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1 Chapter 05 Time Value of Money 2: Analyzing Annuity Cash Flows McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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Introduction Time Value of Money calculations – Can deal with either single cash flows (Chapter 4) – or multiple cash flows over time (Chapter 5) 5-2

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Future Value of Multiple Cash Flows Multiple cash flows – Regular, evenly-spaced Car loans and home mortgage loans Saving for retirement Companies paying interest on debt Companies paying dividends 5-3

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Future Value – Several Cash Flows Concept: Compounding –Value in the future –Different cash flows paid in at different times

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Finding FV – Several Cash Flows Example... Assumptions –Invest $100 today (compounds for 3 years) –Invest $125 at end of year 2 (compounds for 2 years) –Invest $150 at end of year 3 (compounds for 1 year) –Interest rates: 7% 5-5

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FV Several Cash Flows Time Line Example... Several (Different) Cash Flow Values 5-6

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Future Value – Level Cash Flows Concept: Compounding Also called “annuities” –Value in the future –Same cash flows paid in every period

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Finding FV – Level Cash Flows/Annuity Example... Assumptions: –Invest $100 at the end of each year for 5 years –Interest rates: 8% 5-8

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Level Cash Flows Time Line Example... Level (same) Cash Flows Each Period 5-9

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Future Value – Multiple Annuities Concept: Compounding – annuity equation to compute future value – two levels of cash flows To solve for multiple annuities, compute FV for each separately and add them together

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Finding FV – Multiple Annuities Example... Assumptions: –Invest $100 at end of years at 8% –Invest $150 at end of years at 8% 5-11

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Future Value – Multiple Annuities Step 1 (same as FV of Level Cash Flows Calculation) Step 2 Add two sums together – FV of both is $ Step

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Present Value of Multiple Cash Flows Multiple cash flows: – Car loans and home mortgage loans – Determining value of business opportunities 5-13

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Present Value – Several Cash Flows Concept: Discounting –Value of future sum today –Different cash flows paid in at different times

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Finding PV – Several Cash Flows Example... Assumptions: –Deposit $100 today –Deposit $125 next year –Deposit $150 at end of year 2 –Interest rates: 7% 5-15

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PV Several Cash Flows Time Line Example

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PV Several Cash Flows $150/ (1.07) 2 $0 / (1.07) $100$125$150$0 $ $ $0.00 $125/(1.07) $

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Present Value – Level Cash Flows Concept: Discounting –Value of future sum today –Level cash flows paid in at different times Most loans set up with even payments throughout life of loan

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Finding PV – Level Cash Flows Example... Assumptions: –$100 payments at end of each year for 5 years –Interest rates: 8% per year 5-19

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PV Level Cash Flows Time Line Example

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Present Value – Multiple Annuities Concept: Discounting –Changing level cash flows –Ex: Alex Rodriguez’s baseball contract

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PV – Multiple Annuities Example... Assumptions (Alex Rodriguez’s Contract): –$10 million signing bonus –$21 million per year from 2001 – 2004 –$25 million per year in 2005 and 2006 –$27 million per year in –Interest rates: 8% per year 5-22

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PV Multiple Annuities Example (cont.) 5-23

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Perpetuity – Special Annuity Concept: Discounting –Stream of level cash flows paid forever –Preferred stocks are an example –Value of investment is present value of all future annuity payments

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Ordinary Annuities vs. Annuities Due Ordinary Annuity – Payment occurs at the end of each period Annuity Due – Payment occurs at the beginning of each period 5-25

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Annuity Due Time Line Example Cash flows at beginning, not at end of period Five annuity-due cash flows basically same as payment today plus 4-year ordinary annuity Payments occur one period sooner than ordinary annuity -- earn extra period of interest

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Future Value of Annuity Due Concept: Compounding –Value of future sum today –Cash flows at beginning of each period

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Future Value of Annuity Due... – Assumptions: Assumes cash flows at the beginning of each period 5 annuity-due cash flows of $100 each – First cash flow compounds for 5 years – Last cash flow compounds for 1 year Interest rates: 8% 5-28

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Present Value of Annuity Due Concept: Discounting –Today’s value of future sum –Cash flows at beginning of each period

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Present Value of Annuity Due... Assumptions: –Cash flows at beginning of period –5 annuity-due cash flows of $100 First cash flow paid today – not discounted Last cash flow discounted 4 years All cash flows discounted for one year less than ordinary annuity Interest rates: 8% 5-30

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Compounding Frequency Used in situations that do not use yearly time periods – Semiannual bond payments – Quarterly stock dividends – Consumer loans – monthly payments 5-31

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Effect of Compounding Frequency... Assumptions: –$100 deposit today –12% annual interest rate –Bank compounds interest at six months instead of end of year –Interest is earned on interest 5-32

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EARS and APRS... Quoted, or nominal rate called annual percentage rate (APR) Rate that incorporates compounding called effective annual rate (EAR) Relationship between APR and EAR: 5-33

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EARS vs. APR Example... – Assumptions: Borrow $100 today 12% annual interest rate APR: Loan compounds annually -- you pay 12.00% EARS: Loan compounds monthly -- you pay 12.68% – Formula to convert APR to EAR: 5-34

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Annuity Loans Compares payments Compares implied interest rate 5-35

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Finding Payments on Amortized Loan Concept: –Rearrange PV of annuity formula to solve for payment

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Payments on Amortized Loan Example... Assumptions: –Need $10,000 to buy car –Loan term: 4 years –Interest rate: 9% Use interest rate of 0.75 % (=9%/12) and 48 periods (=4 X 12) 5-37

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