HCM 570 Financial Management in Healthcare August 14-December 18, 2004 Joseph Callaghan, Ph.D. Oakland University Accounting and Finance

Mathematics and Tools of Finance Basics Intermediate Case 11

Future and present values –Lump sums –Annuities –Uneven cash flow streams Solving for i and n Investment returns Amortization Time Value Analysis

Time Value of Money Time value analysis is necessary because money has time value. –A dollar in hand today is worth more than a dollar to be received in the future. Why? –Because of time value, the values of future dollars must be adjusted before they can be compared to current dollars. Time value analysis constitutes the techniques that are used to account for the time value of money.

Time Lines CF 0 CF 1 CF 3 CF i% Tick marks designate ends of periods. Time 0 is the starting point (the beginning of Period 1); Time 1 is the end of Period 1 (the beginning of Period 2); and so on.

Time Line Illustration 1 $ %  What does this time line show? 5%

Time Line Illustration 2 -$ %  What does this time line show? ? 10%

What is the FV after 3 years of a $100 lump sum invested at 10%? FV = ? % -$100 Finding future values (moving to the right along the time line) is called compounding. For now, assume interest is paid annually.

After 1 year: FV 1 = PV + INT 1 = PV + (PV x i) = PV x (1 + i) = $100 x 1.10 = $ After 2 years: FV 2 = FV 1 + INT 2 = FV 1 + (FV 1 x i) = FV 1 x (1 + i) = PV x (1 + i) x (1 + i) = PV x (1 + i) 2 = $100 x (1.10) 2 = $

After 3 years: FV 3 = FV 2 + I 3 = PV x (1 + i) 3 = 100 x (1.10) 3 = $ In general, FV n = PV x (1 + i) n.

Three Primary Methods to Find FVs Solve the FV equation using a regular (non- financial) calculator. Use a financial calculator; that is, one with financial functions. Use a computer with a spreadsheet program such as Excel, Lotus 1-2-3, or Quattro Pro.

Non-Financial Calculator Solution $ % -$100 $110.00$ $100 x 1.10 x 1.10 x 1.10 = $

Financial Calculator Solution Financial calculators are pre- programmed to solve the FV equation: FV N = PV x (1 + i) n. There are four variables in the equation: FV, PV, i and n. If any three are known, the calculator can solve for the fourth (unknown).

NI/YR PV PMT FV Using a calculator to find FV (lump sum): (1) For lump sums, the PMT key is not used. Either clear before the calculation or enter PMT = 0. (2) Set your calculator on P/YR = 1, END. INPUTS OUTPUT Notes:

10% What is the PV of $100 due in 3 years if i= 10%? $ PV = ? Finding present values (moving to the left along the time line) is called discounting.

Solve FV n = PV x (1 + i) n for PV PV = $100 / (1.10) 3 = $100(0.7513) = $ PV = FV N / (1 + i) n.

Financial Calculator Solution Either PV or FV must be negative on most calculators. Here, PV = Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT NI/YRPVPMTFV

Opportunity Cost Rate On the last illustration we needed to apply a discount rate. Where did it come from? –The discount rate is the opportunity cost rate. –It is the rate that could be earned on alternative investments of similar risk. –It does not depend on the source of the investment funds. We will apply this concept over and over in this course.

Opportunity Cost Rate (Cont.) The opportunity cost rate is found (at least in theory) as follows. –Assess the riskiness of the cash flow(s) to be discounted. –Identify security investments that have the same risk. –Estimate the expected return that could be earned on the security investment. When applied, the resulting PV provides a return equal to the opportunity cost rate. In most capital finance situations, bench-mark opportunity cost rates are known.

NI/YR PV PMTFV 21.7 INPUTS OUTPUT Solving for i Assume that a bank offers an account that will pay $200 after 5 years on each $75 invested. What is the implied interest rate?

INPUTS OUTPUT Solving for n Assume an investment earns 20 percent per year. How long will it take for the investment to double? NI/YRPVPMTFV

Three Year Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Three Year Annuity Due Types of Annuities

What is the FV of a 3-year ordinary annuity of $100 invested at 10%? $ % FV = $331

Financial Calculator Solution Here there are payments rather than a lump sum present value, so enter 0 for PV. INPUTS OUTPUT I/YRNPMTFVPV

What is the PV of the annuity? $ % $ $ = PV

INPUTS OUTPUT NI/YRPVPMTFV Financial Calculator Solution

What is the FV and PV if the annuity were an annuity due? $ % $100 ? ?

Switch from End to Begin mode on a financial calculator. Repeat the annuity calculations. First find PVA 3 = $ Then enter PV = 0 and press FV to find FV = $ INPUTS OUTPUT NI/YRPVPMTFV

Perpetuities A perpetuity is an annuity that lasts forever. What is the present value of a perpetuity? PV (Perpetuity) =.  What is the future value of a perpetuity? PMT i Perp fv =

Uneven Cash Flow Streams: Setup 0 $100 1 $ % -$50 4 $ $ = PV

Input into the cash flow register: CF 0 = 0 CF 1 = 100 CF 2 = 300 CF 3 = 300 CF 4 = -50 Enter i= 10%, then press NPV button to get $ NPV means “net present value.” Uneven Cash Flow Streams: Financial Calculator Solution

Investment Returns The financial performance of an investment is measured by its return. –Time value analysis is used to calculate investment returns. –Returns can be measured either in dollar terms or in rate of return terms. Assume that a hospital is evaluating a new MRI. The project’s expected cash flows are given on the next slide.

MRI Investment Expected Cash Flows (in thousands of dollars) 0 $310 1 $400 2 $500 3 $ $1,500  Where do these numbers come from?

Simple Dollar Return 0 $310 1 $400 2 $500 3 $ $ 460 = Simple dollar return -$1,500  Is this a good measure?

Discounted Cash Flow (DCF) Dollar Return 0 $310 1 $400 2 $500 3 $ $ 78 = net present value (NPV) -$1,500 Where did the 8% come from? 8%

DCF Dollar Return (Cont.) The key to the effectiveness of this measure is that the discounting process automatically recognizes the opportunity cost of capital. An NPV of zero means the project just earns its opportunity cost rate. A positive NPV indicates that the project has positive financial value after opportunity costs are considered.

Rate of (Percentage) Return 0 $310 1 $400 2 $500 3 $ $ 0.00 = NPV, so rate of return = 10.0%. -$1,500 10%

Rate of Return (Cont.) In capital investment analyses, the rate of return often is called internal rate of return (IRR). In essence, it is the percentage return expected on the investment. To interpret the rate of return, it must be compared to the opportunity cost of capital. In this case 10% versus 8%.

Intra-Year Compounding Thus far, all examples have assumed annual compounding. When compounding occurs intra-year, the following occurs. –Interest is earned on interest more frequently. –The future value of an investment is larger than under annual compounding. –The present value of an investment is smaller than under annual compounding.

% % Annual: FV 3 = 100 x (1.10) 3 = Semiannual: FV 6 = 100 x (1.05) 6 =

EAR is the annual rate which causes the PV to grow to the same FV as under intra-year compounding. What is the EAR for 10%, semiannual compounding? –Consider the FV of $1 invested for one year. FV = $1 x (1.05) 2 = $ –EAR = 10.25%, because this rate would produce the same ending amount ($1.1025) under annual compounding. Effective Annual Rate (EAR)

The EAR Formula EAR = i Stated q q = = (1.05) = = 10.25%. Or, use the EFF% key on a financial calculator.

EAR of 10% at Various Compounding EAR Annual = 10%. EAR Q =( /4) = 10.38%. EAR M =( /12) = 10.47%. EAR D(360) =( /360) = 10.52%.

Using the EAR 0 1 $ % month periods $100 Here, payments occur annually, but compounding occurs semiannually, so we cannot use normal annuity valuation techniques.

First Method: Compound Each CF 01 $ % 4 56 $100$ $331.80

Second Method: Treat as an Annuity EAR = ( 1 + ) - 1 = 10.25% Find the EAR for the stated rate: Then use standard annuity techniques: INPUTS OUTPUT NI/YRPVFV PMT

Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

Step 1: Find the required payments. PMT % -$1, INPUTS OUTPUT NI/YRPVFV PMT

Step 2: Find interest charge for Year 1. INT t = Beginning balance x i. INT 1 = $1,000 x 0.10 = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = $ $100 = $

Step 4: Find ending balance at end of Year 1. End bal= Beg balance - Repayment = $1,000 - $ = $ Repeat these steps for Years 2 and 3 to complete the amortization table.

Note that annual interest declines over time while the principal payment increases. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$ TOTAL $1,206.34$206.34$1,000

$ Interest Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Principal Payments

Context in MS Office Word processing (text-oriented) –MS Word Database (data storage-oriented) –MS Access Presentation (slide-oriented) –MS PowerPoint Web Development (web-page-oriented) –MS FrontPage Numerical Analysis (processing-oriented) –MS Excel

Context in MS Office (cont.) Oriented is key since all perform similar tasks but to different degrees Integration across applications important Need should drive choice among apps In business setting, planning and setup are important Training is costly, so cost/benefit prevails

Purposes of Excel Data entry/storage –Access to data by: File Import Database Query Data entry Data analysis –What if analysis –Functions/Wizards/Macros (Programming) Presentation/Reports/Graphics

Grid/Cells Referencing the grid: columns/row intersect –Fundamental unit is the cell –A1, B3, etc.; relative referencing is default –$ in front of either is absolute reference –Affects what happens when you copy formulas Formulas =$A$1*B1 *, /, +, -, ^ Typical precedence Functions –E.g. =Sum(cell reference to range affected)

Data Entry Cells contain –Text –Numbers –Formulas, referencing other cells –Functions, referencing other cells Divide work (and spreadsheet) –Data –Analysis –Presentation

Objects Workbook (corresponds to *.xls file) Worksheet (many per Workbook) –Special ones for Graphical and statistical output Cells (many per Worksheet)

Practice (d-click table)

Microsoft Office Excel 2003 Tutorial 2 – Working With Formulas and Functions

Use Excel’s functions You can easily calculate the sum of a large number of cells by using a function. A function is a predefined, or built-in, formula for a commonly used calculation. Each Excel function has a name and syntax. –The syntax specifies the order in which you must enter the different parts of the function and the location in which you must insert commas, parentheses, and other punctuation –Arguments are numbers, text, or cell references used by the function to calculate a value –Some arguments are optional

Work with the Insert Function button Excel supplies more than 350 functions organized into 10 categories: –Database, Date and Time, Engineering, Financial, Information, Logical, Lookup, Math, Text and Data, and Statistical functions You can use the Insert Function button on the Formula bar to select from a list of functions. A series of dialog boxes will assist you in filling in the arguments of the function and this process also enforces the use of proper syntax.

Math and Statistical functions

Define functions, and functions within functions The SUM function is a very commonly used math function in Excel. A basic formula example to add up a small number of cells is =A1+A2+A3+A4, but that method would be cumbersome if there were 100 cells to add up. Use Excel's SUM function to total the values in a range of cells like this: SUM(A1:A100). You can also use functions within functions. Consider the expression =ROUND(AVERAGE(A1:A100),1). –This expression would first compute the average of all the values from cell A1 through A100 and then round that result to 1 digit to the right of the decimal point

Copy and paste formulas and functions Copying and pasting a cell or range of cells is a simple, but highly effective means for quickly filling out a large worksheet. To copy and paste a cell or range: –Select the cell or range to be copied and then click the Copy button on the standard toolbar –Select the cell or range into which you want to copy the selection and then click the Paste button on the standard toolbar –Once you are finished pasting, press the Esc key to deselect the selection

Copy and paste effects on cell references Copied formulas or functions that have cell references are adjusted for the target cell or range of cells. For example, if cell G5 contains the formula =F5*B5/B7, and you copy and paste this formula to cell G6, the formula in cell G6 will be =F6*B6/B8. This may or may not be correct for your worksheet, depending upon what you are trying to do. You can control this automatic adjusting of cell references through the use of relative and absolute references.

Problems using copy and paste with formulas When Excel does not have enough room to display an entire value in a cell, it uses a string of these # symbols to represent that value. For example, the formula in cell J5 is =F5-(H5+I5) and this was pasted into cell J6 by updating the cell references there to =F6-(H6+I6). Cell G5 has the formula =F5*B5/B7 and cell G6 contains =F6*B6/B8. This is where things went wrong. Sometimes this automatic update is very useful and other times it does not give you the desired result for your worksheet. In this case, cells B5 and B7 should be referenced in the formula in column G in all 240 payment period rows, but in column J, you want the cell references to be automatically updated. You can control this result using relative and absolute references.

Use relative references A relative reference is a cell reference that shifts when you copy it to a new location on a worksheet. A relative reference changes in relation to the change of location. If you copy a formula to a cell three rows down and five columns to the right, a relative reference to cell B5 in the source cell would become G8 in the destination cell.

Use absolute references An absolute reference is a cell reference that does not change when you copy the formula to a new location. To create an absolute reference, you preface the column and row designations with a dollar sign ($). For example, the absolute reference for B5 would be $B$5. This cell reference would stay the same no matter where you copied the formula.

Use mixed references A mixed reference combines both relative and absolute cell references. You can effectively lock either the row or the column in a mixed reference. –For example, in the case of $B5, the row reference would shift, but the column reference would not –In the case of B$5, the column reference would shift, but the row reference would not You can switch between absolute, relative and mixed references in the formula easily in the edit mode or on the formula bar by selecting the cell reference in your formula and then pressing the F4 key repeatedly to toggle through the reference options.

Open the Insert Function dialog box To get help from Excel to insert a function, first click the cell in which you wish to insert the function. Click the Insert Function button. This action will open the Insert Function dialog box. If you do not see the Insert Function button, you may need to select the appropriate toolbar or add the button to an existing toolbar.

The Average Function The average function is necessary to calculate the average of a range of cells. Like any other formula, the average function may be copied across cells.

Date Functions

Excel's date functions Excel stores dates as integers, where the integer value represents the number of days since January 1, –For example, the integer value for the date January 1, 2008 is because that date is 39,448 days after January 1, 1900 You typically do not see these numbers, because Excel automatically formats them to appear in a date format. This method of storing dates allows you to work with dates the same way you work with numbers. Excel's commonly used date functions are DATE, DAY, MONTH, NOW, TODAY, WEEKDAY and YEAR.

The TODAY and Now functions The TODAY and NOW functions always display the current date and time. You will not normally see the time portion unless you have formatted the cell to display it. If you use the TODAY or NOW function in a cell, the date in the cell is updated to reflect the current date and time of your computer each time you open the workbook.

Use a formula to enter the date

Excel's financial functions Financial functions are very useful to calculate information about loans. Common functions are FV, IPMT, PMT, PPMT and PV. All these financial functions will use similar arguments that differ based upon which function you are using. –Think of the arguments as members of an equation –The arguments represent the values of the equation that are known and the function provides the solution for a single variable, or unknown, value

Use the financial functions The FV function calculates the future value of an investment based on periodic, constant payments and a constant interest rate per period. The IPMT function provides the interest payment portion of the overall periodic loan payment. The PMT function calculates the entire periodic payment of the loan. The PPMT function calculates just the principal payment portion of the overall periodic payment. The PV function calculates the present value of an investment.

Financial Function descriptions

Recognize optional arguments In the preceding figure, note how rate and nper are arguments for each function. For some of the functions, the final two arguments of each function are in brackets. These represent optional arguments, meaning if you do not enter anything, the default values for these arguments will be used. –For example, note the PMT function has fv and type as its final two arguments, which are optional. The assumed values, if no others are supplied, are 0 for both Arguments without brackets do not have default values, so you must supply values or cell references in order for the function to be able to return a value.

Use the Insert Function dialog box to enter function arguments

Create logical functions A function that determines whether a condition is true or false is called a logical function. Excel supports several logical functions such as AND, FALSE, IF, NOT, OR and TRUE. A very common function is the IF function, which uses a logical test to determine whether an expression is true or false, and then returns one value if true or another value if false. The logical test is constructed using a comparison operator that compares two expressions to determine if they are equal, not equal, if one is greater than the other, and so forth. –The comparison operators are =, >, >=, You can also make comparisons with text strings. You must enclose text strings within quotation marks.

Using the If function The arguments for the IF function are: –IF(logical_test,value_if_true,value_if_false) –For example, the function =IF(A1=10,20,30) tests whether the value in cell A1 is equal to 10 –If it is, the function returns the value 20, otherwise the function returns the value 30 –Cell A1 could be empty or contain anything else besides the value 10 and the logical test would be false; therefore, the function returns the value 30 To insert an IF function, click the Insert Function button and search for the IF function, then click OK. When the Function Arguments dialog box appears, simply fill in the arguments.

Case 11 Review Questions