1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a...
-
Upload
eden-harral -
Category
Documents
-
view
213 -
download
1
Transcript of 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a...
![Page 1: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/1.jpg)
1
The Time Value of MoneyThe Time Value of Money
![Page 2: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/2.jpg)
2
What is Time Value?What is Time Value?
“a dollar received today is worth more than a dollar to be received tomorrow”
That is because today’s dollar can be invested so that we have more than one dollar tomorrow
![Page 3: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/3.jpg)
3
The Terminology of Time The Terminology of Time Value of MoneyValue of Money
0 1 2 3 4 5
PV FV
Today
![Page 4: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/4.jpg)
4
The Terminology of Time The Terminology of Time Value of MoneyValue of Money
0 1 2 3 4 5
PV FV
Today
• A timeline is a graphical device used to clarify the timing of the cash flows for an investment
![Page 5: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/5.jpg)
5
The Terminology of Time The Terminology of Time Value of MoneyValue of Money
0 1 2 3 4 5
PV FV
Today
• A timeline is a graphical device used to clarify the timing of the cash flows for an investment
time period
![Page 6: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/6.jpg)
6
The Terminology of Time The Terminology of Time Value of MoneyValue of Money
0 1 2 3 4 5
PV FV
Today
Present Value - An amount of money today, or the current value of a future cash flow
![Page 7: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/7.jpg)
7
The Terminology of Time The Terminology of Time Value of MoneyValue of Money
0 1 2 3 4 5
PV FV
Today
Present Value - An amount of money today, or the current value of a future cash flow
Future Value - An amount of money at some future time period
![Page 8: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/8.jpg)
8
Calculating the Future Calculating the Future ValueValue
Suppose that you have an extra $100 today that you wish to invest for one year. If you can earn 10% per year on your investment, how much will you have in one year?
0 1 2 3 4 5
-100 ?
![Page 9: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/9.jpg)
9
Calculating the Future Calculating the Future Value (cont.)Value (cont.)
Suppose that at the end of year 1 you decide to extend the investment for a second year. How much will you have accumulated at the end of year 2?
0 1 2 3 4 5
-100 ?
![Page 10: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/10.jpg)
10
Calculating the Future Calculating the Future Value (cont.)Value (cont.)
If you extended the investment for a third year, you would have:
0 1 2 3 4 5
-100 ?
![Page 11: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/11.jpg)
11
Calculating the Future Calculating the Future Value (cont.)Value (cont.)
If you extended the investment for a third year, you would have:
![Page 12: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/12.jpg)
12
Generalizing the Future Generalizing the Future ValueValue
Recognizing the pattern that is developing, we can generalize the future value calculations as follows:
![Page 13: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/13.jpg)
13
Compound InterestCompound Interest
Note from the example that the future value is increasing at an increasing rate
In other words, the amount of interest earned each year is increasing• Year 1: $10• Year 2: $11• Year 3: $12.10
The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amount
![Page 14: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/14.jpg)
14
The Magic of The Magic of CompoundingCompounding
On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for $24 worth of beads and other trinkets. Was this a good deal for the Indians?
This happened about 371 years ago, so if they could earn 5% per year they would in 1997 have:
![Page 15: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/15.jpg)
15
The Magic of The Magic of CompoundingCompounding
On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for $24 worth of beads and other trinkets. Was this a good deal for the Indians?
This happened about 371 years ago, so if they could earn 5% per year they would in 1997 have:
FV PV iN
N 1
![Page 16: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/16.jpg)
16
The Magic of The Magic of CompoundingCompounding
If they could have earned 10% per year, they would have:
$1,743,577,261.65 = 24(1.05)371
FV PV iN
N 1
![Page 17: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/17.jpg)
17
The Magic of The Magic of CompoundingCompounding
If they could have earned 10% per year, they would have:
$54,562,898,811,973,500.00 = 24(1.10)371
$1,743,577,261.65 = 24(1.05)371
That’s about 54,563 Trillion dollars!
FV PV iN
N 1
![Page 18: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/18.jpg)
18
Calculating the Present Calculating the Present ValueValue
Example:A cash flow, the value at Year 1 is $2,000, discount rate is 6%. What is the present value?
0 1 2 3 4 5
$2,000?
PV =
2000
(1.06)= 1,887
![Page 19: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/19.jpg)
19
Calculating the Present Calculating the Present ValueValue
Example:A cash flow, the value at Year 2 is $2,000, discount rate is 6%. What is the present value?
0 1 2 3 4 5
$2,000?
PV =
2000
(1.06)(1.06)
= 1,780 (1.06)
2
![Page 20: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/20.jpg)
20
Calculating the Present Calculating the Present ValueValue
Example:A cash flow, the value at Year 3 is $2,000, discount rate is 6%. What is the present value?
0 1 2 3 4 5
$2,000?
![Page 21: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/21.jpg)
21
Calculating the Present Calculating the Present ValueValue
Example:A cash flow, the value at Year 3 is $2,000, discount rate is 6%. What is the present value?
0 1 2 3 4 5
$2,000?
PV =
2000
(1.06)3= 1,679
![Page 22: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/22.jpg)
22
Present ValuePresent Value
But we can turn this around to find the amount that needs to be invested to achieve some desired future value:
![Page 23: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/23.jpg)
23
Why do we discount?Why do we discount?
Opportunities foregone• Borrow & lend (financial markets,
banks)• Productive assets (plant, machinery,
etc.)• Consumption
If cash flow is risky
![Page 24: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/24.jpg)
24
Present Value: An Present Value: An ExampleExample
Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about $100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal?
![Page 25: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/25.jpg)
25
Present Value: An Present Value: An ExampleExample
Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about $100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal?
![Page 26: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/26.jpg)
26
AnnuitiesAnnuities
An annuity is a series of nominally equal payments equally spaced in time
Annuities are very common:• Rent• Mortgage payments• Car payment• Pension income
The timeline shows an example of a 5-year, $100 annuity
0 1 2 3 4 5
100 100 100 100 100
![Page 27: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/27.jpg)
27
Present Value of an Present Value of an Annuity (cont.)Annuity (cont.)
0 1 2 3 4 5
100 100 100 100 100
?????
PV=?
![Page 28: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/28.jpg)
28
Present Value of an Present Value of an Annuity (cont.)Annuity (cont.)
Using the example, and assuming a discount rate of 10% per year, we find that the present value is:
0 1 2 3 4 5
100(1.10)
62.0968.3075.1382.6490.91379.0
8100
(1.10)2
100(1.10)3
100(1.10)4
100(1.10)5
![Page 29: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/29.jpg)
29
Present Value of an Present Value of an AnnuityAnnuity
We can use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components:
![Page 30: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/30.jpg)
30
Present Value of an Present Value of an Annuity (cont.)Annuity (cont.)
Using the example, and assuming a discount rate of 10% per year, we find that the present value is:
0 1 2 3 4 5
62.0968.3075.1382.6490.91379.0
8
100(1.10)
100(1.10)2
100(1.10)3
100(1.10)4
100(1.10)5
![Page 31: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/31.jpg)
31
Present Value of an Present Value of an Annuity (cont.)Annuity (cont.)
Actually, there is no need to take the present value of each cash flow separately
We can use a closed-form of the PVA equation instead:
![Page 32: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/32.jpg)
32
Present Value of an Present Value of an Annuity (cont.)Annuity (cont.)
We can use this equation to find the present value of our example annuity as follows:
This equation works for all regular annuities, regardless of the number of payments
![Page 33: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/33.jpg)
33
The Future Value of an The Future Value of an AnnuityAnnuity
We can also use the principle of value additivity to find the future value of an annuity, by simply summing the future values of each of the components:
![Page 34: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/34.jpg)
34
The Future Value of an The Future Value of an Annuity (cont.)Annuity (cont.)
Using the example, and assuming a discount rate of 10% per year, we find that the future value is:
0 1 2 3 4 5
146.41133.10121.00110.00}=
610.51at year 5
100(1.10)100(1.10)2100 (1.10)3100(1.10)4 100
![Page 35: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/35.jpg)
35
The Future Value of an The Future Value of an Annuity (cont.)Annuity (cont.)
Using the example, and assuming a discount rate of 10% per year, we find that the future value is:
0 1 2 3 4 5
146.41133.10121.00110.00}=
610.51at year 5
100(1.10)100(1.10)2100 (1.10)3100(1.10)4 100
![Page 36: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/36.jpg)
36
The Future Value of an The Future Value of an Annuity (cont.)Annuity (cont.)
Just as we did for the PVA equation, we could instead use a closed-form of the FVA equation:
This equation works for all regular annuities, regardless of the number of payments
![Page 37: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/37.jpg)
37
The Future Value of an The Future Value of an Annuity (cont.)Annuity (cont.)
We can use this equation to find the future value of the example annuity:
![Page 38: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/38.jpg)
38
Uneven Cash FlowsUneven Cash Flows
Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity
We can find the present or future value of such a stream by using the principle of value additivity
![Page 39: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/39.jpg)
39
Uneven Cash Flows: An Uneven Cash Flows: An Example (1)Example (1)
Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment?
0 1 2 3 4 5
100 200 300
![Page 40: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/40.jpg)
40
Uneven Cash Flows: An Uneven Cash Flows: An Example (1)Example (1)
Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment?
0 1 2 3 4 5
100 200 300
![Page 41: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/41.jpg)
41
Uneven Cash Flows: An Uneven Cash Flows: An Example (2)Example (2)
Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year?
0 1 2 3 4 5
300 500 700
![Page 42: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/42.jpg)
42
Uneven Cash Flows: An Uneven Cash Flows: An Example (2)Example (2)
Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year?
0 1 2 3 4 5
300 500 700
![Page 43: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/43.jpg)
43
PerpetuityPerpetuity
A series of cash flows with infinite life.
. . . . . . .
CCC
t = 3t = 2t = 1t = 0
1 )1(0t tr
cV
r
cV 0
![Page 44: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/44.jpg)
44
PerpetuityPerpetuity
Example: Stock dividend valuation model stock price = present value of future dividends
If a stock pays constant dividends of $3 per share beginning in year one, r = 10% , what is the price of the stock?
P =
3
0.10= $30
![Page 45: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/45.jpg)
45
A Perpetuity That Grows @ gA Perpetuity That Grows @ g
. . . . . . .
C(1+g)2C(1+g)C
t = 3t = 2t = 1t = 0
1
1
0 )1(
)1(
tt
t
r
gcV
)(0 gr
cV
![Page 46: 1 The Time Value of Money. 2 What is Time Value? v a dollar received today is worth more than a dollar to be received tomorrow v That is because todays.](https://reader035.fdocuments.in/reader035/viewer/2022070306/5519f91a55034619378b4637/html5/thumbnails/46.jpg)
46
A Perpetuity That Grows @ gA Perpetuity That Grows @ g
Example:A stock pays dividends $3 in year 1, r=10% , then the dividends will grow @ 4% from year 2, what is the price of this stock?
)(0 gr
cV
= $50
)04.10(.
30 V = $50