AMI 4203 Engin Econ Monbey Time Relationship Lecture 2 (1).pptx

Busitema University
Dept. of Agricultural Mechanisation &
Irrigation Engineering
AMI 4203: Engineering Economics:
Money Time Relationship
Prof. Titus Bitek WATMON PhD MSc BEng(Hons) MIET AMIMechE IAENG member
Email. btwatmon@gmail.com
Phone / WhatsApp: 0756 726 101

Lecture Outline
1) Money Time Relationship (Return on Capital Employed
and Interest & Interest Rate)
2) Single Payment Simple Interest Formulas
3) Single Payment Compound Interest Formulas (Interest
& Equivalence)
4) Time Value Money
5) Discounted Cash Flows
6) Concepts of Equivalence & Principles
Thursday, March 30, 2023 Prof. Titus Bitek Watmon 2

Money Time Relationship
Introduction
• A fundamental concept underlies much of the
material covered in the text:
• Money has a time value.
• In other words, Time Value of Money (TVOM).
• The value of a given sum of money depends on both
the amount of money and the point in time when the
money is received or paid.

Money Time Relationships
Return on Capital Employed
• Return on capital employed –
sometimes referred to as the
‘primary ratio’ –
• It is a financial ratio that is used
to measure profitability of a
company and the efficiency with
which it uses its capital.
• Put simply, it measures how
good a business is at generating
profits from capital.
Interest and Interest Rate
• Interest is the cost of borrowing
money, and the money you
earn from your savings.
• While Interest rates indicate
this cost or return as
a percentage of the amount you
are borrowing from the Bank or
lending (since you are “lending”
your savings to the bank)

Single payment simple interest formulas
• Simple interest, as opposed to
compound interest, is rare.
• With an investment that pays
simple interest,
• the amount of interest accumulated
each period depends solely on the
amount invested.
• The following single payment
equation applies to simple interest:
F = P (1 + i * n)
Example: If $100 is invested at 6% interest
(compound interest) for four (4) years, the amount
accumulated at the end of four years is:
F = P (1 + i) n = $100 (F/P,6%,4)
= $100 (1.262)
= $126.20
If, however, interest is simple rather than
compound, then the amount accumulated at the end
of four (4) years is only:
F = P (1 + i n) = $100 [ 1 + (0.06)(4) ]
= $100 (1.24)
= $124.00

Single Payment Compouond Interest Formulas
Interest and Equivalence
Given a present dollar amount P,
interest rate i % per year,
compounded annually, and a future
amount F that occurs n years after
the present, the relationship
between these terms is
F = P (1 + i) n
In equations, the interest rate i
must be in decimal form, not
percent.
Example: If $100 is invested at 6% interest per year,
compounded annually,
then the future value of this investment after 4 years is
F = P (1 + i ) n = $100 (1 + 0.06) 4
= $100 (1.06) 4
= $100 (1.2625)
= $126.25
• Solving the above equation for P yields:
P = F (1 + i ) -n
• The factors F/P and P/F are available in interest tables,
simplifying somewhat the calculations.
• The common notation for these factors is
(F/P,i%,n) = (1 + i ) n
(P/F,i%,n) = (1 + i ) – n

Time Value Money Concept
1. The concept of Time Value Money is based on the idealogy that the same amount of
money spent or received at different times
2. The money has different values because opportunities are available to invest the money
in various enterprises to produce a return over a period of time.
3. Therefore, financial institutions are willing to pay interest on deposits because they can
lend the money to the investors.
4. Based on the specified interest rate, deposits or investments will accumulate interest
over time.
5. As a result, the future value of a present amount of money will be larger than the
existing amount because of the accumulated interest.
6. the present value of a future amount of money to be received some time later would be
smaller than the indicated amount after making a discount for interest that could have
been accumulated if the money were available at present.
7. Hence, the interest rate plays a significant role in determining the time value of money.

Time Value Money Concept Cont…
For instance, if an amount of money is deposited in a bank,
interest accrues at regular time intervals.
1. Each time interval represents an interest period at the end of which the earned
interest on the original amount will be calculated according to a specified
interest rate.
2. The interest accrued in a single interest period is referred to as simple interest.
3. If the earned interest is not withdrawn at the end of an interest period and is
automatically redeposited with the original sum in the next interest period,
4. the interest thus accrued is referred to as compound interest.
5. Thus, the interest rate may be interpreted as the rate at which money increases
in value from present to future.
6. Conversely, the discount rate refers to the rate by which the value of money is
discounted from future to present.

This basic concept of the time value of money is
illustrated by simple examples.
Example 3.1
• Department of Agricultural Mechanization & Irrigation
Engineering of Busitema University has recently received
a bequest from an Alumina of $1 million to establish a
trust for providing annual scholarships in perpetuity.
• The trust fund is deposited at Stanbic bank that pays 7%
interest per annum, and
• only the annual interest will be spent for the designated
purpose.

What is the annual amount that is
available for scholarships?
• Since the interest will be withdrawn
at the end of the interest period,
the simple interest per annum is
($l,000,000)(0.07) = $70,000
• Note that at the end of each year,
the trust fund remains intact after
the interest is withdrawn from the
bank.
• Thus, $70,000 is available annually
for scholarships in perpetuity.
What is the total amount to be
received at the end of 2 years?
The principal and interest at the end
of each year for the 2 years are as
follows:
End of year 1
1,000 + (l,000)(0.08) = 1,080.00
End of year 2
1.080 + (l,080)(0.08) = 1,166.40
Hence, the total amount to be
received at the end of two years is
$1,166.40.

Discounted Cash Flows
• Another term used in financial circles is discounted cash
flows, often referred to as DCF.
• Originally, DCF referred to the process of using the TVOM
or discount rate to convert all future cash flows to a
present single sum equivalent.
• The fact that money has a time value changes how
mathematical operations involving money should be
performed.
• Simply stated, because money has a time value, one
should not add or subtract money unless it occurs at the
same point in time.

Discounted Cash Flows Cont…
In summary the four Discounted Cash Flows rules are:
1. Money has a time value.
2. Quantities of money cannot be added or subtracted
unless they occur at the same point in time.
3. To move money forward one time unit, multiply by 1
plus the discount or interest rate.
4. To move money backward one time unit, divide by 1
plus the discount or interest rate.

Discounted Cash Flows Cont….
Example 3.3 A manufacturer expects to receive $20,200 one
month after the shipment of goods to a retailer.
The manufacturer needs the cash and has arranged with a bank
for a loan of $20,000 upon the shipment of goods on the
condition that 40 Compound Interest Formulas the bank will
collect all of the $20,200 from the retailer a month later.
• What is the monthly interest rate charged by the bank?
• Since there is only one interest period, the interest rate per
month is
𝟐𝟎, 𝟐𝟎𝟎 − 𝟐𝟎, 𝟎𝟎𝟎
𝟐𝟎, 𝟎𝟎𝟎
= 𝟎. 𝟎𝟏 = 𝟏%
• This is also the discount rate by which the future sum of
$20,200 is discounted to a present value of $20,000

Concepts of Equivalence
1. To compare alternatives that
provide the same service over
extended periods of time when
interest is involved,
2. we must reduce them to an
equivalent basis that is
dependent on:
3. If two alternatives are
economically equivalent, then
they are equally desirable.
• Equivalence factors are needed in engineering
economy to make cash flows (CF) at different
points in time comparable.
• Example, a cash payment that has to be made
today cannot be compared directly to a cash flow
that must be made in 5 years
• Since the time value of money changes according
to:
 The interest rate,
 The amount of money involved,
 The timing of receipt or payment,
 The manner in which interest is compounded,
• We need a way to reduce CF's at different times to
an equivalent basis.
• Concepts of Equivalence factors allow us to do so.

Principles of Equivalence
1. Equivalent cash flows have the same economic value at the
same point in time.
2. Cash flows that are equivalent at one point in time are
equivalent at any point in time.
3. Conversion of a cash flow to its equivalent, at another point
in time must reflect the interest rate(s) in effect for each
period between the equivalent cash flows.
4. Equivalence between receipts and disbursements:
the interest rate that sets the receipts equivalent to the
disbursements is the actual interest rate (IRR).
5. Economic equivalence is established, in general, when we are
indifferent between a future payment, or series of payments,
and a present sum of money.

Cash Flow Diagram
• Cash flow is the difference between total cash coming in
(inflows or cash receipts) and total cash going out (outflows or
cash disbursements) for a given period of time.
• Cash flow provides a means for planning the most
effective use of your cash.
• A cash flow diagram is a picture of a financial problem
that shows all cash inflows and outflows plotted along a
horizontal time line.
• It is used to visualize cash flow:
 individual cash flows are presented as vertical arrows along a
horizontal time scale.

Notation and Cash Flow Diagrams
The following notation is utilized in formulas for compound
interest calculations:
i = effective interest rate per interest period
N = number of compounding periods
P = present sum of money; the equivalent value of one or
more cash flows at a reference point in time called present
F = future sum of money; the equivalent value of one or
more cash flows at a reference point in time called future
A = end-of-period cash flows (or equivalent end-of-period
values) in a uniform series continuing for a specified
number of periods,
starting at the end of the first period and continuing through
the last period

Notation and Cash Flow Diagrams Cont...
1. The Horizontal line is a time scale, with progression of time moving from left to
right.
2. The period (e.g., year, quarter, month) labels can be applied to intervals of time
rather than to points on the time scale.
3. The arrows signify cash flows and are placed at the end of the period.
4. If a distinction needs to be made, downward arrows represent expenses
(negative cash flows or cash outflows) and upward arrows represent receipts
(positive cash flows or cash inflows).
5. The cash flow diagram is dependent on the point of view.
6. The situations shown in the figure were based on the cash flows as seen by the
lender.
7. If the directions of all arrows had been reversed, the problem will have to be
diagrammed from borrower's viewpoint.

Cash Flow Diagram Conventions
Example:
• You are 40 years old and have accumulated $50,000
in your savings account.
• You can add $100 at the end of each month to your
account which pays an annual interest rate of 6%
compounded monthly.
• Will you be able to retire in 20 years?

Identifying TVM Problems
For a financial problem to be solved with time value of money formulas:
a) the periods must be of equal length
b) payments, if present, must all be equal and be all inflows or all
outflows
c) payments must all occur either at the beginning or end of a period
d) the interest rate cannot vary along the time line
• The time line is divided into 240 monthly periods (20 years times 12
payments per year) since the payments are made monthly and the interest
is also compounded monthly.
• The $50,000 that you have now (present value) is a negative cash outflow
since you will treat it as though you were just now depositing it into the
bank account.

Future Value of Annuities
• An annuity is a series of equal payments or receipts that occur
at evenly spaced intervals.
Example: Leases and Rental Payments.
• The payments or receipts occur at the end of each period for
an ordinary annuity
• while they occur at the beginning of each period for an annuity
due.
Future Value of an Ordinary Annuity
• The Future Value of an Ordinary Annuity (FVoa) is the value
that a stream of expected or promised future payments will
grow to after a given number of periods at a specific
compounded interest.

• The Future Value of an Ordinary Annuity could be solved by
calculating the future value of each individual payment in the
series using the future value formula and then summing the
results.
• A more direct formula is:
𝐅𝐕𝐨𝐚 = 𝐏𝐌𝐓[
((𝟏 + 𝐢)𝐧−𝟏)
𝐢
]
Where:
FVoa = Future Value of an Ordinary Annuity
PMT = Amount of each payment
i = Interest Rate Per Period
n = Number of Periods

Year 1 2 3 4 5
Begin 0 5,000.00 10,300.00 15,918.00 21,873.08
Interest 0 300.00 618.00 955.08 1,312.38
Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00
End of Year 5,000.00 10,300.00 15,918.00 21,873.08 28,185.46
Example 1: What amount will accumulate if we deposit $5,000 at the end of each
year for the next 5 years?
• Assume an interest of 6% compounded annually.
• PV = 5,000
i = 0.06
n = 5
FVoa = 5,000 [(1.3382255776 – 1) /0.06] = 5,000 (56.37092) = 28,185.46

Example 2
• In practical problems, you may need to calculate both the
future value of an annuity (a stream of future periodic
payments) and the future value of a single amount that you
have today:
• For example, you are 40 years old and have accumulated
$50,000 in your savings account.
• You can add $100 at the end of each month to your account
which pays an interest rate of 6% per year.
• Will you have enough money to retire in 20 years?

• You can treat this as the sum of two separate calculations:
• the future value of 240 monthly payments of $100 Plus
• the future value of the $50,000 now in your account.
PMT = $100 per period
i = 0.06 /12 = 0.005 Interest per period (6% annual rate / 12 payments per year)
n = 240 periods
FVoa = 100 [ (3.3102 - 1) /0.005 ] = 46,204.00
• Whereby,
FV = Future Value
PV = 50,000 Present value (the amount you have today)
i = 0.005 Interest per period
n = 240 Number of periods
• FVad = PV (1+i)240 = 50,000 (1.005)240 = 165,510.22
After 20 years you will have accumulated $211,714.22 (46,204.00 + 165,510.22).
Future Value of an Annuity Due (FVad)

Future Value of an Annuity Due
• The Future Value of an Annuity Due is identical to an
ordinary annuity except that each payment occurs at the
beginning of a period rather than at the end.
• Since each payment occurs one period earlier, we can
calculate the present value of an ordinary annuity and
then multiply the result by (1 + i ).
𝐅𝐕𝐚𝐝. = 𝐅𝐕𝐨𝐚 (𝟏 + 𝒊)
Where:
FVad = Future Value of an Annuity Due
FVoa = Future Value of an Ordinary Annuity
i = Interest Rate Per Period

Future Value of Accumulated Annuity
Example 3: What amount will accumulate if we deposit $5,000 at
the beginning of each year for the next 5 or 7 years?
Assume an interest of 6% compounded annually.
PV = 5,000, i = 0.06, n = 5 or 7 years
Year 1 2 3 4 5 6 7
Begin 0 5,300.00 10,918.00 16,873.08 23,185.46 29,876.59 36,969.19
Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00
Interest 300.00 618.00 955.08 1,312.38 1,691.13 2,092.60 2,518.15
Year
End
5,300.00 10,918.00 16,873.08 23,185.46 29,876.59 36,969.19 44,487.34
Fvoa = 28,185.46(1.06) = 29,876.59 or 44,487.34 respectively

The End
Thank You!

AMI 4203 Engin Econ Monbey Time Relationship Lecture 2 (1).pptx

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