NOTES ON COMPOUND INTEREST AND INTEREST TABLES

Transcription

1 B A P P E N D I X NOTES ON COMPOUND INTEREST AND INTEREST TABLES Interest is the cost of using money. It is the rental charge for funds, just as renting a building and equipment entails a rental charge. When the funds are used for a period of time, it is necessary to recognize interest as a cost of using the borrowed ( rented ) funds. This requirement applies even if the funds represent ownership capital and if interest does not entail an outlay of cash. Why must interest be considered? Because the selection of one alternative automatically commits a given amount of funds that could otherwise be invested in some other alternative. Interest is generally important, even when short-term projects are under consideration. Interest looms correspondingly larger when long-run plans are studied. The rate of interest has significant enough impact to influence decisions regarding borrowing and investing funds. For example, $100,000 invested now and compounded annually for 10 years at 8% will accumulate to $215,900; at 20%, the $100,000 will accumulate to $619,200. interest tables Many computer programs and pocket calculators are available that handle computations involving the time value of money. You may also turn to the following four basic tables to compute interest. Table 1 Future Amount of $1 Table 1 shows how much $1 invested now will accumulate in a given number of periods at a given compounded interest rate per period. Consider investing $1,000 now for three years at 8% compound interest. A tabular presentation of how this $1,000 would accumulate to $1, follows: Cumulative Interest Called Compound Total at End of Year Interest per Year Interest Year 0 $ $ $1, , , ,259.70

2 This tabular presentation is a series of computations that could appear as follows: S 1 = $1,000(1.08) 1 S 2 = $1,000(1.08) 2 S 3 = $1,000(1.08) 3 The formula for the amount of 1, often called the future value of $1 or future amount of $1, can be written S = P(1 + r) n S = $1,000(1 +.08) 3 = $1, S is the future value amount; P is the present value, $1,000 in this case; r is the rate of interest; and n is the number of time periods. Fortunately, tables make key computations readily available. A facility in selecting the proper table will minimize computations. Check the accuracy of the preceding answer using Table 1. Table 2 Present Value of $1 In the previous example, if $1,000 compounded at 8% per year will accumulate to $1, in 3 years, then $1,000 must be the present value of $1, due at the end of 3 years. The formula for the present value can be derived by reversing the process of accumulation (finding the future amount) that we just finished. If P = S = P(1 + r) n S (1 + r) n $1, then P = = $1,000 (1.08) 3 Use Table 2 to check this calculation. When accumulating, we advance or roll forward in time. The difference between our original amount and our accumulated amount is called compound interest. When discounting, we retreat or roll back in time. The difference between the future amount and the present value is called compound discount. Note the following formulas (where P = $1,000): Compound interest = P[(1 + r) n 1] = $ Compound discount = S[ 1 ] = $ Table 3 Amount of Annuity of $1 1 (1 + r) n An (ordinary) annuity is a series of equal payments (receipts) to be paid (or received) at the end of successive periods of equal length. Assume that $1,000 is invested at the end of each of 3 years at 8%: End of Year Amount 1st payment $1, $1, $1,166.40, which is $1,000(1.08) 2 2nd payment $1, ,080.00, which is $1,000(1.08) 1 3rd payment 1, Accumulation (future amount) $3, NOTES ON COMPOUND INTEREST AND INTEREST TABLES A-17

3 The preceding arithmetic may be expressed algebraically as the amount of an ordinary annuity of $1,000 for 3 years = $1,000(1 + r) 2 + $1,000(1 + r) 1 + $1,000. We can develop the general formula for S n, the amount of an ordinary annuity of $1, by using the example above as a basis: 1. S n = 1 + (1 + r) 1 + (1 + r) 2 2. Substitute: S n = 1 + (1.08) 1 + (1.08) 2 3. Multiply (2) by (1 + r): (1.08)S n = (1.08) 1 + (1.08) 2 + (1.08) 3 4. Subtract (2) from (3): 1.08S n S n (1.08) 3 1 Note that all terms on the right-hand side are removed except (1.08) 3 in equation (3) and 1 in equation (2). 5. Factor (4): S n (1.08 1) = (1.08) 3 1 (1.08) (1.08) 6. Divide (5) by (1.08 1): S n = 3 1) = The general formula for the amount of an ordinary (1 + r) annuity of $1 becomes: S n = n 1 or r Compound interest Rate This formula is the basis for Table 3. Look at Table 3 or use the formula itself to check the calculations. Table 4 Present Value of an Ordinary Annuity of $1 Using the same example as for Table 3, we can show how the formula of P n, the present value of an ordinary annuity, is developed. End of Year ,000 1st payment = $ $1,000 (1.08) 1 1,000 2nd payment = $ $1,000 (1.08) 2 1,000 3rd payment = $ $1,000 (1.08) 3 Total present value $2, For the general case, the present value of an ordinary annuity of $1 may be expressed: A-18 APPENDIX B P n = r (1 + r) 2 (1 + r) 3 2. Substitute P n = (1.08) 2 (1.08) Multiply by : P n = (1.08) 2 (1.08) 3 (1.08) Subtract (3) from (2): P n P n = (1.08) 4 5. Factor: P n ( 1 1 ) = 1 [ 1 1 (1.08) 1.08 (1.08) ] (1.08) 3 6. or P n ( ) = [ 1 ]

4 Multiple by : P n = [ (1.08) ] 3 Solving, The general formula for the present value of an annuity of $1.00 is: 1 r 1 (1 + r) n P n = [ 1 ] =.2062 P n = = Compound discount Rate The formula is the basis for Table 4. Check the answer in the table. The present value tables, Tables 2 and 4, are used most frequently in capital budgeting. The tables for annuities are not essential. With Tables 1 and 2, compound interest and compound discount can readily be computed. It is simply a matter of dividing either of these by the rate to get values equivalent to those shown in Tables 3 and 4. CPA and CMA Designations Many American readers may eventually take the Certified Public Accountant (CPA) examination or the Certified Management Accountant (CMA) examination. Certification is important to professional accountants for many reasons, such as: 1. Recognition of achievement and technical competence by fellow accountants and by users of accounting services 2. Increased self-confidence in one s professional abilities 3. Membership in professional organizations offering programs of career-long education 4. Enhancement of career opportunities 5. Personal satisfaction The CPA certificate is issued by individual states; it is necessary for obtaining a state s license to practice as a Certified Public Accountant. A prominent feature of public accounting is the use of independent (external) auditors to give assurance about the reliability of the financial statements supplied by managers. These auditors are called Certified Public Accountants in the United States and Chartered Accountants in many other English-speaking nations. The major U.S. professional association in the private sector that regulates the quality of external auditing is the American Institute of Certified Public Accountants (AICPA). The CMA designation is offered by the Institute of Management Accountants (IMA). The IMA is the largest association of management accountants in the world. 2 The major objective of the CMA certification is to enhance the development of the management accounting profession. In particular, focus is placed on the modern role of the management accountant as an active contributor to and a participant in management. The CMA designation is gaining increased stature in the business community as a credential parallel to the CPA designation. The CMA examination consists of 4 parts taken during 2 days (16 hours): Part 1: Economics, finance, and management Part 2: Financial accounting and reporting Part 3: Management reporting, analysis, and behavioral issues Part 4: Decision analysis and information systems Questions regarding ethical issues will appear on any part of the examination. A person who has successfully completed the U.S. CPA examination is exempt from Part 2. Cost/management accounting questions are prominent in the CMA examination. The CPA examination also includes such questions, although they are less extensive than questions regarding financial accounting, auditing, and business law. On NOTES ON COMPOUND INTEREST AND INTEREST TABLES A-19

5 TABLE 1 Compound Amount of $1.00 (The Future Value of $1.00) S = P(1 + r) n. In this table P = $1.00 Periods 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30% 32% 40% Periods , , , , ,

6 TABLE 2 (Place a clip on this page for easy reference.) Present Value of $1.00. S P =. In this table S = $1.00. (1 + r) n Periods 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30% 32% 40% Periods

7 TABLE 3 Compound Amount of Annuity of $1.00 in Arrears* (Future Value of Annuity) S n = (1 + r)n 1 r Periods 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30% 32% 40% Periods ll *Payments (or receipts) at the end of each period.

8 TABLE 4 (Place a clip on this page for easy reference.) Present Value of Annuity $1.00 in Arrears*. P = 1 [ ] 1 1 n r (1 + r) n Periods 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30% 32% 40% Periods , *Payments (or receipts) at the end of each period.

9 C A P P E N D I X COST ACCOUNTING IN PROFESSIONAL EXAMINATIONS This appendix describes the role of cost accounting in professional examinations. We use professional examinations in the United States, Canada, Australia, Japan, and the United Kingdom to illustrate the role. 1 A conscientious reader who has solved a representative sample of the problems at the end of the chapters will be well prepared for the professional examination questions dealing with cost accounting. This appendix aims to provide perspective, install confidence, and encourage readers to take the examinations. Canadian Professional Examinations Three professional accounting designations are available in Canada: Designation Certified Management Accountant (CMA) Certified General Accountant (CGA) Chartered Accountant (CA) Sponsoring Organization Society of Management Accountants (SMA) Certified General Accountants Association (CGA) Canadian Institute of Chartered Accountants (CICA) The SMA represents over 27,000 certified management accountants employed throughout Canadian business, industry, and government. The CMA Entrance Examination is a two-day examination, divided into three broad categories: 1. Management accounting area 50% 60% 2. Financial accounting area 20% 30% 3. Management studies 15% 25% Objective questions comprise 40% to 50% and cases 50% to 60% of the exam. Topics covered on recent examinations in the management accounting area include relevant costing, transfer pricing, capital budgeting, performance measures, activity-based costing, cost allocation, and productivity. The Society of Management Accountants publishes CMA: The Management Accounting Magazine monthly. This magazine includes details of courses that assist students in preparing for the CMA examination. 1 We appreciate help from Bill Langdon (Canada), Tom Craven (United States), John Goodwin (Australia), Michi Sakurai (Japan), and Louise Drysdale and Andrea Jeffries (U.K.).

10 CPA and CMA Designations American Professional Examinations Many American readers may eventually take the Certified Public Accountant (CPA) examination or the Certified Management Accountant (CMA) examination. Certification is important to professional accountants for many reasons, such as: 1. Recognition of achievement and technical competence by fellow accountants and by users of accounting services 2. Increased self-confidence in one s professional abilities 3. Membership in professional organizations offering programs of career-long education 4. Enhancement of career opportunities 5. Personal satisfaction The CPA certificate is issued by individual states; it is necessary for obtaining a state s licence to practise as a Certified Public Accountant. A prominent feature of public accounting is the use of independent (external) auditors to give assurance about the reliability of the financial statements supplied by managers. These auditors are called Certified Public Accountants in the United States and Chartered Accountants in many other English-speaking nations. The major U.S. professional association in the private sector that regulates the quality of external auditing is the American Institute of Certified Public Accountants (AICPA). The CMA designation is offered by the Institute of Management Accountants (IMA). The IMA is the largest association of management accountants in the world. 2 The major objective of the CMA certification is to enhance the development of the management accounting profession. In particular, focus is placed on the modern role of the management accountant as an active contributor to and a participant in management. The CMA designation is gaining increased stature in the business community as a credential parallel to the CPA designation. The CMA examination consists of 4 parts taken during 2 days (16 hours): Part 1: Economics, finance, and management Part 2: Financial accounting and reporting Part 3: Management reporting, analysis, and behavioral issues Part 4: Decision analysis and information systems Questions regarding ethical issues will appear on any part of the examination. A person who has successfully completed the U.S. CPA examination is exempt from Part 2. Cost/management accounting questions are prominent in the CMA examination. The CPA examination also includes such questions, although they are less extensive than questions regarding financial accounting, auditing, and business law. On the average, cost/managerial accounting represents 35% to 40% of the CMA examination and 5% of the CPA examination. This book includes many questions and problems used in past CMA and CPA examinations. In addition, a supplement to this book, Student Guide and Review Manual [ John K. Harris and Dudley W. Curry (Englewood Cliffs, NJ: Prentice Hall, 1997)], contains over one hundred CMA and CPA questions and explanatory answers. Careful study of appropriate topics in this book will give candidates sufficient background for succeeding in the cost accounting portions of the professional examinations. The IMA publishes Management Accounting monthly. Each issue includes advertisements for courses that help students prepare for the CMA examination. 3 2 The IMA has a wide range of activities driven by many committees. For example, the Management Accounting Practices Committee issues statements on both financial accounting and management accounting. The IMA also has an extensive continuing-education program. 3 Other U.S. professional associations also require detailed knowledge of cost accounting. For example, the Certified Cost Estimator/Analyst (CCEA) program is administered by the Society of Cost Estimating and Analysis, 101 South Whiting Street, Suite 313, Alexandria, VA The society s primary purpose is to improve the effectiveness of cost estimation and price analysis. Special attention is given to contract cost estimation. COST ACCOUNTING IN PROFESSIONAL EXAMINATIONS A-25

11 Australian Professional Examinations The Australian Society of Certified Practising Accountants is the largest body repre senting accountants in Australia. Their professional designation is termed a CPA (Certified Practising Accountant). The basic entry requirements for Associate membership of the Society are having an approved Bachelors degree. Associates of the Society can advance to CPA status by passing the CPA program and having the required amount of relevant work experience. There are two compulsory core segments in the program. Core I covers the practical application of the more common accounting standards and ethics, while more technical standards (such as foreign currency translation) are covered in the Core II segment. Candidates are then required to take three segments from seven elective subjects. These subjects are: (1) external reporting, (2) insolvency and reconstruction, (3) management accounting, (4) management of information systems, (5) auditing, (6) treasury, and (7) taxation. Personal Financial Planning and Superannuation is a new elective subject which will soon be added to the electives. The management accounting segment topics include: 1. Management accounting in the contemporary business environment 2. Accounting for strategic management 3. Long-term project planning and management 4. Costing for decision making 5. Performance measurement and reward systems The Australian Accountant, published each month (except January), includes advertisements for courses that help students prepare for the CPA examination. The Institute of Chartered Accountants in Australia (ICAA) has membership requirements that include passing four core modules (Taxation, Accounting I, Accounting II, and Ethics) and one elective module (one of which is Advanced Management Accounting). Management related topics are in both the Accounting 2 and Advanced Management Accounting modules. These include: purpose and perspective (including strategic and operational management; organizations, goals, ethics; operational environments; cost concepts); strategic management accounting (including strategic applications, project evaluation and capital budgeting); operational management accounting (including decision analysis, financial planning and management, product and service costing, control and performance evaluation). Japanese Professional Examinations There are two major management accounting organizations Japanese Industrial Management and Accounting Association and Enterprise Management Association. The JIMAA is the oldest, largest, and most authoritative accounting organization of its kind in Japan. It directs a School of Cost Control and a School of Corporate Tax Accounting. There are two courses in the School of Cost Control Preparatory Course and Cost Control Course. These courses are taught by university professors and executives from member corporations. The Enterprise Management Association is the Japanese chapter of the U.S.-based Institute of Management Accountants. United Kingdom Professional Examinations A-26 APPENDIX C The Chartered Institute of Management Accountants (CIMA) is the largest professional management accounting body in the United Kingdom. CIMA provides a wide range of services to members in commerce, education, government, and the accounting profession.

12 The syllabus for the CIMA examination consists of four stages: 1. Preparation for business and accounting (including foundation costing ) 2. The tools of management accounting (including operational cost accounting ) 3. The rules of a profession (including management accounting applications ) 4. The application of knowledge to business management and finance (including strategic management accounting and management accounting control systems ) Management Accounting, published monthly by CIMA, includes details of courses assisting students in preparing for their examinations. Management accounting topics are also covered by several other professional bodies. The syllabus for the examinations of the Chartered Association of Certified Accountants (ACCA) has three stages: I (Foundation), II (Certificate), and III (Professional). Skills examined in III include information for control and decision making, management and strategy, and financial strategy. Other accounting bodies include the Institute of Chartered Accountants in England and Wales (ICAEW ) and the Institute for Chartered Accountants of Scotland (ICAS). Both institutes have requirements that cover proficiency in general management topics as well as professional accounting topics. COST ACCOUNTING IN PROFESSIONAL EXAMINATIONS APPENDIX C A-27