1. Warm-Up Review Time Value of Money Calculation of Future Value
Calculation of Current Value
Simple interests and compound interests
Continuous compounding

2. Basics of Time Value of Money
Interest rate
reward for use of capital$
usually expressed in % per year
Simple Interest (self-study)
Only the principal earns interest
Interest amount =P • i • n
Future value = P + P • i • n = P (1 + i • n)

3. Basics of Time Value of Money
Compound Interest
Interest on interest
dependant on compounding period
(yearly, semi-annually, monthly)
For 2 years:
Future value = P ( 1+i) + i • P (1+i) = P (1+ i)2
For n years:
Future value = P (1+ i)n
see column 2 of interest tables

4. What is Time Value?
We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return
In other words, “a dollar received today is worth more than a dollar to be received tomorrow”
That is because today’s dollar can be invested so that we have more than one dollar tomorrow

5. Compound Interest
Note from the example that the future value is increasing at an increasing rate
In other words, the amount of interest earned each year is increasing
Year 1: $10
Year 2: $11
Year 3: $12.10
The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amountchange

6. Interest Formulation Simple Interest Compound Interest
After N periods, the total accumulated value F will grow to

7. Continuous Compounding
There is no reason why we need to stop increasing the compounding frequency at daily
We could compound every hour, minute, or second
We can also compound every instant (i.e., continuously):
Here, F is the future value, P is the present value, r is the annual rate of interest, t is the total number of years, and e is a constant

8. Topics Today
Cash Flow Diagrams
Equivalent Issues
Engineer Decision

9. Cash Flow- expenses and receipts
Engineering projects generally have economic consequences that occur over an extended period of time
For example, if an expensive piece of machinery is installed in a plant were brought on credit, the simple process of paying for it may take several years
Each project is described as cash receipts or expenses at different points in time

10. Categories of Cash Flows
The expenses and receipts due to engineering projects usually fall into one of the following categories:
First cost: expense to build or to buy and install
Operations and maintenance (O&M): annual expense, such as electricity, labor, and minor repairs
Salvage value: receipt at project termination for sale or transfer of the equipment (can be a salvage cost)
Revenues: annual receipts due to sale of products or services
Overhaul: major capital expenditure that occurs during the asset’s life

11. Examples of Cash Inflows & Outflows
Receipts from customers--operating activity
Loans made to other firms--investing activity
Dividend payments--financing activity
Payments to investing activity
Payments of taxes--operating activity
Slide 14.8

12. Types of Cash Flows Single cash flow Uniform series
Linear gradient series
Geometric gradient series
Irregular series

13. Cash Flow Diagrams
The costs and benefits of engineering projects over time are summarized on a cash flow diagram.
Cash flow diagram illustrates the size, sign, and timing of individual cash flows, and forms the basis for engineering economic analysis
Tool! To show expenses and receipts

14. Cash Flow Diagrams
Pictorial representation of engineering economic problem
incomes and expenditures
time period
interest rate

15. Cash Flow diagrams--How
A cash flow diagram is created by first drawing a segmented time-based horizontal line, divided into appropriate time unit. Each time when there is a cash flow, a vertical arrow is added  pointing down for costs and up for revenues or benefits. The cost flows are drawn to relative scale

16. Cash Flow Diagrams P-Pattern “present” F-Pattern “future” A-Pattern1 2 3 n
“present”
F-Pattern 1 2 3 n
“future”
A-Pattern 1 2 3 n
“annual”
G-Pattern 1 2 3 n
“gradient”

17. $13,000 is net positive cash flow Time (# of interest periods)
Cash Flow Diagrams
$15,000
Positive net Cash flow
(receipts)
$2000
$13,000 is net positive cash flow 1 2 3 4 5
Time (# of interest periods)
Negative net Cash Flow
(payments)

18. Single Cash Flow P=Present equivalent value A=Annual equivalent value
Compounding Process P
Discounting Process
P=Present equivalent value
A=Annual equivalent value
F= Future equivalent value

19. Example: Value and Interest
The “value” of money depends on the amount and when it is received or spent.
Example: What amount must be paid to settle a current debt of $1000 in two years at an interest rate of 8% ?
Solution: $1000 ( ) ( ) = $1166 1 2
$1000
$1166

20. An Example of Cash Flow Diagram
Boney (right) borrowed $1,000 from a bank at 8% interest. Two end-of-year payments: at the end of the first year, he will repay half of the $1000 principal plus the interest that is due. At the end of the second year, he will repay the remaining half plus the interest for the second year.

21. An Example of Cash Flow Diagram
Cash flow for this problem is:
End of year Cash flow
$1000
$580 (-$500 - $80)
$540 (-$500 - $40)

22. Time (# of interest periods)
Cash Flow Diagram
$1,000
Time (# of interest periods) 1 2
$540
$580

23. Uneven Payment Series Find the present worth of any uneven stream of
payments by calculating the present value of each
individual payment and summing the results
Future worth can then be calculated by using the
interest formula P5 P1 P6 P2 P3 P0 P4
Years

24. Equal Payment Series F N-1 1 2 3 N A A A A A A1 2 3 N A A A A A A
Subtracting two above equations from each other yields:

25. Linear Gradient Series
(N-1)G A1 2G G
Uniform Series 1 2
N-1 N
Composite Series: uniform series + linear gradient
Find P, given A1, G, I, N

26. Geometric Gradient Series
Particularly relevant to construction costs
Cash flows increase by a constant %(g); compound growth
Example: price changes due to inflation
Present Worth, Pn, of any Cash Flow An
A1(1+g)N-1
A1(1+g) A1
If i=g, then P=? 1 2 3
N-1 N
g > 0 P
Find P, given A1, g, i, N

27. Principal Uses of A Statement of Cash Flows
Evaluate a business’s ability to produce positive cash flows in the future.
Determine whether a company can satisfy its financial obligations.
Identify sources of differences between a business’s net income and its related (net) cash flow from revenue and expense transactions.
Analyze the impact on a business’s financial condition of its major investing and financing transactions.

28. Cash-Flow Data Can Be Used to Address
Will a company generate sufficient cash to retire a long-term debt that matures soon?
Why doesn’t a company’s record profits translate into positive cash flows?
Is a company likely to suspend (or increase) its dividend payments?
How does the composition of a company’s cash flows compare to that of its competitors?

29. Three Major Types of Business Activities
Operating activities: those transactions and events related to the production and delivery of goods and services.
Investing activities: include the making and collecting of loans, the acquisition and disposal of PP&E assets, and the purchase and sale of securities other than trading securities and cash equivalents.
Financing activities: involve obtaining cash from lenders and repaying those amounts and obtaining cash from investors and providing them with a return of and a return on their investments.
Slide 14.7

30. Economic Equivalence Which one would you prefer? $20,000 today
$50,000 ten years from now
$ 8,000 each year for the next ten years
We need to compare their economic worth!
Economic equivalence exists between cash flows if
they have the same economic effect.
Convert cash flows into an equivalent cash flow at
any point in time

31. Equivalence Principles
Use a common time basis
Equivalent cash flows are equivalent at any common point in time
Use the present time = present worth
Use some future point in time = future worth
Equivalence depends on interest rate
Changing the interest rate destroys equivalence

32. Equivalence Principles
Requires conversion of multiple payment cash flows to a single cash flow
Equivalence is maintained regardless of the point of view

33. The Decision Making Process
Define problem
Choose objectives
Identify alternatives
Evaluate consequences
Select the best
Implement
Audit results

34. Making Decisions Facts Market research Expert opinion Costs …
Preferences
Politics
People
Facts
Market research
Expert opinion
Costs …

35. Example: buying a car 57 Chevy 97 Neon 93 Mercedes Purchase $12,000
$7,000
$20,000
Operation
200/mth
50/mth
150/mth
Resale
$13,000
$5,000

36. Modeling
Real World
Information for decision making
Analysis
The Model