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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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