Download The Time Value of Money

Chapter 5 The Time Value of
Money
Pr. Zoubida SAMLAL
1
The Interest Rate
Which would you prefer -- $10,000 today
or $10,000 in 5 years?
Obviously, $10,000 today.
You already recognize that there is TIME
VALUE TO MONEY!!
2
Why TIME?
Why is TIME such an important element
in your decision?
TIME allows you the opportunity to
postpone consumption and earn
INTEREST.
3
Types of Interest
 Simple
Interest
Interest paid (earned) on only the original
amount, or principal, borrowed (lent).
• Compound Interest
Interest paid (earned) on any previous interest
earned, as well as on the principal borrowed
(lent).
4
Simple Interest Formula
Formula
SI = P0(i)(n)
SI: Simple Interest
P0:
Deposit today (t=0)
i:
Interest Rate per Period
n:
Number of Time Periods
5
Simple Interest (FV)
• What is the Future Value (FV) of the deposit?
FV
= P0 + SI
= $1,000 + $140
= $1,140
• Future Value is the value at some future time of a
present amount of money, or a series of
payments, evaluated at a given interest rate.
6
Simple Interest (PV)
• What is the Present Value (PV) of the previous
problem?
The Present Value is simply the
$1,000 you originally deposited.
That is the value today!
• Present Value is the current value of a future
amount of money, or a series of payments,
evaluated at a given interest rate.
7
Types of TVM Calculations
• There are many types of TVM calculations
• The basic types will be covered in this review module
and include:
–
–
–
–
Present value of a lump sum
Future value of a lump sum
Present and future value of cash flow streams
Present and future value of annuities
• Keep in mind that these forms can, should, and will
be used in combination to solve more complex TVM
problems
8
Future Value Single Deposit
FV1 = P0 (1+i)1
= $1,000 (1.07)
= $1,070
Compound Interest
You earned $70 interest on your $1,000
deposit over the first year.
This is the same amount of interest you would
earn under simple interest.
9
Future Value Single Deposit
FV1 = P0 (1+i)1
= $1,000 (1.07)
= $1,070
FV2 = FV1 (1+i)1
= P0 (1+i)(1+i) = $1,000(1.07)(1.07)
P0 (1+i)2
= $1,000(1.07)2
= $1,144.90
=
You earned an EXTRA $4.90 in Year 2 with
compound over simple interest.
10
General Future Value
Formula
FV1 = P0(1+i)1
FV2 = P0(1+i)2
etc.
General Future Value Formula:
FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n)
11
Single-Sum Future Value
Number
of
Periods
2%
1
2
3
4
5
1,02000
1,04040
1,06121
1,08243
1,10408
Table A-1
Discount Rate
4%
6%
8%
1,04000
1,08160
1,12486
1,16986
1,21665
1,06000
1,12360
1,19102
1,26248
1,33823
1,08000
1,16640
1,25971
1,36049
1,46933
10%
1,10000
1,21000
1,33100
1,46410
1,61051
What factor do we use?
12
Single-Sum Future Value
Table A-1- FV of 1$
Number
of
Periods
2%
1
2
3
4
5
1.02000
1.04040
1.06121
1.08243
1.10408
1.04000
1.08160
1.12486
1.16986
1.21665
1.06000
1.12360
1.19102
1.26248
1.33823
1.08000
1.16640
1.25971
1.36049
1.46933
$10,000
x
1.25971
=
$12,597
Present Value
Discount Rate
8%
6%
4%
Factor
10%
1.10000
1.21000
1.33100
1.46410
1.61051
Future Value
13
Example of FV of a Lump Sum
•
1.
How much money will you have in 5 years if you invest $100
today at a 10% rate of return?
Draw a timeline
i = 10%
$100
0
1
2
3
?
4
5
2. Write out the formula using symbols:
FVt = CF0 * (1+r)t
14
Example of FV of a Lump Sum
3. Substitute the numbers into the formula:
FV = $100 * (1+.1)5
4. Solve for the future value:
FV = $161.05
5. Check answer using a financial calculator:
i = 10%
n=5
PV = $100
PMT = $0
FV = ?
15
General Present Value
Formula
PV0 = FV1 (1+i)-1
PV0 = FV2(1+i)-2
etc.
General Future Value Formula:
PV0 = FVn (1+i)-n
or PV0 = FVn (PVIFi,n)
16
Single-Sum Present Value
Table A-2- PV of 1$
Number
of
Periods
4%
2
.92456
.89000
.85734
.82645
.79719
4
.85480
.79209
.73503
.68301
.63552
6
.79031
.70496
.63017
.56447
.50663
8
.73069
.62741
.54027
.46651
.40388
Discount Rate
6%
8%
10%
12%
What factor do we use?
17
Single-Sum Problems
Table
Table
A-2-A-2
PV of 1$
Number
of
Periods
4%
2
.92456
.89000
.85734
.82645
.79719
4
.85480
.79209
.73503
.68301
.63552
6
.79031
.70496
.63017
.56447
.50663
8
.73069
.62741
.54027
.46651
.40388
$20,000
x
.63552
Future Value
Discount Rate
6%
8%
10%
Factor
=
12%
$12,710
Present Value
18
Example of PV of a Lump Sum
•
1.
How much would $100 received five years from now be worth today
if the current interest rate is 10%?
Draw a timeline
i = 10%
$100
?
0
1
2
3
4
5
The arrow represents the flow of money and the
numbers under the timeline represent the time period.
Note that time period zero is today.
19
Example of PV of a Lump Sum
2.
Write out the formula using symbols:
PV = CFt / (1+r)t
3.
Insert the appropriate numbers:
PV = 100 / (1 + .1)5
4.
Solve the formula:
PV = $62.09
5.
Check using a financial calculator:
FV = $100
n=5
PMT = 0
i = 10%
PV = ?
20
Annuities
Annuity requires the following:
(1) Periodic payments or receipts (called
rents) of the same amount,
(2) The same-length interval between such
rents, and
(3) Compounding of interest once each
interval.
Two
Types
Ordinary annuity - rents occur at the end of each period.
Annuity Due - rents occur at the beginning of each period.
21
Annuities
Future Value of an Ordinary Annuity
Rents occur at the end of each period.
No interest during 1st period.
Future Value
Present Value
$20,000 20,000
0
1
2
20,000
3
20,000
4
20,000
5
20,000
6
20,000
7
20,000
8
22
Future Value of an Ordinary
Annuity
Future Value
Present Value
$20,000 20,000
0
1
2
20,000
3
20,000
4
20,000
5
20,000
6
20,000
7
20,000
8
Bayou Inc. will deposit $20,000 in a 12% fund at the
end of each year for 8 years beginning December 31,
Year 1. What amount will be in the fund immediately
after the last deposit?
What table do we use?
23
Future Value of an Ordinary
Annuity
Table A-3- FV of an annuity payment of 1$ per year
Number
of
Periods
4%
6%
2
4
6
8
10
2.04000
4.24646
6.63298
9.21423
12.00611
2.06000
4.37462
6.97532
9.89747
13.18079
Discount Rate
8%
10%
12%
2.08000
4.50611
7.33592
10.63663
14.48656
2.10000
4.64100
7.71561
11.43589
15.93743
2.12000
4.77933
8.11519
12.29969
17.54874
What factor do we use?
24
Future Value of an Ordinary
Annuity
Table A-3- FV ofTable
an annuity
A-3 payment of 1$ per year
Number
of
Periods
4%
6%
2
4
6
8
10
2.04000
4.24646
6.63298
9.21423
12.00611
2.06000
4.37462
6.97532
9.89747
13.18079
$20,000
Deposit
x
Discount Rate
8%
10%
12%
2.08000
4.50611
7.33592
10.63663
14.48656
2.10000
4.64100
7.71561
11.43589
15.93743
2.12000
4.77933
8.11519
12.29969
17.54874
12.29969
Factor
=
$245,994
Future Value
25
Example of FV of an Annuity
2. Write out the formula using symbols:
FVAt = PMT * {[(1+r)t –1]/r}
3. Substitute the appropriate numbers:
FVA20 = $100 * {[(1+.15)20 –1]/.15
4. Solve for the FV:
FVA20 = $100 * 102.4436
FVA20 = $10,244.36
26
Example of FV of an Annuity
5.
Check using calculator:
Make sure that the calculator is set to one period per year
PMT = $100
n = 20
i = 15%
FV = ?
27
Present Value of an Ordinary
Annuity
Present Value
$100,000
100,000
100,000
100,000
100,000
100,000
.....
0
1
2
3
4
19
20
Jaime Yuen wins $2,000,000 in the state lottery. She
will be paid $100,000 at the end of each year for the
next 20 years. How much has she actually won?
Assume an appropriate interest rate of 8%.
What table do we use?
28
Present Value of an Ordinary
Annuity
A-4 payment of 1$ per year
Table A-4- PV ofTable
an annuity
Number
of
Periods
4%
6%
1
5
10
15
20
0.96154
4.45183
8.11090
11.11839
13.59033
0.94340
4.21236
7.36009
9.71225
11.46992
Discount Rate
8%
0.92593
3.99271
6.71008
8.55948
9.81815
10%
0.90900
3.79079
6.14457
7.60608
8.51356
12%
0.89286
3.60478
5.65022
6.81086
7.46944
What factor do we use?
29
Present Value of an Ordinary
Annuity
6-4
Table A-4- PV ofTable
an annuity
payment of 1$ per year
Number
of
Periods
4%
6%
1
5
10
15
20
0.96154
4.45183
8.11090
11.11839
13.59033
0.94340
4.21236
7.36009
9.71225
11.46992
$100,000
Receipt
x
Discount Rate
8%
0.92593
3.99271
6.71008
8.55948
9.81815
9.81815
Factor
=
10%
0.90900
3.79079
6.14457
7.60608
8.51356
12%
0.89286
3.60478
5.65022
6.81086
7.46944
$981,815
Present Value
30
Formulas of Annuities
Present value of an annuity:
PVA = PMT * {[1-(1+r)-t]/r}
Future value of an annuity:
FVA t = PMT * {[(1+r)t –1]/r}
31
How about a stream of payments that
are NOT equal??
32
Formulas of Cash Flow stream
• Future value of a cash flow stream:
n
FV = S [CFt * (1+r)n-t]
t=0
• Present value of a cash flow stream:
–
n
– PV = S [CFt / (1+r)n-t]
t=0
33
Example of PV of a Cash Flow
Stream
•
1.
Joe made an investment that will pay $100 the first year, $300 the
second year, $500 the third year and $1000 the fourth year. If the
interest rate is ten percent, what is the present value of this cash flow
stream?
Draw a timeline:
$100
0
A-2 r=10% , n= 1
A-2 r=10% , n= 2
1
$300
2
$500
3
$1000
4
i = 10%
A-2 r=10% , n= 3
A-2 r=10% , n= 4
34
Example of PV of a Cash Flow
Stream
Number
of
Periods
CF1 * Factor 1
+
CF2 * Factor 2
+
CF3 * Factor 3
+
CF4* Factor 4
Table A-2- PV of 1$
4%
Discount Rate
6%
8%
10%
12%
1
.92456
.89000
.85734
.82645
.79719
2
.85480
.79209
.73503
.68301
.63552
3
.79031
.70496
.63017
.56447
.50663
4
.73069
.62741
.54027
.46651
.40388
=
Present value of a cash flow stream:
n
PV = S [CFt / (1+r)n-t]
t=0
35
Example of PV of a Cash Flow
Stream
2.
Write out the formula using symbols:
n
PV = S [CFt / (1+r)t]
t=0
OR
PV = [CF1/(1+r)1]+[CF2/(1+r)2]+[CF3/(1+r)3]+[CF4/(1+r)4]
3.
Substitute the appropriate numbers:
PV = [100/(1+.1)1]+[$300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4]
36
Example of PV of a Cash Flow
Stream
4.
Solve for the present value:
PV = $90.91 + $247.93 + $375.66 + $683.01
PV = $1397.51
5.
Check using a calculator:
–
Make sure to use the appropriate rate of return, number of periods, and
future value for each of the calculations. To illustrate, for the first cash flow,
you should enter FV=100, n=1, i=10, PMT=0, PV=?. Note that you will have
to do four separate calculations.
37
Example of FV of a Cash Flow
Stream
•
1.
Joe made a decision to start saving money. He will pay $100 now year,
$300 the first year, $500 the second year and $1000 the third year. If
the interest rate is ten percent, what is the future value of this cash
flow stream?
Draw a timeline:
i = 10%
$100
0
$300
1
$500
2
$1000
3
4
A-1 r=10% , n= 4
+
A-1 r=10% , n= 3
+
A-1 r=10% , n= 2
+
A-1 r=10% , n= 138
Example of FV of a Cash Flow
Stream
CF3* Factor 1
+
CF2 * Factor 2
+
CF1 * Factor 3
+
CF0* Factor 4
Table A-1- FV of 1$
Number
of
Periods
2%
1
2
3
4
1,02000
1,04040
1,06121
1,08243
Discount Rate
4%
6%
8%
1,04000
1,08160
1,12486
1,16986
1,06000
1,12360
1,19102
1,26248
1,08000
1,16640
1,25971
1,36049
10%
1,10000
1,21000
1,33100
1,46410
=
Future value of a cash flow stream:
n
FV = S [CFt * (1+r)n-t]
39
Rule of Thumb
•
•
The following are simple rules that you should always use no
matter what type of TVM problem you are trying to solve:
1. Stop and think: Make sure you understand what the problem is
asking. You will get the wrong answer if you are answering the
wrong question.
2. Draw a representative timeline and label the cash flows and
time periods appropriately.
3. Write out the complete formula using symbols first and then
substitute the actual numbers to solve.
4. Check your answers using a calculator.
While these may seem like trivial and time consuming tasks, they
will significantly increase your understanding of the material and
your accuracy rate.
40
Double Your Money!!!
Quick! How long does it take to double
$5,000 at a compound rate of 12% per
year (approx.)?
We will use the “Rule-of-72”.
41
The “Rule-of-72”
Quick! How long does it take to double
$5,000 at a compound rate of 12% per
year (approx.)?
Approx. Years to Double = 72 / i%
72 / 12% = 6 Years
[Actual Time is 6.12 Years]
42
Steps to Solve Time Value of
Money Problems
1.
2.
3.
4.
5.
Read problem thoroughly
Create a time line
Put cash flows and arrows on time line
Determine if it is a PV or FV problem
Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)
43