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Dr Nebil Achour February 2013 CVC019 Topic 2 - Element 4 INTEREST-TIME RELATIONSHIPS
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Topic 2A: Interest-Time Relationships

Sep 26, 2015

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Project Management Lecture Note
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  • Dr Nebil Achour February 2013

    CVC019 Topic 2 - Element 4

    INTEREST-TIME RELATIONSHIPS

  • Introduction

    Purchasing power

    Earning power

    If put to work, it can produce more money if the right decision is made!

    2

    because investment decisions extend into the future i.e.

    there is an element of uncertainty attached to them!

  • 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

    3

  • 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.

    4

  • Cash Flow - Definitions

    5

    Regular payment at end of each

    period Compound

    amount of money after n periods of

    interest i

    Principal sum of money at the start

  • Cash flow Question 1

    6

    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

  • 7

    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

  • 8

    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

  • 9

    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

  • 10

    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

  • 11

    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

  • 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.

    12

    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

  • EQUATION 1 Compound Amount of a single sum S

    13

  • 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

    14

  • 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

    15

    Compound Amount of a single sum S

    = (+ )

  • EQUATION 2 Present Worth of a Single Sum P

    16

  • 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=?

    17

    S = P(1 + i)n

  • 18

    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

  • 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.

    19

  • EQUATION 3 Compound Amount S of a Uniform Series R

    20

  • 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.

    21

    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

    = (+ ) +(+ ) +(+ ) + (+ ) +

  • 22

    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

  • EQUATION 4 Sinking Fund Deposit Factor

    23

  • Sinking Fund Deposit Factor

    24

    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

  • EQUATION 5 Present Worth P of a uniform series R

    25

  • Present Worth P of a uniform series R

    26

    R R R 0 1 2 3 Present worth

    P1 P2 P3 P1+ P2+ P3 = Present worth (P) of a uniform series (R)

  • 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).

    27

    Present Worth P of a uniform series R

    S = R (1 + i)n1i P = S(1 + i)n

    = (+ )(+ ) Equation 5

  • EQUATION 6 Capital Recovery Factor

    28

  • 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).

    29

    P = R (1 + i)n1i(1 + i)n R = P i(1 + i)n(1 + )1

  • Calculating Present Worth Factor (PWF)

    30

  • 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.

    31

  • 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%.

    32

    Calculating PW Factors (PWFs)

  • 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)

    33

    S

    P

    R

    R R R R R

    R R R R

    Calculating PW Factors (PWFs)

  • 34

    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)

  • 35

    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)

  • 36

    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

  • 37

    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)

  • Terminology

    38

    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