Proline Promag 50P, 53P-Electromagnetic Flowmeter. Flow measurement of liquids in chemical or process applications. Email: lam.nguyen@vietan-enviro.com HP: 0945 293292
Proline Promag 50P, 53P-Electromagnetic Flowmeter. Flow measurement of liquids in chemical or process applications. Email: lam.nguyen@vietan-enviro.com HP: 0945 293292
Broadside Array vs end-fire array
Higher directivity.
Provide increased directivity in
elevation and azimuth planes.
Generally used for reception.
Impedance match difficulty in
high power transmissions.
Variants are:
Horizontal Array of Dipoles
RCA Fishborne Antenna
Series Phase Array
Basics of electronics for biomedical engineering can be viewed through the link http://bit.ly/2PPv0mv
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core.
Diodes and its application encapsulate the different characteristics of different type of diodes. Also, define its different biases and how it works.
It provides shortcut method in analyzing Clamper and clipper.
At the end of the powerpoint, there has a review question to answer with answer key provided.
By completing this presentation will be have a clear idea about Antenna's working principles, Antenna's Types & Antenna's Parameters. At the end to this document you'll have a brief idea about Antenna's Tilt vs Distance Calculation & Cluster wise optimum Antenna Selection procedure. Impact of antenna PIM & VSWR have been described elaborately in this document as well.
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
Broadside Array vs end-fire array
Higher directivity.
Provide increased directivity in
elevation and azimuth planes.
Generally used for reception.
Impedance match difficulty in
high power transmissions.
Variants are:
Horizontal Array of Dipoles
RCA Fishborne Antenna
Series Phase Array
Basics of electronics for biomedical engineering can be viewed through the link http://bit.ly/2PPv0mv
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core.
Diodes and its application encapsulate the different characteristics of different type of diodes. Also, define its different biases and how it works.
It provides shortcut method in analyzing Clamper and clipper.
At the end of the powerpoint, there has a review question to answer with answer key provided.
By completing this presentation will be have a clear idea about Antenna's working principles, Antenna's Types & Antenna's Parameters. At the end to this document you'll have a brief idea about Antenna's Tilt vs Distance Calculation & Cluster wise optimum Antenna Selection procedure. Impact of antenna PIM & VSWR have been described elaborately in this document as well.
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
This is the third presentation for the University of New England Graduate School of Business unit GSB711 - Managerial Finance. It explores the time value of money, using examples to help students clarify this concept.
The secret way to sell pi coins effortlessly.DOT TECH
Well as we all know pi isn't launched yet. But you can still sell your pi coins effortlessly because some whales in China are interested in holding massive pi coins. And they are willing to pay good money for it. If you are interested in selling I will leave a contact for you. Just telegram this number below. I sold about 3000 pi coins to him and he paid me immediately.
Telegram: @Pi_vendor_247
how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
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NO1 Uk Black Magic Specialist Expert In Sahiwal, Okara, Hafizabad, Mandi Bah...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
Message: @Pi_vendor_247 on telegram if u want to sell PI COINS.
1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
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Introduction to Indian Financial System ()Avanish Goel
The financial system of a country is an important tool for economic development of the country, as it helps in creation of wealth by linking savings with investments.
It facilitates the flow of funds form the households (savers) to business firms (investors) to aid in wealth creation and development of both the parties
Latino Buying Power - May 2024 Presentation for Latino CaucusDanay Escanaverino
Unlock the potential of Latino Buying Power with this in-depth SlideShare presentation. Explore how the Latino consumer market is transforming the American economy, driven by their significant buying power, entrepreneurial contributions, and growing influence across various sectors.
**Key Sections Covered:**
1. **Economic Impact:** Understand the profound economic impact of Latino consumers on the U.S. economy. Discover how their increasing purchasing power is fueling growth in key industries and contributing to national economic prosperity.
2. **Buying Power:** Dive into detailed analyses of Latino buying power, including its growth trends, key drivers, and projections for the future. Learn how this influential group’s spending habits are shaping market dynamics and creating opportunities for businesses.
3. **Entrepreneurial Contributions:** Explore the entrepreneurial spirit within the Latino community. Examine how Latino-owned businesses are thriving and contributing to job creation, innovation, and economic diversification.
4. **Workforce Statistics:** Gain insights into the role of Latino workers in the American labor market. Review statistics on employment rates, occupational distribution, and the economic contributions of Latino professionals across various industries.
5. **Media Consumption:** Understand the media consumption habits of Latino audiences. Discover their preferences for digital platforms, television, radio, and social media. Learn how these consumption patterns are influencing advertising strategies and media content.
6. **Education:** Examine the educational achievements and challenges within the Latino community. Review statistics on enrollment, graduation rates, and fields of study. Understand the implications of education on economic mobility and workforce readiness.
7. **Home Ownership:** Explore trends in Latino home ownership. Understand the factors driving home buying decisions, the challenges faced by Latino homeowners, and the impact of home ownership on community stability and economic growth.
This SlideShare provides valuable insights for marketers, business owners, policymakers, and anyone interested in the economic influence of the Latino community. By understanding the various facets of Latino buying power, you can effectively engage with this dynamic and growing market segment.
Equip yourself with the knowledge to leverage Latino buying power, tap into their entrepreneurial spirit, and connect with their unique cultural and consumer preferences. Drive your business success by embracing the economic potential of Latino consumers.
**Keywords:** Latino buying power, economic impact, entrepreneurial contributions, workforce statistics, media consumption, education, home ownership, Latino market, Hispanic buying power, Latino purchasing power.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
US Economic Outlook - Being Decided - M Capital Group August 2021.pdfpchutichetpong
The U.S. economy is continuing its impressive recovery from the COVID-19 pandemic and not slowing down despite re-occurring bumps. The U.S. savings rate reached its highest ever recorded level at 34% in April 2020 and Americans seem ready to spend. The sectors that had been hurt the most by the pandemic specifically reduced consumer spending, like retail, leisure, hospitality, and travel, are now experiencing massive growth in revenue and job openings.
Could this growth lead to a “Roaring Twenties”? As quickly as the U.S. economy contracted, experiencing a 9.1% drop in economic output relative to the business cycle in Q2 2020, the largest in recorded history, it has rebounded beyond expectations. This surprising growth seems to be fueled by the U.S. government’s aggressive fiscal and monetary policies, and an increase in consumer spending as mobility restrictions are lifted. Unemployment rates between June 2020 and June 2021 decreased by 5.2%, while the demand for labor is increasing, coupled with increasing wages to incentivize Americans to rejoin the labor force. Schools and businesses are expected to fully reopen soon. In parallel, vaccination rates across the country and the world continue to rise, with full vaccination rates of 50% and 14.8% respectively.
However, it is not completely smooth sailing from here. According to M Capital Group, the main risks that threaten the continued growth of the U.S. economy are inflation, unsettled trade relations, and another wave of Covid-19 mutations that could shut down the world again. Have we learned from the past year of COVID-19 and adapted our economy accordingly?
“In order for the U.S. economy to continue growing, whether there is another wave or not, the U.S. needs to focus on diversifying supply chains, supporting business investment, and maintaining consumer spending,” says Grace Feeley, a research analyst at M Capital Group.
While the economic indicators are positive, the risks are coming closer to manifesting and threatening such growth. The new variants spreading throughout the world, Delta, Lambda, and Gamma, are vaccine-resistant and muddy the predictions made about the economy and health of the country. These variants bring back the feeling of uncertainty that has wreaked havoc not only on the stock market but the mindset of people around the world. MCG provides unique insight on how to mitigate these risks to possibly ensure a bright economic future.
2. The objective of Chapter 4 is to
explain time value of money
calculations and to illustrate
economic equivalence.
3. Money has a time value.
Capital refers to wealth in the form of money or
property that can be used to produce more wealth.
Engineering economy studies involve the
commitment of capital for extended periods of
time.
A dollar today is worth more than a dollar one or
more years from now (for several reasons).
4. Simple interest is used infrequently.
When the total interest earned or charged is
linearly proportional to the initial amount of the
loan (principal), the interest rate, and the number
of interest periods, the interest and interest rate
are said to be simple.
5. Computation of simple interest
The total interest, I, earned or paid may be computed
using the formula below.
P = principal amount lent or borrowed
N = number of interest periods (e.g., years)
i = interest rate per interest period
The total amount repaid at the end of N interest
periods is P + I.
6. If $5,000 were loaned for five years at a
simple interest rate of 7% per year, the
interest earned would be
So, the total amount repaid at the end
of five years would be the original
amount ($5,000) plus the interest
($1,750), or $6,750.
7. Ice Breaking
In 2005, total debit (credit cards, auto loans, home
mortgages, etc.) amounted to more than 100% of total
disposable income for the average U.S household. If
your total disposable income is $50,000, how much
sum can you expect to pay at the end of one year
period when the average interest rate on your debit is
12% per year?
Simple Interest Calculation,
I = P*N* I
I= $50,000*1*(0.12)
Therefore, I = $ 6,000
8. Compound interest reflects both the remaining principal
and any accumulated interest. For $1,000 at 10%…
Period
(1)
Amount owed
at beginning of
period
(2)=(1)x10%
Interest
amount for
period
(3)=(1)+(2)
Amount
owed at end
of period
1 $1,000 $100 $1,100
2 $1,100 $110 $1,210
3 $1,210 $121 $1,331
Compound interest is commonly used in personal and
professional financial transactions.
9. Economic equivalence allows us to compare
alternatives on a common basis.
Each alternative can be reduced to an equivalent
basis dependent on
interest rate,
amount of money involved, and
timing of monetary receipts or expenses.
Using these elements we can “move” cash flows so
that we can compare them at particular points in
time.
10. We need some tools to find economic
equivalence.
Notation used in formulas for compound
interest calculations.
i = effective interest rate per interest period
N = number of compounding (interest) periods
P = present sum of money; equivalent value of one or
more cash flows at a reference point in time; the present
F = future sum of money; equivalent value of one or more
cash flows at a reference point in time; the future
A = end-of-period cash flows in a uniform series
continuing for a certain number of periods, starting at the
end of the first period and continuing through the last
11. A cash flow diagram is an indispensable tool
for clarifying and visualizing a series of cash
flows.
12. Cash flow tables are essential to modeling
engineering economy problems in a
spreadsheet
13. Active Learning-1
Before evaluating the economic merits of a proposed
investment, the XYZ Corporation insists that its engineers
develop a cash-flow diagram of the proposal. An
investment of $10,000 can be made that will produce
uniform annual revenue of $ 5,310 for five years and then
have a market (recovery) value of $ 2000 at the end of
the year five. Annual expenses will be $3000 at the end
of each year for operating and maintaining the project.
Draw a cash-flow diagram for the five year life of the
project.
14. We can apply compound interest formulas to
“move” cash flows along the cash flow diagram.
Using the standard notation, we find that a
present amount, P, can grow into a future
amount, F, in N time periods at interest rate
i according to the formula below.
In a similar way we can find P given F by
15. It is common to use standard notation for
interest factors.
This is also known as the single payment
compound amount factor. The term on the
right read as “F given P at i% interest per
period for N interest periods.”
is called the single payment present worth
factor.
16. Active Learning- 2
$2,500 at time zero is equivalent to how much after six years if
the interest rate is 8% per year?
$3,000 at the end of year seven is equivalent to how much today
(time zero) if the interest rate is 6% per year?
17. Betty will need $12,000 in five years to pay for a major
overhaul on her tractor engine. She has found an
investment that will provide a 5% return on her
invested funds. How much does Betty need to invest
today so she will have her overhaul funds in five
years?
Active Learning- 3
18. Uniform Series Annuity and Future
Equivalent Values
How much will you have in 40 years if you
save $3,000 each year and your account
earns 8% interest each year?
19. Active Learning - 4
A recent government study reported that a college
degree is worth an extra $23,000 per year in income (A)
compared to what a high school graduate makes. If the
interest rate (i) is 6 % per year and you work for 40
years (N) , What is the future compound amount (F) of
this extra income ? Please draw Cash Flow Diagram
and Solve.
20. Uniform Series Annuity and Present
Equivalent Values
How much would be needed today to provide
an annual amount of $50,000 each year for 20
years, at 9% interest each year?
21. Active Learning – 5
If a certain machine undergoes a major overhaul now,
its output can be increased by 20%- which translates
into additional cash flow of $20,000 at the end of each
year for five years. If i= 15% per year, how much can
we afford to invest to overhaul this machine? Draw a
Cash Flow Diagram and Solve?
22. Finding A when given F.
How much would you need to set aside each
year for 25 years, at 10% interest, to have
accumulated $1,000,000 at the end of the 25
years?
23. Finding A when given P.
If you had $500,000 today in an account/fund
earning 10% each year, how much could you
withdraw each year for 25 years?
24. Active Learning- 6
You borrow $15,000 from your credit union to purchase
a used car. The interest rate on your loan is 0.25% per
month and you will make a total of 36 monthly
payments. What is your monthly payment? Make a
Cash Flow Diagram from the perspective of a bank and
Solve.
25. Acme Steamer purchased a new pump for $75,000.
They borrowed the money for the pump from their bank
at an interest rate of 0.5% per month and will make a
total of 24 equal, monthly payments. How much will
Acme’s monthly payments be? Make a Cash Flow
Diagram from the perspective of Acme Steamer and
Solve.
Active Learning- 7
26. It can be challenging to solve for N or i.
We may know P, A, and i and want to find N.
We may know P, A, and N and want to find i.
These problems present special challenges that are
best handled on a spreadsheet.
27. Finding N
Acme borrowed $100,000 from a local bank, which
charges them an interest rate of 7% per year. If Acme
pays the bank $8,000 per year, now many years will it
take to pay off the loan?
So,
This can be solved by using the interest tables and interpolation,
but we generally resort to a computer solution.
(P/A, 7%, 30) = 12.40 (P/A, 7%, 31) = 12.53
28. Finding i
The car you want to buy will cost $60,000 in eight years. You
are going to put aside $6,000 each year for eight years. At what
interest rate must you invest your money now to achieve your
goal ?
So,
Again, this can be solved using the interest tables and
interpolation, but we generally resort to a computer solution.
(F/A, i%, 8) = 10
(F/A, 6%, 8) = 9.8975
(F/A, 7%, 8) = 10.2598
30. Spreadsheet/Excel functions to find N and i.
The Excel function used to solve for N is
NPER(rate, pmt, pv), which will compute the
number of payments of magnitude pmt required to
pay off a present amount (pv) at a fixed interest
rate (rate).
One Excel function used to solve for i is
RATE(nper, pmt, pv, fv), which returns a fixed
interest rate for an annuity of pmt that lasts for nper
periods to either its present value (pv) or future value
(fv).
31.
32. Deferred Annuities (Uniform Series)
Deferred annuities are uniform series that do not
begin until some time in the future.
If the annuity is deferred J periods then the first
payment (cash flow) begins at the end of period
J+1.
33. Finding the value at time 0 of a deferred
annuity is a two-step process.
1. Use (P/A, i%, N-J) find the value of the deferred
annuity at the end of period J (where there are N-
J cash flows in the annuity).
2. Use (P/F, i%, J) to find the value of the deferred
annuity at time zero.
34. Irene just purchased a new sports car and wants to also set
aside cash for future maintenance expenses. The car has a
bumper-to-bumper warranty for the first five years. Irene
estimates that she will need approximately $2,000 per year in
maintenance expenses for years 6-10, at which time she will
sell the vehicle. How much money should Irene deposit into
an account today, at 8% per year, so that she will have
sufficient funds in that account to cover her projected
maintenance expenses?
Active Learning-8
P5 = A(P/A, 8%, 5) = $2,000*(3.9927)= $7,985.40
P0 = F5 (P/F, 8%, 5)= $7,985.40*(0.6806)= $5,434.86
35. Active Learning 9
A father on the day his son is born, wishes to determine
what lump sum amount would have to be paid into an
account bearing interest of 12% per year to provide
withdrawals of $2000 on each of the son’s 18th, 19th and
20th and 21st birthday?
P17 = A(P/A, 12%, 4) = %2,000*(3.0373)= $6,074.60
P0 = F17 (P/F, 12%, 17)= $6,074.60*(0.1456) = $884.46
36. Active Learning 10
When you take your first job, you decide to start saving right
away for your retirement. You put $5000 per year into the
company’s 410(k) plan, which averages 8% per year. Five
years later, you move to another job and start a new 401(k)
plan. You never get around to merging the funds in the two
plans. If the first plan continued to earn interest at the rate of
8% per year for 35 years after you stopped making
contributions, how much is the account worth.
F5 = $5,000(F/A, 8%,5)= $5,000*(5.8666)= $29,333
F40 = P5 (F/P, 8%, 35) = $ 29,333*(14.7853)= $433,697
P0= $5,000(P/A,8%,5)= $5,000*(3.9927)= $19,963.5
F40 = P0 (F/P, 8%,40) = $ 19,963.50*(21.7245)= $433,697
Alternatively,
37. Active Learning 11
The cash flows extends over 8 years. The amounts are $100
for the first year, $200 for the second year, $500 for the third
year and $400 for each year from the fourth year through the
eighth. These cash flows represent something like the
expenditures for a certain piece of equipment or payments
into a fund. The annual interest rate is 20%. Find;
a. The present equivalent expenditure, P0
b. The future equivalent expenditure, F8
c. The annual equivalent expenditure, A
(P0= $ 1203.82, F8=$5176.19 & A=$313.73)
38. Active Learning 12
Two receipts of $1,000 each are desired at the EOYs 10 and
11. To make these receipts possible, four equal yearly annuity
amounts will be deposited in a bank. At EOYs 2,3,4, and 5.
The bank’s interest rate I = 12% per year.
a. Draw a cash-flow diagram for this situation.
b. Determine the value of A that establishes equivalence in
your cash flow diagram.
c. Determine the lump-sum value at the end of year 11 of the
completed cash flow diagram based on your answers to
Part(a) and Part(b).
(A=$224.75 & F= 0)
40. Sometimes cash flows change by a constant
amount each period: Arithmetic Gradient
We can model these situations as a uniform
gradient of cash flows. The table below shows
such a gradient.
End of Period Cash Flows
1 0
2 G
3 2G
: :
N (N-1)G
41. It is easy to find the present value of a
uniform gradient series.
Similar to the other types of cash flows, there is a
formula (albeit quite complicated) we can use to find
the present value, and a set of factors developed for
interest tables.
Present Value = G* (P/G, i%, N)
42. We can also find A or F equivalent to a
uniform gradient series.
Annuity = G (A/G, i%, N)
Future Value= G (F/G, i%, N)
43. Active Learning 13
Suppose that the cash flows are expected to be $1000 for the
second year, $2000 for the third year and $3000 for the fourth
year and that if the interest is 15% per year, it is desire to find
a. Present equivalent value
b. Uniform annual equivalent value at the end of the four
years.
P0 = G(P/G, 15%, 4)= $1000(3.79) = $3,790.00
A = G(A/G, 15%, 4)= $1000(1.3263) = $1,326.30
44. End of Year Cash Flows ($)
1 5,000
2 6,000
3 7,000
4 8,000
Find the Annual Equivalent
of this arithmetic gradient
series of cash flows . i =
15%
End of Year Annuity ($) Gradient ($)
1 5,000 0
2 5,000 1,000
3 5,000 2,000
4 5,000 3,000
A = $5,000 + $1,000 (A/G, 15%,4) = $6326.30
P=A(P/A,15%,4)+G(P/G,15%,4) = $18,065
45. End of Year Cash Flows ($)
1 8,000
2 7,000
3 6,000
4 5,000
Active Learning 15
The cash flows are timed in
reverse gradient. Calculate the
present equivalent at i= 15%
per year using arithmetic
gradient interest factors.
End of Year Annuity ($) Gradient ($)
1 8,000 0
2 8,000 -1,000
3 8,000 -2,000
4 8,000 -3,000
A = $8,000 - $1,000 (A/G, 15% ,4) = $ 6,673.70
P= $8,000(P/A, 15% ,4) -1000(P/G, 15%, 4) = Solve
46. Sometimes cash flows change by a
constant rate, , each period- Geometric
Gradient
End of Year Cash Flows ($)
1 1,000
2 1,200
3 1,440
4 1,728
This table presents a
geometric gradient series. It
begins at the end of year 1
and has a rate of growth, ,
of 20%.
47.
48. Present value of a Geometric Series
Where is the initial cash flow in the series.
i
49. Acme Miracle projects good things for their new weight
loss pill, LoseIt. Revenues this year are expected to
be $1.1 million, and Acme believes they will increase
15% per year for the next 5 years. What are the
present value and equivalent annual amount for the
anticipated revenues? Acme uses an interest rate of
20%.
Active Learning - 16
P = $1.1million 1- (P/F, 20%, 5) (F/P, 15%, 5) ] / (0.20 – 0.15)
50. Active Learning 17
Consider the Geometric sequence of cash flows and
determine P, A and F equivalent values. The rate of
decrease is 20% per year after the first year which
continues for 4th year, and the rate of interest is 25%
per year
P = $1000 1- (P/F, 25%,4) (F/P, 20%, 4) ] / (0.25 – (-0.20))
Therefore, P = $1,849.38
A = $1,849.38 (A/P, 25%, 4) = $783.03
F = $1,849.38 (F/P, 25%, 4) = $4,515.08
51. When interest rates vary with time
different procedures are necessary.
Interest rates often change with time (e.g., a
variable rate mortgage).
We often must resort to moving cash flows one
period at a time, reflecting the interest rate for that
single period.
52. Activity Learning 18
Mr. X borrowed $4000 four years ago when the interest rate
was 4.06% per year. $5,000 was borrowed three years ago at
3.42%. Two years ago she borrowed $6,000 at 5.23% and last
year $7,000 was borrowed at 6.03% per year. Now Mr. X
would like to consolidate her debt into a single 20 year loan
with a 5% fixed annual interest rate. If Mr. X makes annual
payments (starting in first year) to repay her total loan, what
is the amount of each payment?
53. The present equivalent of a cash flow occurring at
the end of period N can be computed with the
equation below, where ik is the interest rate for the
kth period.
If F4 = $2,500 and i1=8%, i2=10%, and i3=11%, then
54. Nominal and effective interest rates.
More often than not, the time between successive
compounding, or the interest period, is less than one
year (e.g., daily, monthly, quarterly).
The annual rate is known as a nominal rate.
A nominal rate of 12%, compounded monthly, means an
interest of 1% (12%/12) would accrue each month, and
the annual rate would be effectively somewhat greater
than 12%.
The more frequent the compounding the greater the
effective interest.
55. The effect of more frequent
compounding can be easily determined.
Let r be the nominal, annual interest rate and M the
number of compounding periods per year. We can
find, i, the effective interest by using the formula
below.
56. Effective Interest Rate: Some Examples
Compounding Frequency Number of Compounding
Periods per year, M
Annually 1
Semi Annually 2
Quarterly 4
Bi Monthly 6
Monthly 12
Daily 365
For a 6% nominal interest rate, compounded monthly,
the effective interest is?
I = 6.18%
57. Active Learning 19
Suppose that $100 amount is invested for 10 years at
a nominal interest rate of 18% compounded
quarterly, How much is the worth at the end of the
10th year?
For a 7% nominal rate, compounded monthly, the
effective interest is.
F = $100(F/P, 19.25%, 10) = Find F
58. Interest can be compounded continuously.
Interest is typically compounded at the end of
discrete periods.
In most companies cash is always flowing, and
should be immediately put to use.
We can allow compounding to occur continuously
throughout the period.
The effect of this compared to discrete
compounding is small in most cases.
59. Continuous Compounding
Continuous compounding is the mathematical limit
that compound interest can reach. It is an extreme
case of compounding since most interest is
compounded on a monthly, quarterly
or semiannual basis. Hypothetically, with continuous
compounding, interest is calculated and added to the
account's balance every infinitesimally small instant.
While this is not possible in practice, the concept of
continuously compounded interest is important in
finance.
60. We can use the effective interest formula
to derive the interest factors: Continuous
Compounding
As the number of compounding periods gets
larger (M gets larger), we find that
61. Continuous compounding interest factors.
The other factors can be found from these. Please refer book
( F/P, r%, N ) = e r N
62.
63. Active Learning 20
You have $1000 to invest for two years. Your bank offers 5%
interest, compounded continuously for funds in a money
market account. Assuming no additional deposits or
withdrawals, how much money will be in that account at the
end of the two years?
F = $10,000 (F/P, 5%, 2) = $10,000*e (0.05) (2) = $11,052
If 5% were not compounded continuously, then the value of F
would have been
F = $10,000 (F/P, 5%, 2) = $10,000*(1.1025) = $11,025