BUS-221 Quantitative Methods LECTURE 6

Learning Outcome Knowledge – Be familiar with basic mathematical techniques including: linear programming, systems of linear equations, calculus (differential and integral) Argument - Justify the interpretation of data under various quantitative analyses, and justify the use of tools chosen. Communication - Present analyses of business situations from a quantitative point of view. The analysis will demonstrate clarity of expression, use of terminology, knowledge of format, aptness for the user group

Topics Application: Finance, cash flow equations. (compound interest, discount, net present value)

Applications of Equations (1 of 12) Modeling is the translating of stated relationships into mathematical symbols. Example 1 – Mixture A chemist must prepare 350 ml of a chemical solution made up of two parts alcohol and three parts acid. How much of each should be used?

Applications of Equations (2 of 12) Example 1 – Continued

Applications of Equations (3 of 12) We refer below to some business terms relative to a manufacturing firm: Fixed cost is the sum of all costs that are independent of the level of production. Variable cost is the sum of all costs that are dependent on the level of output. Total cost = variable cost + fixed cost Total revenue = (price per unit)(number of units sold) Profit = total revenue − total cost

Applications of Equations (4 of 12) Example – Profit The Acme Company produces a product for which the variable cost per unit is $6 and the fixed cost is $80,000. Each unit has a selling price of $10. Determine the number of units that must be sold for the company to earn a profit of $60,000.

Applications of Equations (5 of 12) Example – Continued

Applications of Equations (6 of 12) Example – Investment A total of $10,000 was invested in two business ventures, A and B. At the end of the first year, A and B yielded returns of 6% and 5¾%, respectively, on the original investments. How was the original amount allocated if the total amount earned was $588.75?

Applications of Equations (7 of 12) Example – Continued

Applications of Equations (8 of 12) Example – Apartment Rent A real-estate firm owns the Parklane Garden Apartments, which consist of 96 apartments. At $550 per month, every apartment can be rented. However, for each $25 per month increase, there will be three vacancies with no possibility of filling them. The firm wants to receive $54,600 per month from rent. What rent should be charged for each apartment?

Applications of Equations (9 of 12) Example – Continued

Applications of Equations (10 of 12) Example – Continued

Applications of Equations (11 of 12) Example – Continued

Applications of Equations (12 of 12) Example – Continued

Applications of Inequalities (1 of 4) Solving word problems may involve inequalities. Example 1 – Profit For a company that manufactures aquarium heaters, the combined cost for labor and material is $21 per heater. Fixed costs (costs incurred in a given period, regardless of output) are $70,000. If the selling price of a heater is $35, how many must be sold for the company to earn a profit?

Applications of Inequalities (2 of 4) Example 1 – Continued

Applications of Inequalities (3 of 4) Example – Current Ratio After consulting with the comptroller, the president of the Ace Sports Equipment Company decides to take out a short-term loan to build up inventory. The company has current assets of $350,000 and current liabilities of $80,000. How much can the company borrow if the current ratio is to be no less than 2.5? (Note: The funds received are considered as current assets and the loan as a current liability.)

Applications of Inequalities (4 of 4) Example – Current Ratio

Compound Interest (1 of 6) Example 1 – Compound Interest Suppose that $500 amounted to $588.38 in a savings account after three years. If interest was compounded semiannually, find the nominal rate of interest, compounded semiannually, that was earned by the money.

Compound Interest (2 of 6) Example 1 – Continued

Compound Interest (3 of 6) Example – Compound Interest

Compound Interest (4 of 6) Example – Continued

Compound Interest (5 of 6) Effective Rate Example – Effective Rate To what amount will $12,000 accumulate in 15 years if it is invested at an effective rate of 5%?

Compound Interest (6 of 6) Example – Comparing Interest Rates If an investor has a choice of investing money at 6% compounded daily or 6⅛ % compounded quarterly, which is the better choice?

Present Value (1 of 6) Example 1 – Present Value Find the present value of $1000 due after three years if the interest rate is 9% compounded monthly.

Present Value (2 of 6) Example 1 – Continued Example – Equation of Value A debt of $3000 due six years from now is instead to be paid off by three payments: $500 now, $1500 in three years, and a final payment at the end of five years. What would this payment be if an interest rate of 6% compounded annually is assumed?

Present Value (3 of 6) Example – Continued

Present Value (4 of 6) If an initial investment will bring in payments at future times, the payments are called cash flows. The net present value, denoted NPV, of the cash flows is defined to be the sum of the present values of the cash flows, minus the initial investment. If NPV > 0, then the investment is profitable; if NPV < 0, the investment is not profitable.

Present Value (5 of 6) Example – Net Present Value You can invest $20,000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table below. Assume an interest rate of 7% compounded annually and find the net present value of the cash flows. Year Cash Flow 2 10,000 3 8000 5 6000

Present Value (6 of 6) Example – Continued

Interest Compounded Continuously (1 of 3) Example 1 – Compound Amount

Interest Compounded Continuously (2 of 3) Example – Trust Fund A trust fund is being set up by a single payment so that at the end of 20 years there will be $25,000 in the fund. If interest compounded continuously at an annual rate of 7%, how much money should be paid into the fund initially?

Interest Compounded Continuously (3 of 3) Example – Trust Fund

Annuities (1 of 6) An annuity is any finite sequence of payments made at fixed periods of time, of equal length, over a given interval. The fixed periods of time are referred to as the payment period. The given interval is the term of the annuity.

Annuities (2 of 6) Example 1 – Present Value of Annuity Find the present value of an annuity of $100 per month for 3½ years at an interest rate of 6% compounded monthly.

Annuities (3 of 6) Example – Periodic Payment of Annuity If $10,000 is used to purchase an annuity consisting of equal payments at the end of each year for the next four years and the interest rate is 6% compounded annually, find the amount of each payment.

Annuities (4 of 6) Example – Future Value of an Annuity Find the future value of an annuity consisting of payments of $50 at the end of every three months for three years at the rate of 6% compounded quarterly. Also, find the compound interest.

Annuities (5 of 6) Example – Sinking Fund A sinking fund is a fund into which periodic payments are made in order to satisfy a future obligation. Suppose a machine costing $7000 is to be replaced at the end of eight years, at which time it will have a salvage value of $700. In order to provide money at that time for a new machine costing the same amount, a sinking fund is set up. The amount in the fund at the end of eight years is to be the difference between the replacement cost and the salvage value. If equal payments are placed in the fund at the end of each quarter and the fund earns 8% compounded quarterly, what should each payment be?

Annuities (6 of 6) Example – Continued

Amortization of Loans (1 of 3) The formulas below describe the amortization of the general loan: Table 5.2 Amortization Formulas

Amortization of Loans (2 of 3) Example 1 – Amortizing a Loan A person amortizes a loan of $170,000 by obtaining a 20- year mortgage at 7.5% compounded monthly. Find (a) the monthly payment, (b) the total interest charges, and (c) the principal remaining after five years.

Amortization of Loans (3 of 3) Example 1 – Amortizing a Loan

Perpetuities (1 of 4) We consider briefly the possibility of an infinite sequence of payments. We measure time in payment periods starting now – at time 0 – and consider equal payments, to continue indefinitely. We call such an infinite sequence of payments a perpetuity, visualized on a timeline below.

Perpetuities (2 of 4)

Perpetuities (3 of 4) Example 1 – Present Value of a Perpetuity Dalhousie University would like to establish a scholarship worth $15,000 to be awarded to the first year Business student who attains the highest grade in MATH 1115, Commerce Mathematics. The award is to be made annually, and the Vice President Finance believes that, for the foreseeable future, the university will be able to earn at least 2% a year on investments. What principle is needed to ensure the viability of the scholarship?

Perpetuities (4 of 4) Example 1 – Continued Limits

Matrices (1 of 4)

Matrices (2 of 4) Example 1 – Size of a Matrix

Matrices (3 of 4) Example 3 – Transpose of a Matrix

Matrices (4 of 4) Special Matrices

Matrix Addition and Scalar Multiplication (1 of 5) Example 1 – Matrix Addition

Matrix Addition and Scalar Multiplication (2 of 5) Example – Demand Vectors for an Economy

Matrix Addition and Scalar Multiplication (3 of 5) Properties of Scalar Multiplication 1. k(A + B) = kA + kB 2. (k + l)A = kA + lA 3. k(lA) = (kl)A 4. 0A = 0 5. k0 = 0

Matrix Addition and Scalar Multiplication (4 of 5) Subtraction of Matrices

Matrix Addition and Scalar Multiplication (5 of 5) Example – Matrix Subtraction

Matrix Multiplication Example 1 – Sizes of Matrices and Their Product

Matrix Multiplication Example – Matrix Products

Matrix Multiplication Example – Cost Vector