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BBA/BCOM 2 All right are reserved with CU-IDOL Business Mathematics & Statistics Course Code: BBA102/BCM 102 Semester: 1 SLM Unit: 3-4 E-Lesson: 2 www.cuidol.in Unit 3-4(BBA 102 /BCM 102)

Business Mathematics & Statistics 33 OBJECTIVES INTRODUCTION To make students aware of the concept of annuity In this unit we are going to learn about annuity. To develop an understanding of Matrices. Under this you will learn how to calculate annuity. To make students understand about the In this unit you will learn the concept of operations on matrices. matrices and the operations on matrices. . www.cuidol.in Unit 3-4(BBBA 1/0B2Co/BmC1M021)02) INSTITUTE OF DAIlSlTAriNghCEt aArNeDreOsNerLvINedE LwEiAthRNCIUN-GIDOL

Topics To Be Covered 4  Introduction of Basic concepts of Annuity  Uses of annuity in business  Introduction to matrices  Introduction to various types of matrices  Various operations on Matrices www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Introduction of Annuity 5 An annuity is a contract between you and an insurance company in which you make a lump-sum payment or series of payments and, in return, receive regular disbursements, beginning either immediately or at some point in the future. Many aspects of an annuity can be tailored to the specific needs of the buyer. In addition to choosing between a lump- sum payment or a series of payments to the insurer, you can choose when you want to annuitize your contributions—that is, start receiving payments. An annuity that begins paying out immediately is referred to as an immediate annuity, while one that starts at a predetermined date in the future is called a deferred annuity. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Types of Annuities 6 • Annuities come in three main varieties: fixed, variable, and indexed. Each type has its own level of risk and payout potential. Fixed annuities pay out a guaranteed amount. The downside of this predictability is a relatively modest annual return, generally slightly higher than a CD from a bank. • Variable annuities provide an opportunity for a potentially higher return, accompanied by greater risk. In this case, you pick from a menu of mutual funds that go into your personal \"sub-account.\" Here, your payments in retirement are based on the performance of investments in your sub- account. • Indexed annuities fall somewhere in between when it comes to risk and potential reward. You receive a guaranteed minimum payout, although a portion of your return is tied to the performance of a market index, such as the S&P 500. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Example of Annuity 7 • When you’re accumulating funds for a goal like retirement there are a couple of examples of annuities. • The first example is a fixed annuity. When you put money into a fixed annuity, you’ll know the interest rate up front and how long you will have to leave your money in the annuity before you can take it out without penalty. If you’re the type of person who would skip to read the last page in the story about your financial life — i.e. you want to know exactly what’s going to happen — then a fixed annuity may be a good choice for you. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Example of Annuity 8 • Another example of an annuity that can accumulate value over time is a variable annuity. Unlike its fixed cousin, a variable annuity allows you to choose subaccounts for the money that you contribute. The subaccounts typically include a variety of choices, including money-market funds, bond funds, and funds that are tied to market-based investments. This can allow your money to grow more over time than it might in a fixed annuity (although it could also lose value, including the loss of principal). A variable annuity can be an efficient way to take advantage of the growth that market-based investments can provide to accumulate funds that you plan to turn into income in retirement. When looking at a variable annuity it’s important to ask about the cost of guarantees it may offer, as you will pay fees for certainty. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Advantages of Annuity 9 • Annuities can be helpful in some situations. In general, some of the advantages and benefits include: • Tax-deferred growth and compounding within the annuity contract. Which means you only get taxed on the interest you earn once you start receiving payments, not while it's building up. • Guaranteed rates of return on your dollars • Guaranteed lifetime payments if you annuitize (in some cases you don’t even have to annuitize to receive this benefit) • Other features that may be important to you. These are various bells and whistles that do very specific things, ask your annuitant about these options before signing on any dotted lines. • Note that the guarantees are only as strong as the insurance company that issued the annuity. In other words, if the insurance company fails, the promise is no good. You should mitigate this risk by using only the strongest insurance companies out there. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Uses of Business Mathematics 10  Determine Product Pricing To ensure you can operate your business and produce enough cash flow to invest into your enterprise, you must charge enough for your product to be profitable. Markup is the difference between your merchandise cost and the selling price, giving you gross profit. If your operations require a large markup, such as 70 percent, you may not be competitive in your industry if other companies sell the same items for less. Once you have determined your markup, one way to calculate the retail price is to divide using percents or decimals. For example, if a product costs Rs.10 to produce and your markup is 35 percent, subtract .35 from 1 (or 100 percent), which gives you .65, which is 65 percent. To calculate the price of your product, divide 10 by .65, which rounds to Rs.15.38. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Annuity Calculation 11 www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Annuity Calculation 12  The formula based on PV of an ordinary annuity is calculated based on PV of an ordinary annuity, effective interest rate and a number of periods. Annuity = r * PVA Ordinary / [1 – (1 + r)-n] where, PVA Ordinary = Present value of an ordinary annuity r = Effective interest rate n = Number of periods www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Annuity Calculation 13  Mathematically, the equation for annuity due is represented as, Annuity = r * PVA Due / [{1 – (1 + r)-n} * (1 + r)] where, PVA Due = Present value of an annuity due r = Effective interest rate n = Number of periods www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Annuity Calculation 14 The calculation of annuity payment can be derived by using the PV of ordinary annuity in the following steps: • Step 1: Firstly, determine the PV of the annuity and confirm that the payment will be done at the end of each period. It is denoted by PVA Ordinary. • Step 2: Next, determine the interest rate on the basis of the current market return. Then, the effective rate of interest is computed by dividing the annualized interest rate by the number of periodic payments in a year and it is denoted by r. r = Annualized interest rate / Number periodic payments in a year • Step 3: Next, determine the number of periods by multiplying the number of periodic payments in a year and the number of years, and it is denoted by n. n = Number of periodic payments in a year * Number of years • Step 4: Finally, the annuity payment based on PV of an ordinary annuity is calculated based on PV of ordinary annuity (step 1), effective interest rate (step 2) and a number of periods (step 3) as shown above. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Matrices 15 • A matrix is a rectangular arrangement of numbers into rows and columns. • For example, matrix AAAA has two rows and three columns. • The dimensions of a matrix tells its size: the number of rows and columns of the matrix, in that order. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Matrices Introduction 16 • A matrix is a collection of numbers arranged into a fixed number of rows and columns. The top row is row 1. The leftmost column is column 1. This matrix is a 3x3 matrix because it has three rows and three columns. In describing matrices, the format is: rows x columns www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Types of Matrices 17 • Row Matrix: Row matrix is a type of matrix which has just one row. A=[ 3 2 1 ] which has just one row but has three columns. • Column Matrix: Column matrix is a type of matrix which has just one column. A= A zero matrix or a null matrix is a matrix that has all its elements zero. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Types of Matrices 18 A square matrix is a matrix with an equal number of rows and columns. T is a square matrix of order 2 × 2 V is a square matrix of order 3 × 3 www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Types of Matrices 19 A diagonal matrix is a square matrix that has all its elements zero except for those in the diagonal from top left to bottom right A unit matrix is a diagonal matrix whose elements in the diagonal are all ones. Rectangular matrix is a matrix that is not a square matrix or a matrix not having equal number of rows and column www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Types of Matrices 20 • Triangular matrix is a square matrix having all the elements 0 either below the diagonal or above the diagonal A triangular matrix having elements 0 below diagonal is called upper triangular matrix A triangular matrix having elements 0 above diagonal is called lower triangular matrix www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Types of Matrices 21 Symmetric matrix: if all of its elements follow the rule that : any element “a” is equal to another element whose row and column position is equal to the column and row position of the element “a” respectively. For example: Skew-symmetric matrix: if all of its elements follow the rule that : any element “a” is equal to the negative value of another element whose row and column position is equal to the column and row position of the element “a” respectively. Note: Every skew-symmetric matrix have its diagonal elements 0. For example: www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Operations on Matrices 22 Addition and Subtraction of Matrices • Addition and subtraction is only possible if the order of the matrices is same. mxn and mxn For example www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Operations on Matrices 23 Multiplication of Matrices Scalar multiplication is multiplying all the elements with constant value. Multiplication is possible between two matrices if the number of columns of first matrix is equal to number of rows of second matrix mxn x nxj = mxj 3x2 x 2x4 = 3x4 www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Properties of Matrix operation • A+B = B+A 24 commutative • A + (B + C) = (A + B) + C associative • There is a unique m x n matrix O with A+O = A additive identity • For any m x n matrix A there is an m x n matrix B (called -A) with A+B = O additive inverse • A(BC) = (AB)C associative • A(B + C) = AB + AC distributive • (A + B)C = AC + BC distributive www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Applications of Matrices • In computer based applications, matrices play a vital role in the projection of three dimensional image into 25 a two dimensional screen, creating the realistic seeming motions. Stochastic matrices and Eigen vector solvers are used in the page rank algorithms which are used in the ranking of web pages in Google search • The Matrix calculus is used in the generalization of analytical notions like exponentials and derivatives to their higher dimensions • One of the most important usages of matrices in Computer side applications are encryption of Message codes. Matrices and their inverse matrices are used for a programmer for coding or encrypting a message. A message is made as a sequence of numbers in a binary format for communication and it follows code theory for solving • Hence with the help of Matrices, those equations are solved. With these encryptions only, internet functions are working and even banks could work with transmission of sensitive and private data’s. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Applications of Matrices 26 • In Geology, Matrices are used for taking seismic surveys. They are used for plotting graphs, statistics and also to do scientific studies in almost different fields. • Matrices are used in representing the Real World data like the traits of people’s population, habits, etc. They are best representation methods for plotting the common survey things • Matrices are used in calculating the Gross domestic products in economics which eventually helps in calculating the goods production efficiently. • In Robotics and Automation, matrices are the base elements for the robot movements. The movements of the robots are programmed with the calculation of matrices’ rows and columns. The inputs for controlling robots are given based on the calculations from matrices. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

SUMMARY 27  Annuity- An annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates.  Matrices- Matrix is an arrangement of numbers into rows and columns. Make your first introduction with matrices and learn about their dimensions and elements. A matrix is a rectangular arrangement of numbers into rows and columns. For example, matrix A has two rows and three columns. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Multiple Choice Questions 28 1.If the order of matrix A is m×p. And the order of B is p×n. Then the order of matrix AB is a) n × p b) m × n c) n × p d) n × m 2. A square matrix in which all elements except at least one element in diagonal are zeros is said to be a a) identical matrix b) null/zero matrix c) column matrix d) diagonal matrix Answers: 1. b) 2.d) www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Multiple Choice Questions 29 3. In matrices, columns are denoted by a) A b) B c) R d) C 4. The number of non-zero rows in an echlon form is called ? a) rank of a matrix b) cofactor of the matrix c) reduced echlon form d) conjugate of the matrix Answers: 3. d) 4. a) www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

Frequently Asked Questions 30 Q.1 What is annuity?  Ans: An annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. Q.2 What is matrices? Ans: Matrix is an arrangement of numbers into rows and columns. Make your first introduction with matrices and learn about their dimensions and elements. A matrix is a rectangular arrangement of numbers into rows and columns. For example, matrix A has two rows and three columns. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

REFERENCES 31 1. I Grattan-Guinness and W Ledermann, Matrix theory, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 775- 786. 2. T W Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29. 3. T W Hawkins, Another look at Cayley and the theory of matrices, Archives Internationales d'Histoire des Sciences 26 (100) (1977), 82-112. 4. T W Hawkins, Weierstrass and the theory of matrices, Archive for History of Exact Sciences 17 (1977), 119-163. www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL

32 THANK YOU For queries Email: [email protected] www.cuidol.in Unit 3-4(BBA 102 /BCM 102) All right are reserved with CU-IDOL