There are three main reasons why a dollar in the future is worth less than a dollar today:
1. People prefer present consumption over future consumption
2. Inflation decreases the value of currency over time
3. Uncertainty associated with future cash flows decreases their value
The discount rate incorporates these factors and is used to discount future cash flows to their present value. A higher discount rate leads to a lower present value for future cash flows. Discounting future cash flows converts them to present value dollars.
time value of money
,
concept of time value of money
,
significance of time value of money
,
present value vs future value
,
solve for the present value
,
simple vs compound interest rate
,
nominal vs effective annual interest rates
,
future value of a lump sum
,
solve for the future value
,
present value of a lump sum
,
types of annuity
,
future value of an annuity
What is the 'Time Value of Money - TVM'
The time value of money (TVM) is the idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. TVM is also referred to as present discounted value.
BREAKING DOWN 'Time Value of Money - TVM'
Money deposited in a savings account earns a certain interest rate. Rational investors prefer to receive money today rather than the same amount of money in the future because of money's potential to grow in value over a given period of time. Money earning an interest rate is said to be compounding in value.
BREAKING DOWN 'Compound Interest'
Compound Interest Formula
Compound interest is calculated by multiplying the principal amount by one plus the annual interest rate raised to the number of compound periods minus one.The total initial amount of the loan is then subtracted from the resulting value.
time value of money
,
concept of time value of money
,
significance of time value of money
,
present value vs future value
,
solve for the present value
,
simple vs compound interest rate
,
nominal vs effective annual interest rates
,
future value of a lump sum
,
solve for the future value
,
present value of a lump sum
,
types of annuity
,
future value of an annuity
What is the 'Time Value of Money - TVM'
The time value of money (TVM) is the idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. TVM is also referred to as present discounted value.
BREAKING DOWN 'Time Value of Money - TVM'
Money deposited in a savings account earns a certain interest rate. Rational investors prefer to receive money today rather than the same amount of money in the future because of money's potential to grow in value over a given period of time. Money earning an interest rate is said to be compounding in value.
BREAKING DOWN 'Compound Interest'
Compound Interest Formula
Compound interest is calculated by multiplying the principal amount by one plus the annual interest rate raised to the number of compound periods minus one.The total initial amount of the loan is then subtracted from the resulting value.
Government bonds are fixed interest securities
This means that a bond pays a fixed annual interest – this is known as the coupon
The coupon (paid in £s, $s, Euros etc.) is fixed but the yield on a bond will vary
The yield is effectively the interest rate on a bond
The yield will vary inversely with the market price of a bond
When bond prices are rising, the yield will fall
When bond prices are falling, the yield will rise
Factor Affecting exchange rate and Theories of exchange rate Jatin Goyal
It explains the following topics
Factor Affecting the exchange rate
CURRENCY DEPRECIATION VS.CURRENCY APPRECIATION
Foreign exchange
Theories of exchange rate
Corporate Valuations “Techniques & Application”: A compilation of research oriented valuation articles.
Contents: Business valuation, Relative valuation, Sum of the parts valuation and value creation, ESOP valuation, Discounted Cash Flow Valuation, Enterprise Valuation etc.
Government bonds are fixed interest securities
This means that a bond pays a fixed annual interest – this is known as the coupon
The coupon (paid in £s, $s, Euros etc.) is fixed but the yield on a bond will vary
The yield is effectively the interest rate on a bond
The yield will vary inversely with the market price of a bond
When bond prices are rising, the yield will fall
When bond prices are falling, the yield will rise
Factor Affecting exchange rate and Theories of exchange rate Jatin Goyal
It explains the following topics
Factor Affecting the exchange rate
CURRENCY DEPRECIATION VS.CURRENCY APPRECIATION
Foreign exchange
Theories of exchange rate
Corporate Valuations “Techniques & Application”: A compilation of research oriented valuation articles.
Contents: Business valuation, Relative valuation, Sum of the parts valuation and value creation, ESOP valuation, Discounted Cash Flow Valuation, Enterprise Valuation etc.
Time Value of Money (TVM), also known as present discounted value, refers to the notion that money available now is worth more than the same amount in the future, because of its ability to grow.
The term is similar to the concept of ‘time is money’, in the sense of the money itself, rather than one’s own time that is invested. As long as money can earn interest (which it can), it is worth more the sooner you get it.
Introduction to Financial Analytics -Fundamentals of Finance Class I
by Reuben Ray; reuben@pexitics.com
• Time value of money.
• Present value & future value of money.
• Applications of TVM (Time Value of Money)
• Annuity & perpetuity concepts.
• Introduction to financial statements.
TVM, Future Value Interest Factor (FVIF), Present Value Interest Factor (PVIF), present value interest factor of an annuity (PVIFA)
Using estimated rates of return, you can compare the value of the annuity payments to the lump sum.
The present value interest factor may only be calculated if the annuity payments are for a predetermined amount spanning a predetermined range of time.
Time Value of Money Formula
FV = PV x [ 1 + (i / n) ] (n x t)
Formula for Future Value Interest factor:
FVIF = (1+r)n
Formula for PVIF
PVIF = 1 / (1 + r)n
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2. Intuition Behind Present Value
n There are three reasons why a dollar tomorrow is worth less than a
dollar today
• Individuals prefer present consumption to future consumption. To
induce people to give up present consumption you have to offer them
more in the future.
• When there is monetary inflation, the value of currency decreases over
time. The greater the inflation, the greater the difference in value between
a dollar today and a dollar tomorrow.
• If there is any uncertainty (risk) associated with the cash flow in the
future, the less that cash flow will be valued.
n Other things remaining equal, the value of cash flows in future time
periods will decrease as
• the preference for current consumption increases.
• expected inflation increases.
• the uncertainty in the cash flow increases.
Aswath Damodaran 2
3. Discounting and Compounding
• The mechanism for factoring in these elements is the discount rate.
• Discount Rate: The discount rate is a rate at which present and future
cash flows are traded off. It incorporates -
(1) Preference for current consumption (Greater ....Higher Discount Rate)
(2) expected inflation (Higher inflation .... Higher Discount Rate)
(3) the uncertainty in the future cash flows (Higher Risk....Higher Discount Rate)
• A higher discount rate will lead to a lower value for cash flows in the
future.
• The discount rate is also an opportunity cost, since it captures the returns
that an individual would have made on the next best opportunity.
• Discounting future cash flows converts them into cash flows in present
value dollars. Just a discounting converts future cash flows into present
cash flows,
• Compounding converts present cash flows into future cash flows.
Aswath Damodaran 3
4. Present Value Principle 1
n Cash flows at different points in time cannot be compared and
aggregated. All cash flows have to be brought to the same point in
time, before comparisons and aggregations are made.
Aswath Damodaran 4
5. Cash Flow Types and Discounting Mechanics
n There are five types of cash flows -
• simple cash flows,
• annuities,
• growing annuities
• perpetuities and
• growing perpetuities
Aswath Damodaran 5
6. I.Simple Cash Flows
n A simple cash flow is a single cash flow in a specified future time
period.
Cash Flow: CFt
_______________________________________________|
Time Period: t
n The present value of this cash flow is-
PV of Simple Cash Flow = CFt / (1+r)t
n The future value of a cash flow is -
FV of Simple Cash Flow = CF0 (1+ r)t
Aswath Damodaran 6
7. Application 1: The power of compounding -
Stocks, Bonds and Bills
n Ibbotson and Sinquefield, in a study of returns on stocks and bonds
between 1926-92 found that stocks on the average made 12.4%,
treasury bonds made 5.2% and treasury bills made 3.6%.
n The following table provides the future values of $ 100 invested in
each category at the end of a number of holding periods - 1, 5 , 10 , 20,
30 and 40 years.
Holding Period Stocks T. Bonds T.Bills
1 $112.40 $105.20 $103.60
5 $179.40 $128.85 $119.34
10 $321.86 $166.02 $142.43
20 $1,035.92 $275.62 $202.86
30 $3,334.18 $457.59 $288.93
40 $10,731.30 $759.68 $411.52
Aswath Damodaran 7
8. Concept Check
n Most pension plans allow individuals to decide where their pensions
funds will be invested - stocks, bonds or money market accounts.
n Where would you choose to invest your pension funds?
o Predominantly or all equity
o Predominantly or all bonds and money market accounts
o A Mix of Bonds and Stocks
n Will your allocation change as you get older?
o Yes
o No
Aswath Damodaran 8
9. The Frequency of Compounding
n The frequency of compounding affects the future and present values of
cash flows. The stated interest rate can deviate significantly from the
true interest rate –
• For instance, a 10% annual interest rate, if there is semiannual
compounding, works out to-
Effective Interest Rate = 1.052 - 1 = .10125 or 10.25%
Frequency Rate t Formula Effective Annual Rate
Annual 10% 1 r 10.00%
Semi-Annual 10% 2 (1+r/2)2-1 10.25%
Monthly 10% 12 (1+r/12)12-1 10.47%
Daily 10% 365 (1+r/365)365-1 10.5156%
Continuous 10% expr-1 10.5171%
Aswath Damodaran 9
10. II. Annuities
n An annuity is a constant cash flow that occurs at regular intervals for a
fixed period of time. Defining A to be the annuity,
A A A A
| | | |
0 1 2 3 4
Aswath Damodaran 10
11. Present Value of an Annuity
n The present value of an annuity can be calculated by taking each cash
flow and discounting it back to the present, and adding up the present
values. Alternatively, there is a short cut that can be used in the
calculation [A = Annuity; r = Discount Rate; n = Number of years]
1 - 1
(1 + r)n
PV of an Annuity = PV(A,r, ) = A
n
r
Aswath Damodaran 11
12. Example: PV of an Annuity
n The present value of an annuity of $1,000 for the next five years,
assuming a discount rate of 10% is -
1
1 -
(1.10)
5
PV of $1000 each year for next 5 years = $1000 = $3,791
.10
n The notation that will be used in the rest of these lecture notes for the
present value of an annuity will be PV(A,r,n).
Aswath Damodaran 12
13. Annuity, given Present Value
n The reverse of this problem, is when the present value is known and
the annuity is to be estimated - A(PV,r,n).
r
Annuity given Present Value = A(PV, r,n) = PV
1 - 1
(1 + r)n
Aswath Damodaran 13
14. Future Value of an Annuity
n The future value of an end-of-the-period annuity can also be calculated
as follows-
( + r n) - 1
1
FV of an Annuity = F V ( Ar,,n ) = A
r
Aswath Damodaran 14
15. An Example
n Thus, the future value of $1,000 each year for the next five years, at
the end of the fifth year is (assuming a 10% discount rate) -
(1.10) - 1
5
FV of $1,000 each year for next 5 years = $1000 = $6,105
.10
n The notation that will be used for the future value of an annuity will be
FV(A,r,n).
Aswath Damodaran 15
16. Annuity, given Future Value
n if you are given the future value and you are looking for an annuity -
A(FV,r,n) in terms of notation -
r
Annuity given Future Value = A(FV,r,n)= FV
( 1 +r) - 1
n
Aswath Damodaran 16
17. Application 2: Saving for College Tuition
n Assume that you want to send your newborn child to a private college
(when he gets to be 18 years old). The tuition costs are $ 16000/year
now and that these costs are expected to rise 5% a year for the next 18
years. Assume that you can invest, after taxes, at 8%.
• Expected tuition cost/year 18 years from now = 16000*(1.05)18 = $38,506
• PV of four years of tuition costs at $38,506/year = $38,506 * PV(A ,8%,4
years)= $127,537
n If you need to set aside a lump sum now, the amount you would need
to set aside would be -
• Amount one needs to set apart now = $127,357/(1.08)18 = $31,916
n If set aside as an annuity each year, starting one year from now -
• If set apart as an annuity = $127,537 * A(FV,8%,18 years) = $3,405
Aswath Damodaran 17
18. Application 3: How much is an MBA worth?
n Assume that you were earning $40,000/year before entering program
and that tuition costs are $16000/year. Expected salary is $
54,000/year after graduation. You can invest money at 8%.
For simplicity, assume that the first payment of $16,000 has to be made at
the start of the program and the second payment one year later.
• PV Of Cost Of MBA = $16,000+16,000/1.08 + 40000 * PV(A,8%,2
years) = $102,145
n Assume that you will work 30 years after graduation, and that the
salary differential ($14000 = $54000-$40000) will continue through
this period.
• PV of Benefits Before Taxes = $14,000 * PV(A,8%,30 years) = $157,609
• This has to be discounted back two years - $157,609/1.082 = $135,124
• The present value of getting an MBA is = $135,124 - $102,145 = $32,979
Aswath Damodaran 18
19. Some Follow-up Questions
1. How much would your salary increment have to be for you to break
even on your MBA?
2. Keeping the increment constant, how many years would you have to
work to break even?
Aswath Damodaran 19
20. Application 4: Savings from Refinancing Your
Mortgage
n Assume that you have a thirty-year mortgage for $200,000 that carries
an interest rate of 9.00%. The mortgage was taken three years ago.
Since then, assume that interest rates have come down to 7.50%, and
that you are thinking of refinancing. The cost of refinancing is
expected to be 2.50% of the loan. (This cost includes the points on the
loan.) Assume also that you can invest your funds at 6%.
Monthly payment based upon 9% mortgage rate (0.75% monthly rate)
= $200,000 * A(PV,0.75%,360 months)
= $1,609
Monthly payment based upon 7.50% mortgage rate (0.625% monthly rate)
= $200,000 * A(PV,0.625%,360 months)
= $1,398
n Monthly Savings from refinancing = $1,609 - $1,398 = $211
Aswath Damodaran 20
21. Refinancing: The Trade Off
n If you plan to remain in this house indefinitely,
Present Value of Savings (at 6% annually; 0.5% a month)
= $211 * PV(A,0.5%,324 months)
= $33,815
• The savings will last for 27 years - the remaining life of the existing
mortgage.
• You will need to make payments for three additional years as a
consequence of the refinancing -
Present Value of Additional Mortgage payments - years 28,29 and 30
= $1,398 * PV(A,0.5%,36 months)/1.0627
= $9,532
n Refinancing Cost = 2.5% of $200,000 = $5,000
n Total Refinancing Cost = $9,532 + $5,000 = $14,532
n Net Effect = $ 33,815 - $ 9,532 - $ 14,532 = $9,751: Refinance
Aswath Damodaran 21
22. Follow-up Questions
1. How many years would you have to live in this house for you break
even on this refinancing?
2. We've ignored taxes in this analysis. How would it impact your
decision?
Aswath Damodaran 22
23. Application 5: Valuing a Straight Bond
n You are trying to value a straight bond with a fifteen year maturity and
a 10.75% coupon rate. The current interest rate on bonds of this risk
level is 8.5%.
PV of cash flows on bond = 107.50* PV(A,8.5%,15 years) + 1000/1.08515 =
$ 1186.85
n If interest rates rise to 10%,
PV of cash flows on bond = 107.50* PV(A,10%,15 years)+ 1000/1.1015 =
$1,057.05
Percentage change in price = -10.94%
n If interest rate fall to 7%,
PV of cash flows on bond = 107.50* PV(A,7%,15 years)+ 1000/1.0715 =
$1,341.55
Percentage change in price = +13.03%
n This asymmetric response to interest rate changes is called convexity.
Aswath Damodaran 23
24. Application 6: Contrasting Short Term and
Long Term Bonds
Price Changes as a function of Bond Maturities
20.00%
15.00%
% Change in Price
10.00%
% Change if rate drops
5.00% to 7%
0.00% % Change if rate
increases to 10%
-5.00%
-10.00%
-15.00%
1 5 15 30
Bond Maturity
Aswath Damodaran 24
25. Bond Pricing Proposition 1
n The longer the maturity of a bond, the more sensitive it is to changes in
interest rates.
Aswath Damodaran 25
26. Application 7: Contrasting Low-coupon and
High-coupon Bonds
Bond Price Changes as a function of Coupon Rates
25.00%
20.00%
15.00%
% Price Change
10.00% % Change if rate
5.00% drops to 7%
0.00% % Change if rate
-5.00% increases to 10%
-10.00%
-15.00%
-20.00%
0% 5% 10.75% 12%
Coupon Rate
Aswath Damodaran 26
27. Bond Pricing Proposition 2
n The lower the coupon rate on the bond, the more sensitive it is to
changes in interest rates.
Aswath Damodaran 27
28. III. Growing Annuity
n A growing annuity is a cash flow growing at a constant rate for a
specified period of time. If A is the current cash flow, and g is the
expected growth rate, the time line for a growing annuity looks as
follows –
Figure 3.8: A Growing Annuity
A(1+g) A(1+g)2 A(1+g)3 A(1+g)n
0 1 2 3 ........... n
Aswath Damodaran 28
29. Present Value of a Growing Annuity
• The present value of a growing annuity can be estimated in all cases,
but one - where the growth rate is equal to the discount rate, using the
following model:
1
1 -
(1 + r)
n
PV of an Annuity = PV(A,r, ) = A
n
r
• In that specific case, the present value is equal to the nominal sums of
the annuities over the period, without the growth effect.
Aswath Damodaran 29
30. Appendix 8: The Value of a Gold Mine
n Consider the example of a gold mine, where you have the rights to the
mine for the next 20 years, over which period you plan to extract 5,000
ounces of gold every year. The price per ounce is $300 currently, but it
is expected to increase 3% a year. The appropriate discount rate is
10%. The present value of the gold that will be extracted from this
mine can be estimated as follows –
(1.03)20
1 -
(1.10)20
PV of extracted gold = $300*5000 * (1.03) = $16,145,980
.10 - .03
Aswath Damodaran 30
31. PV of Extracted Gold as a Function of
Expected Growth Rate
Present Value of Extracted Gold as a function of Growth Rate
$50,000,000
$45,000,000
$40,000,000
Present Value of Extracted Gold
$35,000,000
$30,000,000
$25,000,000
$20,000,000
$15,000,000
$10,000,000
$5,000,000
$-
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
Growth Rate in Gold Prices
Aswath Damodaran 31
32. PV of Extracted Gold as a Function of
Expected Growth Rate
Present Value of Extracted Gold as a function of Growth Rate
$50,000,000
$45,000,000
$40,000,000
Present Value of Extracted Gold
$35,000,000
$30,000,000
$25,000,000
$20,000,000
$15,000,000
$10,000,000
$5,000,000
$-
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
Growth Rate in Gold Prices
Aswath Damodaran 32
33. Concept Check
n If both the growth rate and the discount rate go up by 1%, will the
present value of the gold to be extracted from this mine increase or
decrease?
Aswath Damodaran 33
34. IV. Perpetuity
n A perpetuity is a constant cash flow at regular intervals forever. The
present value of a perpetuity is-
A
PV of Perpetuity =
r
Aswath Damodaran 34
35. Application 9: Valuing a Console Bond
n A console bond is a bond that has no maturity and pays a fixed
coupon. Assume that you have a 6% coupon console bond. The value
of this bond, if the interest rate is 9%, is as follows -
Value of Console Bond = $60 / .09 = $667
Aswath Damodaran 35
36. V. Growing Perpetuities
n A growing perpetuity is a cash flow that is expected to grow at a
constant rate forever. The present value of a growing perpetuity is -
CF1
PV of Growing Perpetuity =
(r - g)
where
• CF1 is the expected cash flow next year,
• g is the constant growth rate and
• r is the discount rate.
Aswath Damodaran 36
37. Application: Valuing a Stock with Growing
Dividends
n Southwestern Bell paid dividends per share of $2.73 in 1992. Its
earnings and dividends have grown at 6% a year between 1988 and
1992, and are expected to grow at the same rate in the long term. The
rate of return required by investors on stocks of equivalent risk is
12.23%.
Current Dividends per share = $2.73
Expected Growth Rate in Earnings and Dividends = 6%
Discount Rate = 12.23%
Value of Stock = $2.73 *1.06 / (.1223 -.06) = $46.45
Aswath Damodaran 37