BUS-221 Quantitative Methods LECTURE 7

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 and Linear Functions (1 of 5) Example 1 – Production Levels

Applications and Linear Functions (2 of 5) Demand and Supply Curves

Applications and Linear Functions (3 of 5) Demand and Supply Curves

Applications and Linear Functions (4 of 5) Example 3 – Graphing Linear Functions

Applications and Linear Functions (5 of 5) Example – Determining a Linear Function

Applications of Systems of Equations (1 of 7) Equilibrium When the demand and supply curves of a product are represented on the same coordinate plane, the point where the curves intersect is called the point of equilibrium.

Applications of Systems of Equations (2 of 7) Example 1 – Tax Effect on Equilibrium

Applications of Systems of Equations (3 of 7) Example 1 – Continued

Applications of Systems of Equations (4 of 7) Example 1 – Continued

Applications of Systems of Equations (5 of 7) Break-Even Points profit = total revenue(TR) – total cost(TC) total cost(TC) = variable cost(VC) + fixed cost(FC) The break-even point is where TR = TC.

Applications of Systems of Equations (6 of 7) Example – Break-Even Point, Profit, and Loss

Applications of Systems of Equations (7 of 7) Example – Break-Even Point, Profit, and Loss

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 5 – 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

Integration by Tables (1 of 5)

Integration by Tables (2 of 5) Example 1 – Integration by Tables

Integration by Tables (3 of 5) Example – Integration by Tables

Integration by Tables (4 of 5) Example – Integration by Tables

Integration by Tables (5 of 5) Example – Finding a Definite Integral by Using Tables

Approximate Integration (1 of 4)

Approximate Integration (2 of 4) Example 1 – Trapezoidal Rule

Approximate Integration (3 of 4)

Approximate Integration (4 of 4) Example – Simpson’s Rule

Area between Curves (1 of 8) Example 1 – An Area Requiring Two Different Integrals

Area between Curves (2 of 8)

Area between Curves (3 of 8) Example – Finding an Area between Two Curves

Area between Curves (4 of 8) Example – Area of a Region Having Two Different Upper Curves

Area between Curves (5 of 8) Example – Continued

Area between Curves (6 of 8)

Area between Curves (7 of 8) Example – Advantage of Horizontal Elements

Area between Curves (8 of 8) Example – Continued

Consumers’ and Producers’ Surplus (1 of 3)

Consumers’ and Producers’ Surplus (2 of 3)

Consumers’ and Producers’ Surplus (3 of 3) Example 1 – Finding Consumers’ Surplus and Producers’ Surplus