Chapter 18 Financial Mathematics

Learning Objectives Explain why a dollar today is worth more than a dollar in the future. Define future value and present value. Explain the difference between an ordinary annuity and an annuity due. Calculate the future value and present value of an annuity.

Decisions for Financial Managers 1.Where should we invest our funds? i.e. What types of equipment, buildings, services, programs, or other opportunities should we invest? 2.How will we finance our investment needs? Will we use debt, equity, or both?

Investment and Financing Investment decisions involve expending funds today while expecting to realize returns in the future Financing decisions involve the receipt of funds today in return for a promise to make payments in the future Therefore, manager’s major task is evaluating the relative attractiveness of alternative investment and financing opportunities

Time Value of Money Concept of time value money: –Payment that is made or received in the first year has a greater value than an identical payment made or received in the tenth year –“A dollar today is worth more than a dollar tomorrow.” The idea that the value of money changes over time is absolutely critical to investment and financing decisions.

Time Value of Money If we accept that a dollar today is worth more than a dollar received in the future, how can we determine what that difference is? –We must determine the cost for money –e.g. the interest rate associated with borrowing, the opportunity cost of investment, etc. Interest rate - the price for the commodity called money (e.g. 10% per year)

Present Value & Future Value If the interest rate is 10%, a dollar received 1 year from now is only worth $ in today’s dollar. –This concept is called “Present Value” Conversely, if the interest rate is 10%, $ today will be worth exactly $1 one year from now. –This concept is called “Future Value”

Simple vs. Compound Interest Simple interest - interest is only calculated on the principal Compound interest - interest is calculated on the principal and the interest. –Compound interest is the basis for calculating future values over multiple time periods.

Future Value – Single Sum Example – nursing home may want to invest $100,000 today in a fund to be used in 2 years for replacement Question – what sum of money would be available 2 years from now? Solution – set up time graph using: –# of periods during which compounding occurs (n) –Present value of future sum (p) –Future value (f) of present sum –Interest rate per period (i)

Future Value, cont. Knowing values of any three variables can solve for the fourth Time line - helps conceptualize the problem and identify the known values Value 0 – represents present time (today), Values 1 and 2 represents Year 1 and Year 2

Figure 18–1 Future Value of Present Sum

Future Value, cont. We can solve for future value through substitution of following formula: Factor (i, n) – future value of $1 invested today for n periods at i rate of interest per period:

Present Value – Single Sum Same table of values used Formula: Factor p (i, n) represents present value of $1 received in n periods at interest rate of i

Present Value, cont. Example – assume that an HMO has $100,000 debt obligation due in 2 years Question – how much money to set aside today to meet the obligation at the expected yield of 12%? Calculation to solve the problem:

Figure 18–3 Present Value of a Future Sum

Present and Future Value Future and present value tables are reciprocals of each other as shown in the formula: Only one of the two tables is necessary to solve present or future value problem

Annuities Often, there is more than one payment or receipt If each payment or receipt is constant per time period, there is an annuity situation Basic formula: Consider example: –F – future value of invested annuity deposits at the end of period n –R – periodic deposits that are constant for each period ($50,000) invested for 3 years at 8% –Deposits are made at the end of each period => ordinary annuity

Figure 18–5 Determining the Future Value of an Annuity

Future Value of Annuity, cont. To solve the problem:

Alternative Annuity Calculation Values in annuity future value table could be determined through addition for a single sum of future values:

Future Annuity Formula In general, a future value annuity factor can be expressed as follows:

Annuity and Single Sum A financial mathematics problem may be part annuity and part single sum In this case, time graph is helpful Consider example: –Hospital has sinking fund payment requirement for last 10 years of a bond’s life –$45M must be available at the end to retire debt –$5M available today –Time line – 20 years before debt retirement –Investment yield – 10% per year –Question – what annual deposit is required?

Figure 18–6 Determining the Annual Sinking Fund Deposit

Annuity Sinking Fund, cont. Future value: Final solution:

Present Value of Annuity Procedure analogous to future value of annuity Payments at the end of the period General equation: Factor P (i, n) - present value of $1 received at the end of each period for n periods when i is the rate of interest

Present Value Annuity Problem Consider example: –Purchase of an older hospital –Obligation – payments of $100,000 per year for 4 years to vested employees –Discount rate – 12% –Question - what the present value of this obligation is so that it can be subtracted from the negotiated purchase price?

Figure 18–8 Present Value of the Pension Obligation

Present Value of Pension Obligation, cont. To solve the problem: P = $100,000 x P(12%,4) or P = $100,000 x P = $303,730

Alternative Annuity Present Value Calculation Present-value annuity problems can be thought of as a series of individual single-sum problems The present-value annuity factor P(i,n) is the sum of the individual single-sum present values:

Present Annuity Formula The present-value annuity factor [P(i,n)] can be expressed as follows:

Lease Liability Example In most business situations, ordinary annuity problems do not arise Classic exception – lease with front-end payments Example: –Computer lease for 5 years –Quarterly payments of $1,000 due at the beginning of each quarter => annuity due –Discount rate – 16% per year –Question – what is the present value of the lease liability?

Figure 18–9 Present Value of a Lease Liability

Lease Liability Example, cont. The present value of the lease liability can be calculated as follows: