Money-Time Relationships and Equivalence PDF

IT2007 Money-Time Relationships and Equivalence Money is used as a means to store value. Before countries had viable currencies, the value was stored using gold, silver, and other precious metals....

IT2007 Money-Time Relationships and Equivalence Money is used as a means to store value. Before countries had viable currencies, the value was stored using gold, silver, and other precious metals. Eventually, the governments of many countries started using paper money backed by gold rather than using precious metals as a means of exchange. Time Value of Money (Sharma, 2015) It accounts for the interest an investment earns, and it indicates that an amount of money with a certain value now will increase in value in the future due to the interest the money earns during the intervening time period. Capital is wealth in the form of money or property that can be used to produce more wealth. o Equity Capital is owned by individuals who have invested their money or property in a business project or venture o Debt Capital (Borrowed Capital) is obtained from lenders for investment. Interest o It is defined as a commodity such as rent on money borrowed or loaned from one individual to another, from one institution to another institution, or from an institution to an individual. (Yates, J., 2017) o It is the manifestation of the time value of money. (Blank, L., 2018) Interest and Rate of Return (Blank, L., 2018 & Khan, Z., 2018) Computationally, interest is the difference between an ending amount of money and the beginning amount. There are always two perspectives to an amount of interest: interest paid and interest earned. From the perspective of the borrower, Interest paid on borrowed funds (a loan) is 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐩𝐚𝐢𝐝 = 𝐚𝐦𝐨𝐮𝐧𝐭 𝐨𝐰𝐞𝐝 𝐧𝐨𝐰 − 𝐩𝐫𝐢𝐧𝐜𝐢𝐩𝐚𝐥 𝐚𝐦𝐨𝐮𝐧𝐭 determined using the original amount, also called the principal. When interest paid over a specific 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐚𝐜𝐜𝐫𝐮𝐞𝐝 𝐩𝐞𝐫 𝐭𝐢𝐦𝐞 𝐮𝐧𝐢𝐭 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞 (%) = × 𝟏𝟎𝟎% time unit is expressed as a 𝐎𝐫𝐢𝐠𝐢𝐧𝐚𝐥 𝐚𝐦𝐨𝐮𝐧𝐭 percentage of the principal, the The time unit of the rate is called the interest period. By far, the result is called the interest rate. most common interest period used to state an interest rate is 1 year. Example 1: An employee at LaserKinetics.com borrows Php 10,000 on May 1 and must repay a total of Php 10,700 exactly 1 year later. Determine the interest amount and the interest rate paid. ₱𝟕𝟎𝟎 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 = ₱𝟏𝟎, 𝟕𝟎𝟎 − 𝟏𝟎, 𝟎𝟎𝟎 = 𝟕𝟎𝟎 𝐑𝐚𝐭𝐞 𝐨𝐟 𝐑𝐞𝐭𝐮𝐫𝐧 = × 𝟏𝟎𝟎% = 𝟕% 𝐩𝐞𝐫 𝐲𝐞𝐚𝐫 ₱𝟏𝟎𝟎𝟎𝟎 Example 2: You borrow Php 11,000 today and must repay a total of Php 11,550 exactly 1 year later. What are the interest amount and the interest rate paid? ₱𝟓𝟓𝟎 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 = ₱𝟏𝟏, 𝟕𝟎𝟎 − ₱𝟏𝟏, 𝟎𝟎𝟎 = ₱𝟓𝟓𝟎 𝐑𝐚𝐭𝐞 𝐨𝐟 𝐑𝐞𝐭𝐮𝐫𝐧 = × 𝟏𝟎𝟎% = 𝟓% 𝐩𝐞𝐫 𝐲𝐞𝐚𝐫 ₱𝟏𝟏, 𝟎𝟎𝟎 02 Handout 1 *Property of STI  [email protected] Page 1 of 5  IT2007 From the perspective of an investor, interest earned is the 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐞𝐚𝐫𝐧𝐞𝐝 = 𝐭𝐨𝐭𝐚𝐥 𝐚𝐦𝐨𝐮𝐧𝐭 𝐧𝐨𝐰 − 𝐩𝐫𝐢𝐧𝐜𝐢𝐩𝐚𝐥 final amount minus the initial amount or principal. 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐚𝐜𝐜𝐫𝐮𝐞𝐝 𝐩𝐞𝐫 𝐮𝐧𝐢𝐭 𝐭𝐢𝐦𝐞 Interest earned over a specific 𝐑𝐚𝐭𝐞 𝐨𝐟 𝐫𝐞𝐭𝐮𝐫𝐧 (%) = × 𝟏𝟎𝟎% 𝐩𝐫𝐢𝐧𝐜𝐢𝐩𝐚𝐥 period of time is expressed as a The term return on investment (ROI) is used equivalently with ROR percentage of the original amount in different industries and settings, especially where large capital and is called rate of return (ROR). funds are committed to engineering-oriented programs. Example: Calculate the amount deposited 1 year ago to have ₱1000 now at an interest rate of 5% per year. Also, calculate the amount of interest earned during this time period. The total amount accrued (₱1000) is the sum of the original deposit and the earned interest. If X is the original deposit, 𝐓𝐨𝐭𝐚𝐥 𝐚𝐜𝐜𝐫𝐮𝐞𝐝 = 𝐝𝐞𝐩𝐨𝐬𝐢𝐭 + 𝐝𝐞𝐩𝐨𝐬𝐢𝐭(𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞) ₱𝟏𝟎𝟎𝟎 = 𝐗 + 𝐗(𝟎. 𝟎𝟓) = 𝐗(𝟏 + 𝟎. 𝟎𝟓) 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 = ₱𝟏𝟎𝟎𝟎 − 𝟗𝟓𝟐. 𝟑𝟖 = 𝟏. 𝟎𝟓𝐗 = ₱𝟒𝟕. 𝟔𝟐 The original deposit is 𝟏𝟎𝟎𝟎 𝑿= = ₱𝟗𝟓𝟐. 𝟑𝟖 𝟏. 𝟎𝟓 Simple and Compound Interest Simple interest is the total interest earned or charged is linearly proportional to the initial amount of the loan (principal), the interest rate, and the number of interest periods. where: I - total interest P - principal amount lent or borrowed 𝐈 = 𝐏𝐧𝐢 n - number of interest periods (annually, semi-annually, monthly, weekly, daily, etc.) i - interest rate per interest period Example: You borrow ₱1,500 from your friend for three years at a simple 𝐈 = 𝐏𝐍𝐢 interest rate of 8% per year. How much interest will you pay after three 𝐈 = (₱𝟏, 𝟓𝟎𝟎)(𝟑)(𝟎. 𝟎𝟖) years, and what is the value of the total amount that will you will pay? 𝐈 = 𝟑𝟔𝟎 where: F - total amount to be paid P - principal amount lent or borrowed 𝐅 = 𝐏(𝟏 + 𝐧𝐢) n - number of interest periods (annually, semi-annually, monthly, weekly, daily, etc.) i - interest rate per interest period Example: What is the total amount to be paid if you 𝐅 = 𝐏(𝟏 + 𝐧𝐢) borrowed Php 20,000 from your parents for tuition, = 𝐏𝐡𝐩 𝟐𝟎, 𝟎𝟎𝟎(𝟏 + (𝟒 × 𝟎. 𝟎𝟓)) for a period of 4 years at 5% simple interest? = 𝐏𝐡𝐩 𝟐𝟒, 𝟎𝟎𝟎 02 Handout 1 *Property of STI  [email protected] Page 2 of 5  IT2007 Compound Interest is the interest charge for any interest period, e.g., a year, based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period. where: A - total amount to be paid 𝐧 𝐀 = 𝐏(𝟏 + 𝐢) P - principal amount lent or borrowed n - number of interest periods (annually, semi- 𝐂𝐨𝐦𝐩𝐨𝐮𝐧𝐝 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 = 𝐀 − 𝐏 annually, monthly, weekly, daily, etc.) i - interest rate per interest period Example 1: The City Bank has issued a loan of ₱1,000 (Calculation in Yearly Basis) to a sole proprietor for a period of 5-years. The 𝑨 = 𝑷 (𝟏 + 𝒊 ) 𝒏 interest rate for this loan is 5%, and the interest is 𝑨 = ₱𝟏, 𝟎𝟎𝟎(𝟏 + 𝟎. 𝟎𝟓)𝟓 compounded annually. Compute for the compound 𝑨 = ₱𝟏, 𝟐𝟕𝟔 amount and interest. 𝐂𝐨𝐦𝐩𝐨𝐮𝐧𝐝 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 = 𝐀 − 𝐏 𝐂𝐨𝐦𝐩𝐨𝐮𝐧𝐝 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 = 𝟏, 𝟐𝟕𝟔 − 𝟏, 𝟎𝟎𝟎 𝐂𝐨𝐦𝐩𝐨𝐮𝐧𝐝 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭 = 𝟐𝟕𝟔 Example 2: Your local bank branch recently (Calculation in Daily Basis) announced a new savings plan with an interest rate of 𝐀 = 𝐏(𝟏 + 𝐢)𝐧 0.0274% compounded daily. What is the compound 𝐀 = ₱𝟓𝟐, 𝟎𝟎𝟎(𝟏 + 𝟎. 𝟎𝟎𝟎𝟐𝟕𝟒)𝟑𝟔𝟓 payment on ₱52,000 at the end of one year? 𝐀 = ₱𝟓𝟕, 𝟒𝟔𝟖. 𝟔𝟖 Concept of Equivalence The time value of money and the interest rate considered together helps in developing the concept of economic equivalence, which means that different sums of money at different times would be equal in economic value. Cash Flow Diagrams The cash flow diagram is a graphical representation of cash flows drawn on a time scale. The horizontal line is a time scale, with the progression of time moving from left to right. The period (e.g., year, quarter, and month) can be applied to intervals of time. Cash flow diagram time t = 0 is the present, and the end of interval 1 is the end of time period 1. The arrows signify cash flows and are placed at the end of the period when the end-of-period convention is used. The end-of-period convention means that all cash flows are assumed to occur at the end of an interest period. Downward arrows represent expenses (negative cash flows or cash outflows). o Negative cash flows: These can be the first cost of assets, engineering design cost, annual operating costs, periodic maintenance and rebuild costs, loan interest and principal payments, major expected/unexpected upgrade costs, income taxes, etc. 02 Handout 1 *Property of STI  [email protected] Page 3 of 5  IT2007 Upward arrows represent receipts (positive cash flows or cash inflows). o Positive cash flows: These can be revenues, operating cost reductions, asset salvage value, receipt of loan principal, income tax savings, receipts from stock and bond sales, construction and facility costs savings, saving or return of corporate capital funds, etc. The cash flow diagram is dependent on the point of view. Your point of view when you borrowed from an entity. The entity’s perspective when lending money. When cash inflows and cash outflows occur at the end of a given interest period, the net cash flow can be determined from the following relationship: 𝐍𝐞𝐭 𝐜𝐚𝐬𝐡 𝐟𝐥𝐨𝐰 = 𝐑𝐞𝐜𝐞𝐢𝐩𝐭𝐬 – 𝐃𝐢𝐬𝐛𝐮𝐫𝐬𝐞𝐦𝐞𝐧𝐭𝐬 Example: = 𝐂𝐚𝐬𝐡 𝐢𝐧𝐟𝐥𝐨𝐰𝐬 – 𝐂𝐚𝐬𝐡 𝐨𝐮𝐭𝐟𝐥𝐨𝐰𝐬 Period Cash Cash Net Cash Inflow Outflow flow 1 1,000 500 500 2 800 1,200 -400 3 900 0 900 4 0 600 -600 5 400 1,500 -1,100 Notations help identify the variables in given problems. P - present sum of money; the equivalent value of one or more cash flows at the present time reference point F - future sum of money; the equivalent value of one or more cash flows at a future time reference point A - end-of-period cash flows (or equivalent end-of-period values) in a uniform series continuing for a specified number of periods, starting at the end of the first period and continuing through the last period G - uniform gradient amounts are used if cash flows increase by a constant amount in each period. n - number of compounding periods (e.g., years, month, or days) i - interest rate or return per time period 02 Handout 1 *Property of STI  [email protected] Page 4 of 5  IT2007 Example 1: Before evaluating the economic merits of a proposed investment, the XYZ Corporation ₱80,000 insists that its engineers develop a cash flow diagram of the proposal. An investment of ₱210,000 ₱210,000 ₱210,000 ₱210,000 ₱210,000 ₱400,000 can be made that will produce uniform annual revenue of ₱210,000 for five years and then 1 2 3 4 5 =N have a market (recovery) value of ₱80,000 at the end of year five. Annual expenses will be ₱120,000 at the end of each year for operating and ₱120,000 ₱120,000 ₱120,000 ₱120,000 ₱120,000 maintaining the project. Draw the cash flow Years diagram for the five-year life of the project. Use the ₱400,000 corporation’s viewpoint. Example 2: You plan to borrow ₱15,000 to help in buying a two-wheeler. You have arranged to repay P = ₱15,000 𝑭 = 𝑷 (𝟏 + 𝒊 ) 𝒏 the entire principal plus interest of 8.5% per year i = 8.5% per year 𝑭 = ₱𝟏𝟓, 𝟎𝟎𝟎(𝟏+. 𝟎𝟖𝟓)𝟓 after 5 years. Identify the symbols involved and their n = 5 years 𝑭 = ₱𝟐𝟐, 𝟓𝟓𝟒. 𝟖𝟓 values for the total amount owed after 5 years. Also, F=? draw the cash flow diagram. References: Blank, L. & Tarquin A. (2018). Engineering economy (8th ed.). McGraw-Hill. Khan, Z., Siddiquee, A., Kumar, B., & Abidi, M. (2018). Principles of engineering economics with applications (2nd ed.). Cambridge University Press. Sharma, K.R. (2015). An introduction to engineering economics. Momentum Press Engineering. Yates, J.K. (2017). Engineering economics. CRC Press. 02 Handout 1 *Property of STI  [email protected] Page 5 of 5