college of biomedical engineering and applied...
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College Of Biomedical Engineering And Applied Sciences
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Time value of money
How Time and Interest Affect Money? The same amount of money flowing at different time has
different value. That is, a Rupee in hand today worth more than a Rupee to
be received in next period. The concept is that money has time value: the same
amount of money flowing in or out of the business at two different points of time has different value.
The flowing of money in a business is given by cash flow
diagram
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Cash Flow
A payment or receipt of cash represented with amount and flow direction( in and out) in a business
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Cash Out Flow
A payment, or disbursement, of cash for
expenses, investments, and so on.
Cash In Flow A receipt of cash from an investment, an
employer, or other sources.
Type Of Cash Flow
Even Cash Flow.
Un-even Cash Flow.
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Even Cash Flow
Cash flow at different time period has same value.
Year 4
0
1 3 5 6 n-1 n 2
A
Uneven Cash Flow
Cash flow at different time period has different value.
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0
1 2 3 4 5 6 7 8 9
Time value of money
Value of money can be added to the principal value in two ways
No compounding of any principal value and interested amount Simply interest is added to the principal value once a year and continued for a years.
Money is compounded with suitable time period and at the end of each period new principal amount is introduced with addition of interested amount. In engineering analysis and project evaluation as well as business propose compounding principal is used .
Our main focus is on compounding method of value addition
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Future Value:
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The value of present cash flow or series of cash
flows at any future date compounded with a
suitable rate.
Present Value:
The value of future cash flow or series of cash
flows at present (or, today’s date ) discounted
with a suitable rate.
( ) n F=P 1+i
( ) n F
P= 1+i
Time value of money
Method of value addition 1. Compounding process
The process of adding money to the principal value is called compounding process. In this process money is added to the principal value for time N years with interest rate of I% . In a simple words we can say calculating of future value of present money with certain interest rate for N no of year from now is compounding process
2. Discounting process
It is the process of deducting money from supposed future value so as to get its present worth at certain interest rate for N years from now. In another word the process of calculating present value of future amount with certain interest rate I% for N years from now is called discounting process 8
Contd………
Future
Present
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Discounting process
Compounding process
Annuity:
( )
( )
( )
( )
n n
n n
i 1+ i i 1+ iA = P A/P
1+ i -1 1+ i -1
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A series of equal / constant / fixed amount of
cash flow in a regular time interval over a given
period of time. The equivalence in present
value, future value and annuity is given as
follow
-1 A
n (1+i)
i F *
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( ) F
n 1+i
i A
- 1 *
( ) P
n 1+i
i A
- 1 *
( 1+i ) n
Factor: What is Factor?
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•A factor is defined as the parameter that
establishes an equivalence between two
cash flows or among several Cash flows.
Factor: Type? • Single Payment Factors:
Single Payment Compound Amount Factor
Single Payment Present Worth Factor
• Uniform Series Present Worth Factor.
• Capital Recovery Factor.
• Sinking Fund Factor.
• Uniform Series Compound Amount Factor
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Single Payment Factor
•It is the process of calculating the present
and future worth of given amount for N no
of years with Interest rate I% •The process of finding present value or future value of given single amount for a given period N at Interest rate I% is called single payment present worth factor written as (P/F,I%,N)
•The process of calculating future value of present amount for a given period N with interest rate of I% is called single payment compound factor (F/P,I%,N) find F with given P,I and N
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Single Payment Compound Amount Factor
Year
0 1
n-1
2
P = Given
F ?
given i n
( )
( ) ( )
( )
......................................................
......................................................
.......................
1
2
2 1 1 1
n
n
F = P + Pi = P 1+ i
F = F + F i = F 1+ i = P 1+ i
F = = P 1+ i
( )n
F/P = 1+ i
SPCAF = Single
payment compound
amount factor.
SPCAF
Single Payment Factor
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Single Payment Factor
P = ?
F Given
Year
0 1
n-1
2
given i n
( )n
1P/F =
1+ i
SPPWF
SPPWF = Single
payment present
worth factor.
Single Payment Present Worth Factor
If you deposit 12000 compounding annually for 17 years what will be in your account with 12% interest at the end
For abroad study your father would like to deposit some amount of money in your account. After 10 year you need total sum of 2400000 then what will be required amount to deposit now with interest 7%
You are asked to convert 2000 to 32000 by 5 years . What will be the required interest rate .
Two bank gives you a offer in cash deposit . Bank A offer 15% interest on your deposit and bank B offer 18% interest in your deposit. At the end of 12th year you receive equal amount from the both bank. What more amount that you deposited in bank A
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Questions
Uniform Series Present Worth Factor
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Year
0
1 3 4 5 6 n-1 n 2
?P
A = Given
given i
When uniform value of annuity is given along with annual interest rate for the given period of time then we can find the present equivalence value of whole cash distribution by using this factor represented as (P/A,I,N) read as find P for given A,I, N
Uniform Series Present Worth Factor
( ) ( ) ( )
1 2 n
1 1 1P = A + A + ... + A
1+ i 1+ i 1+ i
( )
( )
( )
( )
n n
n n
1+ i -1 1+ i -1P = A P/A
i 1+ i i 1+ i
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USPWF USPWF= Uniform Series
Present Worth Factor.
questions
If a women sell his properties and deposit the whole amount in saving account with interest rate of 10% at the end of each year she wish to withdraw 200000 for her expense. At the end of 11th year she has just finish her deposit. How much she earn on selling her properties.
Ram got an offer to receive 1ooooo0 in 10 installment (payment made at the end of year) every year he got Rs 100000. what is the present value of amount he receive with the rate of 5%
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Capital Recovery Factor This is the factor responsible to find the annuity as a
payback for N no of years of present amount of money.
If we need to withdraw equal amount of money for N no of Years from deposited amount or want to get return of whole investment by N no of years with certain interest rate of I% then we have to use capital recovery factor
In this case we are trying to get return of whole amount that we deposited or invested so it is termed as capital recovery factor
Functionally it is written as (A/P,I%,N)
read as find annuity value for given present worth with interest rate of I and time N years
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Capital Recovery Factor
( )
( )
( )
( )
n n
n n
i 1+ i i 1+ iA = P A/P
1+ i -1 1+ i -1
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Year
0
1 3 4 5 6 n-1 n 2
P Given
?A =
given i
CRF CRF= Capital
Recovery Factor.
Questions you are allowed to spend from your dad account for
your study expenses. He deposited Rs 250000 with interest of 13%. In four years how much you can withdraw each year
Samundra deposit 120000 in a saving account . He had to spend Rs 45000 annually for his household. For how many years he will receive the money from bank if he got interest of 10% compounding annually.
Ramila has to withdraw 65000 annually for her expense so her husband deposit 300000 on her account at the end of fourth year her balance is null. How much interest she was paid
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Sinking Fund Factor
It is factor which can established a equivalence between future value and uniformly distributed annuity for a span of time .
If we need the certain amount in future and has to deposit from now to get the required amount then we have to use sinking fund factor.
Generally sinking factor is used to calculate annuity amount for required future amount
Functionally it is written as (A/F,I%,N)
Read as find annuity with given future value ,interest rate for N no of years
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Sinking Fund Factor
F = Given
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?A =
Year 0
1 3 4 5 6 n-1 n 2
given i
A / F
SFF
SFF = Sinking
fund factor.
Sinking Fund Factor
( )
( )
( )
( )
( )
( )
n
n
n
n n
n
i 1+ iA = P
1+ i -1
i 1+ i1= F
1+ i 1+ i -1
i= F
1+ i -1
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questions If I had to deposit 100000 yearly compounding
annually for 13 year what will be the amount I am gone receive at the end if I was paid 8% interest.
To get Foard car after five year Shyam has decided to pay from now. If the car cost 4500000 at that time and he had deposited 200000 per year at the rate of 17%. How much he has to pay more on buying date to get car with full payment
For your son’s higher study you wish to deposit some amount from now so open a account and deposit 15000 yearly . Your son enrolled to university on his 22 year. How much you have deposited till the date for his study. Rate of return is 5% on your saving.
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This is the factor which gives the future value of uniformly distributed annuity for a span of time
If you deposit equal amount of cash per year in your bank account and wish to know how much you have in your account after N no of years with given interest I% from bank then we need to use uniform series compound amount factor to get exact amount in our account.
This is the method to established a equivalence between future amount and annuity distribution for a span of time.
Functionally it is written as (F/A,I%,N) spoken as find Future value of equally distributed annuity A for N years with interest rate of I%
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Uniform Series Compound Amount Factor
Uniform Series Compound Amount Factor
( )F A
n1+ i -1
i
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Year 0
1 3 4 5 6 n-1 n 2
?F =
A = Given
given i
USCAF = Uniform
Series Compound
Amount factor.
F/A
USCAF
Questions If I had deposited 4500 per month in my account and
given interest of 18% then what will be my projected balance in 10 year if the compounding of money is once a year.
Rosika has to pay her loan of 120000 with interest 23% compounding annually and she can pay 20000 per year. How long it takes her to clear her loan
Safal has to pay 300000 in 1oth year from now to a bank so he wish to collect money from now to make easier in future so deposited 70000 . What will be the minimum rate of return on his saving
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Uniform Series Compound Amount Factor
( )F/A,i,n
F/A
( )F/A,i,n
A/F
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F/A & A/F Factors: Notation and Equations
Factor Notation
Name
Find/Given
Factor
Formula
Standard Notations
Equation
Excel
Function
Uniform Series
Compound amount
Sinking Fund ( )A/F, i,n
( )F = A F/A, i,n
( )A = F A/F, i,n
FV(i%,n,A)
PMT(i%,n, ,F)
Arithmetic Gradient Factor:
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Increase / decrease by a constant amount.
1 2 3 4 5 6 7 8 9 0
A G [ ] - n
(1+i) n -1 i
1
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( ) F
n 1+i
i
G - 1
i [ ] - Gn
i
( ) P
n 1+i
i
G - 1
i [ ] -
n
( 1+i ) n (1+i)
n
Geometric Gradient Factor:
Increase / decrease by a constant percentage.
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1 2 3 4 n-1 n
0
A(1+g) A
A(1+g) 2
A(1+g) 3
A(1+g) n-2
A(1+g) n-1
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( ) P
n 1+g 1-
A [ ] g
1+i
(i – g) If, g ≠ i
P nA g (1+i)
If, g = i
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Fig. : Cash Flow Diagram
Year
0 1
3 4 5 6 7 8
2
' ?AP ?AP
?TP
$500A
8%i
Po=$5000
Example 1: An engineering technology group just
purchased New Cad software for $5000 now and annual
payments of $500 Per year for 6 years starting 3 years
from now for Annual Upgrades. What is the present worth
of the payments if the interest Rate is 8% per year?
Solution:
( )
( )
'
A
'
A A
P = $500 P/A,8%,6
P = P P/F,8%,2
T 0 AP = P + P
= 5000 + 500(P/A,8%,6)(P/F,8%,2)
= 5000 + 500(4.6299)(0.8573)
= $6981.60
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Hand Calculation:
The total present worth PA is determined by adding and the
initial payment P0 in year 0
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Excel Calculation:
Go to Excel
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Example 2: An engineering company in Wyoming that own
50 hectares of valuable lands has decided to lease the
mineral Rights to a mining company. The primary
objective is to obtain long term income to finance ongoing
projects 6 and 16 years from the present time. The
engineering company makes a proposal to the mining
company that it pay $20000 per year for 20 years
beginning 1 year from now, plus $10,000 six years from
now and $15,000 sixteen years from now. If the mining
company wants to pay off its lease immediately, how
much should it pay now if the investment should make
16% per year?
Answer: P= $ 124075.00
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Fig. : Cash
Flow Diagram
P = ? 16% i
0 1 3 4 5 6 7 2
$10,000
Year 15
$15,000
A = $20,000
16 17 18 19 20
The total present worth:
Hand Calculation:
Solution:
= 20000(5.9288) + 10000(0.4104) + 15000 (0.0930)
P = P + P + P
= 20000(P/A,16%,20)+10000(P/F,16%,6)+15000(P/F,16%,16)
= $124075.00
T A 6 16
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Example 3 : Three contiguous counties in Florida have
agreed to pool tax resources, already designated for
county-maintained bridge refurbishment. At a recent
meeting, the county engineers estimated that a total of
$500,000 will be deposited at the end of net year into an
account for the repair of old and safety-questionable
bridges throughout the three-county area. Further, they
estimate that the deposits will increase by $100,000 per
year for only 9 years thereafter, then cease. Determine the
equivalent (a) Present Worth and (b) annual series
amounts if country funds earn interest at a rate of 5% per
year.
Answer: P= $ 7026.05’000 and A= $ 909.91’000
Factor:
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1 2 3 4 5 6 9 0
7 8 10
$500
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Example 4: Engineers at Sea World, a division of Busch
Gardens, Inc., have completed an innovation on an
existing water sports ride to make it more exciting. The
modification costs only $8000 and is expected to last six
years with a $1300 salvage value for the solenoid
mechanisms. The maintenance cost is expected to be
high at $ 1700 the first year, increasing by 11% per year
thereafter. Determine the equivalent present worth of the
modification and maintenance cost. The interest rate is
8% per year.
Answer: P= $ 17305.852
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Example 5:Gerri, an engineer at Fujitsu, Inc., has tracked
the average inspection cost on a robotics manufacturing
line for 8 years. Cost averages were steady at $100 per
completed unit for the first four years, but have increased
consistently by $50 per unit for each of the last four
years. Gerri plans to analyze the gradient increase using
the P/G factor. Where is the present worth located for the
gradient? What is the general relation used to calculate
total present worth in year 0? Calculate total present
worth at 8% interest rate.
Answer: P3= $ 767.89
Po= 767.89(P/A,8%,3)+100(P/A,8%,3)
Po= $ 867.26
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Example 6: Chemical engineers at a Coleman Industries
plant in the Midwest have determined that a small
amount of newly available chemical additive will increase
the water repellency of Coleman’s tent fabric by 20%. The
plant superintendent has arranged to purchase the
additive through a 5 year contract at $7000 per year,
starting 1 year from now. He expects the annual price to
increase by 12% per year there after for the next 18
years. Additionally, an initial investment of $35000 was
made now to prepare a site suitable for the contractor to
deliver the additive. Use i =15% to determine the total
present worth for all these cash flow.
Answer: P= $ 107652.18
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Example 7 : Assume that you are planning to invest
money at 7% per year as shown by the increasing
gradient in the figure. Further, you expect to withdraw
according to the decreasing gradient shown. Find the net
present worth and equivalent annual series for the entire
cash flow sequence.
Answer: Rs. 61267.96
Example 8 : Calculate the equivalent PW at i=15% using
uniform gradient formula.
End of year Cash Flow in Rs.
1
2
3
4
5
20000
19000
18000
17000
16000
Note: Requires Cash Flow.
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Example 9 : Professional Engineering, Inc., requires that
$500 per year be placed into a sinking fund account to
cover any unexpected major rework on field equipment.
In one case, $500 is deposited for 15 years and covered a
rework costing $10000 in year 15. What rate of return did
this practice provide to the company?
? i
0 1 3 4 5 6 7 2 Year
10 11 12 13 14 15
Fig. : Cash Flow Diagram
A=$500
$10000
Solution:
( )F A
n1+ i -1
i
10000 =
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500 [(1+i) -1]
i *
20i = (1+i) -1
i = (1+i) -1
20
Then; from hit and trial method, calculate the value of i.
Therefore, i = 4%
15
15
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Example 10 : How long will it take for $1000 to double if the
interest rate is 5% per year?
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Solution:
5% i
0 1 3 4 5 6 7 2 Year
n-2 n-1 n= ?
Fig. : Cash Flow Diagram P=$1000
F=$2000
F= P(F/P,5%,n)
2000=1000(1+i)
(1+0.05) = 2
Then; from hit and trial method, calculate the value of n.
Therefore, n = 14.2 years
n
n