Transcript 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 = P 0 (i)(n) SI : Simple Interest P 0 : i : n : Deposit today (t=0) Interest Rate per Period Number of Time Periods 5
Simple Interest (FV)
• What is the Future Value ( FV ) of the deposit?
•
FV = = P 0 + SI = $1,000 $1,140 + $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.
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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
FV 1 = P 0 (1+ i ) 1 = $1,000 = $1,070 (1 .07
) 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.
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Future Value Single Deposit
FV 1 = P 0 (1+ i ) 1 = $1,000 = $1,070 (1 .07
) FV 2 = FV 1 (1+ i ) 1 P = P 0 (1+ i )(1+ i ) = $1,000 (1 .07
)(1 .07
) 0 (1+ i ) 2 = $1,000 (1 .07
) 2 = $1,144.90
You earned an EXTRA
$4.90
in Year 2 with compound over simple interest. 10 =
General Future Value Formula
FV 1 FV 2 = P 0 (1+ i ) 1 = P 0 (1+ i ) 2 etc.
General Future Value Formula: or FV n = P 0 (1+ i ) n FV n = P 0 ( FVIF i , n ) 11
Single-Sum Future Value
Number of Periods 1 2 3 4 5 2% Table A-1 4% Discount Rate 6% 8% 10% 1,02000 1,04040 1,06121 1,08243 1,10408 1,04000 1,08160 1,12486 1,16986 1,21665 1,06000 1,12360 1,19102 1,26248 1,33823 1,08000 1,16640 1,25971 1,36049 1,46933 1,10000 1,21000 1,33100 1,46410 1,61051 What factor do we use?
12
Single-Sum Future Value
Table A-1- FV of 1$ Number of Periods 2% 4% Discount Rate 6% 8% 10% 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
1.10000
1.21000
1.33100
1.46410
1.61051
$10,000 x 1.25971 = $12,597 Present Value Factor Future Value
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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 2.
Write out the formula using symbols: FV t = CF 0 * (1+r) t 4 5 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 = $0 FV = ?
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General Present Value Formula
PV PV 0 0 = FV 1 = FV 2 (1+ (1+ i i ) ) -2 -1 etc.
General Future Value Formula: or PV 0 = FV n PV 0 = FV n (1+ i ) -n ( PVIF i , n ) 16
Single-Sum Present Value
Number of Periods 2 4 6 8 4% .92456
.85480
.79031
.73069
Table A-2- PV of 1$ 6% Discount Rate 8% 10% .89000
.79209
.70496
.62741
.85734
.73503
.63017
.54027
.82645
.68301
.56447
.46651
12% .79719
.63552
.50663
.40388
What factor do we use?
17
Single-Sum Problems
Number of Periods 2 4 6 8 4% .92456
.85480
.79031
.73069
6% Discount Rate 8% 10% .89000
.79209
.70496
.62741
.85734
.73503
.63017
.54027
.82645
.68301
.56447
.46651
12% .79719
.63552
.50663
.40388
$20,000 x .63552 = $12,710 Future Value Factor Present Value
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• 1.
Example of PV of a Lump Sum
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.
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Example of PV of a Lump Sum
2.
Write out the formula using symbols: PV = CF t / (1+r) t 3.
Insert the appropriate numbers: PV = 100 / (1 + .1) 5 4.
Solve the formula: PV = $62.09
5.
FV = $100 n = 5 PMT = 0 i = 10% PV = ?
Check using a financial calculator: 20
Annuities
Two Types Annuity requires the following:
(1) (2) (3) Periodic payments or receipts (called rents) of the same amount, The same-length interval between such rents, and Compounding of interest once each interval.
Ordinary annuity
- rents occur at the end of each period.
Annuity Due
- rents occur at the beginning of each period.
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Annuities
Future Value of an Ordinary Annuity
Rents occur at the end of each period.
No interest during 1 st period.
Present Value Future Value
$20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 0 1 2 3 4 5 6 7 8
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Present Value
Future Value of an Ordinary Annuity
Future Value
$20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 0 1 2 3 4 5 6 7 8
Bayou Inc. will deposit $20,000 in a 12% fund at the end of each year for 8 years beginning December 31, Year 1. What amount will be in the fund immediately after the last deposit?
What table do we use?
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Future Value of an Ordinary Annuity
Table A-3- FV of an annuity payment of 1$ per year Number of Periods 4% 6% Discount Rate 8% 10% 12% 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
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?
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Future Value of an Ordinary Annuity
Number of Periods 2 4 6 8 10 4% 6% Discount Rate 8% 10% 12% 2.04000
4.24646
6.63298
9.21423
12.00611
2.06000
4.37462
6.97532
9.89747
13.18079
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
$20,000 x 12.29969 = $245,994 Deposit Factor Future Value
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Example of FV of an Annuity
2. Write out the formula using symbols: FVA t = PMT * {[(1+r) t –1]/r} 3. Substitute the appropriate numbers: FVA 20 = $100 * {[(1+.15) 20 –1]/.15 4. Solve for the FV: FVA 20 = $100 * 102.4436
FVA 20 = $10,244.36
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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 = ?
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Present Value of an Ordinary Annuity
Present Value
0 $100,000 1 100,000 2 100,000 100,000 . . . . .
3 4 100,000 100,000 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?
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Present Value of an Ordinary Annuity
Number of Periods 1 5 10 15 20 4% 6% Discount Rate 8% 10% 12% 0.96154
4.45183
8.11090
11.11839
13.59033
0.94340
4.21236
7.36009
9.71225
11.46992
0.92593
3.99271
6.71008
8.55948
9.81815
0.90900
3.79079
6.14457
7.60608
8.51356
0.89286
3.60478
5.65022
6.81086
7.46944
What factor do we use?
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Present Value of an Ordinary Annuity
Number of Periods 1 5 10 15 20 4% 6% Discount Rate 8% 10% 12% 0.96154
4.45183
8.11090
11.11839
13.59033
0.94340
4.21236
7.36009
9.71225
11.46992
0.92593
3.99271
6.71008
8.55948
9.81815
0.90900
3.79079
6.14457
7.60608
8.51356
0.89286
3.60478
5.65022
6.81086
7.46944
$100,000 x 9.81815 = $981,815 Receipt Factor Present Value
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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??
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Formulas of Cash Flow stream
• Future value of a cash flow stream: •
n
FV = S [CF t
t=0
* (1+r) n-t ] Present value of a cash flow stream: – –
n
PV = S [CF t / (1+r) n-t ]
t=0
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• 1.
Example of PV of a Cash Flow Stream
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 $300 $500 $1000 0 A-2 r=10% , n= 1 A-2 r=10% , n= 2 A-2 r=10% , n= 3 A-2 r=10% , n= 4 1 2 3 4 i = 10% 34
Example of PV of a Cash Flow Stream
Number of CF1 * Factor 1 Periods + 1 CF2 * Factor + 2 2 CF3 * Factor 3 + CF4* Factor 4 3 4 = 4% .92456
.85480
.79031
.73069
Table A-2- PV of 1$ 6% Discount Rate 8% 10% .89000
.79209
.70496
.62741
.85734
.73503
.63017
.54027
Present value of a cash flow stream:
n
PV = S [CF t
t=0
/ (1+r) n-t ]
.82645
.68301
.56447
.46651
12% .79719
.63552
.50663
.40388
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Example of PV of a Cash Flow Stream
2.
Write out the formula using symbols:
n
PV = S [CF t
t=0
/ (1+r) t ] OR PV = [CF 1 /(1+r) 1 ]+[CF 2 /(1+r) 2 ]+[CF 3 /(1+r) 3 ]+[CF 4 /(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.
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0 • 1.
$100
Example of FV of a Cash Flow Stream
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% $300 $500 $1000 1 2 3 4 A-1 r=10% , n= 4 + A-1 r=10% , n= 3 + A-1 r=10% , n= 2 + A-1 r=10% , n= 1
Example of FV of a Cash Flow Stream
CF3* Factor 1 CF2 * + Factor 2 + CF1 * Factor 3 + CF0* Factor 4 Number of Periods 1 2 3 4 2% Table A-1- FV of 1$ 4% Discount Rate 6% 8% 10% 1,02000 1,04040 1,06121 1,08243 1,04000 1,08160 1,12486 1,16986 1,06000 1,12360 1,19102 1,26248 1,08000 1,16640 1,25971 1,36049 1,10000 1,21000 1,33100 1,46410 =
Future value of a cash flow stream:
n FV =
S
[CF t * (1+r) n-t ]
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Rule of Thumb
• • 1.
2.
The following are simple rules that you should always use no matter what type of TVM problem you are trying to solve: 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.
Draw a representative timeline and label the cash flows and time periods appropriately.
3.
4.
Write out the complete formula using symbols first and then substitute the actual numbers to solve.
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.
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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” .
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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. Read problem thoroughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a PV or FV problem 5.
Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) 43