# CHAPTER 9 Time Value Analysis

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1 Copyright 2008 by the Foundation of the American College of Healthcare Executives 6/11/07 Version 9-1 CHAPTER 9 Time Value Analysis Future and present values Lump sums Annuities Uneven cash flow streams Solving for I and N Investment returns Amortization

2 9-2 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.

3 9-3 Time Lines I% CF 0 CF 1 CF 2 CF 3 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.

4 9-4 What is the FV after 3 years of a \$100 lump sum invested at 10%? -\$ % FV =? Finding future values (moving to the right along the time line) is called compounding. For now, assume interest is paid annually.

5 9-5 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 x I) = FV 1 x (1 + I) = PV x (1 + I) x (1 + I) = PV x (1 + I) 2 = \$100 x (1.10) 2 = \$

6 9-6 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.

7 9-7 Three Primary Methods to Find FVs Solve the FV equation using a regular (nonfinancial) 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.

8 Nonfinancial Calculator Solution % -\$100 \$ \$ \$ \$100 x 1.10 x 1.10 x 1.10 = \$

9 9-9 Financial Calculator Solution INPUTS OUTPUT N I/YR PV PMT FV Notes: (1) Set your calculator on P/YR = 1, END. (2) For lump sums, the PMT key is not used. Either clear the calculator before you start or enter PMT = 0.

10 9-10 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 \$ Pv Present value % Rate Interest rate 5 6 \$ =100*(1.10)^3 (entered into Cell A6) 7 8 \$ =A3*(1+A4)^A2 (entered into Cell A8) 9 10 \$ =FV(A4,A2,,-A3) (entered into Cell A10)

11 9-11 What is the PV of \$100 due in 3 years if I = 10%? % PV =? \$100 Finding present values (moving to the left along the time line) is called discounting.

12 9-12 Solve FV N = PV x (1 + I ) N for PV PV = FV N / (1 + I ) N. PV = \$100 / (1.10) 3 = \$100(0.7513) = \$75.13.

13 Time Line Solution % \$75.13 \$82.64 \$90.91 \$100 \$ = \$ Note that the calculated present value (\$75.13), when invested at 10 percent for three years, will produce the starting future value (\$100).

14 9-14 Financial Calculator Solution INPUTS OUTPUT N I/YR PV PMT FV Either PV or FV must be negative on most calculators. Here, PV = Put in \$75.13 today, take out \$100 after three years.

15 9-15 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 \$ Fv Future value % Rate Interest rate 5 6 \$ =A3/(1+A4)^A2 (entered into Cell A6) 7 8 \$ =PV(A4,A2,,-A3) (entered into Cell A8) 9 10

16 9-16 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.

17 9-17 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. Why securities? Estimate the return expected on these similarrisk investments. When applied, the resulting PV provides a return equal to the opportunity cost rate. In most time value situations, benchmark opportunity cost rates are known.

18 9-18 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? INPUTS OUTPUT N I/YR PV PMT FV 21.7

19 9-19 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 \$ (75.00) Pv Present value 4 \$ Fv Future value % =RATE(A2,,A3,A4) (entered into Cell A8) 9 10

20 9-20 Solving for N Assume an investment earns 20 percent per year. How long will it take for the investment to double? INPUTS OUTPUT N I/YR PV PMT FV 3.8? What is the Rule of 72?

21 9-21 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 \$ (1.00) Pv Present value 4 \$ 2.00 Fv Future value =NPER(A2,,A3,A4) (entered into Cell A8) 9 10

22 9-22 Types of Annuities Three-Year Ordinary Annuity I% PMT Three-Year Annuity Due PMT PMT I% PMT PMT PMT

23 9-23 What is the FV of a three-year ordinary annuity of \$100 invested at 10%? % \$100 \$100 \$ FV = \$331

24 9-24 Financial Calculator Solution INPUTS OUTPUT N I/YR PV PMT FV Here there are payments rather than a lump sum present value, so enter 0 for PV.

25 9-25 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 \$ (100.00) Pmt Payment % Rate Interest rate \$ =FV(A4,A2,A3) (entered into Cell A8) 9 10

26 9-26 What is the PV of the annuity? % \$ \$ = PV \$100 \$100 \$100

27 9-27 Financial Calculator Solution INPUTS OUTPUT N I/YR PV PMT FV

28 9-28 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 \$ (100.00) Pmt Payment % Rate Interest rate \$ =PV(A4,A2,A3) (entered into Cell A8) 9 10

29 9-29 What is the FV and PV if the annuity were an annuity due? % \$100 \$100 \$100??

30 9-30 Switch from End to Begin mode on a financial calculator. Repeat the annuity calculations. First find PVA 3 = \$ INPUTS OUTPUT N I/YR PV PMT FV Then enter PV = 0 and press FV to find FV = \$

31 9-31 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 \$ (100.00) Pmt Payment % Rate Interest rate 5 6 \$ =PV(A4,A2,A3,,1) (entered into Cell A6) 7 8 \$ =PV(A4,A2,A3)*(1+A4) (entered into Cell A8) 9 10

32 9-32 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 \$ (100.00) Pmt Payment % Rate Interest rate 5 6 \$ =FV(A4,A2,A3,,1) (entered into Cell A6) 7 8 \$ =FV(A4,A2,A3)*(1+A4) (entered into Cell A8) 9 10

33 9-33 Perpetuities A perpetuity is an annuity that lasts forever. What is the present value of a perpetuity? PMT PV (Perpetuity) =. I? What is the future value of a perpetuity?

34 9-34 Uneven Cash Flow Streams 0 10% \$100 \$300 \$300 -\$50 \$ \$ = PV (often called Net PV [NPV])

35 9-35 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 4 \$ 100 Value 1 Year 1 CF Value 1 Year 2 CF Value 1 Year 3 CF 7 (50) Value 1 Year 4 CF \$ =NPV(A2,A4:A7) (entered into Cell A10)

36 9-36 Discussion Items Assume the cash flows on the previous slide are the cash flows from an investment. How much would you be willing to pay for these flows? What would be the financial benefit to you if you could buy it for less, say, \$500?

37 9-37 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.

38 9-38 MRI Investment Expected Cash Flows (in thousands of dollars) \$1,500 \$310 \$400 \$500 \$750? Where do these numbers come from?

39 9-39 Simple Dollar Return \$1, \$310 \$400 \$ 460 = Simple dollar return \$500 \$750? Is this a good measure?

40 9-40 Discounted Cash Flow (DCF) Dollar Return 0 8% \$1, \$310 \$400 \$500 \$ 78 = Net present value (NPV) \$750? Where did the 8% come from?

41 9-41 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 \$ (1,500) Year 0 CF Value 1 Year 1 CF Year 2 CF Year 3 CF Value 1 Year 4 CF \$ 78 =NPV(A2,A4:A7)+A3 (entered into Cell A10)

42 9-42 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.

43 9-43 Rate of (Percentage) Return 0 10% \$1, \$310 \$400 \$500 \$ 0.00 = NPV, so rate of return = 10.0%. \$750

44 9-44 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 \$ (1,500) Values Year 0 CF Year 1 CF Year 2 CF Year 3 CF Values Year 4 CF % =IRR(A3:A7,A2) (entered into Cell A10)

45 9-45 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 with the opportunity cost of capital, in this case 10% versus 8%.

46 9-46 Intrayear Compounding Thus far, all examples have assumed annual compounding. When compounding occurs intrayear, the following occurs: Interest is earned on interest 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.

47 % % Annual: FV 3 = 100 x (1.10) 3 = Semiannual: FV 6 = 100 x (1.05) 6 =

48 9-48 Effective Annual Rate (EAR) EAR is the annual rate that causes the PV to grow to the same FV as under intrayear 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.

49 9-49 The EAR Formula I Stated EAR = M M = = (1.05) = = 10.25%. Or, use the EFF% key on a financial calculator.

50 9-50 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%.

51 9-51 Using the EAR 0 1 5% 2 3 \$ month periods \$100 \$100 Here, payments occur annually, but compounding occurs semiannually, so we can not use normal annuity valuation techniques.

52 First Method: Compound Each CF % 2 3 \$ \$100 \$ \$331.80

53 9-53 Second Method: Treat as an Annuity Find the EAR for the stated rate: 2 ) EAR = ( = 10.25%. 2 Then use standard annuity techniques: INPUTS OUTPUT N I/YR PV PMT FV

54 9-54 Amortization Construct an amortization schedule for a \$1,000, 10% annual rate loan with three equal payments.

55 9-55 Step 1: Find the required payments % -\$1,000 PMT PMT PMT INPUTS OUTPUT N I/YR PV PMT FV

56 9-56 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 = \$

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

58 9-58 BEG PRIN END YR BAL PMT INT PMT BAL 1 \$1,000 \$402 \$100 \$302 \$ TOTAL \$1, \$ \$1,000 Note that annual interest declines over time while the principal payment increases.

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

60 9-60 Conclusion This concludes our discussion of Chapter 9 (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?

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