Chapter 5 Time Value of Money Handout
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Transcript of Chapter 5 Time Value of Money Handout
8/15/2019 Chapter 5 Time Value of Money Handout
http://slidepdf.com/reader/full/chapter-5-time-value-of-money-handout 1/12Page
Chapter 5 - The Time
Value of Money
©©©© 2005, Pearson Prentice Hall
The Time Value of Money
Compounding and
Discounting Single Sums
We know that receiving P 1.00 today is worth
more than P 1.00 in the future. This is due
to opportunity costs.
The opportunity cost of receiving P 1.00 in the
future is the interest we could have earned if
we had received the P 1.00 sooner.
Today Future
If we can measure this opportunity
cost, we can:
Translate P 1.00 today into its equivalent in the
future (compounding).
Translate P 1.00 in the future into its equivalent
today (discounting).
?
Today Future
Today
?
Future
Compound Interest
and Future Value
Future Value - single sums
If you deposit P 100.00 in an account earning 6.0%, how
much would you have in the account after 1 year?
Mathematical Solution:
FV = PV (FVIF i, n )
FV = 100 (FVIF .06, 1 ) (use FVIF table, or)
FV = PV (1 + i)n
FV = 100 (1.06)1 = P 106.00
0 1
PV = -100 FV = 106
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Future Value - single sums
If you deposit P 100.00 in an account earning 6.0%, how
much would you have in the account after 5 years?
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 = P 133.82
0 5
PV = -100 FV = 133.82
Mathematical Solution:
FV = PV (FVIF i, n )
FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)
FV = PV (1 + i/m) m x n
FV = 100 (1.015)20 = P 134.68
0 20
PV = -100 FV = 134.68
Future Value - single sumsIf you deposit P 100.00 in an account earning 6.0% with
quarterly compounding, how much would you have in
the account after 5 years?
Mathematical Solution:
FV = PV (FVIF i, n )
FV = 100 (FVIF .005, 60 ) (can’t use FVIF table)
FV = PV (1 + i/m)
m x n
FV = 100 (1.005)60 = P 134.89
0 60
PV = -100 FV = 134.89
Future Value - single sumsIf you deposit P 100.00 in an account earning 6.0% with
monthly compounding, how much would you have in the
account after 5 years?
0 100
PV = -1000 FV = 2.98M
Future Value - continuous compoundingWhat is the FV of P 1,000.00 earning 8.0% with
continuous compounding, after 100 years?
Mathematical Solution:
FV = PV (e in)
FV = 1000 (e .08x100) = 1000 (e 8)
FV = P 2,980,957.99
Present Value Mathematical Solution:
PV = FV (PVIF i, n )
PV = 100 (PVIF .06, 1 ) (use PVIF table, or)
PV = FV / (1 + i)n
PV = 100 / (1.06)1 = P 94.34
PV = -94.34 FV = 100
0 1
Present Value - single sumsIf you receive P 100.00 one year from now, what is the
PV of that P 100.00 if your opportunity cost is 6.0%?
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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 = P 74.73
Present Value - single sumsIf you receive P 100.00 five years from now, what is the
PV of that P 100.00 if your opportunity cost is 6.0%?
0 5
PV = -74.73 FV = 100
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 = P 362.45
Present Value - single sumsWhat is the PV of P 1,000.00 to be received 15
years from now if your opportunity cost is 7.0%?
0 15
PV = -362.45 FV = 1000
Calculator Solution:
P/Y = 1 N = 5
PV = -5,000 FV = 11,933
I = 19%
0 5
PV = -5,000 FV = 11,933
Present Value - single sumsIf you sold land for P 11,933.00 that you bought 5
years ago for P 5,000.00, 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 P 11,933.00 that you bought 5 years
ago for P 5,000.00, what is your annual rate of return?
Present Value - single sumsSuppose you placed P 100.00 in an account that pays
9.6% interest, compounded monthly. How long will it
take for your account to grow to P 500.00?
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 present value andfuture value problem, there are four
variables:FV, PV, i and n.
When doing problems, you will be giventhree variables and you will solve for thefourth variable.
Keeping this in mind makes solving timevalue problems much easier!
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The Time Value of Money
Compounding and Discounting
Cash Flow Streams
0 1 2 3 4
Annuity: a sequence of equal cash
flows, occurring at the end of each
period.
0 1 2 3 4
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.
Examples of Annuities:
Calculator Solution:
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = P 3,246.40
Future Value - annuityIf you invest P 1,000.00 each year at 8.0%, how
much would you have after 3 years?
0 1 2 3
1000 1000 1000
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 = P 3,246.40
0.08
Future Value - annuityIf you invest P 1,000.00 each year at 8.0%, how
much would you have after 3 years?
Calculator Solution:
P/Y = 1 I = 8 N = 3
PMT = -1,000
PV = P 2,577.10
0 1 2 3
1000 1000 1000
Present Value - annuityWhat is the PV of P 1,000.00 at the end of each of
the next 3 years, if the opportunity cost is 8.0%?
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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 = P 2,577.10
0.08
Present Value - annuityWhat is the PV of P 1,000.00 at the end of each of
the next 3 years, if the opportunity cost is 8.0%?
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:
PV = PMT (PVIFA i, n )
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
(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
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PMT
iPV =
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 P 10,000.00
annually forever, if you require
8.0% per year on the investment?
PMT P 10,000.00
i 0.08PV = =
= P 125,000.00
Ordinary Annuity
vs.
Annuity Due
P 1,000.00 P 1,000.00 P 1,000.00
4 5 6 7 8
Begin Mode vs. End Mode
P 1,000.00 P 1,000.00 P 1,000.00
4 5 6 7 8year year year
5 6 7
PVin
END
Mode
FVin
END
Mode
Begin Mode vs. End Mode
P 1,000.00 P 1,000.00 P 1,000.00
4 5 6 7 8
year year year
6 7 8
PVin
BEGIN
Mode
FVin
BEGIN
Mode
Earlier, we examined this
“ordinary” annuity:
Using an interest rate of 8.0%, we
find that:
The Future Value (at 3) is
P 3,246.40.
The Present Value (at 0) is
P 2,577.10.
0 1 2 3
1,000 1,000 1,000
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What about this annuity?
Same 3-year time line,
Same 3 P 1,000.00 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
1000 1000 1000
Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = -1,000
FV = P 3,506.11
0 1 2 3
-1000 -1000 -1000
Future Value - annuity dueIf you invest P 1,000.00 at the beginning of each of
the next 3 years at 8.0%, how much would you
have at the end of year 3?
0 1 2 3
-1000 -1000 -1000
Future Value - annuity dueIf you invest P 1,000.00 at the beginning of each of
the next 3 years at 8.0%, 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 = P 3,506.11
Future Value - annuity dueIf you invest P 1,000.00 at the beginning of each of
the next 3 years at 8.0%, 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 = P 3,506.11
.08
(1 + i)
(1.08)
Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = 1,000
PV = P 2,783.26
0 1 2 3
1,000 1,000 1,000
Present Value - annuity dueWhat is the PV of P 1,000.00 at the beginning of each of
the next 3 years, if your opportunity cost is 8.0%?
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 = P 2,783.26
.08
(1 + i)
(1.08)
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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
0 1 2 3 4
-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.
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
period CF PV (CF)
0 -P 10,000.00 -P 10,000.00
1 2,000.00 1,818.18
2 4,000.00 3,305.79
3 6,000.00 4,507.89
4 7,000.00 4,781.09PV of Cash Flow Stream: P 4,412.95
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000Annual 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)
Find the APY for the quarterly loan:
The quarterly loan is more expensive than
the 8.0% loan with annual compounding!
APY = ( 1 + ) m - 1quoted rate
m
APY = ( 1 + ) 4 - 1
APY = .0808, or 8.08%
.0785
4
First Group Project
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Instructions
Get the best commercial bank interest rate forthe following initial deposits: P 1,000.00,
P 5,000.00, and P 10,000.00 for one (1) year. The best rate shall come from a survey/canvass
of at least three banks.
List down the following information: name ofbank, branch, the name of the authorizedrepresentative of the bank, and the contactnumber.
Instructions
On the next meeting, each group will reveal
the rates for the three initial deposits.
The group with the best rates will get thehighest score (10 points).
Each group will submit a one page report for
the interest rates obtained to be submitted on
the next meeting.
The next meeting will be next Wednesday.
Practice Problems Example
0 1 2 3 4 5 6 7 8
P 0 0 0 0 40 40 40 40 40
Cash flows from an investment are
expected to be P 40,000.00 per year at
the end of years 4, 5, 6, 7, and 8. If you
require a 20.0% rate of return, what is
the PV of these cash flows?
This type of cash flow sequence is
often called a “deferred annuity.”
0 1 2 3 4 5 6 7 8
P 0 0 0 0 40 40 40 40 40
2) Find the PV of the annuity:
PV3: End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5
PV3= P 119,624.00
0 1 2 3 4 5 6 7 8
P 0 0 0 0 40 40 40 40 40
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Then discount this single sum back to
time 0.
PV: End mode; P/YR = 1; I = 20;
N = 3; FV = 119,624;
Solve: PV = P 69,226.00
P 119,624.00
0 1 2 3 4 5 6 7 8
P 0 0 0 0 40 40 40 40 40
The PV of the cash flow
stream is P 69,226.00.
P 69,226.00
0 1 2 3 4 5 6 7 8
P 0 0 0 0 40 40 40 40 40
P 119,624.00
Retirement Example
After graduation, you plan to invest
P 400.00 per month in the stock
market. If you earn 12.0% per year
on your stocks, how much will you
have accumulated when you retire in
30 years?
0 1 2 3 . . . 360
400 400 400 400
Using your calculator,
P/YR = 12
N = 360
PMT = -400
I%YR = 12
FV = P 1,397,985.65
0 1 2 3 . . . 360
400 400 400 400
Retirement Example
If you invest P 400.00 at the end of each month for
the next 30 years at 12.0%, 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 P 400.00 at the end of each month for
the next 30 years at 12.0%, 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 = P 1,397,985.65
.01
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House Payment Example
If you borrow P 100,000.00 at 7.0%
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 = P 100,000.00PMT = -P 665.30
0 1 2 3 . . . 360
? ? ? ? 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=P 665.30.005833
Team Assignment
Upon retirement, your goal is to spend 5
years traveling around the world. To
travel in style will require P 250,000.00per 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
10.0% annually.
How much do we need to have by
the end of year 30 to finance thetrip?
PV30 = PMT (PVIFA .10, 5) (1.10) =
= 250,000 (3.7908) (1.10) =
= P 1,042,470.00
27 28 29 30 31 32 33 34 35
250 250 250 250 250
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Using your calculator,
Mode = BEGIN
PMT = -P 250,000.00
N = 5
I%YR = 10
P/YR = 1
PV = P 1,042,466.00
27 28 29 30 31 32 33 34 35
250 250 250 250 250
Now, assuming 10.0% annual
compounding, what monthly
payments will be required for you
to have P 1,042,466.00 at the end of
year 30?
27 28 29 30 31 32 33 34 35
250 250 250 250 250
P 1,042,466
• Using your calculator,
Mode = END
N = 360
I%YR = 10
P/YR = 12
FV = P 1,042,466.00PMT = -P 461.17
27 28 29 30 31 32 33 34 35
250 250 250 250 250
P 1,042,466.00
So, you would have to place P 461.17 in
your retirement account, which earns
10.0% annually, at the end of each of the
next 360 months to finance the 5-yearworld tour.