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BACHELOR OF BUSINESS ADMINISTRATION/ BACHELOR OF COMMERCE BUSINESS MATHEMATICS AND STATISTICS BBA102 /BCM102 A.B. Rao S

CHANDIGARH UNIVERSITY Institute of Distance and Online Learning Course Development Committee Chairman Prof. (Dr.) R.S. Bawa Vice Chancellor, Chandigarh University, Punjab Advisors Prof. (Dr.) Bharat Bhushan, Director, IGNOU Prof. (Dr.) Manjulika Srivastava, Director, CIQA, IGNOU Programme Coordinators & Editing Team Master of Business Administration (MBA) Bachelor of Business Administration (BBA) Co-ordinator - Prof. Pragya Sharma Co-ordinator - Dr. Rupali Arora Master of Computer Applications (MCA) Bachelor of Computer Applications (BCA) Co-ordinator - Dr. Deepti Rani Sindhu Co-ordinator - Dr. Raju Kumar Master of Commerce (M.Com.) Bachelor of Commerce (B.Com.) Co-ordinator - Dr. Shashi Singhal Co-ordinator - Dr. Minakshi Garg Master of Arts (Psychology) Bachelor of Science (Travel & Tourism Co-ordinator - Ms. Nitya Mahajan Management) Master of Arts (English) Co-ordinator - Dr. Shikha Sharma Co-ordinator - Dr. Ashita Chadha Bachelor of Arts (General) Master of Arts (Mass Communication and Co-ordinator - Ms. Neeraj Gohlan Journalism) Bachelor of Arts (Mass Communication and Co-ordinator - Dr. Chanchal Sachdeva Suri Journalism) Co-ordinator - Dr. Kamaljit Kaur Academic and Administrative Management Prof. (Dr.) Pranveer Singh Satvat Prof. (Dr.) S.S. Sehgal Pro VC (Academic) Registrar Prof. (Dr.) H. Nagaraja Udupa Prof. (Dr.) Shiv Kumar Tripathi Director - (IDOL) Executive Director - USB © No part of this publication should be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the author and the publisher. SLM SPECIALLY PREPARED FOR CU IDOL STUDENTS Printed and Published by: Himalaya Publishing House Pvt. Ltd., E-mail: [email protected], Website: www.himpub.com For: CHANDIGARH UNIVERSITY Institute of Distance and Online Learning CU IDOL SELF LEARNING MATERIAL (SLM)

Business Mathematics and Statistics Course Code: BBA102 /BCM10 Credits: 4 Course Objectives: z To identify the basic mathematical tools which are used in business. z To develop insights in mathematical concepts towards understanding business problems. z To analyze managerial situations using mathematical concept to formulate problems for synthesis of information. Syllabus Unit 1: Introduction to Business Mathematics: Basic Concepts and Uses. Unit 2: Simple and Compound Interest: Meaning, Basic Concepts and Applications of Simple and Compound Interest. Unit 3: Annuity: Meaning, Basic Concepts and its Applications. Unit 4: Matrices: Introduction, Standard Definitions and Types of Matrices, Matrix Representation of Data and Operations. Unit 5: Determinant: Properties of Determinants, Determinant of 2 × 2 and 3 × 3 Matrices (Direct and Without Expanding), Cramer’s Rule and Inverse Matrix Method. Unit 6: Permutation and Combinations: Concept of Factorial, Sum Rule and Product Rule, Properties of Permutations and Combinations, Applications. Unit 7: Linear Programming: Formulation of Equation: Graphical Method of Solution; Problems Relating to Two Variables including the Case of Mixed Constraints; Cases Having No Solution, Multiple Solutions, Unbounded Solution and Redundant Constraints. Unit 8: Introduction of Statistics: Meaning, Scope, Importance and Limitations, Applications of Statistics in Managerial Decision-making. Unit 9: Classification and Tabulation of Data: Meaning, Types and Classifications. Unit 10: Measures of Central Tendency: Arithmetic Mean, Mathematical Properties of Arithmetic Mean, Median, Quartiles, Mode, Geometric Mean. CU IDOL SELF LEARNING MATERIAL (SLM)

Unit 11: Dispersion: Range, Quartile Deviation, Standard Deviation, Coefficient of Variation. Unit 12: Sampling: Introduction of Sampling, Difference between Census and Sampling. Unit 13: Probability: Types of Probability and Non-probability Sample. Unit 14: Correlation Analysis: Significance, Types, Methods of Correlation Analysis: Karl Pearson’s Correlation Coefficient, Rank Correlation Coefficient, Properties of Correlation. Unit 15: Regression Analysis: Regression Lines; Probable Error, Relationship between Correlation and Regression Coefficients. Unit 16: Time Series Analysis: Introduction, Trend Analysis (Using Moving Averages Method and Least Square Method only). Text Books: 1. Veerarajan, T. (2016), Discrete Mathematics, New Delhi: Tata McGraw-Hill. 2. Singaravelu (2013), Allied Mathematics, Chennai: Meenakshi Agency. Reference Books: 1. Vittal, P.R. (2017), Allied Mathematics, Chennai: Margham Publications. 2. Venkatachalapathy, S.G. (2007), Allied Mathematics, Chennai: Margham Publications. CU IDOL SELF LEARNING MATERIAL (SLM)

CONTENTS Unit 1: Introduction to Business Mathematics 1-8 Unit 2: Simple and Compound Interest 9 - 34 Unit 3: Annuities 35 - 65 Unit 4: Matrices 66 - 93 Unit 5: Determinants 94 - 116 Unit 6: Permutations and Combinations 117 - 132 Unit 7: Linear Programming 133 - 162 Unit 8: Introduction to Statistics 163 - 175 Unit 9: Classification and Tabulation of Data 176 - 198 Unit 10: Measures of Central Tendency 199 - 240 Unit 11: Dispersion 241 - 279 Unit 12: Sampling 280 - 283 Unit 13: Types of Probability 284 - 294 Unit 14: Correlation Analysis 295 - 330 Unit 15: Regression Analysis 331 - 353 Unit 16: Time Series Analysis 354 - 383 References 384 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 1 INTRODUCTION TO BUSINESS MATHEMATICS Structure 1.0 Learning Objectives 1.1 Introduction 1.2 Need and Uses of Mathematics 1.3 Application of Mathematics 1.4 Summary 1.5 Key Words/Abbreviations 1.6 Learning Activity 1.7 Unit End Questions (MCQ and Descriptive) 1.8 References 1.0 Learning Objectives After studying this unit, you will be able to: z Explain the importance of mathematics for business data. z Elaborate the various mathematical aspects and in business data. z Study the references of mathematical approach to decision-making in business and also to learn some topics in operations research. 1.1 Introduction Nowadays, many a business concern maintains its own department of statistics, for evaluating its business policies on profits, sales, consumer preferences, manufacturing costs, quality of products, CU IDOL SELF LEARNING MATERIAL (SLM)

2 Business Mathematics and Statistics productivity and consumers good will. Economic planning without accurate and reliable statistics is unthinkable. Statistical charts, graphs and diagrams pertaining to quantitative data, are visual aids and immediately appeal to the eye and render business figures and facts, more significant and easily intelligible. Success in trade and commerce at present, is largely determined by correct estimates and probabilities and favorable forecasts based on statistics collected systematically, presented accurately and interpreted wisely. Whether a person is a banker, an investor, a manufacturer or a stock exchange broker, he would be benefited immensely; if he possesses statistical knowledge relating to monetary savings, rates of interest, diverse investment markets, costs of different raw materials and their markets, stocks and share markets etc. One of the several reasons for the Insurance Companies to make profits is that, with the knowledge of the operation of the Law of Statistical Regularity, of statistics of births and deaths due to various causes — their actuaries calculate insurance premiums. Statistics is also known as the Arithmetic of Human Welfare. Human Welfare is greatly enhanced by agricultural and industrial progress. Statistical techniques are golden keys to industrial progress for modern statistical techniques such as Quality Control, Market Research etc. when properly used by industrial establishments can lead to increased output, improved quality, reduced costs, encouraging profits, saving in wastage of materials and labour and wise decisions in purchase policy and launching of sales and thereby to the promotion of industrial prosperity. In other words the tools of statistical analysis may not lead us to peaks of precision or to depths of illusion, but they do certainly and invariably show us the practical levels of human expectation in quantitative terms. Large scale and as well as small scale industrial establishments can profit much by maintaining statistical departments and employing expert statisticians. Such departments under wise and expert statistical supervision collect, record and analyse data relating to industry, prepare statistical charts, diagrams and maps, design and execute market research surveys, furnish periodical progress reports and tender the most sensible statistical advice to the business executives. Undoubtedly statistics, and statisticians are indispensable aids in trade, commerce and industry. In the modern times high-speed technology has made it possible to analyse data pertaining to Business, Economics and financial facts and figures systematically, scientifically with accuracy and reliability. CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Business Mathematics 3 Information technology based on high-speed computerisation has led to progressive development in the appropriate interpretation of numerical facts, thereby ensuring the effective usefulness. Business figures relating to capital investments, sales, profits, losses, rates, financial ratios, work, time and distance, percentages, appreciation, depreciation, balance sheets, stocks, shares etc. form the substance of Business Mathematics. The statement ‘Number rules the universe’ can be justified by the impact of computer technology on facts and figures in the modern world of Business, science and operational processes. Mathematical approach to decision making in Business becomes more reliable, and meaningfully significant. Linear Programming, transportation, Models, Assignment Models, PERT, CPM, Queuing Models are effective tools in operations-research and quantitative data. Business Mathematics is an integral part of Commerce and Management courses. Commercial Arithmetic which deals with Rates, Brokerage, Commission, Discount, Dividend, Mensuration, Permutations, Combinations, etc. are all useful to Business Management. 1.2 Need and Uses of Mathematics Mathematics is used in almost every aspect of life. It also plays an important role in business decision making. Business mathematics is used to record and manage business operations. Many Business activities like accounting, financial analysis, sales etc. are done by using business mathematics. When we talk about decision making for business, mathematics plays an important role in it. A businessman has to take thousands of decisions every day for his business, therefore, mathematics helps in rationalizing the information and it gives the accurate results inherent for any decision making. Mathematics has numerous concepts like statistics, probability, algebra, calculus etc. that are used in business mathematics. These concepts provide accurate statistics solutions for various business activities. Mathematics is an important subject which enhances the knowledge of the person and it also improves problem-solving skills. CU IDOL SELF LEARNING MATERIAL (SLM)

4 Business Mathematics and Statistics 1.3 Application of Mathematics Mathematics is used in almost every field of daily life. Business involves the buying and selling of goods in order to earn profit, it uses mathematics to record, classify, summarize and analyse the business transactions. So mathematics is used by commercial enterprises to record and manage the business operations such as, elementary arithmetic involving fractions, decimals, percentage, elementary algebra, statistics and probability. Now a days business management is using advanced mathematics such as calculus matrix algebra and liner programming. Practical applications include checking accounts, forecasting the sales, price discounts, mark-ups, mark- downs, payroll calculations, simple and compound interest, reducing wastage of resources. Some applications of mathematics in business and commerce are listed below: 1. Algebra 2. Operation Research 3. Statistics and Probability 4. Calculus 5. Matrix and Linear Algebra 2.1 Algebra Mathematical principles are needed to study accounting. It incorporates successful exploration of numerical, geometrical and logical relationships. Mathematics benefits accountant in comparison — mathematical formulas help business and commerce to compare income, cost, expenses and profits. The various formulas are derived using various percentage, ratios and equations. The various ratios are derived such as: inventory turnover ratios, profitability ratios, debtor turnover ratio, debt- equity ratio etc. Mathematics is helpful in deriving accounting equation. The basic idea in accounting is that total wealth of business is called Assets. There are two possible claims on assets (A) called liabilities (L) and capital (C). By using mathematical relation A = L + C, accountants use mathematics in order to arrive the total cost and taking decision regarding manufacturing or buying the product. The total cost formula for business is T = a + bx; where ‘T’ is total cost ’a’ is fixed cost, ‘b’ is cost per unit produced and ‘x’ is no. of units produced. Also profits are determined by subtracting total cost from total revenue and helps in analysing the financial health of business and prices are CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Business Mathematics 5 determined by adding some markup to cost. So accountant used addition and percentage to determine the prices of product. 2.2 Operational Research (OR) OR is concerned with determining the maximum (profit, performance, yields) or minimum (loss, cost, risk) of some real world objectives. OR includes game theory, linear programming formulation techniques, PERT, CPM, transportation problems. Linear programming also called linear optimization is a method to achieve the best outcome (as maximum profit or lowest cost or ensuring best use of available resources) in a mathematical model whose requirements are represented by linear relationship. Some of industries that use LP model include transportation, energy, telecommunication and manufacturing. Linear function to be maximized by mathematical function. 2.3 Statistics and Probability Statistics is very indispensable for the businessman. It formulates various plans and policies and forecasts trends of future such as change in demand, market fluctuations using statistical techniques. On the other hand, future events are uncertain and to predict these uncertainties, probability is an effective tool to forecast sales, scenario, future returns and risk evaluation in the business world. Before introducing the product, team of market research analyse data relating to population, income of consumer, tastes, preferences, habits, pricing policy of competitors by using various statistical techniques. We can collect and analysis the data in the field of economy by using statistical methods. Probability theory serves as a useful tool for decision making, estimating number of defective units, sales expected and also in business policies. Through the use of statistical (regression) techniques Levine and Zervos (1998) attempted to find the empirical relationship between various measures of stock market development, banking development, and long-run economic growth. They concluded that even after controlling for many factors associated with growth, stock market liquidity and banking development are both positively and robustly correlated with contemporaneous and future rates of economic growth, capital accumulation, and productivity growth. The small business firms especially those in the fashion industry should learn and apply probability theory since there line of business was more prone to chance occurrences (Orga, &Ogbo, 2012). In their study in Nigeria they observed that the small business firms fail despite the programmes of government directed at their survival. The application of probability theory in small business was examined to find the implications and in restoring the gap between the rich and the CU IDOL SELF LEARNING MATERIAL (SLM)

6 Business Mathematics and Statistics poor through better and informed decisions. The findings indicated that probability theory has wide application in small business firms; probability shows specificity in business situations and is inevitable in this era of information overload caused by ICT. In nutshell, statistics and probability are very useful in taking various decisions relating to material, production, finance, personnel and marketing in an Industry. 2.4 Calculus Calculus is another branch of mathematics made up of two fields — differential calculus and integral calculus. Differential calculus plays valuable role in management and business for decision making in production (e.g., supply of raw material, wage rates and taxes). In calculus, the case when ‘y’ is a function of ‘x’ or we can say one variable (y) is dependent on other variable (x) and the derivative of ‘y’ w.r.t. ‘x’, i.e., dy/dx measures the change of variable ‘y’ w.r.t. change in variable ‘x’. Derivative enables a firm to make important production decisions. It is also called marginal function. Demand can be assumed as a function of price. This operator is also helpful in calculating minimum cost and maximum profit. Also total cost of production and marketing depends on no. of units in mathematical relations, which can be described as c(x) = F + v(x), where c(x) is cost function v(x) is variable cost and F is fixed cost. Revenue function R(x) can be represented as R(x) = xp(x), where ‘x’ is no. of units and ‘p’ is rate per unit. Hence, knowledge of derivative is essential for understanding the economic relations. Another integral operator is used to calculate the total revenue in case of marginal revenue is given. So calculus plays a vital role in taxes, profit and revenue calculations which are very important for any business. 2.5 Matrix and Linear Algebra Matrices play prominent role in developing a solution required for commercial organizations. It has knowledge to deal with unique needs of various sectors of Industry. It gives opportunities to finance and logistics management and customer relationship by providing them a variety of solutions. Also product price matrices are helpful to set bulk purchase discount. Determinants and Cramer rule are helpful in problem solving related to business and economy. It enables oneself in obtaining and optimal solution to maximize profit or minimize cost problems. Linear algebra serves a purpose of powerful tool for its application in business. As total cost, revenue, supply, demand and population are all related with a system of linear equations. Leontief (1987) derived a production equation in input-output analysis and got Noble prize for his contribution. The model given by him was X = CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Business Mathematics 7 CX + d, where x is the production factor, c is consumption matrix and d is demand vector. If matrix I - C is invertible then appropriate production for a given final demand can be computed directly via X = (I - C)-1 d. This basic input-output analysis however is a very powerful tool (Miller & Blair, 2009). It can Predict what happens to an economy when final demand changes. By changing the consumption matrix this can represent what happens to an economy when the relative cost in terms of other goods (a change in one or more entries in internal demand) of producing one good can change both internal and final demand economy. Dyck & Sumaila (2010), has applied the Leontief technological coefficients at total current impact of the fisheries sector at current production and then estimate total output supported throughout the economy at the current level of production. They recognized that the non-linearity of fisheries production could cause problems when doing predictions at various levels of production. 1.4 Summary Mathematics and statistics are of much relevance in the study of business data relating to manufacturing costs, consumer preferences, sales, profits, quality of products progress in trade and commerce can be known through estimates and probabilities. Rates, brokerage, commission, dividend permutation, combinatin are significant in business mathematics. 1.5 Key Words/Abbreviations Business policies, consumer preferences, estimates, probabilities, discounts, manufacturing costs, stock exchange, brokerage, decision-making, linear programming, quality control, PERT, CPM. 1.6 Learning Activity Study the progressive aspects of a business concern, by using percentages, rates, ratios in the interpretation of sales, profits, costs, consumer preferences during the last 3 years. CU IDOL SELF LEARNING MATERIAL (SLM)

8 Business Mathematics and Statistics 1.7 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions 1. Describe briefly the scope and usefulness of Business Mathematics in trade commerce and industry. 2. Explain the significance of Business Mathematics with reference to financial management. 3. Explain the usage of mathematical methods with reference to ‘decision-making’ in Business. 4. Describe briefly the use of certain mathematical techniques in certain spheres of business activity. 5. Discuss the statement ‘Business Mathematics provides accuracy to statements of facts and figures whereas statistical calculations are supportive estimates based on probability.’ 6. “Mathematical methods if used properly, ensure reliability to business transactions.” Discuss. 7. Write an explanatory note on; Use of Mathematics tools in ‘Decision-making’. B. Multiple Choice/Objective Type Questions 1. Business mathematics is concerned with the following except ________. (a) Sales and profits (b) Financial ratios (c) Capital investments (d) State capital (e) Commission and brokerage 2. In which one of these environments the usage of business mathematics is most significant. (a) Sound environment (b) Political environment (c) Economic environment (d) Business environment (e) Culural environment Answers: (1) (d); (2) (d). 1.8 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 2 SIMPLE AND COMPOUND INTEREST Structure 2.0 Learning Objectives 2.1 Introduction 2.2 Simple Interest 2.3 Present Worth and True Discount 2.4 Banker’s Discount 2.5 Average Due Date 2.6 Compound Interest 2.7 Summary 2.8 Key Words/Abbreviations 2.9 Learning Activity 2.10 Unit End Questions (MCQ and Descriptive) 2.11 References 2.0 Learning Objectives After studying this unit, you will be able to: z Elaborate the meanings and the methods of calculation of S.I. and C.I. z Analyse the use and the procedure of calculation of the present worth, true discount, banker’s discount. z Work out various types of examples. z Explain the significance of these in practical business. CU IDOL SELF LEARNING MATERIAL (SLM)

10 Business Mathematics and Statistics 2.1 Introduction The money that is paid for the use of money that is taken as a loan is known as ‘Interest’. The person giving the loan is the money lender and the person taking it is the borrower. The money lender lends money in consideration of which the borrower makes a payment. The sum of money that is borrowed is called the ‘Principal’ and the money that is paid for its use is called ‘interest’. The interest that is charged for a loan of ` 100, for one year, is called the ‘Rate percent per annum’. The total amount that is due from the borrower at any time is: = Principal + Interest due till that time i.e., Amount = Principal + Interest Notations: P = Principal I = Interest accrued r = Rate of interest per annum n = Time in years A = Amount due i = r/100 = rate of interest per annum per unit of money (1 rupee) 2.2 Simple Interest Interest is said to be ‘Simple’ if it is not added to the Principal at the end of each year, but it accumulates in proportion to the time or period of the loan. In other words the ‘Principal’ is fixed but the interest accrues in proportion to time. Calculations Explained: (i) Given P, n, r to find I Procedure: I = Pnr 100 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 11 (ii) Given I, n, r to find P Procedure: P = 100uI nr (iii) Given I, P, N; to find r Procedure: r = 100uI Pn (iv) Given P, n, r; to find A §Pnr· § nr · Procedure A = P ¨ ¸ P¨1 + ¸ ©100¹ © 100¹ ? A = P (1 + ni) where i = r/100 = interest on ` 1 for 1 year. Remark: If the time period = D days D then n = 365 Illustrative Examples: 1. Find the S.I. on ` 2,800 for 5 years at 8% per annum. Pnr 2800 u 5 u 8 = ` 1,120 I= = 100 100 2. Find the rate per cent at which Rs.4600, would give ` 1,150 as interest, in 4 years. 100 × I 100 u 1,150 25 r = Pn = 4, 600 u 4 = 4 = 6¼% 3. Find the principal that is necessary to give ` 560 as interest in 3½ years at 5% S.I. per annum. 100 × I 100 q 560 P = nr = 3,5 q 5 = ` 3,200 4. A sum of money, lent out at simple interest, amounts to ` 2,520 in 2 years and to ` 2,700 in 5 years. Find the sum of money and the rate of interest. [N.D.A.] CU IDOL SELF LEARNING MATERIAL (SLM)

12 Business Mathematics and Statistics Solution: In 5 years the amount = ` 2,700 In 2 years the amount = ` 2,520 ? In 3 years S.I. = ` 180 2 In 2 years S.I. = ` 180 × 3 = ` 120 ? Principal = Amount in 2 years - S.I. in 2 years = ` (2,520 - 120) = ` 2,400 120 u 100 Rate of Interest = 2400 u 2 = ` 2½ Rate of S.I. is 2½%. 2.3 Present Worth and True Discount Suppose a person ‘A’ owes a definite sum of money to another ‘B’ and the payment of that money would be due on some future date. In the mean time, if ‘A’ desires to make an immediate payment of the debt in full then he would be entitled to make a deduction from the amount due, because he makes an immediate payment. The money so deducted is known as ‘discount’ and the remaining amount that is paid after such a deduction is known as the ‘Present Value’ or ‘Present Worth’ of the debt. Usually, in the case of ‘Bills’ such discounts calculated on percentage basis are allowed in consideration of an immediate cash payment. This discount is called ‘true discount’. Thus if ‘A’ is the amount of a bill that would be due for payment after a period of n years and the rate of interest for one rupee for the purpose of discount is ‘i’ then CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 13 A P.W. = 1 ni where P.W. = Present Wroth =1 A § r· ¨i ¸ nr / 100 © 100¹ A u 100 = 100 + nr 100 ? P.W. = Debt × 100 + nr We note the following: (1) Debt P.W. (100 + nr) = 100 (2) Discount nr = P.W. × 100 or Discount = Debt - P.W. 100 = Debt - Debt × 100 + nr § 100 · = Debt¨1 ¸ © 100 + nr¹ §100 + nr 100· = Debt¨ ¸ © 100 + nr ¹ § nr · = Debt ¨ ¸ ©100 + nr¹ § nr · =A¨ ¸ ©100 + nr¹ Ani = 1 + ni § A· = ¨ ¸ ×n×i ©1 + ni¹ CU IDOL SELF LEARNING MATERIAL (SLM)

14 Business Mathematics and Statistics = (P.W.) × n × i = interest on Present Worth The following facts should be noted well: z ‘Debt’ corresponds to ‘Amount’. z ‘True Discount’ corresponds to ‘Interest’. z ‘Present Value’ or ‘Present Worth’ corresponds to ‘Principal’. z ‘Rate of Discount’ corresponds to ‘Rate of Simple Interest’. Now-a-days in practical business calculations ‘true discount’ is not of much use. 2.4 Banker’s Discount The interest that a Banker charges for discounting a Bill before the date on which it falls due is known as ‘Banker’s Discount’. It is obtained by calculating the interest on the face value of the bill for the remaining tenure of the bill. Banker’s Discount = Ani where A = face value of the bill n = number of years after which it is due i = rate of interest The present Value or the Present Worth of a bill is: = (face value of the bill) - (Banker’s Discount) = A - Ani = A (1 - ni) The gain of the Banker in discounting a bill as per this method, is equal to the excess of this ‘Discount’ over the ‘True Discount’. That is, Banker’s Gain = B.D. - T.D. Ani = Ani - 1 + Ani A(ni) 2 = 1 ni CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 15 Ani = 1 ni × ni = Interest on True Discount 2.5 Average Due Date If a number of bills for amounts a1, a2 , a3, ..... are due after n1, n2, n3, ...... days but it is felt necessary to cash these bills on one and the same day for the total a1 + a2 + a3 + .............. of the different amounts of the bills, then the date on which all these bills could be cashed is called the ‘Average Due Date’. If N denotes the number of days after which the full value of the bills would be due, then the following equation enables us to determine the ‘Average Due Date’. an + a n + a n +......... N= 1 1 22 33 a1 + a2 + a3 + ............. Explanation: The above equation is derived as follows: The P.V. of the total of amounts a1 + a2 + a3 + …… = (a1 + a2 + a3 + .........) - (a1 + a2 + a3 + ..........)ni where ‘I’ is the interest per rupee per day. But the present values of the different bills are respectively. a1 - a1n1i a2 - a2n2i a3 - a3n3i ? (a1 + a2 + a3 + ……) - (a1 + a2 + a3 + ……)ni = (a1 + a2 + a3 + ……) - (a1n1 + a2n2 + a3n3 + ……)i ? (a1 + a2 + a3 + .................. ) NI = (a1n1 + a2n2 + a3n3 + ……)i an1 1 + a n2 2 + a n3 3 + … ?n= a2 + a2 + a3 + … CU IDOL SELF LEARNING MATERIAL (SLM)

16 Business Mathematics and Statistics Illustrative Examples Example1: A radio-set is offered for ` 950 cash or ` 1,000 on instalments on the following terms: ` 100 cash down and the balance in 9 equal monthly instalments. What is the average rate of simple interest charged? Solution: Since the first payment is ` 100, the payment of the balance, i.e., ` 1,000 - ` 100 = ` 900 may be considered as the repayment of a loan of ` 900 in 9 monthly instalments of ` 100 each. The equated time of payment is obtained from: 900 × t = 100 (1 + 2 + 3 + ....... + 9) 100 9 (10) ? t = 900 u 2 =5 The interest is ` 50, therefore nr I = p × 12 × 100 5r 50 = 900 × 12 × 100 ?r= 50 u 12 u 100 900 u 5 = 120/9 = 13.33% Example 2: A person desires to create an endownment fund to provide for a prize of ` 300 every year. If the fund can be invested at 4% per annum, find the value of the endownment. Solution: The amount of endownment fund = ` 300/0.04 = 100 × 300/4 = 7,500 Ans.: ` 7,500 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 17 Example 3: A person undertakes to pay back a loan of ` 8,000 in monthly instalments of ` 200 plus interest at 12% on the outstanding balance. What is the average rate of interest earned by the money lender? Solution: The number of instalments = 8,000 = 40 200 The equated time t = 200 (1 + 2 + 3 + ........+ 40) 200 u 40 = 41/2 = 20½ months. ? The interest on ` 8,000 for 20½ months at 12%. 8000 u 0.12 u 41 = 12 u 2 8000 u 12 u 41 = 100 u 12 u 2 = 40 × 41 = ` 1,640 If the interest on ` 8,000 for 40 months is ` 1,640, then the rate of interest is 1640 u 100 u 12 123 = = 6.15% 800 u 40 20 Note: In this case, the amount of each instalment would be: 8000 1640 = 9640 = ` 241 40 40 Example 4: The cash price of a Pressure Cooker is ` 240. A customer desires to take it on the basis of 12 equal monthly instalments. The seller agrees provided the customer pays 14% simple interest. What is the value of each instalment? Solution: Let an instalment be denoted by p, then the toal amount of 12 instalments would be 12p. An instalment of p for 12 months (including the 1st payment) is equivalent to a single payment of 12p at the end of: p (1 + 2 + 3 + .........+ 12) 12p = 6½ months CU IDOL SELF LEARNING MATERIAL (SLM)

18 Business Mathematics and Statistics Interest on ` 240 @ 14% p.a. for 6½ months is: ­240 u 14 u 13½ =`® ¾ ¯ 12 u 100 u 2 ¿ 182 =` 10 = ` 18.20 ? The total amount payable in 12 instalments = ` 240 + ` 18.20 = ` 258.20 ? The value of each instalment 258.20 =` 12 = ` 21.52 Example 5: 10% Debenture Stock purchased at 108 in 1990 is due to be paid off in 2000 at ` 105 for every 100 stock. What is the return to the investor on the basis of simple interest? Solution: Premium = ` 5 Interest = ` 100 Total return = ` 105 ? The purchaser gets ` 105 in 10 years on an investment of ` 108 ? The rate of interest received from the investment is 105 u 100 = 108 u 10 = 9.44% p.a. Example 6: A money lender advances ` 8,400 on the condition that a sum of ` 9,000 should be paid back at the end of a year. Find the return on the lender’s investment. Solution: Interest earned = ` 9,000 - ` 8,400 = ` 600 Interest on ` 8,400 for 1 year is ` 600, therefore rate of interest is 600 u 100 = = 7.14% p.a. 8400 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 19 Example 7: A certain sum of money is deposited with a Banker at a specified rate of interest. An amount of ` 480 was withdrawn at the end of the first year. Again at the end of the second year ` 260 was withdrawn and then the sum remaining at credit was ` 1,000. If no withdrawals had been made the sum deposited would have earned an annual simple interest of ` 80. What is the original sum deposited and the rate of interest? Solution: Let the original amount deposited be ‘p’ and ‘r’ the rate of interest, then pr = 80 … (1) and [ p (1 + r) - 480] (1+ r) - 260 = 1000 … (2) 80 Putting r = p in equation (2), we get ª § 80· ºª 80º «p¨1 ¸ 480»«1 » 1260 ¬© p ¹ ¼ p ¼ ? (p - 400) (p + 80) = 1260 p p2 - 1580p - 32000 = 0 (p - 1600) (p + 20) = 0 ? p = 1600 The sum deposited is ` 1,600 80 Rate of interest = 1600 = 0.05 The rate of interest is ` 5% p.a. Example 8: A loan of ` 8,500 has to be repaid in monthly instalments of ` 200 each. What is the rate of interest charged? Solution: Equating these instalments of ` 200 each for 50 months, with a single payment of ` 8,500 at the end of the period t, where t is the period of the average due date, we get 200 (1 + 2 + 3 +........ + 50) = 10000t 200 u 50 (50 + 1) 10000 u 2 = t CU IDOL SELF LEARNING MATERIAL (SLM)

20 Business Mathematics and Statistics 51 ?t= 2 Interest charged = ` 1,500 ? 1,500 = Pnr t where n = 12 1500 r = Pn 1500 = 1500 = §8500 u 51· p (t/12) ¨ ¸ © 2 u 12 ¹ 2400 = 289 Rate of interest = 8.35% Example 9: A building society advances ` 35,000 to be repaid in 15 equal annual instalments 1 along with Simple Interest 6 4% p.a. Find the amount of each instalment. Solution: Let p = the annual instalment The ‘average due date’ is px1 + px2 + ........+ px15 = 15p 1 + 2 + ....... + 15 = 15 15 u 16 = 2 u 15 = 8 years Interest due on ` 35,000 in 8 years @ 6¼% per annum is: 35000 25 = 100 × 4 × 8 = 17,500 Total sum to be paid = 15p CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 21 = 35,000 + 17,500 = ` 52,500 Each annual instalment = p 52, 000 = 15 = ` 3,500 Example 10: A bill for ` 5,460 was drawn on 12th June 2007 at 6 months date and discounted on 17th July 2007 at the rate of 5%. Find the Banker’s Discount in this bill. Solution: Bill drawn on = 12-06-2007 Adding 6 months = 12-12-2007 Adding 3 days of grace = 15-12-2007 Bill matures on = 15-12-2007 Bill discounted on = 17-07-2007 Remainder of the tenure = 151 days Working Month Days July 14 Aug. 31 Sept. 30 Oct. 31 Nov. 30 Dec. 15 151 days Banker’s Discount = Interest on ` 5,460 at 5% p.a. for 151 days. i.e., B.D. = 5,460 × 151 5 365 × 100 = ` 112.94 CU IDOL SELF LEARNING MATERIAL (SLM)

22 Business Mathematics and Statistics Example 11: A sewing machine is offered for ` 300 cash or 8 monthly instalments of ` 40 each, the payment starting at the end of the first month. What is the rate of interest charged? Solution: At the end of each month the following amounts are outstanding: ` 300 at the end of the 1st month ` 260 at the end of the 2nd month ` 220 at the end of the 3rd month ` 180 at the end of the 4th month ` 140 at the end of the 5th month ` 100 at the end of the 6th month ` 60 at the end of the 7th month ` 20 at the end of the 8th month Now ` 20 being the difference between ` 320 and ` 300 is the interest that is charged on: 300 + 260 + 220 + 180 + 140 + 100 + 60 + 40 + 20 for one month. The sum of the above arithmetic series is: n a = 20, d = 40, n = 8 S = 2 [ 2a + (n - 1) d] 8 = 2 [ 40 + (8 - 1) 40] = 4 (40 + 280) = 4 × 320 = 1280 r1 Now 20 = 1280 × 100 12 150 [Ans: Rate of interest is 18.75%] =8 = 18¾ = 18.75 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 23 2.6 Compound Interest There is much difference between Simple Interest considered earlier and the Compound Interest. The main difference is that in the case of Compound Interest the interest earned at the end of the year is added to the Principal and therefore the Principal becomes a variable quantity and in this manner interest grows. That is, the interest that is left to accumulate earns interest on itself, and the interest is added periodically to the Principal. For example, the calculation of Compound Interest on ` 8,000 for 3 years at 5% per annum is as follows: Principal at the beginning = 8000 = P Interest at the end of the ½ = (8000) (0.05) 1st year ¾ ¿ = 400 Amount at the end of the ½ = 8000 + 8000 (0.05) ½ = P + Pi 1st year ¾ ¾ = 8000 + 400 ¿ = P (1 + i) ¿= 8400 Interest at the end of the ½ = (8400) (0.05) ½ = (P + Pi) i 2nd year ¾ ¾ ¿ = 420 ¿ Amount at the end of the ½ = 8400 + 420 ½ = (P + Pi) + (P + Pi) i 2nd year ¾ ¾ = (P + Pi) (1 + i) ¿ = 8820 ¿ = P (1 + i)2 Interest at the end of the ½ = (8820) (0.05) ½ 3rd year ¾ ¾ = P (1 + i)2 × i ¿ = 441 ¿ Amount at the end of the ½ = 8,820 + 441 ½ = P(1 + Pi)2 + P(1 + i)2 × i 3rd year ¾ ¾ = 9,261 ¿ = P (1 + i)3 ¿ ? The amount at the end of 3 years is ` 9,261 In brief, Amount at the end of the 1st year = P (1 + i) Amount at the end of the 2nd year = P (1 + i)2 CU IDOL SELF LEARNING MATERIAL (SLM)

24 Business Mathematics and Statistics Amount at the end of the 3rd year = P (1 + i)3 Amount at the end of the nth year = P (1 + i)n That is, A = P (1 + i)n Or A = P (1 + r/100)n By logarithmic Calculation, A = Antilog [log P + n log (1 + i)] Further, A/P = (1 + i)n By logarithmic calculation it follows that log A - log P = n log (1 + i) log A log P ? log (1 + i) = n Illustrative Example: Example 1 : Find the time that is necessary for a certain sum of money to double itself, the rate of interest per ` 1/- per annum being i. Solution: Putting A = 2P in the equation log A log P log (1 + i) , we get log 2P log P log 2 + log P log P log (1 + i) = log (1 + i) log 2 = log (1 + i) For instance, if i = 0.05 log 2 log 2 Then n = log (1 + .05) = log (1.05) 0.3010 = 0.0212 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 25 = Antilog [log 3010 - log 0212] = Antilog [3.4786 - 2.3263] = Antilog [1.1523] = 14.20 Money doubles itself in 14.2 years, taking the rate of interest as 0.05 for ` 1/- per annum. The following table shows the different time periods required for money to double itself at different rates of interest. ri n 3 .03 23.44 3.50 .035 20.15 4 .04 17.68 4½ .045 15.74 5 .05 14.20 5½ .055 12.95 6 .06 11.89 6½ .065 11.01 7 .07 10.25 7½ .075 9.58 8 .08 9.01 8½ .085 8.49 9 .09 8.04 9½ .095 7.64 10 .1 7.27 Example 2: Find the compound interest earned from ` 16,000 for 3 year at 12% per annum? Solution: Here, P = ` 16,000 Rate of interest, r = 12% = 0.125 No. of periods, n = 3 Interest compounded annually ? i = r Amount A = P (1 + i)n ? A = 16,000 (1 + 0.125)3 = 16,000 (1·423828) = ` 22,781.25 CU IDOL SELF LEARNING MATERIAL (SLM)

26 Business Mathematics and Statistics Now Compound Interest I = A - P = 22,781.25 - 16,000 = ` 6,781.25 ? Interest paid is ` 6,781.25. Example 3: If ` 1,750 is invested at 9% per annum interest for 10 years and interest is compounded half-year, find the amount and interest. Solution: P = ` 1,750 and r = 9% = 0.09 Interest is calculated half-yearly. r ? i = 2 = 0.045 and n = 10 × 2 = 20 Amount A = P (1 + i)n ? A = 1,750 (1.045)20 Let x = (1.045)20 log x = 20 log 1.045 = 20 × 0.0191 = 0.3820 x = antilog (0.03820) = 2.41 Now, A = 1,750 × 2.41 = 4,217.50 Interest, I = S - P = 4,217.50 - 1,750 = ` 2,467.50 ? The amount is ` 4,217.50, and interest is ` 2,467.50 Example 4: To what amount ` 10,000 accumulate in 6 year, if invested at 8% compounded quarterly? [(1.02)24 = 1.6084]. Solution: P = ` 10,000, and r = 8% = 0.08 Rate of interest as per conversion period r 0.08 i = 4 = 4 = 0.02 and n = 6 × 4.24 Amount A = P (1 + i)n ? A = 10,000 (1 + 0.02)24 = 10,000 (1.02)24 = 10,000 × 1.6084 = ` 16,084 ? The amount is ` 16,084. CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 27 Example 5: Find the rate of interest, if the sum of money will double itself in 10 years by investing at compound interest. Solution: Let P be the principal. Let the rate of interest be r. Interest is compounded annually, r = i and n = 10 Given that the amount will be doubled in 10 years. i.e., A = 2 P But A = P (1 + i)n ? 2P = P (1 + i)10 ? 2 = (1 + i)10 Taking log on both sides log 2 = 10 log (1 + i) log 2 0.0310 log (1 + i) = 10 10 = 0.0301 Now, 1 + i = antilog (0.0301) 1 + i = 1.072 i = 0.072 = 7.2% ? The required rate of interest is 7.2%. Example 6: The amounts for a certain sum with compound interest at a certain rate in two years and in three years are ` 8,820 and ` 9,261 respectively. Find the rate and sum. Solution: Let P be the principal amount and i be the rate of interest. Amount A = P (1 + i)n At the end of two years, A = ` 8,820 ? 8,820 = P (1 + i )2 … (1) At the end of three years, A = ` 9,261 ? 9,261 = P (1 + i)3 … (2) CU IDOL SELF LEARNING MATERIAL (SLM)

28 Business Mathematics and Statistics Dividing (2) by (1), we get 3 9, 261 P(1 i) =2 8,820 P(1 i) ˆ 1.05 = 1 + i ˆ i = 0.05 = 5% Now, substituting i = 0.05 in equation (1) we get, 8,820 = P (1.05)2 ˆ 8,820 = P × 1.1025 × P = ` 8,000 ? Required rate is 5% and principal is ` 8,000. Example 7: A certain sum of money is invested at 4% p.a. compound annually. The interest for 2nd year is ` 25. Find the interest for 3rd year. Solution: Let P be the principal. A = P (1 + i)n Amount at the end of 1st year = P (1.04) Amount at the end of 2nd year = P (1.04)2 Interest for 2nd year = Amount at the end of 2nd year - Amount at the end of 1st. Given interest for second year is ` 25. 25 = P (1.04)2 - P (1.04) P (1.04) (1.04 - 1) = 25 … (1) P (1.04) (0.04) = 25 Amount at the end of 3rd year = P (1.04)3 Interest for 3rd year = Amount at the end of 3rd year - Amount at the end of 2nd year ? Interest for 3rd year = P (1.04)3 - P (1.04)2 = P (1.04)2 [1.04 - 1] = P (1.04)2 (0.04) = P (1.04) (1.04) (0.04) = 25 × 1.04 [using (1)] = ` 26. ? Interest for third year is ` 26. CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 29 Example 8: Lal deposited an amount of ` 50,000 in two different Banks A and B, dividing the amount into two investments. Bank A calculates interest at a rate of 7% per annum and Bank B calculates interest at the rate of 6% per annum convertible semi-annually. At the end of 3 years, he received ` 10,632.35 as the return on his investment. What amount he has deposited in both Banks? Solution: Let ` X and ` Y be the amounts deposited in Bank A and Bank B respectively. Total investment is ` 50,000 ? X + Y = 50,000 … (1) Compound interest is given by I = A - P where A = P (1 + i)n Interest earned from Bank A: I1 = X (1.07)3 - X ? I1 = 1.225043 X - X ? I1 = 0.225043 X Interest earned from Bank B: I2 = (1.03)6 Y - Y I2 = 1.194052 Y - Y I2 = 0.194052 Y Total interest earned I = I1 + I2 … (2) ? I = 0.225043 X + 0.194052 Y It is given that total interest received is ` 10,632.35 ? 0.225043 X + 0.194052 Y = 10632.35 Solving (1) and (2) we get, X = ` 30,000 and Y = ` 20,000. ? ` 30,000 and ` 20,000 are the sums deposited in Bank A and Bank B respectively. CU IDOL SELF LEARNING MATERIAL (SLM)

30 Business Mathematics and Statistics 2.7 Summary Borrowers take loans from money lenders and repay the amount borrowed at a certain rate of interest per annum, for the specific period. This is called ‘Amount of Repayment’, consisting of principle plus simple interest. If interest is charged on interest, then such an interest is called Compound Interest. Pnr S.I. = 100 Pnr § Pnr· Amount = P - S.I. = P + P¨1 ¸ 100 = © 100¹ § n Amount A - [Compound Interest + Principal] = P¨1 r· © ¸ 100¹ § r· n P¨1 ¸ ? C.I. = P © 100¹ 2.8 Key Words/Abbreviations S.I. = Simple Interest, C.I. = Compound Interest r Time period = n, (r = interest); Percentage of Interest = 100 i P.W. = Present Worth; T.D. = True Discount; B.D. = Banker’s Discount; A.D.D. = Average Due Date. 2.9 Learning Activity After thoroughly studying and solving the examples, 1 to 25, fill in the blanks in the table given in Example 6. CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 31 2.10 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions 1. In what time will ` 1.250 amount to ` 1,400 at 6% per annum? [Ans: 2 years] 2. The present worth of a bill due at the end of 4 years is ` 575; and if the bills were due at the end of 2½ years its present worth would be ` 620. Find the rate per cent. [Ans: 6%] 3. If the discount on ` 161 due 2½ years hence be ` 21, at what rate percent is the interest claculated? [Ans: 6%] 4. What sum will amount to ` 1,000 in 50 years at 4% p.a. compound interest, the interest being paid half-yearly? [Ans: ` 138] 5. Find the difference between simple and compound interest on ` 1,500 for 5 years at 4½ years at 4½ p.a. [Ans. 31.50] 6. A sum of money at simple interest amount to ` 2,800 in two years and ` 3,250 in five years. Find the sum and the rate of interest. [Ans: 1,500, 6%] 7. The amount of ` 3,000 at 3% p.a. compound interest for 10 years is equal to the amount of ` 2,000 at 6% p.a. compound interest for a certain period. Find this period. [Ans: 12 years] 8. A sum of money put out at Compound Interest amounts in 2 years to ` 672 and in 3 years to ` 714. Find the rate of interest. [Ans: 61/4%] 9. A man buys a house on condition that he shall pay ` 8,820 now and equal sums at the end of one year and two years. What would be the cash value of the house, if Compound Interest payable yearly be calculated at 5% per annum? [Ans: ` 25,220] 10. A simple interest on a certain sum of money for 3 years at 4% is ` 303.60 p. Find the Compound Interest on the sum for the same period and at the same rate. (Fractions of a paisa to be ignored). [Ans: 315.90] 11. The Compound Interest on a certain sum of money for 2 years is ` 920.25 and the simple interest is ` 900. Find the sum and the rate per cent. [Ans: ` 10,000, 4½] CU IDOL SELF LEARNING MATERIAL (SLM)

32 Business Mathematics and Statistics 12. A certain sum of money put out at Compound Interest at 5 percent yields ` 20 more in two years than an equal sum put out at simple interest at the same rate. Find the sum of money. [Ans: ` 8,000] 13. Divide the sum of ` 3,903 between a brother and a sister who are 18 years and 16 years old respectively, in such a way that their shares invested at 4% Compound Interest should be equal when they attain the age of 21 years. [Ans: ` 1,875, ` 2,028] 14. A sum of ` 5,150 lent for 2 years at 6% Compound Interest (compounded annually). If the borrower prefers to pay the money back by two equal annual payments, one at the end of the first year and the other at the end of the second, how much should be pay at the end of each year? [Ans: ` 2,809] 15. A money lender borrows a certain sum of money at 3% per annum simple interest and invests the same at 5 per annum Compound Interest (compounded annually). After 3 years he makes a profit of ` 1,082. Find the amount he borrowed. [Ans: ` 16,000] 16. Find the Simple Interest on ` 75,500 at 81/3% per annum for 2.5 years. [Ans: ` 15,722.87] 17. Find the Simple Interest on ` 8,450 at 12% per annum for 100 days. [Ans: 277.80] 18. Find the Simple Interest on ` 1,25,000 at 18% per annum for the period from 11th April 2008 to 23rd May 2008. [Ans: ` 2,589.04] 19. At what rate percent per annum will a sum of money double itself in 6½ years? [Ans: ` 15.38%] 20. A sum of money was invested at Simple Interest at a certain rate for 3 years. Had it been invested at a rate 2% higher than the present rate it would have given the investor ` 360 more. What is the sum of money invested? [Ans: ` 6,000] 21. ` 800 amounts to ` 920 in 3 years at Simple Interest. Had the interest rate been increased by 3% then what would be the amount? [Ans: ` 992] 22. If a sum of money doubles itself in 8 years at Simple Interest, then what is the rate of interest per annum? [Ans: 12.5] 23. If the difference between Compound Interest and Simple Interest on a certain sum of money at 10% per annum for 2 years is ` 631, then what is the sum? [Ans: ` 6,100] CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 33 24. Find the Compound Interest on ` 2,800 at 16% per annum for 9 months compounded quarterly. [Ans: ` 349.60] 25. A sum of money at Compound Interest amounts to thrice itself in 3 years. In how many years will it be 9 times itself. [Ans: 6 years] 26. Fill in the blanks in the following table. Ex. No. Principal (P) r (%) n (Yrs) § nr · § Pnr· P¨1 + ¸ ¨¸ ©100¹ © 100¹ Simple Amount Interest 1. 18.250 634 — 21,957.031 3,707.031 2. 29,540 8 13 534 43,694.58 — 6½ — 32,449.21 3. 44,375 11¼ 4. 73,295 — 11 1,31,427.09 58,132.096 34 5. 1,23,984 8 13 125 days — 3,538.36 [Ans: (1) 3¼, (2) 14,154.58, (3) 76,824.21, (4) 6¾%, (5) 2,398.36] B. Multiple Choice/Objective Type Questions 1. If the rate of interest is 8%, money doubles itself in ________. (a) 51/2 years (b) 61/3 years (c) 71/2 years (d) 9.01 years (e) None of these 2. If the S.I. on ` 4,600 is 4 years is ` 1,150, then the rate of S.I. is ________. (a) 91/2 % (b) 51/3 % (c) 61/4 % (d) 7.2% (d) None of these CU IDOL SELF LEARNING MATERIAL (SLM)

34 Business Mathematics and Statistics 3. If the difference between C.I. and S.I. for 2 years at 10% per annum is ` 631, then the principle is ________. (a) 4,100 (b) 5,300 (c) 6,100 (d) 7,400 (e) None of these 4. The C.I. on ` 2,800 for 9 months compounded quarterly is ` 349.6, then the rate of interest per annum is ________. (a) 111/2% (b) 14% (c) 15% (d) 16% (e) None of these 5. A steel cupboard worth ` 10,000 is purchased on instalment basis for ` 200 down and of monthly instalments of ` 1,000 each. The rate of interest charged is ________. (a) 23% p.a. (b) 24% p.a. (c) 25% p.a. (d) 26% p.a. (e) None of these Ans wers: (1) (d); (2) (c); (3) (c); (4) (d); (5) (b); 2.11 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 3 ANNUITIES Structure 3.0 Learning Objectives 3.1 Introduction 3.2 Immediate Annuity 3.3 Present Worth 3.4 Perpetuity 3.5 Annuity Due 3.6 Present Value of an Annuity Due 3.7 Deferred Annuity 3.8 Present Value of a Deferred Annuity 3.9 Endowment Fund 3.10 Annuity in the Case of which Payments are made other than Annually 3.11 Sinking Fund 3.12 Repayment of Loan by Instalments 3.13 Summary 3.14 Key Words/Abbreviations 3.15 Learning Activity 3.16 Unit End Questions (MCQ and Descriptive) 3.17 References CU IDOL SELF LEARNING MATERIAL (SLM)

36 Business Mathematics and Statistics 3.0 Learning Objectives After studying this unit, you will be able to: z Explain the meaning of various types of annuities and the importance of each. z Elaborate the methods of calculation by noting down the various formulae. z Work out the examples given in the exercise. 3.1 Introduction An annuity is a fixed sum paid regularly at equal intervals of time. In other words, an annuity stands for a series of equal payments regularly paid at equal intervals of time. The payments may be made yearly, half-yearly, quarterly or monthly depending upon the conditions of agreement. The person who receives an annuity is known as ‘annuitant’. When an annuity is payable unconditionally for a certain specified period, then it is said to be ‘Annuity Certain’. There are three types of ‘Annuity Certain’ viz : 1. Immediate Annuity 2. Annuity Due 3. Deferred Annuity Immediate Annuity: According to this type, the annuity payment falls due at the end of the first interval. Therefore payment is made regularly at the end of each of the successive intervals of time. Annuity Due: In the case of this type the annuity payment falls due at the beginning of the first interval. Hence regular payment is made at the beginning of each of the successive intervals of time. Deferred Annuity: This type of annuity is also known as ‘Reversion’. According to this, the annuity begins after the lapse of a certain period. Therefore the annuity that is deferred for n years commences only after n years, the first payment being made at the end of n + 1 years. We note the following: z A “Life Annuity” is the type in the case of which, annuity is payable only during the lifetime of a person or of the surviors of a number of persons. CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 37 z An ‘Annuity Contingent” is the type where annuity is payable till some contingency, that is, the happening of some event such as marriage of a girl, commencement of education of a child, or the death of the annuitant. z An annuity is said to be a “Perpetuity” when it is to continue for ever. It is said to be “Deferred Perpetuity” if it does not commence at once. The annuity that is left unpaid for a certain number of years is said to be ‘Forborne’ for that number of years. Application of Annuities Annuities can be purchased from insurance providers, banks, mutual fund companies, stockbrokers, and other financial institutions. They come in several different forms, including immediate and deferred annuities, and fixed and variable annuities. Each form has different properties and involves different costs. Although the money placed in an annuity is first subject to taxation at the same rate as ordinary income, it is then invested and allowed to grow tax-deferred until it is withdrawn. Distribution is flexible and can take the form of a lump sum, a systematic payout over a specified period, or a guaranteed income spread over the remainder of a person’s life. In most cases annuities are a long-term investment vehicle, since the costs involved make it necessary to hold an annuity for a number of years in order to reap financial benefits. Because of their flexibility, annuities can be a good choice for small business owners in planning for their own retirement or in providing an extra reward or incentive for valued employees. An annuity is an interest-bearing financial contract that combines the tax-deferred savings and investment properties of retirement accounts with the guaranteed-income aspects of insurance. Annuities can be described as the flip side of life insurance. Life insurance is designed to provide financial protection against dying too soon. Annuities provide a hedge against outliving your retirement savings. While life insurance plans are designed to create principal, an annuity is designed to liquidate principal that has been created, usually in the form of regular payments over a number of years. Annuities can be purchased from insurance providers, banks, mutual fund companies, stockbrokers, and other financial institutions. They come in several different forms, including immediate and deferred annuities, and fixed and variable annuities. Each form has different properties and involves different costs. Although the money placed in an annuity is first subject to taxation at the same rate as ordinary income, it is then invested and allowed to grow tax-deferred until it is CU IDOL SELF LEARNING MATERIAL (SLM)

38 Business Mathematics and Statistics withdrawn. Distribution is flexible and can take the form of a lump sum, a systematic payout over a specified period, or a guaranteed income spread over the remainder of a person’s life. In most cases annuities are a long-term investment vehicle, since the costs involved make it necessary to hold an annuity for a number of years in order to reap financial benefits. Because of their flexibility, annuities can be a good choice for small business owners in planning for their own retirement or in providing an extra reward or incentive for valued employees. Types of Annuities There are several different types of annuities available, each with different properties and costs that should be taken into consideration as business owners put together their retirement investment portfolio. The two basic forms that annuities take are immediate and deferred. An immediate annuity, as the name suggests, begins providing payouts at once. Payouts may continue either for a specific period or for life, depending on the contract terms. Immediate annuities — which are generally purchased with a one-time deposit, with a minimum of around $10,000 — are not very common. They tend to appeal to people who wish to roll over a lump-sum amount from a pension or inheritance and begin drawing income from it. The immediate annuity would be preferable to a regular bank account because the principal grows more quickly through investment and because the amount and duration of payouts are guaranteed by contract. Immediate annuities are also known by the name income annuities. What is important to remember when considering an immediate annuity is that “at the end of the day, you’ve got to remember what you’re buying is insurance, not an investment vehicle like a stock or mutual fund,” explains Rob Nestor in an article by Murray Coleman in Investor’s Business Daily. Deferred annuities delay payouts until a specific future date. The principal amount is invested and allowed to grow tax-deferred over time. More common than immediate annuities, deferred annuities appeal to people who want a tax-deferred investment vehicle in order to save for retirement. There are also two basic types of deferred annuity: fixed and variable. Fixed annuities provide a guaranteed interest rate over a certain period, usually between one and five years. In this way, fixed annuities are comparable to certificates of deposit (CDs) and bonds, with the main benefit that the sponsor guarantees the return of the principal. Fixed annuities generally offer a slightly higher interest rate than CDs and bonds, while the risk is also slightly higher. In addition, like other types of annuities, the principal is allowed to grow tax-deferred until it is withdrawn. CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 39 The more popular of the deferred annuity types is the variable annuity which offers an interest rate that changes based on the value of the underlying investment. Purchasers of variable annuities can usually choose from a range of stock, bond, and money market funds for investment purposes in order to diversify their portfolios and manage risk. Some of these funds are created and managed specifically for the annuity, while others are similar to those that may be purchased directly from mutual fund companies. The minimum investment usually ranges from $500 to $5,000, depending on the sponsor, and the investments (or subaccounts) usually feature varying levels of risk, from aggressive growth to conservative fixed income. In most cases, the annuity principal ca n be transferred from one investment to another without being subject to taxation. Variable annuities are subject to market fluctuations, however, and investors also must accept a slight risk of losing their principal if the sponsor company encounters financial difficulties. Features of Annuities Variable annuities have a number of features that differentiate them from common retirement accounts, such as 401(k)s and IRAs, and from common equity investments, such as mutual funds. One of the main points of differentiation involves tax deferral. Unlike 401(k)s or IRAs, variable annuities are funded with after-tax money — meaning that contributions are subject to taxation at the same rate as ordinary income prior to being placed in the annuity. In contrast, individuals are allowed to make contributions to the other types of retirement accounts using pre-tax dollars. That is why financial specialists usually instruct people to first maximize their contributions to 401(k) plans and IRAs before considering annuities. On the plus side, there is no limit on the amount that an individual may contribute to a variable annuity, while contributions to the other types of accounts are limited by the federal government. Unlike the dividends and capital gains that accrue to mutual funds, however, which are taxable in the year they are received, the money invested in annuities is allowed to grow tax free until it is withdrawn. Another feature that differentiates variable annuities from other types of financial products is the death benefit. Most annuity contracts include a clause guaranteeing that the investor’s heirs will receive either the full amount of principal invested or the current market value of the contract, whichever is greater, in the event that the investor dies before receiving full distribution of the assets. However, any earnings are taxable for the heirs. CU IDOL SELF LEARNING MATERIAL (SLM)

40 Business Mathematics and Statistics Another benefit of variable annuities is that they offer greater withdrawal flexibility than other retirement accounts. Investors are able to customize the distribution of their assets in a number of ways, ranging from a lump-sum payment to a guaranteed lifetime income. Some limitations, however, do apply. For example, the federal government imposes a 10 percent penalty on withdrawals taken by anyone before they reach the age of 59 1/2 years. But contributors to variable annuities are not required to begin taking distributions until age 85, whereas contributors to IRAs and 401(k)s are required to begin taking distributions by age 70 1/2. Costs Associated with Annuities In exchange for the various features offered by annuities, investors must pay a number of costs. Many of the costs are due to the insurance aspects of annuities, although they vary among different sponsors. One common type of cost associated with annuities is the insurance cost, which averages 1.25 per cent and pays for the guaranteed death benefit in addition to the insurance agent’s commission. There are also usually management fees, averaging 1 percent, which compensate the sponsor for taking care of the investments and generating reports. Many annuities also charge modest administrative or contract fees. One of the more problematic costs of annuities, in the eyes of their critics, is the surrender charge for early removal of the principal. In most cases, this fee begins at around 7 percent but then phases out over time. However, the surrender fee is charged in addition to the 10 percent government penalty for early withdrawal if the investor is under age 59 1/2. All of the costs associated with variable annuities detract somewhat from their attractiveness as a financial product when one compares them to mutual funds. The costs also mean that there are no quick profits associated with annuities; instead, they must be held as a long-term investment. In fact, it can take as long as 17 years for the benefits of tax deferral to outpace the administrative expenses of an annuity. For investors who wish to put money away for an extended period, a variable annuity may be a very good investment vehicle. Distribution Options On the positive side, investors in annuities have a number of options for receiving the distribution of their funds. The three most common forms of distribution — all of which have various costs deducted — are lump sum, lifetime income, and systematic payout. Some investors CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 41 who have contributed to a variable annuity over many years may elect to take a lump-sum withdrawal. The main drawback to this approach is that all the taxes are due immediately. Other investors may decide upon a systematic payout of the accumulated assets over a specified time period. In this approach, the investor can determine the amount of payments as well as the intervals at which payments will be received. Finally, some investors choose the option of receiving a guaranteed lifetime income. This option is the most expensive for the investor, and does not provide any money for heirs, but the sponsor of the annuity must continue to make payouts even if the investor outlives his or her assets. A similar distribution arrangement is joint-and-last-survivor, which is an annuity that keeps providing income as long as one person in a couple is alive. Annuities are rather complex financial products, and as such they have become the subject of considerable debate among experts in financial planning. As mentioned earlier, many experts claim that the special features of annuities are not great enough to make up for their cost as compared to other investment options. As a result, financial advisors commonly suggest that individuals maximize their contributions to IRAs, 401(k)s, or other pre-tax retirement accounts before considering annuities (investors should avoid placing annuities into IRAs or other tax-sheltered accounts because the tax shelter then becomes redundant and the investor pays large annuity fees for nothing). Some experts also prefer mutual funds tied to a stock market index to annuities, because such funds typically cost less and often provide a more favorable tax situation. Contributions to annuities are taxed at the same rate as ordinary income, for instance, while long-term capital gains from stock investments are taxed at a special, lower rate — usually 20 percent. Still other financial advisors note that, given the costs involved, annuities require a very long-term financial commitment in order to provide benefits. It may not be possible for some individuals to tie up funds for the 17 to 20 years it takes to benefit from the purchase of an annuity. Despite the drawbacks, however, annuities can be beneficial for individuals in a number of different situations. For example, annuities provide an extra source of income and an added margin of safety for individuals who have contributed the limit allowed under other retirement savings options. In addition, some kinds of annuities can be valuable for individuals who want to protect their assets from creditors in the event of bankruptcy. An annuity can provide a good shelter for a retirement nest egg for someone in a risky profession, such as medicine. Annuities are also recommended for people who plan to spend the principal during their lifetime rather than leaving it CU IDOL SELF LEARNING MATERIAL (SLM)

42 Business Mathematics and Statistics for their heirs. Finally, annuities may be more beneficial for individuals who expect that their tax bracket will be 28 per cent or lower at the time they begin making withdrawals. Annuities may also hold a great deal of appeal for small businesses. For example, annuities can be used as a retirement savings plan on top of a 401(k). They can be structured in various ways to reward employees for meeting company goals. In addition, annuities can provide a nice counterpart to life insurance, since the longer the investor lives, the better an annuity will turn out to be as an investment. Finally, some annuities allow investors to take out loans against the principal without paying penalties for early withdrawal. Overall, some financial experts claim that annuities are actually worth more than comparable investments because of such features as the death benefit, guaranteed lifetime income, and investment services. A small business owner considering setting up an annuity should consider all options, look carefully at both the costs and the returns, and be prepared to put money away for many years. It is also important to shop around for the best possible product and sponsor before committing funds. “Before investing in an annuity, make sure the insurance company that will sponsor the contract is financially healthy,” counseled Mel Poteshman in Los Angeles Business Journal. Also, find out from the sponsoring company the interest rates that have been paid out over the last five to ten years and how interest rate changes are calculated. This will give you an idea of the annuity’s overall performance and help you identify the annuity that provides the best long-range financial security. 3.2 Immediate Annuity To find the amount of an immediate annuity certain at the end of n years. Let A be the annuity, r the rate per ` l for one year, n the number; of years and M the amount. The first payment is made at the end of the first year. The amount of this sum A in the remaining (n - 1) years is ARn-l, where R = I + r/100; r% being the effective rate of interest per annum. At the end of the second year another sum A is due and the amount of this sum in the remaining (n - 2) years is ARn-2; and proceeding in this manner; we get M = ARn-1 + ARn-2 + …AR2 + AR + A = A (1 + R + R2 + … + Rn-1) CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 43 ?M= n … (1) Putting i = (R 1) A R1 r , we get 100 ? R= I + r =l+i 100 [(1 i)n 1] M= A 111 = = A [(1+i)” -1] … (2) i Example 1: How much will an annuity of ` 100 amount to in 20 years at 4½% interest? Answer to the nearest paisa. Given that log 1.045 = .0191163, log 24.117 = 1.3823260 Solution: Using formula (1) and substituting A = 100 and R = 1 + 4½/100 = 1 + .045 = 1.045. We get, n M= (R 1) = A R1 20 [(1.045) 1] 100 1.045 1 = 20000 (1.045)20 - 1] 19 Now, (1.045)20 = Antilog [20 log (1.045)] = Antilog [20 (.0191163] = Antilog [.382326] = 2.4117 20000 u 1.4117 M= 9 = 3137.11 Amount is ` 3137.11 CU IDOL SELF LEARNING MATERIAL (SLM)

44 Business Mathematics and Statistics Example 2: How much will an annuity of ` 500 amount to in 20 years at 3½% interest? Answer to the nearest paisa. Given that log 103.5 = 2.0149403 and log 19.89784 = 1.2988060 [I.C.W.A.] Solution: Using formula (1) n M = A (R 1) R1 20 [(1.035) 1] = 500 1.035 1 20 = 500 [(1.035) 1] .035 1 Now, (1.035)20 = Antilog [20 log (l.035)] = Antilog [20 (.0149403)] = Antilog (0.2988060) = 1.989784 500(1.989784 1) ?M= 0.035 500 u 989.784 = 35 = 100 × 141.397714 = 14,139.7714 ? Amount is ` 14,139.77 3.3 Present Worth To find the present worth (value) of an immediate annuity of ` A, payable yearly for n years, at r% per annum compound interest. Suppose P1 is the present value of the first payment (say A) that is made at the end of the first year, then CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 45 A = P1R P1 = A/R Putting V = 1/R, we get P1 = AV Similarly, if P2 is the present value of the second payment of ` A that is made at the end of two years, then A = P2R2 P2 = A/R2 = AV2 Proceeding in this manner, we get P3 = A/R3 = AV3 Pn = A/Rn = AVn The present value of the annuity is P = P1 + P2 + …+Pn = A/R + A/R2 + ....... + Pn = A/R + A/R2 +................. + A/Rn = AV + AV2 + ....... + AVn = AV(1 + V + V2 + ...... + Vn-1) n §1 V · ? P = AV¨ ¸ … (3) ©1 V ¹ … (4) Also P = A§1 R n· ¨ ¸ 1 R©1 R ¹ = §1 n A¨ R· ¸ ©R 1 ¹ n A §R 1· ? P =n ¨ ¸ R © R 1 ¹ CU IDOL SELF LEARNING MATERIAL (SLM)