Transcript Ch. 5 - The Time Value of Money

Chapter 9
The Time Value of Money
Notes:
• Although it is easiest to use your
financial calculator to solve time value
problems, you MUST understand what
you are doing. This will require a lot of
practice to eliminate mistakes.
• Understanding the concept of Time
Value of Money NOW is extremely
important because all the remaining
chapters will require TVM concept
application.
Notes:
• In your Test, you will be REQUIRED to
show both your financial calculator
solution and Mathematical solution
(either using the formula or the Financial
Tables)
• Only in multiple choice questions will you
not be required to show your calculation.
Therefore, you can use the financial
calculator alone. You must however, make
sure that you know how to use your
calculator properly; otherwise you will
easily make mistakes with the use of a
financial calculator.
Using your Financial Calculators (TI BAII Plus and
Sharp EL-733A)
• For the first time that you are using your calculator,
perform the following:
– TI BAII Plus users, Set Payment Frequency and
Compounding Frequency to 1 (press 2nd, press P/Y,
press 1, press ENTER, press down arrow, press 1,
press ENTER, press 2nd, press QUIT)
– Sharp EL-733A users, Set to FIN Mode if not yet in
FIN Mode ( press 2nd F, press Mode). You should
see FIN on the Display.
• Set Decimal to 4 places
– TI BAII Plus users, press 2nd, press Format, press 4,
press ENTER , press 2nd, press QUIT.
– Sharp EL-733A users, press 2ndF, press TAB, press
4.
Using your Financial Calculators (TI BAII Plus and Sharp
EL-733A)
• To start each calculation,
– TI BAII Plus users, press CE/C, press 2nd, press CLR
TVM, press 2nd, press CLR Work. BAII Plus has a
continuous memory. Turning-off the calculator
does not erase what was previously stored in its
memory, although turning it on again resets the
display to zero. Therefore, it is extremely
important to clear memory before each
calculation.
– Sharp EL-733A users, press 2ndF, press CA.
• To erase the previously entered number,
– TI BAII Plus users, simply press CE/C.
– Sharp EL-733A users, simply press C CE.
• Enter Outflow Value as negative. To enter it as
negative enter the value/s, press +/-. DO NOT use
the minus sign key.
Using your Financial Calculators (TI BAII Plus and
Sharp EL-733A)
• The order in which data (PV, n, I, etc) are entered does not
matter.
• To compute for the result, press CPT for TI BAII Plus
users, press COMP for Sharp EL-733A users. Then press
whatever variable you are computing for (PV. FV, etc)
• To perform calculations involving annuity dues, payment
must be set to the begin mode.
– TI BAII Plus users, press 2nd, press BGN, press 2nd,press
SET, press 2nd, press QUIT.
– Sharp EL-733A users, press BGN.
• It is important to reset the mode back to END. Most
payment problems are made at the end of each year
(ordinary annuities).
We know that receiving $1 today is worth
more than $1 in the future. This is due
to the time value of money.
The cost of receiving $1 in the future is the
interest we could have earned if we had
received the $1 sooner.
Today
Future
If we can MEASURE this interest
cost, we can:
• Translate $1 today into its equivalent in
the future (COMPOUNDING).
Today
Future
?
If we can MEASURE this interest
cost, we can:
• Translate $1 today into its equivalent in
the future (COMPOUNDING).
Today
Future
?
• Translate $1 in the future into its
equivalent today (DISCOUNTING).
Today
?
Future
Future Value –
single sum
Future Value - single sums
If you deposit $100 in an account earning 6%, how much
would you have in the account after 5 years?
PV = -100
0
Calculator Solution:
I/Y = i = 6
N=n=5
PV = -100
FV = $133.82
FV =
5
Future Value - single sums
If you deposit $100 in an account earning 6%, how much
would you have in the account after 5 years?
PV = -100
FV = 133.82
0
Calculator Solution:
I/Y = i = 6
N=n=5
FV = $133.82
5
PV = -100
Future Value - single sums
If you deposit $100 in an account earning 6%, how much
would you have in the account after 5 years?
PV = -100
FV = 133.82
0
5
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 = $133.82
Future Value - single sums
If you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have in the
account after 5 years?
PV = -100
FV =
0
Calculator Solution:
I/Y = i = 1.5
N = n = 20
FV = $134.68
20
PV = -100
Future Value - single sums
If you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have in the
account after 5 years?
PV = -100
FV = 134.68
0
Calculator Solution:
I/Y = i = 1.5
N = n = 20
FV = $134.68
20
PV = -100
Future Value - single sums
If you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have in the
account after 5 years?
PV = -100
FV = 134.68
0
20
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 = $134.68
Present Value –
single sum
Present Value - single sums
If you will receive $100 5 years from now, what is
the PV of that $100 if the interest rate is 6%?
PV =
FV = 100
0
Calculator Solution:
I/Y = i =6
N=n=5
PV = -74.73
5
FV = 100
Present Value - single sums
If you will receive $100 5 years from now, what is
the PV of that $100 if the interest rate is 6%?
PV = -74.73
FV = 100
0
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
5
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 = -5,000
FV = 11,933
0
Calculator Solution:
N=n=5
PV = -5,000
FV = 11,933
I/Y = i =19%
5
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
.419 = ((1/ (1+i)5)
2.3866 = (1+i)5
(2.3866)1/5 = (1+i)
i = .19
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 =
0
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 = -100
FV = 500
0
Calculator Solution:
• FV = 500
• I/Y = i = 0.8
PV = -100
• N = 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?
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.
0
1
2
3
4
Examples of Annuities:
• If you buy a bond, you will receive
equal 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.
Future Value - annuity
If you invest $1,000 at the end of the next 3 years, at
8%, how much would you have after 3 years?
0
1
2
3
Future Value - annuity
If you invest $1,000 at the end of the next 3 years, at
8%, how much would you have after 3 years?
0
1000
1000
1000
1
2
3
Calculator Solution:
I/Y= i = 8
N=n=3
PMT = -1,000
FV = $3,246.40
Future Value - annuity
If you invest $1,000 at the end of the next 3 years, at
8%, how much would you have after 3 years?
0
1000
1000
1000
1
2
3
Calculator Solution:
I/Y= i = 8
N=n=3
PMT = -1,000
FV = $3,246.40
Future Value - annuity
If you invest $1,000 at the end of the next 3 years, 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 - annuity
If you invest $1,000 at the end of the next 3 years, 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 - annuity
If you invest $1,000 at the end of the next 3 years, 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
.08
= $3246.40
Present Value - annuity
What is the PV of $1,000 at the end of each of the
next 3 years, if the discount rate is 8%?
0
1
2
3
Present Value - annuity
What is the PV of $1,000 at the end of each of the
next 3 years, if the discount rate is 8%?
0
1000
1000
1000
1
2
3
Calculator Solution:
I/Y = i = 8
N=n=3
PMT = -1,000
PV = $2,577.10
Present Value - annuity
What is the PV of $1,000 at the end of each of the
next 3 years, if the discount rate is 8%?
0
1000
1000
1000
1
2
3
Calculator Solution:
I/Y = i = 8
N=n=3
PMT = -1,000
PV = $2,577.10
Present Value - annuity
What is the PV of $1,000 at the end of each of the
next 3 years, if the discount rate is 8%?
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 discount rate is 8%?
Mathematical Solution:
PV = PMT (PVIFA i, n )
PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)
PV = PMT
1
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 discount rate is 8%?
Mathematical Solution:
PV = PMT (PVIFA i, n )
PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)
PV = PMT
1
1 - (1 + i)n
i
PV = 1000
1
1 - (1.08 )3
.08
= $2,577.10
Ordinary Annuity
vs.
Annuity Due
Earlier, we examined this
“ordinary” annuity:
0
1000
1000
1000
1
2
3
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.
What about this annuity?
1000
1000
1000
0
1
2
3
• 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.”
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
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?
-1000
-1000
-1000
0
1
2
Calculator Solution:
Mode = BEGIN I/Y = i = 8
N=n=3
PMT = -1,000
FV = $3,506.11
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?
-1000
-1000
-1000
0
1
2
Calculator Solution:
Mode = BEGIN I/Y = i = 8
N=n=3
PMT = -1,000
FV = $3,506.11
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?
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)
FV = PMT (1 + i)n - 1
i
(1 + i)
(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)
FV = 1,000 (1.08)3 - 1
.08
= $3,506.11
(1.08)
Present Value - annuity due
What is the PV of $1,000 at the beginning of each of
the next 3 years, if the discount rate is 8%?
0
1
2
3
Present Value - annuity due
What is the PV of $1,000 at the beginning of each of
the next 3 years, if the discount rate is 8%?
1000
1000
1000
0
1
2
Calculator Solution:
Mode = BEGIN I/Y = i = 8
N = n= 3
PMT = 1,000
PV = $2,783.26
3
Present Value - annuity due
What is the PV of $1,000 at the beginning of each of
the next 3 years, if the discount rate is 8%?
1000
1000
1000
0
1
2
Calculator Solution:
Mode = BEGIN I/Y = i = 8
N = n= 3
PMT = 1,000
PV = $2,783.26
3
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)
PV = PMT
1
1 - (1 + i)n
i
(1 + i)
(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)
PV = PMT
PV = 1000
1
1 - (1 + i)n
i
1
1 - (1.08 )3
.08
(use PVIFA table, or)
(1 + i)
(1.08)
= $2,783.26
Other Cash Flow Patterns
0
1
2
3
Uneven Cash Flows
-10,000
2,000
4,000
6,000
7,000
0
1
2
3
4
• 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
-10,000 2,000
0
1
4,000
6,000
7,000
2
3
4
• Sorry! There’s no quickie for this one.
We have to discount each cash flow
back separately.
Uneven Cash Flows
-10,000 2,000
0
1
4,000
6,000
7,000
2
3
4
• Sorry! There’s no quickie for this one.
We have to discount each cash flow
back separately.
Uneven Cash Flows
-10,000 2,000
0
1
4,000
6,000
7,000
2
3
4
• Sorry! There’s no quickie for this one.
We have to discount each cash flow
back separately.
Uneven Cash Flows
-10,000 2,000
0
1
4,000
6,000
7,000
2
3
4
• Sorry! There’s no quickie for this one.
We have to discount each cash flow
back separately.
Uneven Cash Flows
-10,000 2,000
0
1
4,000
6,000
7,000
2
3
4
• Sorry! There’s no quickie for this one.
We have to discount each cash flow
back separately.
-10,000
2,000
4,000
6,000
7,000
0
1
2
3
4
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
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
• 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?
0
0
0
0
40 40 40 40 40
0
1
2
3
4
5
6
7
8
0
0
0
0
40 40 40 40 40
0
1
2
3
4
5
6
7
• This type of cash flow sequence is
often called a “deferred annuity.”
8
0
0
0
0
40 40 40 40 40
0
1
2
3
4
5
6
How to solve:
1) Discount each cash flow back to
time 0 separately.
Or,
7
8
0
0
0
0
40
40
40
0
1
2
3
4
5
6
2) Find the PV of the annuity:
PV3: End mode; I/YR = i = 20;
PMT = 40,000; N = n = 5
PV3= $119,624
40 40
7
8
0
0
0
0
40
40
40
0
1
2
3
4
5
6
119,624
40 40
7
8
0
0
0
0
40
40
40
0
1
2
3
4
5
6
40 40
7
119,624
Then discount this single sum back to
time 0.
PV: End mode; I/YR = i = 20;
N = n = 3; FV = 119,624;
Solve: PV = $69,226
8
0
0
0
0
40
40
40
0
1
2
3
4
5
6
69,226
119,624
40 40
7
8
0
0
0
0
40
40
40
0
1
2
3
4
5
6
69,226
119,624
• The PV of the cash flow
stream is $69,226.
40 40
7
8
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?
0
400
400
400
1
2
3
400
. . . 360
0
400
400
400
1
2
3
400
. . . 360
0
400
400
400
1
2
3
• Using your calculator,
N = n = 360
PMT = -400
I/Y = i = 1
FV = $1,397,985.65
400
. . . 360
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 )
FV = PMT (1 + i)n - 1
i
(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
FV = 400 (1.01)360 - 1
.01
= $1,397,985.65
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?
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
0
?
?
?
1
2
3
• Using your calculator,
N = n = 360
I/Y = i = 0.5833
PV = $100,000
PMT = -$665.30
?
. . . 360
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 )
PV = PMT
1
1 - (1 + i)n
i
(can’t use PVIFA table)
House Payment Example
Mathematical Solution:
PV = PMT (PVIFA i, n )
100,000 = PMT (PVIFA .07, 360 )
PV = PMT
(can’t use PVIFA table)
1
1 - (1 + i)n
i
1
100,000 = PMT 1 - (1.005833 )360
.005833
PMT=$665.30