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- 1. Lecture # 2 Time Value of Money – Interest theory 1-1 Dr. A. Alim
- 2. Time Value of Money The “time value” of money is the most important concept in engineering economy All firms make use of investment of funds Investments are expected to earn a return Money possesses a “time value” 1-2Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 3. Equivalence Different sums of money at different times may be equal in economic value 0 1 $100 now $106 one year from now Interest rate = 6% per year $100 now is said to be equivalent to $106 one year from now, if the $100 is invested at the interest rate of 6% per year. 1-3Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 4. Interest Rate and Rate of Return Interest – the manifestation of the time value of money Rental fee that one pays to use someone else’s money Difference between an ending amount of money and a beginning amount of money Interest rate (%) = interest accrued per time unit x 100% original amount 1-4Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 5. Rate of Return Interest earned over a period of time is expressed as a percentage of the original amount, specifically; interest accrued per time unit Rate of return (%) = x 100% original amount Borrower’s perspective – interest rate paid Lender’s perspective – interest rate earned 1-5Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 6. Simple and Compound Interest Simple Interest: Interest = (principal)(number of periods)(interest rate) Future value (F) = principal(P) + P x n x i Compound Interest: Interest earns interest on interest Compounds over time Interest = (principal + all accrued interest) (interest rate) Future value (F) = principal(P) x (1 + i)n 1-6Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 7. Terminology and Symbols • Specific symbols and their respective definitions have been developed for use in engineering economy • Symbols tend to be standard in most engineering economy texts world-wide • Mastery of the symbols and their respective meanings is most important in understanding of the subsequent material! 1-7Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 8. Terminology and Symbols • P = value or amount of money at a time designated as the present or time 0. • Also P is referred to as present worth (PW), present value (PV), net present value (NPV), discounted cash flow (DCF), and capitalized cost (CC); dollars 1-8Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 9. Terminology and Symbols • F = value or amount of money at some future time. • Also F is called future worth (FW) and future value (FV); dollars 1-9Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 10. Terminology and Symbols • A = series of consecutive, equal, end-of-period amounts of money. • Also A is called the annual worth (AW) , equivalent uniform annual worth (EUAW); dollars per year, dollars per month • n = number of interest periods; years, months, days 1-10Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 11. Terminology and Symbols • i = interest rate or rate of return per time period; percent per year, percent per month • t = time, stated in periods; years, months, days, etc 1-11Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 12. Cash Flows: Their Estimation and Diagramming Definition of terms Cash Inflows - amount of funds flowing into the firm Cash Outflows – amount of funds flowing out of the firm Net Cash Flow equals cash inflows – cash outflows Assumption for analysis – end of period Funds flow at the end of a given (interest) period 1-12Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 13. Cash Flow Diagrams A typical cash flow diagram might look like: 0 1 2 … … … n-1 n 1. Draw a time line One time period 0 1 2 … … … n-1 n 2. Show the cash flows Cash flows are shown as directed arrows (+ for up or – for down) --- (+) inflow; (-) outflow Always assume end-of-period cash flows! 1-13Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 14. IMPORTANT • Cash flow diagrams are not limited to P, F, and/or A. For example, this could be a very real diagram: P F 0 1 2 … … n-1 n 1-14Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 15. P and F • The symbols P and F represent one-time occurrences: • Specifically: $P $F 0 1 2 … … n-1 n 1-15Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 16. P and F • It should be clear that a present value P represents a single sum of money at some time prior to a future value F • This is an important basic point to remember 1-16Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 17. Annual Amounts • It is important to note that the symbol A always represents a uniform mount (i.e., the same amount each period) that extends through consecutive interest periods. 1-17Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 18. Annual Amounts • Cash Flow diagram for annual amounts might look like the following: ………… $A $A $A $A $A A = equal, end of period cash flow amounts $P 0 1 2 3 .. N-1 N 1-18Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 19. Remember ! A = equal payments, end of each period F = single payment, end of last period P = single payment, beginning of first period 1-19Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 20. Interest Rate – i% per period • The interest rate i is assumed to be a compound rate, unless specifically stated as “simple interest” • The rate i is expressed in percent per interest period, for example, 12% per year. 1-20Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 21. Standard Notation for Interest Factors Standard notation has been adopted to represent the various interest factors Consists of two cash flow symbols, the interest rate, and the number of time periods General form: (X/Y,i%,n) X represents what is unknown Y represents what is known i and n represent input parameters; can be known or unknown depending upon the problem 1-21Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 22. Notation - continued Example: (F/P,6%,20) is read as: To find F, given P when the interest rate is 6% and the number of time periods equals 20. In problem formulation, the standard notation is often used. Example : F = P (F/P,6%,20) 1-22Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 23. Notation - continued There are 3 sources to determine the values of these factors: Tables at the back of many books provide tabulations of common values for i% and n. Using EXCEL functions. Calculating the factors from formulas. 1-23Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 24. 1-24Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 25. Introduction To Solution By Computer Application of Microsoft’s Excel© spreadsheet program Good review in Appendix “A”, Blank’s Book. Excel financial functions Present Value P: =PV(i%,n,A,F) e.g. =PV(i%,n,A) or PV(i%,n,,F) Future Value F: =FV(i%,n,A,P) Equal, periodic value A: =PMT(i%,n,P,F) No. of periods: =NPER((i%,A,P,F) Compound interest rate: =RATE(n,A,P,F) Compound interest rate: =IRR(first_cell:last_cell) Present value of a series: =NPV(i%,second_cell:last_cell) + first_cell 1-25Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 26. 1-26Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved Calculating factors From formulas:
- 27. Example: 1-27Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved P F 5 years i = 8% Cash flow diagram from the point of view of the graduate.
- 28. Example: P = 10000 dollars i = 8% n = 5 years 1) Using tables: From Blank's book table 12, for 8% interest rate and 5 years, we find: F/P = 1.4693 then F= $10000 x 1.4693 = $14,693 2) Using formula: F/P = (1+i)^5 = 1.46932808 then F = $14,693.28 3) Using EXCEL function FV EXCEL assigns a negative value to cash outflow and a positive value to cash inflow P in this case is a positive cash flow, hence F calculated will be negative. F = ($14,693.28) 1-28Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 29. Example: 1-29Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 30. Summary: Compounding Factors 1. Single-Payment compound amount factor (F from P) 2. Single-Payment present worth factor (P from F) 3. Uniform-series present worth factor (P from A) 4. Capital recovery factor (A from P) 5. Uniform-series compound amount factor (F from A) 6. Sinking fund factor (A from F) Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 1-30 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 31. Single-Payment Factors(F/P and P/F) Objective: Derive factors to determine the present or future worth of a cash flow Cash Flow Diagram – basic format 0 1 2 3 n-1 n P0 Fn i% / period P0 = Fn1/(1+i)n →(P/F,i%,n) factor: Excel: =PV(i%,n,,F) Fn = P0(1+i)n →(F/P,i%,n) factor: Excel: =FV(i%,n,,P) Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 1-31 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 32. Uniform-Series: Present Worth Factor (P/A) and Capital Recovery Factor(A/P) Cash flow profile for P/A factor . . . . 0 1 2 3 n-2 n-1 n $A per interest period i% per interest period Required: To find P given A Cash flows are equal, uninterrupted and flow at the end of each interest period Find P Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 1-32 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 33. (P/A) and (A/P) Factor Formulas (1 ) 1 0 (1 ) n n i P A for i i i (1 ) (1 ) 1 n n i i A P i (P/A,i%,n) factor Excel: =PV(i%,n,A) (A/P,i%,n) factor Excel: =PMT(i%,n,P) Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 1-33 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 34. Sinking Fund Factor and Uniform Series Compound Amount Factor (A/F and F/A) Cash flow diagram for (A/F) factor Start with what has already been developed 1 (1 ) (1 ) (1 ) 1 n n n i i A F i i . . . . 0 1 2 3 n-2 n-1 n A=? per interest period i% per interest period F = given Find A, given F (1 ) 1n i A F i Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 1-34 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 35. (F/A) factor from (A/F) Given: Solve for F in terms of A to yield (1 ) 1n i A F i (1 ) 1n i F A i (A/F,i%,n) factor Excel: =PMT(i%,n,,F) (F/A,i%,n) factor Excel: =FV(i%,n,A) Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 1-35 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 36. Minimum Attractive Rate of Return Investors expect to earn a return on their investment (commitment of funds) over time A profitable investment should earn (return) funds in excess of the investment amounts Economic projects should earn a reasonable return, which is termed: MARR – Minimum attractive rate of return Also termed the “hurdle” rate for an investment 1-36Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 37. The MARR The MARR is established by the financial managers of the firm The MARR is expressed as a percent value Most, if not all, projects should earn at a rate equal to or greater than the established MARR MARR is set based upon: The cost of all types of capital Allowance for risk 1-37Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 38. Types of Financing Equity Financing – the firm uses funds either from retained earnings, new stock issues, or owner’s infusion of money Debt Financing – the firm borrows funds from outside sources The cost of debt financing = the interest rate charged on the debt (loan) amounts Weighted average cost of capital (WACC) = Σ Xi (int. rate)i The MARR is approximated from the weighted average cost of all sources of capital to the firm A firm’s ROR > MARR > WACC 1-38Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 39. Graphical Presentation: MARR 0% ROR - % MARR - % Safe Investment e.g. bank WACC - % Acceptable range for new projects 1-39Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
- 40. 1-40 Source: Plant design and economics for chemical engineers, 5th edition By M.S. Peters et. al., McGraw Hill 2005. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 7th Edition, 2012 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved

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