9780273713654_pp03b_time value of money.ppt

3b.1 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.
Chapter 3 -
Support
The Time Value
of Money

Assume that you deposit $1,000 at
a compound interest rate of 7% for
2 years.
Remember? Future Value
Single Deposit (Graphic)
0 1 2
$1,000
FV2
7%

Future Value Excel Formula
[Calculates a single value in the future based on current
expectations]
• Excel function is =FV(rate,nper,pmt,pv,type)
• rate: the interest rate per period
• nper: the total number of compounding periods
• pmt: the payment made each period and cannot
change over the life of the annuity ($0 in a single
cash flow)
• pv: the present value you begin with
• type: is the number 0 (normal period-end) or 1
(beginning of the period) and indicates when
payments are due/occur

Future Value Example
Students may refer to the supporting Excel file for Chapter 3 (VW13E-
03.xlsx) to use in developing their own solutions to other present value
problems on the “Future Value” tab. Students may find this much easier!

Story Problem Revisited
Julie Miller wants to know how large her deposit
of $10,000 today will become at a compound
annual interest rate of 10% for 5 years.
0 1 2 3 4 5
$10,000
FV5
10%

Future Value Example
Refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) on the
“Future Value” tab. This is the identical answer as our other methods in
one formula shown in the boxed area above!

Another Future
Value Story Problem
• John and Mary are saving for retirement
and currently have $127,833.56 as a nest
egg.
• John indicates that they plan to retire 25
years from today while Mary expects that a
6% rate of return is appropriate for their risk
level given historical returns.
• Calculate how large the account is expected
to grow.

“New” FV Story Problem
Refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) on the
“Future Value” tab.
B C D E F
2 Explanations
3 6.00%  Compound 6% per year
4 25  25 periods in the problem
5 -
$  No payment as single flow
6 (127,833.56)
$  Invests $127,833.56 today
7 0  Not relevant in single flow
8
9 $548,645.11  =FV(D3,D4,D5,D6,D7)
pv:
type:
Outputs
Future Value (FV):
Inputs
rate:
nper:
pmt:

“New” FV Story Problem
• John and Mary will have their $ 127,833.56
investment grow to $ 548,645.11 in 25 years
if they earn exactly 6% each year.
• Note that the Excel answer is a ‘positive’
amount. This indicates that John and Mary
DEPOSITED $ 127,833.56 (the negative
amount as they have less cash) to receive
the positive $548,645.11 (when they receive
cash at retirement when they WITHDRAW the
funds).

Remember? Present Value
Single Deposit (Graphic)
Assume that you need $1,000 in 2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded annually.
0 1 2
$1,000
7%
PV1
PV0

Present Value Excel Formula
[Calculates a single current value based on future
expectations]
• Excel function is =PV(rate,nper,pmt,fv,type)
• nper: the total number of discounted periods
• pmt: the payment made each period and cannot
change over the life of the annuity ($0 in a single
cash flow)
• fv: the future value you expect to attain

Present Value Example
problems on the “Present Value” tab. Students may find this much easier!

Story Problem Revisited
Julie Miller wants to know how large of a
deposit to make so that the money will
grow to $10,000 in 5 years at a discount
rate of 10%.
0 1 2 3 4 5
$10,000
PV0
10%

Present Value Example
problems on the “Present Value” tab. Students may find this much easier!

Another Present Value
Story Problem
• John and Mary are expecting to build a
$100,000 nest egg to use to travel the world
upon retirement. They would like to know
how much they need to set aside today to
reach this goal.
• John indicates that they will retire 20 years
from today while Mary thinks that a 6% rate
of return is appropriate for their risk level.
Calculate how much they need to set aside
today.

B C D E F
2 Explanations
3 6.00%  Discount 6% per period
5 -
$  No payment as single flow
6 100,000
$  Want $100,000 in future
8
9 ($31,180.47)  =PV(D3,D4,D5,D6,D7)
type:
nper:
Inputs
Outputs
rate:
present value (pv)
pmt:
fv:
“New” PV Story Problem
03.xls) to use in developing their own solutions to other present value
problems on the “Present Value” tab.

“New” PV Story Problem
• John and Mary need to set aside $31,180.47
today if they earn exactly 6% each year for
the next 20 years to reach their goal.
• Note that the Excel answer is a ‘negative’
amount. This indicates that John and Mary
will need to DEPOSIT this amount of money
(they have less cash) to receive the positive
$100,000 (when they receive cash) they would
WITHDRAW monies.

We will use the “Rule-of-72”.
Remember? Double
Your Money!!!
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?

Output: The Periods!
03.xlsx) to use in developing their own solutions to other interest rate
problems on the “Periods” tab. Students may find this much easier!
NOTE: The same answer! Solved in only one cell in Excel!!

Remember? Double
Your Money!!!
What if you were given the periods
(years) and wanted to solve for the
interest rate? Wouldn’t it be the same
concept for the rule of 72?
Yes! We simply need to solve using a
different function in Excel called “rate”.

Output: The Interest Rate!
03.xlsx) to use in developing their own solutions to other interest rate
problems on the “Rate” tab.
Note that we “guessed 12% as 12 x 6 = 72. The answer was very close!

PVA3 = $1,000/(1.07)1 +
$1,000/(1.07)2 +
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32
Remember? Example of an
Ordinary Annuity -- PVA
$1,000 $1,000 $1,000
0 1 2 3 4
$2,624.32 = PVA3
7%
$934.58
$873.44
$816.30
Cash flows occur at the end of the period

Present Value of an Annuity
[Calculates a single current value based on future expectations]
• Excel function is =PV(rate,nper,pmt,fv,type)
• nper: the total number of payments or periods
• pmt: the payment that is made/received each period
and cannot change over the life of the annuity
• fv: a single future value you expect to receive (can be
$0)

PV Annuity Example
(Ordinary Annuity)
problems on the “PV Annuity” tab. Students will find this much easier!

“New” PV Annuity Example
(Ordinary Annuity)
• John and Mary are trying to build a nest egg to use
in the future. They would like to know how much
they need to set aside in a single lump sum today to
be equivalent to investing $10,000 each year starting
one year from today to reach this goal.
• John indicates that they will use the money 20 years
from today while Mary thinks that a 6% rate of return
is appropriate for their risk level.
• Calculate the equivalent present value of this
ordinary annuity stream.

(Ordinary Annuity)
B C D E F
2 Explanations
5 (10,000)
$  $10,000 invested per yr
6 -
$  No future amount in additon
7 0  Ordinary Annuity
8
9 $127,833.56  =PV(D3,D4,D5,D6,D7)
fv:
type:
Outputs
present value (pv)
Inputs
rate:
nper:
pmt:
problems on the “PV Annuity” tab. Students will find this much easier!

(Ordinary Annuity)
today to be equivalent to setting aside
$10,000 per year at exactly 6% each year for
the next 25 years.
• In this case, John and Mary need to decide
which is their preference. This sum will grow
to exactly the same as the future value of an
ordinary annuity (see slides 7 to 9).

Remember? Example of an
Ordinary Annuity -- FVA
FVA3 = $1,000(1.07)2 +
$1,000(1.07)1 + $1,000(1.07)0
= $1,145 + $1,070 + $1,000
= $3,215
$1,000 $1,000 $1,000
0 1 2 3 4
$3,215 = FVA3
7%
$1,070
$1,145
Cash flows occur at the end of the period

• Excel function is = FV(rate,nper,pmt,pv,type)
• nper: the total number of payments or periods
• pmt: the payment that is made/received each
period and cannot change over the life of the
annuity
• pv: a single present amount you begin with (can
be $0)
Future Value of an Annuity
[Calculates a single current value based on future expectations]

Future Value Annuity
Example (Ordinary Annuity)
problems on the “FV Annuity” tab. The same answer as before is arrived at
in a single step in Excel - one formula ‘=FV(.07,3,-1000,0,0)’!

“New” Future Value Annuity
• John and Mary are trying to build a nest egg to use
in the future. They would like to know how much
they need to set aside in a single lump sum today to
be equivalent to investing $10,000 each year
starting one year from today to reach this goal. (See
slides 21 to 23 and also 7 to 9)
• John indicates that they will use the money 20 years
from today while Mary thinks that a 6% rate of
return is appropriate for their risk level.
ordinary annuity stream.

B C D E F
2 Explanations
3 6.00%  Compound 6% per year
5 (10,000)
$  $10,000 per year invested
6 -
$  No additional monies today
8
9 $548,645.12  =FV(D3,D4,D5,D6,D7)
pv:
type:
Outputs
Future Value (FV):
Inputs
rate:
nper:
pmt:
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use
in developing their own solutions to other present value problems on the “FV
Annuity” tab. Students will find this is the exact same answer as those derived using
PV of an annuity and then Future Value!

• John and Mary will accumulate nearly
$550,000 by investing $10,000 per year
at exactly 6% each year for the next 25
years.
• In this case, note that this result is
equivalent to the future value of a single
sum where John and Mary needed to
set aside over $127,000 to generate this
sum.

Remember? Example of
an Annuity Due – PVAD
PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 +
$1,000/(1.07)2 = $2,808.02
$1,000.00 $1,000 $1,000
0 1 2 3 4
$2,808.02 = PVADn
7%
$ 934.58
$ 873.44
Cash flows occur at the beginning of the period

PV Annuity
Example (Annuity Due)
Students may refer to the supporting Excel file for Chapter 3 to
use in developing their own solutions to other present value
annuity problems.

“New” PV Annuity
• John and Mary are trying to build a nest egg to
use in the future. They would like to know how
much they need to set aside in a single lump
sum today to be equivalent to investing $10,000
each year starting today to reach this goal.
• John indicates that they will use the money 25
years from today while Mary thinks that a 6%
rate of return is appropriate for their risk level.
annuity due stream.

Students may refer to the supporting Excel file for Chapter 3 to
use in developing their own solutions to other present value
annuity problems.
B C D E F
2 Explanations
5 (10,000)
$  $10,000 invested per yr
6 -
$  No future amount in additon
7 1  Ordinary Annuity
8
9 $135,503.58  =PV(D3,D4,D5,D6,D7)
fv:
type:
Outputs
present value (pv)
Inputs
rate:
nper:
pmt:

today to be equivalent to setting aside $10,000
per year at exactly 6% each year for the next 25
years.
• In this case, John and Mary need to decide which
is their preference. This sum will grow to exactly
the same as the future value of an ordinary
annuity (see slides 24 to 26) plus EXTRA interest.
• EXTRA INTEREST: So the amount is one years
interest higher for each payment or
$127,833.56*6% = $7,670.01 higher PV!

Annuity Due
• An annuity due is used when the cash flow
occurs at the beginning of the period
• As before, you see the value is higher by an
amount equal to i% times the ordinary annuity
value.
• Present values of annuities will be larger
because each cash flow is “discounted” one
less period. See previous examples.
• Future values of annuities will be larger
because each cash flow gets compounded
one “extra” period. No examples shown here.

Mixed Flows Example
Julie Miller will receive the set of cash
flows below. What is the Present Value
at a discount rate of 10%.
0 1 2 3 4 5
$600 $600 $400 $400 $100
PV0
10%

Solve a “mixed flows” problem
using the NPV function
Students may refer to
the supporting Excel file
for Chapter 3 (VW13E-
03.xlsx) to use in
developing their own
solutions to other
present value problems
on the “Mixed Flows”
tab.
Simply type in the cash
flows in the green
column and the correct
interest rate in yellow
and you get your
answer!
Period Cash Flows
1 600.00
$
Interest Rate:
2 600.00
$
(discount rate)
3 400.00
$
4 400.00
$
Present Value*: $1,677.15
5 100.00
$
=NPV(F3,C3:C22)
6
7
8
* Do NOT include cash
9
flows in period 0. Please
10
add or subtract these flows
11
from the final answer above.
12
(Cell F6)
13
14
15
16
17
18
19
20
10%

Solve a “mixed flows” problem
using the NPV function
Students can also solve
this in one formula step
again!
=npv(.10, 600, 600, 400,
400, 100)
This will generate the
same and correct output.
The model on the right is
designed to be slightly
more flexible in solving
these types of problems.
Remember the NPV
function, because we will
use it later!
Period Cash Flows
1 600.00
$
Interest Rate:
2 600.00
$
(discount rate)
3 400.00
$
4 400.00
$
Present Value*: $1,677.15
5 100.00
$
=NPV(F3,C3:C22)
6
7
8
* Do NOT include cash
9
flows in period 0. Please
10
add or subtract these flows
11
from the final answer above.
12
(Cell F6)
13
14
15
16
17
18
19
20
10%

Remember? BWs Effective
Annual Interest Rate
Basket Wonders (BW) has a $1,000
CD at the bank. The interest rate is
6% compounded quarterly for 1
year. What is the Effective Annual
Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 )4 - 1
= 1.0614 - 1 = .0614 or 6.14%!

BWs Effective
Annual Interest Rate
We can use a single function in Excel again to
solve for the Effective Annual Interest Rate (EAR)
Nominal Periods Effective Formula Used
6% 1 6.0000% =EFFECT(B3,C3)
6% 2 6.0900% =EFFECT(B4,C4)
6% 4 6.1364% =EFFECT(B5,C5)
6% 6 6.1520% =EFFECT(B6,C6)
6% 12 6.1678% =EFFECT(B7,C7)
6% 365 6.1831% =EFFECT(B8,C8)
6% 1,000,000 6.1837% =EFFECT(B9,C9)
So it is very simple to solve for the effective rate using Excel.
Note that a large number of periods per year can be used to
approximate continuous compounding!

Julie Miller is borrowing $10,000 at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.
Step 1: Payment
PV0 = R (PVIFA i%,n)
$10,000 = R (PVIFA 12%,5)
$10,000 = R (3.605)
R = $10,000 / 3.605 = $2,774
Remember? Amortizing
a Loan Example

1. The first step is to use the “PMT” function to determine the
yearly (in this case) payment on the loan
2. Now you can use Excel to easily create the table you see below!
Refer to ‘VW13E-03.xlsx’ on the ‘Effect and Loan’ tab.
Remember? Amortizing
a Loan Example
Amortizing a loan
Step 1: Calculating the loan amount
Rate: 12.00% ï Interest rate per period (year in this case)
nper: 5 ï Number of periods (5 years in this case)
pv 10,000.00
$ ï Beginning loan balance today (positive)
fv -
$ ï Ending loan balance at end of periods
payment: ($2,774.10) ï Payment needed (negative)
Step 2: Create a table
Period Beginning Bal Payment Interest in Period Principal in Period Ending Balance
0 10,000.00
$
1 10,000.00
$ $2,774.10 1,200.00
$ $1,574.10 8,425.90
$
2 8,425.90
$ $2,774.10 1,011.11
$ $1,762.99 6,662.91
$
3 6,662.91
$ $2,774.10 799.55
$ $1,974.55 4,688.37
$
4 4,688.37
$ $2,774.10 562.60
$ $2,211.49 2,476.87
$
5 2,476.87
$ $2,774.10 297.22
$ $2,476.87 0.00
$

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