Upon completion of this chapter you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity; time value of money and the application application of single‐payment interest calculations to single-and multiple-payment problems; time value of money or cash flow diagrams; application of compound, effective, nominal and continuous interest calculations; inflation and the time value of money.
Chapter 4 – Unit 1 Return on Investment and Single‐ Payment Calculations IET 350 Engineering Economics Learning Objectives – Chapter 4 Upon completion of this chapter you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity Time value of money and the application of single‐ Time value of money and the application of single payment interest calculations to single‐ and multiple‐ payment problems Time value of money or cash flow diagrams Application of compound, effective, nominal and continuous interest calculations Inflation and the time value of money Learning Objectives – Unit 1 Upon completion of this unit you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity Time value of money and the application of single‐ Time value of money and the application of single payment interest calculations to single‐ and multiple‐ payment problems Time value of money or cash flow diagrams Application of compound, effective, nominal and continuous interest calculations Inflation and the time value of money Introduction The concept of interest being charged is not new having been traced as far back as 1900 B.C. in Babylon Except when made illegal because of religious objections ( (1200‐1500 A.D.) interest has been a normal part of business ) p in the Western economies Return or return on investment (ROI) may include all types returns: interest on a savings account at the bank; interest on investments such as certificates of deposit; dividends from the ownership of stock; profit from selling stock; etc Return Return is measured by a percentage determined over a specified time period: ⎛ Earnings ⎞ Return = 100 × ⎜ ⎟ ⎝ Investment ⎠ Return can also be used to measure gain from purchase and sale of an investment: ⎛ Selling Price ‐ Purchase Price ⎞ Return = 100 × ⎜ ⎟ Investment ⎝ ⎠ Return Costs associated with an investment are typically deducted: ⎛ Revenue ‐ Expenses ⎞ Return = 100 × ⎜ ⎟ Investment ⎝ ⎠ Shareholders use various return ratios as key measures of a corporation’s financial performance: ⎛ Revenue ‐ Costs ⎞ Return on Assets = 100 × ⎜ ⎟ ⎝ Total Assets ⎠ Return on Investment Return on Investment (ROI) can be used as an analysis and evaluation tool for a multitude of applications. ROI is applied to decision making by using estimated returns ROI is applied to decision making by using estimated returns and the time value of money Refer to the list on page 130 of the Bowman text Estimated returns may use historical data Time value of money places all financial events at a common point or points in time using appropriate interest rates. Return on Investment ROI is an important benchmark used by management to determine if changes (investments) are producing the desired results. ROI determination and how it is used for analysis is important y p to all areas of an organization that employs team‐based decision making. Examples include: Making make vs. buy decisions must consider return implications of investing in equipment to make components Making investment decision on individual products by comparing ROI rather than profit Return on Investment ROI can be thought of as the financial efficiency of the organization. Efficiency is the measure of output over input: Efficiency = Output Input ROI is the measure of profit (output) over investment (input): Profit Return on Investment = Investment Return on Investment ROI can be determined using income before or after taxes Since taxes may represent a significant cost to the organization, after‐tax income is frequently used The affect of inflation should be included in ROI calculations The affect of inflation should be included in ROI calculations since the income associated with the investment will occur over time. Economic periods of high inflation can significantly affect ROI Economic conditions also affect the prevailing interest rate. Assuming the organization is borrowing funds to finance an investment, interest rates will also significantly affect ROI. 10 Microsoft Excel® Hints Excel® has several built‐in ROI functions: IRR(values, guess) → returns the internal rate of return for a series of cash flows XIRR(values dates guess) → returns the internal rate of XIRR(values, dates, guess) → returns the internal rate of return for a schedule of cash flows MIRR(values, finance_rate, reinvest_rate) → returns the internal rate of return for a series of periodic cash flows including both the cost of investment and interest on reinvestment of cash 11 End Unit 1 Material Go to Unit 2 Time Value of Money 12 Chapter 4 – Unit 2 Time Value of Money IET 350 Engineering Economics Learning Objectives – Unit 2 Upon completion of this unit you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity Time value of money and the application of single‐ Time value of money and the application of single payment interest calculations to single‐ and multiple‐ payment problems Time value of money or cash flow diagrams Application of compound, effective, nominal and continuous interest calculations Inflation and the time value of money 14 Time Value of Money The value of a dollar today is different from the value of a dollar in the future. Value of the dollar over time is affect by: Interest paid on borrowed money Interest received on money loaned Interest received on money loaned. Interest can be classified as: Simple – applied to the principle amount only Compound – applied to principle and interest Continuous Compounding – similar to compound interest except the number of time periods = infinity 15 Simple Interest Simple Interest is paid on the amount invested or charged on the amount loaned only. Interest is not applied to accumulated interest. Rarely used F = P × (1 + nisimple ) Where: F = Future Value ($) P = Principle ($) n = Time (#time periods) isimple = Simple Interest (% per year) 16 Simple Interest ‐ Example You loan your brother‐in‐law $2,000 at 5% simple interest per year to be repaid in 6 years How much will you receive if and when payment is made? F = Future Value = ? P = Principle = $2,000 n = Time = 6 years isimple = Simple Interest = 5% per year F = P × (1 + ni simple ) = $2,000 × (1 + 6 × 0.05) = $2,000 × (1 + 0.30) F = $2,600 Remember mathematical priorities require that you multiple before adding. For (1+6x0.05), first multiply 6 by 0.05, then add 1 17 Compound Interest Compound interest is paid on the amount invested and the accumulated interest Ö interest on interest. Compound interest is an exponential function of interest over time. Compounding period can be daily, monthly, quarterly, annually or any other period of time. Compound interest is the basis for engineering economic analysis and will be used through‐out IET 350 unless otherwise indicated 18 Compound Interest Compound interest is determined by the following equation: F = P × (1 + i)n Where: F = Future Value ($) P = Principle ($) n = Time (years) i = Interest (% per time period) 19 Compound Interest ‐ Example You loan your brother‐in‐law $2,000 at 5% compound interest per year to be repaid in 6 years How much will you receive if and when payment is made? F = P × (1 + i ) n F = Future Value = ? P = Principle = $2,000 = $2,000 × (1 + 0.05) n = Time = 6 years = $2,000 × (1.340096) i = Interest = 5% per year F = $2,680.19 Interest has increased $80.19 compared to simple interest. This amount is the interest paid on the accumulated interest Ư interest on interest. 20 Single Payments Single payment calculations cover situations where a single amount of money is borrowed or invested for some period of time with interest compounded Using the compound interest formula, if three of the four g p , factors are known, the fourth factor can be determined. For example: The amount to be invested today (P) can be found for a desired future amount (F) for a given interest and time F = P × (1 + i) n 21 Single Payment ‐ Example You have $5,000 to invest for 6 years at which time you need $7,500 for graduate school tuition. What interest rate compounded annually is required? F = P × (1 + i)n F = Future Value = $7,500 P = Principle = $5,000 n = Time = 6 years i = Interest = ? per year $7,500 = $5,000 × (1 + i)6 $7,500 = (1 + i)6 $5,000 1.50 = (1 + i)6 1.50 = 1 + i 1.069913 = 1 + i i = 0.069913 = 6.99% 22 Single Payment ‐ Example You have $5,000 to invest in a fund that pays 4.5% compounded annually. When will your investment grow to $7,500? F F t F = Future Value = $7,500 V l $7 500 P = Principle = $5,000 n = Time = ? years i = Interest = 4.5% per year You can use logarithms to simply a factor to a power: ln(xY )= Yln(x) F = P × (1 + i)n $7,500 = $5,000 × (1 + 0.045)n $7,500 = (1.045)n $5,000 1.50 = (1.045)n ln(1.50) = n× ln(1.045) 0.405465 = n(0.044017 ) n = 9.2 years 23 Solution Methods Several methods are available to determine solutions to compound interest problems including: Solving the compound interest equation with any calculator p p y F = P × (1 + i)n with exponential function capability Using interest tables Bowman page 139 24 Solution Methods Several methods are available to determine solutions to compound interest problems including: Using built‐in compound interest functions on a calculator (example from an HP‐39gs) on a calculator (example from an HP 39gs) Using built‐in spreadsheet functions in Excel such as the future value function FV( ) 25 Microsoft Excel® Hints Excel® has several built‐in functions useful for single‐payment: FV(rate, nper, pmt, pv, type) → returns the future value of an investment for a specified time period and interest rate PV(rate nper pmt fv type) → returns the present value of PV(rate, nper, pmt, fv, type) → returns the present value of an investment for a specified time period and interest rate RATE(nper, pmt, pv, fv, type, guess) → returns the interest rate per period for a cash flow NPER(rate, pmt, pv, fv, type) → returns the number of periods for a cash flow with a constant interest rate Note that the PV value must be entered as a negative number 26 Example Problem 4.1 Example Problem 4.1 Solution 27 End Unit 2 Material Additional Reading Ö Financial Functions: http://www.functionx.com/excel/Lesson12.htm Go to Unit 3 Cash Flow Diagrams 28 Chapter 4 – Unit 3 Cash Flow Diagrams IET 350 Engineering Economics Learning Objectives – Unit 3 Upon completion of this unit you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity Time value of money and the application of single Time value of money and the application of single‐ payment interest calculations to single‐ and multiple‐ payment problems Time value of money or cash flow diagrams Application of compound, effective, nominal and continuous interest calculations Inflation and the time value of money 30 10 Time Value of Money Diagrams Visual representations of complex problems are excellent tools to conceptualize the various components and parameters of the problem The time value of money f y diagram visually shows all cash flow g y and noncash financial transactions The time value of money diagram is also known as a cash flow diagram 31 Time Value of Money Diagrams Diagram conventions: Horizontal line represents the project or investment under analysis Horizontal line displays time with present time t0 at left end of line 32 Time Value of Money Diagrams Diagram conventions: Arrows coming from below the line with arrow head pointing toward the line represent cash flowing into the p j project The initial investment, principle, present value or present worth (P) is shown at t0 33 11 Time Value of Money Diagrams Diagram conventions: The future value (F) is shown at the right end of the horizontal line. Arrows coming from above the line with arrow head Arrows coming from above the line with arrow head pointing away from the line represent cash from or out of into the project 34 Time Value of Money Diagrams Diagram conventions: The length of the lines can be drawn to scale to show the relative amount of dollar flow. Multiple cash or noncash inflows and outflows are shown at the appropriate time on the horizontal line 35 End Unit 3 Material Go to Unit 4 Compounding Interest 36 12 Chapter 4 – Unit 4 Compounding Interest IET 350 Engineering Economics Learning Objectives – Unit 4 Upon completion of this unit you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity Time value of money and the application of single Time value of money and the application of single‐ payment interest calculations to single‐ and multiple‐ payment problems Time value of money or cash flow diagrams Application of compound, effective, nominal and continuous interest calculations Inflation and the time value of money 38 Interest As previously discussed, interest can be classified as: Simple – applied to the principle amount only Compound – applied to principle and interest C ti Continuous Compounding – C di similar to compound interest i il t di t t except the number of time periods = infinity Interest can also be classified as: Nominal interest – interest rate without the effect of interest compounding Effective interest – interest rate including the effect of interest compounding. 39 13 Effective Interest Nominal interest is the stated annual interest rate. Effective or actual interest is adjusted by the number of compounding periods to reflect the actual interest rate for p j the duration of the investment or project m r⎞ ⎛ Effective Interest = ieff = ⎜ 1 + ⎟ ‐ 1 ⎝ m⎠ Where: r = nominal interest rate/year m = #compounding periods/year 40 Effective Interest ‐ Example What is the actual interest rate given a 4% annual interest rate compounded monthly? m r⎞ ⎛ ieff = ⎜ 1 + ⎟ ‐ 1 ⎝ m⎠ 12 ⎛ 0.04 ⎞ = ⎜ 1 + ⎟ ‐ 1 12 ⎠ ⎝ = (1.00333)12 ‐ 1 ieff = 0.04074 = 4.074% 41 Effective Interest The effective interest rate can be combined with the compound interest equation to determine the future value of a principle amount: r⎞ ⎛ F = P × ⎜ 1 + ⎟ ⎝ m⎠ nm Where: r = nominal interest rate/year m = #compounding periods/year n = time (years) 42 14 Effective Interest ‐ Example You loan your brother‐in‐law $2,000 at 5% annual interest compounded weekly to be repaid in 6 years How much will you receive if and when payment is made? Interest has increased $19.14 compared to annually compounded interest and $99.33 compared to simple interest. r⎞ ⎛ F = P × ⎜ 1 + ⎟ ⎝ m⎠ nm 6×52 ⎛ 0.05 ⎞ = $2,000 × ⎜ 1 + ⎟ 52 ⎠ ⎝ = $2,000 × (1.000962)312 = $2,000 × (1.349664) F = $2699.33 43 Continuous Compounding Continuous compounding is not typically used for engineering economic analysis. It is used for savings accounts and other investment areas The number of continuous compounding periods is infinity p gp y The equation uses the mathematical constant e (e=2.71828). Scientific calculators have the function as a key (eY) ieff = er ‐ 1 F = P × (er ) n Where: r = nominal interest rate/year n = time (years) 44 Continuous Compounding ‐ Example You loan your brother‐in‐law $2,000 at 5% annual interest compounded continuously to be repaid in 6 years How much will you receive if and when payment is made? Total interest received has increased an additional $0.39 compared to weekly compounded interest. F = P × (er ) n = $2,000 × (2.718280.05 ) = $2,000 × (1.0512716) = $2,000 × (1.349859) F = $2699.72 45 15 Microsoft Excel® Hints Excel® has built‐in effective and nominal interest functions: EFFECT(nominal_rate, npery) → returns the effective annual interest rate given a nominal interest rate NOMINAL(effect_rate, npery) → returns the nominal interest rate given an effective interest rate 46 Example Problem 4.2 Example Problem 4.2 Solution 47 End Unit 4 Material Additional Reading Ö Demystifying the Natural Logarithm: http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/ Go to Unit 5 Inflation and Time Value of Money 48 16 Chapter 4 – Unit 5 Inflation and Time Value of Money IET 350 Engineering Economics Learning Objectives – Unit 5 Upon completion of this unit you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity Time value of money and the application of single Time value of money and the application of single‐ payment interest calculations to single‐ and multiple‐ payment problems Time value of money or cash flow diagrams Application of compound, effective, nominal and continuous interest calculations Inflation and the time value of money 50 Inflation Inflation is a rise in general level of prices of goods and services over time. Although "inflation" is sometimes used to refer to a rise in the prices of a specific set of goods or services, a rise in prices of one set (such as food) without a rise in others (such as wages) is not included in the original meaning of the word. )i i l d di h i i l i f h d Inflation can be thought of as a decrease in the value of the unit of currency. It is measured as the percentage rate of change of a price index but it is not uniquely defined because there are various price indices that can be used Reference: http://en.wikipedia.org/wiki/Inflation 51 17 Inflation General types of inflation: Specific price increase of a good or service over time General increase in the price level at the national level There are many measures of inflation. For example, different price indices can be used to measure changes in prices that affect different people. Two widely known indices for which inflation rates are reported in many countries are the Consumer Price Index (CPI), which measures consumer prices, and the GDP deflator, which measures price variations associated with domestic production of goods and services. Reference: http://en.wikipedia.org/wiki/Inflation 52 Inflation Differential inflation is the difference between inflation of a specific good and general/national inflation ⎛ Differential ⎞ = ⎛ Inflation Rate ⎞ – ⎛ General ⎞ ⎜ Inflation ⎟ ⎜ Specific Good ⎟ ⎜ Inflation Rate ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Negative differential inflation indicates the specific good’s price change is less than the general change in prices 53 Inflation Average inflation over time is determined using a formula similar to the single payment formula F = P × (1 + f )n or F = P × (1 + g )n Where: f = general inflation rate/year g = specific goods inflation rate/year n = time (years) 54 18 Inflation ‐ Importance Assuming all goods, services, wages, power, raw materials, etc. following the same general or national rate of inflation – economic analysis can disregard the effect of inflation The organization can adjust the selling price of the goods g j gp g or services it provides by the general inflation rate However, not all goods and services purchased by an organization follow the general inflation rate. Ignoring specific goods inflation rates can have a negative effect on net income Example: Cost of fuel in 2008 has a higher inflation rate compared to the general inflation rate. 55 Inflation and ROI We can expand the concept of Return on Investment (ROI) to include both the return from the project and the effect of inflation: + ic = (1 + f )(1 + ia ) Where: f = general inflation rate (g can also be used) ic =combined inflation and actual rate of return ia =actual rate of return 56 Inflation and ROI ‐ Example You are investing $500,000 in a project that has an estimated 10% rate of return. Current GDP deflator is 4.5% What is the actual ROI? + ic = (1 + f )(1 + ia ) + 0.10 = (1 + 0.045)(1 + ia ) 1.10 − 1 = ia 1.045 ia = 0.052632 = 5.26% The net effect of inflation on this project is that the projected 10% return or $50,000 will actually be 5.26% or $26,316. 57 19 End Chapter 4 Material Student Study Guide Ö Chapter 4 Homework Assignment Ö Problem Set 4 58 20 ... problems, management goals, efficiency and productivity Time value of money and the application of single Time value of money and the application of single payment interest calculations to single and multiple‐ payment problems... Time value of money and the application of single Time value of money and the application of single payment interest calculations to single and multiple‐ payment problems Time value of money or cash flow diagrams... of single Time value of money and the application of single payment interest calculations to single and multiple‐ payment problems Time value of money or cash flow diagrams Application of compound, effective, nominal and