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# Lecture # 2 time value of money

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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