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

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)

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

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

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

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

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

Topics Today Cash Flow Diagrams Equivalent Issues Engineer Decision

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

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

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

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

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

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

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

Cash Flow Diagrams P-Pattern “present” F-Pattern “future” A-Pattern 1 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”

$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)

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

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 (1 + 0.08) (1 + 0.08) = $1166 1 2 $1000 $1166

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.

An Example of Cash Flow Diagram Cash flow for this problem is: End of year Cash flow 0 +$1000 1 -$580 (-$500 - $80) 2 -$540 (-$500 - $40)

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

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

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

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

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

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.

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?

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

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

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

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

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

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

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

Modeling Real World Information for decision making Analysis The Model