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University of Rizal System College of Engineering - Mechanical Engineering ES10 – Engineering Economics Chapter 2: Money-Time Relationship Equivalence 2.1 Interest and th...

University of Rizal System College of Engineering - Mechanical Engineering ES10 – Engineering Economics Chapter 2: Money-Time Relationship Equivalence 2.1 Interest and the Time Value of Money Learning Objectives: Calculate the impact of interest on the time value of money. Definition of Interest From the viewpoint of the borrower, interest is the amount of money paid for the use of borrowed capital. For the lender, interest is the income produces by the money which the lender has lent. The charging of interest is justified from the standpoint of the lender, because he has to forgo the use of his money during the time it is borrowed, and to compensate him also for the risk which he has to take in lending his money. From the borrower’s viewpoint, it is usually advantageous to borrow money if in so doing he will be able to earn more that the interest which he has to pay. Simple Interest The interest on borrowed money is said to be simple interest if the interest to be paid is directly proportional to the length of time the amount or principal is borrowed. The principal is the amount of money borrowed and on which interest is charged. The rate of interest is the amount earned by one unit of principal during a unit of time. The formula for simple interest is I = Pin where I = total interest earned by the principal P = amount of the principal i = rate of interest (expressed in decimal form) n = number of interest periods The total amount to be repaid is equal to the sum of the principal and the total interest and is given by the formula: F=P+I where I = Pin, so, F = P + Pin therefore, F = P (1 + in) where F = future worth P = amount of the principal i = rate of interest (expressed in decimal form) n = number of interest periods Ordinary and Exact Simple Interest Ordinary simple interest is computed on the basis of one banker's year, which is 1 banker's year = 12 months, each consisting of 30 days 1 banker's year = 360 days Exact simple interest is based on the exact number of days, 365 for an ordinary year and 366 days for a leap year. The leap years are those which are exactly divisible by 4, but excluding the century years such as the years 1900, 2100, etc. Here is the reason why Leap Year Basic Rule: Generally, a year is a leap year if it is divisible by 4. This is why most years like 2016, 2020, and 2024 are leap years, as they are divisible by 4. Century Years (Years ending in "00"): For years that are divisible by 100 (century years), there is an additional condition. A century year is a leap year only if it is also divisible by 400. Why this rule? The Earth doesn't orbit the Sun in exactly 365.25 days—it's actually about 365.2425 days. Over time, if every year divisible by 4 were made a leap year, we would slightly over-correct, adding too many leap days. To compensate, the rule excludes century years unless they are divisible by 400. For example: 1900: Divisible by 4 and 100, but not by 400, so not a leap year. 2000: Divisible by 4, 100, and 400, so is a leap year. 2100: Divisible by 4 and 100, but not by 400, so not a leap year. If d is the number of days in the interest period, then  𝐝 Ordinary simple interest = Pi 𝟑𝟔𝟎 𝐝 Exact simple interest = Pi (for ordinary year) 𝟑𝟔𝟓 𝐝 Exact simple interest = Pi (for leap year) 𝟑𝟔𝟔 where P = amount of the principal i = rate of interest (expressed in decimal form) d = number of days invested Compound Interest In compound interest, the interest earned by the principal is not paid at the end of each interest period, but is considered as added to the principal, and therefore will also earn interest for the succeeding periods. The interest earned by the principal when invested at compound interest is much more than that earned by the same principal when invested at simple interest for the same number of periods. Using the same nomenclature as that for simple interest, the total amount due after n periods for compound interest is given by the formula: 𝐅 = 𝐏 (𝟏 + 𝐢)𝐧 where F = future worth P = amount of the principal i = rate of interest (expressed in decimal form) n = number of interest periods which is derived in accordance with table below: Derivation of Formula in Compound Interest Interest Principal at Interest earned Compound amount at the end of period Period beginning of period during period 1 P Pi P + Pi = P (1 + i) 2 P (1 + i) P (1 + i) i P (1 + i) + P (1 + i) i = P (1 +i) 2 3 P (1 + i) 2 P (1 + i) 2 i P (1 + i) 2 + P (1 + i) 2 i = P (1 +i) 3 … … … … n P (1 + i) n-1 P (1 + i) i n-1 P (1 + i) + P (1 + i) n-1 i = P (1 + i) n = F n-1 The factor (1 + i) n is called the "Single Payment Compound Amount Factor" and is designated F 𝐅 by SPCAF = ( , i%, n). Thus, (1 + i) n = ( , i%, n) P 𝐏 And formula F = P (1+i) n is written 𝐅 F = P (1+i) n = ( , i%, n) 𝐏 Continuous Compounding If r is the nominal annual interest rate and m is the number of interest periods each year, then r the interest rate per interest period in n years is = , and the number of interest periods in n years m is mn. Equation F = P (1+i)n the single payment compound amount factor, may be written as 𝐫 𝐦𝐧 𝐅 = 𝐏 (𝟏 + ) 𝐦 Increasing m, the number of interest periods per year, without limit, it becomes very large and r approaches infinity and approaches zero. This is the situation for continuous compounding: m 𝐅 = 𝐏𝐞𝐫𝐧 which is the continuous compounding single payment compound amount formula. Nominal Rate of Interest For compound interest, the rate of interest usually quoted is nominal rate of interest which specifies the rate of interest and the number of interest periods per year. Thus, a nominal rate of interest of 8% compounded quarterly means that there are 4 interest periods each year, the rate per 8% period being = 2%. In the formulas stated above, in this case i=0.02. 4 Effective Rate of Interest The effective rate of interest is the actual rate of interest on the principal for one year. It is equal to the nominal rate if the interest is compounded annually, but greater than the nominal rate if the number of interest periods per year exceeds one, such as for interest compounded semi-annually, quarterly or monthly. To make this clear, imagine P1.00 to be invested at the nominal rate of 8% compounded quarterly. This amount will become after one year, Php 1(1 + 0.02)4 = P1(1.0824) = P1.0824. Thus, the actual interest earned is P0.0824. This corresponds to an effective rate of 8.24%, which is greater than the nominal rate of 8% compounded quarterly. In general, if F 1 is the amount P1.00 becomes after 1 year, n the number of periods in one year, and i the rate of interest period, then the effective rate in decimal form is Effective rate of interest = F 1 -1 Effective rate of interest = (1 + i) n = 1 Present Value The principal P in the formula F = P (1 + i) n may be considered as the value of the compound amount F at present, or it is the amount which when invested now will become F after n periods. P is called the present value of the amount F, and is given by the formula: 𝐅 P = 𝐅 (𝟏 + 𝐢)−𝐧 = 𝐧 (𝟏 +𝐢 ) The factor (1 + i) -n is called the "Single Payment Present Worth Factor" and is designated SPPWF P = ( , i%, n). Thus F 𝟏 𝐏 (1 + i)-n = = ( , i%, n) (𝟏+𝐢)𝐧 𝐅 𝐅 and Equation P = 𝐅 (𝟏 + 𝐢 )−𝐧 = becomes (𝟏 +𝐢 )𝐧 𝐅 𝐏 P = 𝐅 (𝟏 + 𝐢)−𝐧 = = ( , i%, n) (𝟏 +𝐢 )𝐧 𝐅 Compound Interest Frequency Annually Once a Year Semi-Annually Twice a Year or Every 6 Months Quarterly 4 Times a Year or Every 3 Months Bi-monthly Once Every 2 Months Monthly Once a Month Weekly Once a Week Bi-Weekly Once Every 2 Weeks 2.2 The Concept of Equivalence Learning Objectives: Explain and apply the concept of equivalence in financial analysis. Equivalent terms are used very often in the transfer from one scale to another. Some common equivalencies or conversions are as follows: Length: 100 centimeters = 1 meter 1000 meters = 1 kilometer 12 inches = 1 foot 1 inch = 2.54 centimeters Many equivalent measures are a combination of two or more scales. For example, 110 kilometers per hour (kph) is equivalent to 68 miles per hour (mph) or 1.133 miles per minute, based on the equivalence that 1 mile = 1.6093 kilometers and 1 hour 60 minutes. We can further conclude that driving at approximately 68 mph for 2 hours is equivalent to traveling a total of about 220 kilometers, or 136 miles. Three scales – time in hours, length in miles, and length in kilometers are combined to develop equivalent statements. An additional use of these equivalencies is to estimate driving time in hours between two cities using two maps, one indicating distance in miles, a second showing kilometers. Note that throughout these statements the fundamental relation 1 mile = 1.6093 kilometers is used. If this relation changes, then the other equivalencies would be in error. When considered together, the time value of money and the interest rate develop the concept of economic equivalence, which means that different sums of money at different times would be equal in economic value. For example, if the interest rate is 6% per year, Php100 today (present time) is equivalent to Php106 one year from today. Amount accrued = Php100 + Php100(0.06) = P100(1+0.06) = Php106 So, if someone offered you a gift of Php100 today or Php106 one year from today, it would make no difference which offer you accepted from an economic perspective. In either case you have Php106 one year from today. However, the two sums of money are equivalent to each other only when the interest rate is 6% per year. At a higher or lower interest rate, Php100 today is not equivalent to Php106 one year from today. In addition to future equivalence, we can apply the same logic to determine equivalence for previous years. A total of Php100 now is equivalent to Php100/1.06 = Php94.34 one year ago at an interest rate of 6% per year. From the illustration below, we can state the following: Php94.34 last year, Php100 now, and Php106 one year from now are equivalent at an interest rate of 6% per year. The fact that these sums are equivalent can be verified by computing the two interest rates for 1-year interest periods. Equivalence of Three Amounts at a 6% per Year Interest Time 2.3 Cash Flows Learning Objectives: Analyze and evaluate different types of cash flows in economic studies. Cash flows are described as the inflows and outflows of money. These cash flows may be estimates or observed values. Every person or company has cash receipts – revenue and income (inflows); and cash disbursements expenses, and costs (outflows). These receipts and disbursements are the cash flows, with a plus sign representing cash inflows and a minus sign representing cash outflows. Cash flows occur during specified periods of time, such as 1 month or 1 year. Of all the elements of the engineering economy study approach, cash flow estimation is likely the most difficult and inexact. Cash flow estimates are just that – estimates about an uncertain future. Once estimated, the techniques of engineering economics guide the decision-making process. But the time-proven accuracy of an alternative's estimated cash inflows and outflows clearly dictates the quality of the economic analysis and conclusion. Cash inflows, or receipts, may be comprised of the following, depending upon the nature of the proposed activity and the type of business involved. Samples of Cash Inflow Estimates Revenues (usually incremental resulting from an alternative). Operating cost reductions (resulting from an alternative). Asset salvage value. Receipt of loan principal. Income tax savings. Receipts from stock and bond sales. Construction and facility cost savings. Saving or return of corporate capital funds. Cash outflows, or disbursements, may be comprised of the following, again depending upon the nature of the activity and type of business. Samples of Cash Outflow Estimates First cost of assets. Engineering design costs. Operating costs (annual and incremental). Periodic maintenance and rebuild costs. Loan interest and principal payments. Major expected/unexpected upgrade costs. Income taxes. Expenditure of corporate capital funds. Background information for estimates may be available in departments such as accounting, finance, marketing, sales, engineering, design, manufacturing, production, field services, and computer services. The accuracy of estimates is largely dependent upon the experiences of the person making the estimate with similar situations. Usually point estimates are made; that is, a single-value estimate is developed for each economic element of an alternative. If a statistical approach to the engineering economy study is undertaken, a range estimate or distribution estimate may be developed. Though more involved computationally, a statistical study provides more complete results when key estimates are expected to vary widely. Once the cash inflow and outflow estimates are developed, the net cash flow can be determined. Net cash flow = receipts - disbursements = cash inflows - cash outflows Since cash flows normally take place at varying times within an interest period, a simplifying assumption is made. The end-of-period convention means that all cash flows are assumed to occur at the end of an interest period. When several receipts and disbursements occur within a given interest period, the net cash flow is assumed to occur at the end of the interest period. However, it should be understood that, although future amounts (F) are located at the end of the interest period by convention, the end of the period is not necessarily December 31. Thus, end of the period means end of interest period, not end of calendar year. The cash flow diagram is a very important tool in an economic analysis, especially when the cash flow series is complex. It is a graphical representation of cash flows drawn on a time scale. The diagram includes what is known, what is estimated, and what is needed. That is, once the cash flow diagram is complete, another person should be able to work the problem by looking at the diagram. Cash Flow Time Scale for 5 Years  Cash flow diagram time t = 0 is the present, and t = 1 is the end of time period 1. We assume that the periods are in years for now. The time scale of the figure below is set up for 5 years. Since the end-of-year convention places cash flows at the end of years, the "1" marks the end of year 1. While it is not necessary to use an exact scale on the cash flow diagram, you will probably avoid errors if you make a neat diagram to approximate scale for both time and relative cash flow magnitudes. The direction of the arrows on the cash flow diagram is important. A vertical arrow pointing up indicates a positive cash flow. Conversely, an arrow pointing down indicates a negative cash flow. The figure below illustrates a receipt (cash inflow) at the end of year 1 and equal disbursements (cash outflows) at the end of years 2 and 3. Positive and Negative Cash Flow Example Problem: A new college student has a job with Boeing Aerospace. She plans to borrow $10,000 now to help in buying a car. She has arranged to repay the entire principal plus 8% per year interest after 5 years. Construct the cash flow diagram. Solution: The figure above presents the Cash flow diagram from the vantage point of the borrower. The present sum, P is a cash flow of the loan principal at year 0, and the future sum F is the cash outflow of the repayment at the end of year 5. The interest rate is 8%. Cash Flow Diagram Please take note that this lecture is adapted from our resources and is intended to use during our class for this semester only. Do not disseminate (upload online, etc.). Thank you! Prepared By: JAKKI STACY WAYNE A. SERRA, MSCM Instructor, College of Engineering