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9 - 1 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Future and present values Lump sums Annuities Uneven cash flow streams Solving for I and N Types of interest rates Amortization Time Value of Money Analysis

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9 - 2 Copyright © 1999 by the Foundation of the American College of Healthcare Executives The cost of time -- opportunity cost of money Relationship between asset value and future cash flows Examples - real estate and GE stock Time Value -- What is it??

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9 - 3 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Relationship between future cash flows and asset value changes The role of time value analysis Types of analysis - future value and present value analysis Time Value -- What is it??

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9 - 4 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Present value Future value Interest rate Discount rate Cash flow Terminology of Time Value Analysis

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9 - 5 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Time Line Lump sum Annuity Uneven cash flows Terminology of Time Value Analysis

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9 - 6 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Time Lines CF 0 CF 1 CF 3 CF 2 0 12 3 I% Tick marks designate ends of periods. Time 0 is today (the beginning of Period 1); Time 1 is the end of Period 1 (the beginning of Period 2); and so on.

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9 - 7 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Time Line Illustration 1 (Lump sum) $100 0 1 2 5% What does this time line show?

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9 - 8 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Time Line Illustration 2 (annuity) $100 01 23 10% What does this time line show?

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9 - 9 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Time Line Illustration 3 (uneven cash flows) 100 50 75 0123 6% -50 What does this time line show?

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9 - 10 Copyright © 1999 by the Foundation of the American College of Healthcare Executives What is the FV after 3 years of a $100 lump sum invested at 10%? FV = ? 0123 10% -$100 Finding future values (moving to the right along the time line) is called compounding. For now, assume interest is paid annually.

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9 - 11 Copyright © 1999 by the Foundation of the American College of Healthcare Executives After 1 year: FV 1 = PV + INT 1 = PV + (PV x I) = PV x (1 + I) = $100 x 1.10 = $110.00. 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 = $121.00.

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9 - 12 Copyright © 1999 by the Foundation of the American College of Healthcare Executives After 3 years: FV 3 = FV 2 + I 3 = PV x (1 + I) 3 = 100 x (1.10) 3 = $133.10. In general, FV N = PV x (1 + I) N.

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9 - 13 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 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.

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9 - 14 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Non-Financial Calculator Solution $133.10 0123 10% -$100 $110.00$121.00 $100 x 1.10 x 1.10 x 1.10 = $133.10.

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9 - 15 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 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).

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9 - 16 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 3 10 -100 0 NI/YR PV PMT FV 133.10 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:

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9 - 17 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Ordinary Annuity PMT 0123 I% PMT 0123 I% PMT Annuity Due Types of Annuities

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9 - 18 Copyright © 1999 by the Foundation of the American College of Healthcare Executives What is the FV of a 3-year ordinary annuity of $100 invested at 10%? $100 0123 10% 110 121 FV = $331

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9 - 19 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 310 0 -100 331.00 Financial Calculator Solution Have payments but no lump sum, so enter 0 for present value. INPUTS OUTPUT I/YRNPMTFVPV

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9 - 20 Copyright © 1999 by the Foundation of the American College of Healthcare Executives What is the FV if the annuity were an annuity due? Do ordinary annuity calculation as described before Multiply result by (1 + interest rate)

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9 - 21 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 10% What is the PV of $100 due in 3 years if I = 10%? (lump sum) $100 012 3 PV = ? Finding present values (moving to the left along the time line) is called discounting.

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9 - 22 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Solve FV N = PV x (1 + I ) N for PV: PV = $100 / (1.10) 3 = $100(0.7513) = $75.13. PV = FV N / (1 + I ) N.

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9 - 23 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Financial Calculator Solution 3 10 0 100 -75.13 Either PV or FV must be negative on most calculators. Here, PV = -75.13. Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT NI/YRPVPMTFV

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9 - 24 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Opportunity Costs 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.

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9 - 25 Copyright © 1999 by the Foundation of the American College of Healthcare Executives What is the PV of this ordinary annuity? $100 012 3 10% $90.91 82.64 75.13 $248.68 = PV

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9 - 26 Copyright © 1999 by the Foundation of the American College of Healthcare Executives This problem has payments but no lump sum, so enter 0 for future value. 3 10 100 0 -248.69 INPUTS OUTPUT NI/YRPVPMTFV

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9 - 27 Copyright © 1999 by the Foundation of the American College of Healthcare Executives What is the PV if the annuity were an annuity due? $100 01 23 10% $100 ? ?

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9 - 28 Copyright © 1999 by the Foundation of the American College of Healthcare Executives What is the PV if the annuity were an annuity due? Calculate present value of ordinary annuity as described above Multiply result by (1 + interest rate) Annuities due not as common as ordinary annuities

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9 - 29 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 310 100 0 -273.55 Switch from End to Begin mode on a financial calculator. Repeat the annuity calculations. PV = $273.55. INPUTS OUTPUT NI/YRPVPMTFV

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9 - 30 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Perpetuities A perpetuity is an annuity that lasts forever. What is the present value of a perpetuity? PV (Perpetuity) =. PMT I

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9 - 31 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Uneven Cash Flow Streams: Setup 0 $100 1 $300 2 3 10% -$50 4 $ 90.91 247.93 225.39 -34.15 $530.08 = PV

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9 - 32 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Input into “CF” registers: 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 (PV) = $530.09. Uneven Cash Flow Streams: Financial Calculator Solution

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9 - 33 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

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9 - 34 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 0123 10% 0123 5% 456 134.01 -100133.10 123 0 -100 Annual: FV 3 = 100 x (1.10) 3 = 133.10. Semiannual: FV 6 = 100 x (1.05) 6 = 134.01.

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9 - 35 Copyright © 1999 by the Foundation of the American College of Healthcare Executives We will deal with 3 different rates: I Stated = stated, or nominal, or quoted, rate per year. I Period = periodic rate. EAR= effective annual rate.

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9 - 36 Copyright © 1999 by the Foundation of the American College of Healthcare Executives I Stated is the rate given in contracts. Often an annual rate. Compounding periods (M) may be given: 8% compounded quarterly. 12% compounded monthly. I Period is the rate per period. I Period = I Stated / M. For 8% compounded quarterly: periodic rate = 2%. For 12% compounded monthly: periodic rate = 1%.

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9 - 37 Copyright © 1999 by the Foundation of the American College of Healthcare Executives EAR is the annual rate which causes any 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 = $1.1025. EAR = 10.25%, because this rate would produce the same ending amount ($1.1025) under annual compounding. Effective Annual Rate (EAR)

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9 - 38 Copyright © 1999 by the Foundation of the American College of Healthcare Executives The EAR Formula Or use the EFF% key on a financial calculator. EAR = 1 + - 1.0 I Stated M M = 1 + - 1.0 0.10 2 2 = (1.05) 2 - 1.0 = 0.1025 = 10.25%.

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9 - 39 Copyright © 1999 by the Foundation of the American College of Healthcare Executives EAR of 10% at Various Compounding EAR Annual = 10%. EAR Q =(1 + 0.10/4) 4 - 1.0 = 10.38%. EAR M =(1 + 0.10/12) 12 - 1.0 = 10.47%. EAR D(360) =(1 + 0.10/360) 360 - 1.0 = 10.52%.

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9 - 40 Copyright © 1999 by the Foundation of the American College of Healthcare Executives What’s the value at the end of Year 3 of the following CF stream if the stated interest rate is 10%, compounded semiannually? 0 1 $100 23 5% 45 6 6-month periods $100 Note that payments occur annually, but compounding occurs semiannually, so we can not use normal annuity valuation techniques.

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9 - 41 Copyright © 1999 by the Foundation of the American College of Healthcare Executives First Method: Compound Each CF 01 $100 23 5% 456 $100$100.00 110.25 121.55 $331.80

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9 - 42 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Second Method: Treat as an Annuity EAR = ( 1 + ) - 1 = 10.25%. 0.10 2 2 Find the EAR for the stated rate: Then use standard annuity techniques: 3 10.25 0 -100 INPUTS OUTPUT NI/YRPVFV PMT 331.80

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9 - 43 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

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9 - 44 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Step 1: Find the required payments. PMT 0123 10% -$1,000 3 10 -1000 0 402.11 INPUTS OUTPUT NI/YRPVFV PMT

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9 - 45 Copyright © 1999 by the Foundation of the American College of Healthcare Executives 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 = $402.11 - $100 = $302.11.

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9 - 46 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Step 4: Find ending balance at end of Year 1. End bal= Beg balance - Repayment = $1,000 - $302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

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9 - 47 Copyright © 1999 by the Foundation of the American College of Healthcare Executives Note that annual interest declines over time while the principal payment increases. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$698 269840270332366 3366402373660 TOT$1,206.34$206.34$1,000

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9 - 48 Copyright © 1999 by the Foundation of the American College of Healthcare Executives $ 0123 402.11 Interest 302.11 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Principal Payments

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