TOPIC 1 STOICHIOMETRIC RELATIONSHIPS 1.3 REACTING MASSES AND VOLUMES.
Topic 2A: Interest-Time Relationships
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Transcript of Topic 2A: Interest-Time Relationships
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Dr Nebil Achour February 2013
CVC019 Topic 2 - Element 4
INTEREST-TIME RELATIONSHIPS
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Introduction
Purchasing power
Earning power
If put to work, it can produce more money if the right decision is made!
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because investment decisions extend into the future i.e.
there is an element of uncertainty attached to them!
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Introduction
Fact 1: A Pound received today has a greater value than a Pound received at some future time
Fact 2: The economic value of a sum depends on when it is received
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Aim
This lecture aims to
describe changes in the value of money over time;
investigate relationships between money, time, interest rates, payments etc.;
introduce equations for these relationships;
explain how investment decisions are affected by these
relationships; and
correspond with Element 4 in your hand-outs.
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Cash Flow - Definitions
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Regular payment at end of each
period Compound
amount of money after n periods of
interest i
Principal sum of money at the start
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Cash flow Question 1
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Question 1: If I invest P Pounds today, how much will my money return in n years?
S/P,n,i
R R R R R
0 1 2 3 4 5 Time (n periods) P S
Interest i
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Question 2: How much should I invest today to receive S Pounds money after n years?
S/P, n, i R R R R R
0 1 2 3 4 5 Time (n periods) P S
Interest i
S/P, n, i
Cash flow Question 2
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Question 3: If I invest P Pounds today, how much will my money return every year for n years?
R/P, n, i
S/P, n, i
P/S, n, i R R R R R
0 1 2 3 4 5 Time (n periods) P S
Interest i
Cash flow Question 3
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Question 4: How much should I invest today to receive R Pounds money every year for n years?
P/R, n, i
S/P, n, i
P/S, n, i
R/P, n, i
R R R R R
0 1 2 3 4 5 Time (n periods) P S
Interest i
Cash flow Question 4
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Question 5: If I invest R Pounds every year, how much will my money return after n years?
S/R, n, i
S/P, n, i
P/S, n, i
R/P, n, i
P/R, n, i
R R R R R
0 1 2 3 4 5 Time (n periods) P S
Interest i
Cash flow Question 5
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Question 6: How much should I invest every year to receive S Pounds after n years?
R/S, n, i
S/P, n, i
P/S, n, i
R/P, n, i
P/R, n, i
S/R, n, i
R R R R R
0 1 2 3 4 5 Time (n periods) P S
Interest i
Cash flow Question 6
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These are the base of the six equations
to manipulate cash flow. The following slides show how the six
main equations can be derived. This will help you understand where and
how they can be used.
It will also provide you with a better understanding of their limitations.
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R/S, n, i
S/P, n, i
P/S, n, i
R/P, n, i
P/R, n, i
S/R, n, i
Cash flow Questions summary
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EQUATION 1 Compound Amount of a single sum S
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Invest P= 100,000; n = 7 years; i=5%; What is the Compound Amount of P in 7 years? (i.e. S=?) 0 1 2 3 4 5 6 7 100,000 100,000(1+0.05) 100,000(1+0.05)(1+0.05) 100,000(1+0.05)(1+0.05)(1+0.05) .. 100,000(1+0.05)(1+0.05)(1+0.05)(1+0.05)(1+0.05)(1+0.05)(1+0.05) 100,000(1+0.05)3 100,000(1+0.05)7 If the interest rate is 5% per annum, at the end of year 7 we would therefore have 100,000(1 + 0.05)7.
Compound Amount of a single sum S
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This could be represented by the following mathematical expression where 'P' is the sum invested, 'S' is the sum at the end of the investment, 'n' is the number of periods and 'i' is the interest paid over one period. Equation 1 S = 100,000 x (1 + 0.05)7 S = 100,000 x 1.4071 S = 140,710
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Compound Amount of a single sum S
= (+ )
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EQUATION 2 Present Worth of a Single Sum P
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Present Worth of a Single Sum P
The present worth of a future sum is the sum of money that would have to be invested now to produce that future sum (Equation 2).
Equation 1 used to calculate compound amounts can be rearranged to give Equation 2 and thus
the present worth of a single sum.
P=?
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S = P(1 + i)n
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Equation 2
Example: How much should I invest now to produce a sum of 140,710 in 7 years? i=5%
interest = 5% S=140,710 0 1 2 3 4 5 6 7
= (+ ) P = 140,710(1 + 0.05)7 = 100,000
Present Worth of a Single Sum P
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Discounting process
100,000 (P) in year 0 is equivalent to
140,710 (S) in 7 years time (given an interest rate of 5%).
This is known as Discounting Process! It is often used as it provided a means of
converting future cash flows into a common base.
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EQUATION 3 Compound Amount S of a Uniform Series R
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Compound Amount S of a Uniform Series R
Consider a sum of money R that is to be invested at the end of each year for n years.
The sum invested at the end of year one will earn interest for n-1 years and after n years the sum will have become.
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R R R R R
0 1 2 3 4 5 R R(l + i)n-4
R(l + i)n-3
R(l + i)n-2 R(l + i)n-1
= (+ ) +(+ ) +(+ ) + (+ ) +
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If the above equation is multiplied by (1 + i) we have: If we subtract the first equation from the second we obtain:
= (+ ) +(+ ) +(+ ) + (+ ) + + = + + + + + + (+ )
= + = [(+ )] = (+ )
Compound Amount S of a Uniform Series R
Equation 3
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EQUATION 4 Sinking Fund Deposit Factor
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Sinking Fund Deposit Factor
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How much money should be deposited on a regular basis (R) to generate a specified capital sum (S) at the end of payments? Equation 3 S = R (1 + i)n1i
= (1 + )1 Equation 4
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EQUATION 5 Present Worth P of a uniform series R
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Present Worth P of a uniform series R
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R R R 0 1 2 3 Present worth
P1 P2 P3 P1+ P2+ P3 = Present worth (P) of a uniform series (R)
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Equation 2 (Present Worth for a single sum P) is true for this case! However, S has to be replaced by the Compound Amount of a uniform series (i.e. Equation 3).
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Present Worth P of a uniform series R
S = R (1 + i)n1i P = S(1 + i)n
= (+ )(+ ) Equation 5
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EQUATION 6 Capital Recovery Factor
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Capital Recovery Factor
If we have a lump sum P, which is invested now, we can determine the amount R that can be withdrawn on a regular basis leaving the investment exhausted at the end of a time period n.
The capital recovery factor R is the inverse of the
present worth factor P for a uniform series (i.e. Eq.5).
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P = R (1 + i)n1i(1 + i)n R = P i(1 + i)n(1 + )1
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Calculating Present Worth Factor (PWF)
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Calculating PW Factors (PWFs)
The first step is to transfer the problem into a visual form such as a cash flow diagram.
The next step is to determine from the diagram which factors are to be calculated using the relevant equations.
The final step is to calculate and compare the answers.
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Example: An excavator is purchased for 900,000 and is expected
to yield an income of 600,000 per annum. The annual running costs should be 100,000 per year. It is intended to keep this machine for 5 years when it will be sold for 200,000. Produce a cash flow diagram for the life of this excavator. Interest rate i=10%.
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Calculating PW Factors (PWFs)
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Cash Flow Diagram (Figure 4.6)
Positive (income) 200,000 600,000 600,000 600,000 600,000 600,000 0 1 2 3 4 5 100,000 100,000 100,000 100,000 100,000 900,000 Negative (expenditure)
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S
P
R
R R R R R
R R R R
Calculating PW Factors (PWFs)
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Period Income Expenditure Cash Flow PWF PW0 900000 -9000001 600000 100000 5000002 600000 100000 5000003 600000 100000 5000004 600000 100000 5000005 800000 100000 700000
Total
Cash Flow = Income- Expenditure
P = S(1 + i)n = S 1(1 + i)n Eq. 2 ?
Calculating PW Factors (PWFs)
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Period Income Expenditure Cash Flow PWF PW0 900000 -900000 1.000001 600000 100000 500000 0.909092 600000 100000 500000 0.826453 600000 100000 500000 0.751314 600000 100000 500000 0.683015 800000 100000 700000 0.62092
Total
Calculating PW Factors (PWFs)
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Period Income Expenditure Cash Flow PWF NPW0 900000 -900000 1.00000 -9000001 600000 100000 500000 0.90909 454545.52 600000 100000 500000 0.82645 413223.13 600000 100000 500000 0.75131 375657.44 600000 100000 500000 0.68301 341506.75 800000 100000 700000 0.62092 434644.9
Total
Calculating Net Present Worth (NPW)
NPW = PWF * Cash Flow
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Period Income Expenditure Cash Flow PWF NPW0 900000 -900000 1.00000 -9000001 600000 100000 500000 0.90909 454545.52 600000 100000 500000 0.82645 413223.13 600000 100000 500000 0.75131 375657.44 600000 100000 500000 0.68301 341506.75 800000 100000 700000 0.62092 434644.9
Total 1119578
Calculating Net Present Worth (NPW)
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Terminology
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The following terms are interchangeable: Net Present Cost (NPC) Net Present Worth (NPW) Net Present Value (NPV)
CVC019 Topic 2 - Element 4IntroductionIntroductionAimCash Flow - DefinitionsCash flow Question 1Cash flow Question 2Cash flow Question 3Cash flow Question 4Cash flow Question 5Cash flow Question 6Cash flow Questions summaryEQUATION 1Compound Amount of a single sum SCompound Amount of a single sum SEQUATION 2Present Worth of a Single Sum P Present Worth of a Single Sum P Discounting processEQUATION 3Compound Amount S of a Uniform Series RCompound Amount S of a Uniform Series REQUATION 4Sinking Fund Deposit FactorEQUATION 5Present Worth P of a uniform series RPresent Worth P of a uniform series REQUATION 6Capital Recovery FactorSlide Number 30Calculating PW Factors (PWFs)Calculating PW Factors (PWFs)Cash Flow Diagram (Figure 4.6)Calculating PW Factors (PWFs)Slide Number 35Slide Number 36Slide Number 37Terminology