# 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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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.

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

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

Future Value – Several Cash Flows Concept: Compounding –Value in the future –Different cash flows paid in at different times... 5-4

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

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

Future Value – Level Cash Flows Concept: Compounding Also called “annuities” –Value in the future –Same cash flows paid in every period... 5-7

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

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

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... 5-10

Finding FV – Multiple Annuities Example... Assumptions: –Invest \$100 at end of years 1 - 3 at 8% –Invest \$150 at end of years 4 - 5 at 8% 5-11

Future Value – Multiple Annuities Step 1 (same as FV of Level Cash Flows Calculation) Step 2 Add two sums together – FV of both is \$690.66 Step 3 5-12

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

Present Value – Several Cash Flows Concept: Discounting –Value of future sum today –Different cash flows paid in at different times... 5-14

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

PV Several Cash Flows Time Line Example... 5-16

PV Several Cash Flows \$150/ (1.07) 2 \$0 / (1.07) 3 0 1 2 3 \$100\$125\$150\$0 \$116.82 \$131.02 \$0.00 \$125/(1.07) \$347.84 5-17

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... 5-18

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

PV Level Cash Flows Time Line Example... 5-20

Present Value – Multiple Annuities Concept: Discounting –Changing level cash flows –Ex: Alex Rodriguez’s baseball contract... 5-21

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 2007 - 2010 –Interest rates: 8% per year 5-22

PV Multiple Annuities Example (cont.) 5-23

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... 5-24

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

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... 5-26

Future Value of Annuity Due Concept: Compounding –Value of future sum today –Cash flows at beginning of each period... 5-27

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

Present Value of Annuity Due Concept: Discounting –Today’s value of future sum –Cash flows at beginning of each period... 5-29

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

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

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

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

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

Annuity Loans Compares payments Compares implied interest rate 5-35

Finding Payments on Amortized Loan Concept: –Rearrange PV of annuity formula to solve for payment... 5-36

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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