UNDERSTANDING HEALTHCARE FINANCIAL MANAGEMENT, 5ed. Time Value Analysis

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1 This is a sample of the instructor resources for Understanding Healthcare Financial Management, Fifth Edition, by Louis Gapenski. This sample contains the chapter models, end-of-chapter problems, and end-of-chapter solutions, as well as PowerPoint slides for Chapter 3. The complete instructor resources consist of 39 pages of instructor notes; chapter models, end-of chapter problems, end-of-chapter solutions; and 744 PowerPoint slides. If you adopt this text you will be given access to complete materials. To obtain access, your request to hap1@ache.org and include the following information in your message: Book title Your name and institution name Title of the course for which the book was adopted and season course is taught Course level (graduate, undergraduate, or continuing education) and expected enrollment The use of the text (primary, supplemental, or recommended reading) A contact name and phone number/ address we can use to verify your employment as an instructor You will receive an containing access information after we have verified your instructor status. Thank you for your interest in this text and the accompanying instructor resources.

2 A B C D E F G H Chapter 3 UNDERSTANDING HEALTHCARE FINANCIAL MANAGEMENT, 5ed Time Value Analysis This spreadsheet model performs many of the calculations contained in Chapter 3. We recommend that you use the model in the following manner: 1. First, recognize that you do not have to use this model at all to understand time value analysis concepts. However, using the model will increase your understanding of the relevant concepts, and it will surely help when you use spreadsheet models for other purposes, especially any problem sets or cases assigned for this course. 2. Start by reading the chapter in its entirety. 3. Now place the text alongside your computer with this model on the screen. When you come to an explanation of a calculation in the text, see if the model has a matching calculation. 4. We assume that you know the basics of Excel, but you have not encountered some of its features or you may need a refresher or two. So we have built in explanations of how to do some of the functions in the model. As a result, you will learn more about Excel at the same time you learn about time value analysis. 5. Throughout this model, page numbers of the matching text calculations are provided in pink. Input data are in red on a yellow background, and output data are in green on a beige background. You are encouraged to change the input data to learn more about the calculations in the model. FUTURE VALUE OF A LUMP SUM (COMPOUNDING) (PAGE 81) 33 Find the future value (FV) of $100 after five years at an interest rate of 5% Interest rate 5.0% Note that input values are always Make some changes to the input values Lump sum 100 in red on a yellow background. to see the effect on future value. Time period Note that an FV at year end in green on This problem was solved using the formula, FV = PV x (1+I) N. However, there are a number of ways this problem could have been solved. One of the most valuable features in Excel is the "Function Wizard." To begin, first move the cursor to the cell where the answer is to appear. In this case, H90. Next, select the Function Wizard icon found in the toolbar at the top of the screen. It looks like this This button allows you to enter the Function Wizard. Upon clicking on it, you will see a dialog box entitled "Paste Function." On the left side of the dialog box is a menu entitled "Function Category," and on the right is a menu called "Function Name."

3 A B C D E F G H We will be selecting the "FV" function from the "Financial" category, and we will be using the following dialog box to input our data Notice that we entered a cell reference as the input for the problem instead of the actual value. We do this so that our spreadsheet can automatically reflect any changes to the input data. This is one of the features that make the spreadsheet such a valuable tool. Also note that the FV function can use up to five inputs. To find the future value of a lump sum only three are required, so the resulting formula has double commas, which means that the third input is zero. The final input (Type) is not required. The wizard enters =FV(B35,G38,,-B36) in Cell G90. FV = $ GRAPHIC VIEW OF THE COMPOUNDING (GROWTH) PROCESS (PAGE 85) With a spreadsheet, calculating FVs is a simple operation, and we can use them to graph the relationship between future value, growth, interest rates, and time. A similar graph can be found in the textbook in Figure 3.1. Period (n) 0% 5% 10% 15%

4 A B C D E F G H Future Value of $1 $5.00 $4.00 $3.00 $2.00 $1.00 $0.00 Relationships Among Future Value, Growth, Interest Rate, and Time Periods To create a graph in Excel, you first must access the "Chart Wizard" found in the toolbar near the top of the screen denoted by. Upon selecting the Chart Wizard, the first input dialog box will appear, and it will ask for the "Chart Type." In this case, we want a line graph, so we select "Line," and then we click one of the subtypes, in this case the first one After clicking "Next," we are presented with the "Source Data" box. We first enter the "Data Range" for the chart. These are the 159 data that will comprise the lines in our line graph. For our example, this information can be found in the above table. To select the 160 data range, use the cursor to highlight the cells range from B99 to E104. Note that the data are contained in columns

5 A B C D E F G H Now we must select the "Series" for our graph. Essentially, this is the set of data that will comprise each line in the graph. In this 191 graph, you can name each individual series by selecting it in the Series section and typing a label into the Name box. When an 192 individual series is highlighted, the Values section will let you know what data from the data range make up that particular line. To 193 place appropriate labels on the X axis, you must go to the Category (X) axis labels box and highlight from A99 to A

6 A B C D E F G H At this point, all of the necessary data for the chart have been inserted. From here, the chart just needs to be formatted according to your preferences (e.g., show or hide gridlines, change the numbers on the axes). PRESENT VALUE OF A LUMP SUM (DISCOUNTING) (PAGE 87) Find the present value (PV) of $ discounted back five years at an interest rate of 5%. Interest rate Lump sum 5.0% Make some changes to the input values to see the effect on present value. Time period PV at year end This problem can also be solved using the function wizard using a procedure similar to that for the FV function. Begin by putting the pointer on the cell in which you want to display the result, in this case Cell C264. Then, after selecting the "PV" function from the "Paste Function" box, the input data for the problem must be entered. Then click OK to get the result, $ PV = $ SOLVING FOR INTEREST RATE (I) (PAGE 92) What is the interest rate of a security priced at $78.35 that pays $100 after five years? N 5 Make some changes to the input values 272 PV to see how the variables affect I (Cell C298). 273 FV Once again, Excel has a special function for this calculation. We suggest using either a financial calculator or the Function Wizard 276 to solve this type of problem, because of its complexity. The procedure can be carried out using the Function Wizard, by selecting the 277 "Rate" function from the list of financial functions in the "Paste Function" dialog box. Upon entering the time, present value, and 278 future value, the interest rate can be found. Note that you can either type the data in or activate the menu slot and then click on

7 A B C D E F G H the appropriate cell. I = 5.00% We noted above the difficulty of solving this problem mathematically. This is because it involves taking the n th root of a value (an operation that generally requires either a calculator or a computer). However, if you would like to know how to solve the problem mathematically, the formula is: (FV N / PV) 1/N - 1, which is derived from the FV formula. N 5 I = 5.00% PV FV 100 SOLVING FOR TIME (N) (PAGE 93) A security yields 5%, costs $78.35 today, and will return $100 at some future date. What is the security's term to maturity? I 5.0% Make some changes to see the impact 316 PV on N (Cell C338). 317 FV The Function Wizard can be used to solve for time, or N. This operation can be performed by selecting the "Nper" option from the li 320 of financial functions, and then entering the input data into the dialog box

8 A B C D E F G H N = 5.00 Solving for N mathematically is very complicated. The formula for finding N involves using natural logarithms, which is a complex operation. For this reason, we highly suggest the use of the Function Wizard or financial calculator to solve this type of problem. However, here is the formula needed to solve for N: N = (ln (FV N / PV ) / (ln (1+ I)). This formula is applied in Cell C345. N = 5.00 ORDINARY ANNUITIES (PAGE 94) (Future Value Calculation Only, See Row 427 for Present Value) If the interest rate is 5%, what is the future value of an ordinary annuity that pays $100 at the end of each of the next three years? As explained below, one way to solve this problem is to find the future value of each of the annuity payments. However, this is somewhat tedious, especially if a lot of years are involved. In the following example, we use the input data of the interest rate and time to calculate the future value in time period 3 of each individual cash flow. Lastly, we take the sum of all the future values, which gives us the future value of the entire annuity. N 3 Change the I and PMT inputs to see their I 5.0% effect on FV. Note that N cannot be changed. PMT 100 Time period Annuity pmt Annuity FV FV = $ An easier procedure is to use the Function Wizard to solve for the future value of an annuity. This procedure is similar to that of a lump sum future value. Whereas before we left the "Pmt" field blank, now we insert the annuity payment ($100 in this case). First, we access the "FV" function box from the list of financial functions. Then, we input our new data. A key 369 thing to watch is the "Type" input box. Previously, we left this box alone. A "0" or no entry in the box indicates an ordinary annuity, and a "1" indicates an annuity due. Though we can leave the box blank, it is a good habit to enter a "0" in the field. FV = $ ANNUITIES DUE (PAGE 96) (Future Value Calculation Only, See Row 463 for Present Value) If the interest rate is 5%, what is the future value of an annuity due that pays $100 at the beginning of each of the next three years? The procedure for solving this problem follows the previous example with one notable exception. Because the payments occur at the

9 A B C D E F G H beginning of each year, the first annuity payment occurs in time period 0, and the last occurs in time period 2. N 3 Change the I and PMT inputs to see their I 5.0% effect on FV. Note that N cannot be changed. PMT 100 Time period Annuity pmt Annuity FV FV = $ Additionally, using the Function Wizard for this problem is exactly like above, but we enter a "1" instead of a "0" into the "Type" field. FV = $ ORDINARY ANNUITIES (PAGE 94) N 436 I 437 PMT (Present Value) If you were given the option of receiving a lump sum of money today or an annuity that pays $100 at the end of each of the next three years, at what price should you be indifferent to the two options, if the interest rate is 5%? The way to solve this problem is to find the PV of the annuity and then compare it with the lump sum. First, we consider each payment separately. 3 Change the I and PMT inputs to see their 5.0% effect on PV. Note that N cannot be changed. Time period Annuity pmt Annuity PV PV = $ Or, you could use the Function Wizard for this ordinary annuity.

10 A B C D E F G H PV = $ ANNUITIES DUE (PRESENT VALUE) (PAGE 96) (Present Value) What if the payments occurred at the beginning of each year? This problem is solved just like the previous problem, except that the payments occur in periods 0 through 2. N 3 Change the I and PMT inputs to see their I 5.0% effect on PV. Note that N cannot be changed. PMT 100 Time period Annuity pmt Annuity PV PV = $ Using the Function Wizard, we follow the same procedure as above, except remember to enter a "1" to tell Excel that this problem ha payments occurring at the beginning of the periods PV = $ PERPETUITIES (PAGE 99) What is a perpetuity worth if it pays $100 every year and the discount rate is 5%? PMT 100 Change the payment and discount rate 503 I 5.0% to see their impact on present value PV = $2, UNEVEN CASH FLOW STREAMS

11 A B C D E F G H Present Value (Page 100) Calculate the present value of the following cash flow stream, discounted at 10%. I 10.0% Change the discount rate (I) to examine its effect on PV Change the cash flows to examine their impact on PV PV of Stream NPV = $ As we show above, the first way to solve for the present value of this uneven cash flow stream is to use the timeline to find the present value of each of the cash flows in the periods in which they occur, then sum all the present values. This procedure will yield the correct present value. This problem could also be set up in a column format; it is a matter of personal preference as to which format is easier to interpret and use. Once we have placed the data into columns, we can solve for the present value of each of the cash flows (like we did previously) and add all of the present values together to get the final answer. I 10.0% N CF PV Try entering a $500 outflow at Year NPV = $ With, the financial calculator, we could enter each of these cash flows and the discount rate, and simply press NPV for the present value of the cash flow stream. In Excel, we can perform a similar calculation by using the "NPV" function. While this function is very similar, there is a key distinction. In the cash flow register of your calculator, the first entry you make would be the cash flow to occur in time period zero. However, the "NPV" function interprets the first data entry as being the cash flow in time period one. Therefore, the initial cash flow must be added separately. In this particular example, the initial cash flow is zero. Data from either the timeline or the columns could be entered here.

12 A B C D E F G H NPV = $ Both methods will yield the same result, so use the one that you are most comfortable with. In the event that you have a problem consisting of a cash flow at time period zero, you will have to manually add this value to the NPV of the remaining cash flows. Also note that when using the Function Wizard, we used the cash flow data from columns. However, we could have just as easily used the timeline data that we initially presented. Future Value (Page 101) Calculate the future value of the cash flow stream illustrated above in the previous question. First, we will solve this problem by adding the future values of all the cash flows in time period 7. I 10.0% Change the discount rate (I) to examine its effect on FV. N # compoundings CF FV FV = $ Excel does not have a net future value function, but the above procedure works for this type of problem. 601 An alternative is to calculate the NPV and then compound that value out to the end of the cash flow 602 stream, but that procedure is not as easy conceptually as that used above USING TIME VALUE ANALYSIS TO MEASURE FINANCIAL RETURNS Dollar Return (Page 103) Calculate the net present value (NPV) of the following cash flow stream, discounted at 10% I 10.0% Change the discount rate (I) to 611 examine its effect on PV Change the cash flows to 616 examine their impact on PV 617 PV of Stream NPV = $ As we show above, the first way to solve for the NPV of this investment is to use 623 the timeline to find the present value of each of the cash flows in the periods in which they occur, then 624 sum all the present values. This procedure will yield the correct present value This problem could also be set up in a column format; it is a matter of personal preference as to which format is easier to

13 A B C D E F G H interpret and use. Once we have placed the data into columns, we can solve for the present value of each of the cash flows (like we did previously) and add all of the present values together to get the final answer. I 10.0% N CF PV NPV = $80.95 With, the financial calculator, we could enter each of these cash flows and the discount rate, and simply press NPV for the present value of the cash flow stream. In Excel, we can perform a similar calculation by using the "NPV" function. While this function is very similar, there is a key distinction. In the cash flow register of your calculator, the first entry you make would be the cash flow to occur in time period zero. However, the "NPV" function interprets the first data entry as being the cash flow in time period one. Therefore, the initial cash flow must be added separately. In this particular example, the initial cash flow is Rate of Return (Page 104) NPV = $80.95 Calculate the internal rate of return (IRR) of the cash flow stream illustrated above in the previous question. Here we use the IRR function: IRR = 15.3% Note that the IRR function has this format: IRR(range, starting guess). The starting guess is required to "begin" the iterative calculation procedure used by Excel. We used the discount rate as the guess, but any reasonable value could have been entered. SEMIANNUAL AND OTHER COMPOUNDING PERIODS (PAGE 105) If $100 is invested in an account at an interest rate of 6%, annual compounding, for three years, what is the FV? N 3 FV = $ I 6.0% PV 100 What is the FV with semiannual compounding? N (years x 2) 6 FV = $ I (I per year/2) 3.0% PV 100 $0.30 is the difference. What is the PV of an ordinary annuity of $100 per year for three years when the interest rate is 8%, compounded annually? N 3 PV = $ I 8.0%

14 A B C D E F G H PMT 100 What is the PV of an ordinary annuity of $100 per year for three years when the interest rate is 8%, compounded semiannually? N 6 FV = $ I 4.0% PMT 50 Remember that in cases of non-annual compounding, all input variables (N, I, PMT) must reflect the number of compounding period AMORTIZED LOANS (PAGE 108) What would the required payment be on a $1,000 loan that is to be repaid in three equal installments at the end of each of the next three years if the interest rate is 6%? N I PV PMT = $374,110 3 Change the inputs to see the impact 6.0% on the payment amount and the amortization table. Now, construct an amortization table for the loan described above. N Loan amount Payment Interest Principal Balance 1 $1,000,000 $374,110 $60,000 $314,110 $685, , ,110 41, , , , ,110 21, ,934 0 Totals $1,122,329 $122,329 $1,000,000

15 UNDERSTANDING HEALTHCARE FINANCIAL MANAGEMENT, 5ed Chapter 3 -- Time Value Analysis PROBLEM 1 Find the following values for a lump sum assuming annual compounding a. The future value of $500 invested at 8 percent for one year b. The future value of $500 invested at 8 percent for five years c. The present value of $500 to be received in one year when the opportunity cost rate is 8 percent d. The present value of $500 to be received in five years when the opportunity cost rate is 8 percent

16 A B C D E F G H I UNDERSTANDING HEALTHCARE FINANCIAL MANAGEMENT, 5ed Chapter 3 -- Time Value Analysis PROBLEM 1 Find the following values for a lump sum assuming annual compounding: a. The future value of $500 invested at 8 percent for one year b. The future value of $500 invested at 8 percent for five years c. The present value of $500 to be received in one year when the opportunity cost rate is 8 percent d. The present value of $500 to be received in five years when the opportunity cost rate is 8 percent ANSWER Spreadsheet solution: a. FV = $500 x (1.08)1 = $ ($540.00) =FV(0.08,1,,500) b. FV = $500 x (1.08)5 = $ ($734.66) =FV(0.08,5,,500) c. PV = $500 / (1.08)1 = $ ($462.96) =PV(0.08,1,,500) d. PV = $500 / (1.08)5 = $ ($340.29) =PV(0.08,5,,500)

17 3-1 CHAPTER 3 Time Value Analysis Future F t and present values Solving g for I and N Investment returns Opportunity cost Amortization 8/22/06

18 3-2 Time Value of Money Time value analysis is necessary because money has time value. A dollar in hand today is worth ot more oethan 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.

19 3-3 Time Lines I% CF 0 CF 1 CF 2 CF 3 Tick marks designate ends of periods. Time 0 is the starting gp point (the beginning g of Period 1); Time 1 is the end of Period 1 (the beginning of Period 2); and so on.

20 3-4 What is the FV after three years of a $100 lump sum invested at 10%? % -$100 FV =? Finding g future values (moving to the right along the time line) is called compounding. For now, assume interest is paid annually.

21 3-5 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 xi)= FV 1 x(1+i) = PV x (1 + I) x (1 + I) = PV x (1 + I) 2 = $100 x (1.10) 10) 2 = $121.00

22 3-6 After 3 years: FV 3 = FV 2 +I 3 = PV x (1 + I) 3 = 100 x (1.10) 10) 3 = $ In general, FV N = PV x (1 + I) N

23 3-7 Three Primary Methods to Find FVs Solve l the FV equation using a regular calculator. Use a financial calculator that is, one with financial functions. Use a computer with a spreadsheet program such as Microsoft Excel.

24 3-8 Regular Calculator l Solution % -$100 $ $ $ $100 x 1.10 x 1.10 x 1.10 = $133.10

25 3-9 Spreadsheet Solution Function = FV(rate,nper,pmt,pv,type) pmt pv type) Cell formula = FV(0.1,3,,100) Cell display = (133.10) 10)

26 3-10 Spreadsheet Solution (Cont.) rate = interest rate per period nper = total number of payment periods in an annuity pmt = payment made each period; cannot change over the life of the annuity pv = present value or lump sum amount that a series of future payments is worth now; if omitted, pv=0 type = payment at beginning of period = 1; payment at end of period = 0 or omitted

27 3-11 What is the PV of $100 due in three years if I = 10%? % PV =? $100 Finding present values (moving to the left along the time line) is called discounting.

28 3-12 Solve FV N = PV x (1 + I ) N for PV PV = FV /(1+I) N N PV = $100 / (1.10) 3 = $100(0.7513) = $75.13? If I offer you $75.13 today or $100 three years from now, which would you prefer?

29 3-13 Spreadsheet Solution Function = PV(rate,nper,pmt,fv,type) pmt type) Cell formula = PV(0.1,3,,100) Cell display = (75.13)

30 3-14 Solving for I Assume that a bank offers an account that will pay $200 after five years on each $75 invested. What is the implied interest rate? Function = RATE(nper,pmt,pv,fv,type) Cell formula = RATE(5,0,-75,200) 00) Cell display = 22%

31 3-15 Solving for N Assume an investment earns 20 percent per year. How long will it take for the investment to double? Function = NPER(rate,pmt,pv,fv,type) pv type) Cell formula = NPER(0.2,0,-1,2) Cell display = ? What is the Rule of 72?

32 3-16 Types of Annuities Three Year Ordinary Annuity I% PMT Three Year Annuity Due PMT PMT I% PMT PMT PMT

33 3-17 What is the FV of a three-year ordinary annuity of $100 invested at 10%? % $100 $100 $ FV = $331

34 3-18 Spreadsheet Solution Function= FV(rate,nper,pmt,pv,type) pmt pv type) Cell formula = FV(0.1,3,100) Cell display = (331.00)

35 3-19 What tis the PV of fthe annuity? % $100 $100 $100 $ $ = PV

36 3-20 Spreadsheet Solution Function = PV(rate,nper,pmt,fv,type) pmt type) Cell formula = PV(0.1,3,100) Cell display = (248.69)

37 3-21 What is the FV and PV if the annuity were an annuity due? % $100 $100 $100??

38 3-22 What is the FV of the annuity due? % $100 $100 $ FV = $364

39 What tis the PV of fthe annuity due? % $ $100 $100 $ $ = PV

40 3-24 Perpetuities A perpetuity is an annuity that lasts forever. What t is the present value of a perpetuity? PV (Perpetuity) = PMT I? What is the future value of a perpetuity?

41 Uneven Cash Flow Streams: Setup % 4 $ $ = PV $100 $300 $300 -$50

42 3-26 Spreadsheet Solution Function = NPV(rate,value1,value2, ) value2 Cell formula =NPV(0.1,100,300,300,-50) Cell display =

43 3-27 Return on Investment t (ROI) The financial performance of an investment is measured by its return on investment. 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.

44 3-28 MRI Investment t Expected Cash Flows (thousands) $1,500 $310 $400 $500 $750? Where do these numbers come from?

45 3-29 Simple Dollar Return $1,500 $310 $400 $500 $ $ 460 = Simple dollar return? Is this a good measure?

46 3-30 Discounted Cash Flow (DCF) Dollar Return -$1, % $310 $400 $500 $750 $ 78 = net present value (NPV)

47 3-31 Spreadsheet Solution A B C D % Interest rate 3 $ (1,500) Year 0 CF Year 1 CF Year 2 CF Year 3 CF Year 4 CF $ 78 =NPV(A2,A4:A7)+A3 (entered into Cell A10)

48 3-32 Opportunity Cost Rate To find an investment s dollar return (NPV), we need to apply a discount rate. Where does 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.

49 3-33 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 alternative investments (usually securities) that have the same risk. Estimate t the expected return on the similar-risk il i alternative investment. When applied, the resulting PV provides a return equal to the opportunity cost rate. In most time value situations, s, benchmark opportunity cost rates are known.

50 3-34 Opportunity Cost Rate (Cont.) When calculating lating NPV, the discounting process automatically recognizes the opportunity cost of capital. Thus, A positive NPV means that the investment is expected to create value for the investor. A negative NPV means that the investment is expected to lose value for the investor.

51 3-35 -$1, Rate of f(percentage) )Return % $310 $400 $500 $750 $ 0.00 = NPV, so E(R) = 10.0%.

52 3-36 Spreadsheet Solution A B C D % Interest rate guess 3 $ (1,500) Year 0 CF Year 1 CF Year 2 CF Year 3 CF Year 4 CF % =IRR(A3:A7,A2) (entered into Cell A10)

53 3-37 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 ty cost of capital. In this case, 10 percent versus 8 percent.

54 3-38 Intra-Year Compounding Thus far, all examples have assumed annual compounding. When compounding occurs intra-year, the following occurs: Interest t t is earned on interest t during the year (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.

55 % Annual: FV 10) 3 3 = 100 x (1.10) = % Semiannual: FV = x (1.05) =

56 3-40 Effective Annual Rate (EAR) EAR is the annual rate that causes the PV to grow to the same FV as under intra-year compounding. What is the EAR for 10 percent, 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) 1025) under annual compounding.

57 3-41 The EAR Formula I Stated EAR = M M = = (1.05) = = 10.25%

58 3-42 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%

59 3-43 Spreadsheet Solution Function = EFFECT(nominal_rate,npery) nper Cell formula = EFFECT(0.10,4) Cell display = Function = EFFECT(nominal_rate,npery) Cell formula = EFFECT(0.10,12) 12) Cell display = Function = EFFECT(nominal_rate,npery) Cell formula = EFFECT(0.10,365) Cell display =

60 3-44 Spreadsheet Solution nominal_rate = nominal interest rate npery = number of compounding periods per year

61 3-45 Using the EAR month 5% periods $100 $100 $100 Here, payments occur annually, but compounding occurs semiannually, so we can not use normal annuity valuation techniques.

62 First Method: Compound Each CF % $100 $100 $ $331.80

63 3-47 Second Method: Treat as an Annuity Find the EAR for the stated rate: Function = EFFECT(nominal_rate,npery) Cell formula = EFFECT(0.10,2) Cell display = Then, use standard annuity techniques: Function = FV(rate,nper,pmt,pv,type) pmt pv type) Cell formula = FV(0.1025,3,100) Cell display = (331.80)

64 3-48 Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with three equal payments.

65 3-49 Step 1: Find the required payments % -$1,000 PMT PMT PMT Function = PMT(rate,nper,pv,fv,type),p,, Cell formula =PMT(0.1,3,1000) Cell display = (402.11)

66 3-50 Step 2: Find interest t 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 = $302.11

67 3-51 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

68 3-52 BEG PRIN END YR BAL PMT INT PMT BAL 1 $1,000 $402 $100 $302 $ TOTAL $1,206 $206 $1,000 Note that annual interest declines over time while the principal payment increases.

69 $ Interest Principal Payments Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, t which h is falling.

70 3-54 Conclusion This concludes our discussion i of Chapter 3 (Time Value Analysis). Although not all concepts were discussed in class, you are responsible for all of the material in the text.? Do you have any questions?