )
Present Value - annuityWhat 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)
Present Value - annuityWhat 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 - (1 + i)n
i
Present Value - annuityWhat 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 - (1 + i)n
i
1
PV = 1000 1 - (1.08 )3 = $2,577.10
.08
Present Value - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
Other Cash Flow Patterns
0 1 2 3
The Time Value of Money
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.
Present Value of a Perpetuity
• When we find the PV of an annuity, we think of the following relationship:
Present Value of a Perpetuity
• When we find the PV of an annuity, we think of the following relationship:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )
Mathematically,
Mathematically,
(PVIFA i, n ) =
Mathematically,
(PVIFA i, n ) = 1 - 1 - 11
(1 + i)(1 + i)nn
ii
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 - 1 - 11
(1 + i)(1 + i)nn
ii
When n gets very large,
When n gets very large,
1 -
1
(1 + i)n
i
When n gets very large,
this becomes zero.1 -
1
(1 + i)n
i
When n gets very large,
this becomes zero.
So we’re left with PVIFA =
1 i
1 - 1
(1 + i)n
i
• So, the PV of a perpetuity is very simple to find:
Present Value of a Perpetuity
PMT i
PV =
• So, the PV of a perpetuity is very simple to find:
Present Value of a Perpetuity
What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?
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 $10,000PMT $10,000 i .08 i .08
PV = =PV = =
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 $10,000PMT $10,000 i .08 i .08
= $125,000= $125,000
PV = =PV = =
Ordinary Annuity vs.
Annuity Due
$1000 $1000 $1000
4 5 6 7 8
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 5 6 7
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 5 6 7
PVPVinin
ENDENDModeMode
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 5 6 7
PVPVinin
ENDENDModeMode
FVFVinin
ENDENDModeMode
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 6 7 8
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 6 7 8
PVPVinin
BEGINBEGINModeMode
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 6 7 8
PVPVinin
BEGINBEGINModeMode
FVFVinin
BEGINBEGINModeMode
Earlier, we examined this “ordinary” annuity:
Earlier, we examined this “ordinary” annuity:
0 1 2 3
10001000 10001000 1000 1000
Earlier, we examined this “ordinary” annuity:
Using an interest rate of 8%, we find that:
0 1 2 3
10001000 10001000 1000 1000
Earlier, we examined this “ordinary” annuity:
Using an interest rate of 8%, we find that:
• The Future Value (at 3) is $3,246.40.
0 1 2 3
10001000 10001000 1000 1000
Earlier, we examined this “ordinary” annuity:
Using an interest rate of 8%, we find that:
• The Future Value (at 3) is $3,246.40.
• The Present Value (at 0) is $2,577.10.
0 1 2 3
10001000 10001000 1000 1000
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.”
0 1 2 3
10001000 1000 1000 1000 1000
0 1 2 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?
Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = -1,000
FV = $3,506.11
0 1 2 3
-1000-1000 -1000 -1000 -1000 -1000
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?
0 1 2 3
-1000-1000 -1000 -1000 -1000 -1000
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 = 3 PMT = -1,000
FV = $3,506.11
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:
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)
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)
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)(1 + i)
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 = $3,506.11
.08
(1 + i)(1 + i)
(1.08)(1.08)
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%?
0 1 2 3
Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
10001000 1000 1000 1000 1000
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 = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
10001000 1000 1000 1000 1000
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%?
Present Value - annuity due
Mathematical Solution:
Present Value - annuity due
Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:
Present Value - annuity due
Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:
PV = PMT (PVIFA i, n ) (1 + i)
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)
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)(1 + i)
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 = 1000 1 - (1.08 )3 = $2,783.26
.08
(1 + i)(1 + i)
(1.08)(1.08)
• 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
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
• Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
• Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
• Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
• Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
• Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
period CF PV (CF)
0 -10,000 -10,000.00
1 2,000 1,818.18
2 4,000 3,305.79
3 6,000 4,507.89
4 7,000 4,781.09
PV of Cash Flow Stream: $ 4,412.95
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
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.
Annual Percentage Yield (APY)
Annual Percentage Yield (APY)
APY = APY = (( 1 + 1 + ) ) m m - 1- 1quoted ratequoted ratemm
Annual Percentage Yield (APY)
• Find the APY for the quarterly loan:
APY = APY = (( 1 + 1 + ) ) m m - 1- 1quoted ratequoted ratemm
Annual Percentage Yield (APY)
• Find the APY for the quarterly loan:
APY = APY = (( 1 + 1 + ) ) m m - 1- 1quoted ratequoted ratemm
APY = APY = (( 1 + 1 + ) ) 4 4 - 1- 1.0785.078544
Annual Percentage Yield (APY)
• Find the APY for the quarterly loan:
APY = APY = (( 1 + 1 + ) ) m m - 1- 1quoted ratequoted ratemm
APY = APY = (( 1 + 1 + ) ) 4 4 - 1- 1
APY = .0808, or 8.08%APY = .0808, or 8.08%
.0785.078544
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 = APY = (( 1 + 1 + ) ) m m - 1- 1quoted ratequoted ratemm
APY = APY = (( 1 + 1 + ) ) 4 4 - 1- 1
APY = .0808, or 8.08%APY = .0808, or 8.08%
.0785.078544
Practice Problems
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?
Example
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
• 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?
• This type of cash flow sequence is often called a “deferred annuity.”
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:
1) Discount each cash flow back to time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:
1) Discount each cash flow back to time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:
1) Discount each cash flow back to time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:
1) Discount each cash flow back to time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:
1) Discount each cash flow back to time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:
1) Discount each cash flow back to time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:
1) Discount each cash flow back to time 0 separately.
Or,
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
2) Find the PV of the annuity:
PV: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5
PV = $119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
2) Find the PV of the annuity:
PV3: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5
PV3= $119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
119,624119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
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
119,624119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
69,22669,226
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
119,624119,624
• The PV of the cash flow stream is $69,226.
69,22669,226
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
119,624119,624
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?
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?
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
• Using your calculator,
P/YR = 12
N = 360
PMT = -400
I%YR = 12
FV = $1,397,985.65
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
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?
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:
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 )
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)
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
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)360 - 1 = $1,397,985.65
.01
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
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?
0 1 2 3 . . . 360
? ? ? ?
• Using your calculator,
P/YR = 12
N = 360
I%YR = 7
PV = $100,000
PMT = -$665.30
00 11 22 33 . . . 360. . . 360
? ? ? ?? ? ? ?
House Payment Example
Mathematical Solution:
House Payment Example
Mathematical Solution:
PV = PMT (PVIFA i, n )
House Payment Example
Mathematical Solution:
PV = PMT (PVIFA i, n )
100,000 = PMT (PVIFA .07, 360 ) (can’t use PVIFA table)
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
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 - (1.005833 )360 PMT=$665.30
.005833
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.
• 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
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
Using your calculator,
Mode = BEGIN
PMT = -$250,000
N = 5
I%YR = 10
P/YR = 1
PV = $1,042,466
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
• Now, assuming 10% annual compounding, what monthly payments will be required for you to have $1,042,466 at the end of year 30?
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
1,042,4661,042,466
Using your calculator,
Mode = END
N = 360
I%YR = 10
P/YR = 12
FV = $1,042,466
PMT = -$461.17
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
1,042,4661,042,466
• So, you would have to place $461.17 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.