CHAPTER 5 TIME VALUE OF MONEY 2 ANALYZING

CHAPTER 5 TIME VALUE OF MONEY 2: ANALYZING ANNUITY CASH FLOWS M: Finance 3 rd Edition Cornett, Adair, and Nofsinger Copyright © 2016 by Mc. Graw-Hill Education. All rights reserved.

Introduction • Time Value of Money calculations • Can deal with either single cash flows • Or multiple cash flows over time

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

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%

FV Several Cash Flows Time Line Example Several (Different) Cash Flow Values

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%

Level Cash Flows Time Line Example Level (Same) Cash Flows Each Period

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 1 - 3 at 8% • Invest $150 at end of years 4 - 5 at 8%

Future Value – Multiple Annuities Step 1 (same as FV of Level Cash Flows Calculation) Step 2 Step 3 Add two sums together – FV of both is $690. 66

Present Value of Multiple Cash Flows • Multiple cash flows • Car loans and home mortgage loans • Determining value of business opportunities

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%

PV Several Cash Flows Time Line Example

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

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 2007 – 2010 • Interest rates: 8% per year

PV – Multiple Annuities Example (cont. )

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

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%

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%

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

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

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

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%

Annuity Loans • Compares payments • Compares implied interest rate

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% APR • Use interest rate of 0. 75% (= 9%/12) and 48 periods (= 4 x 12) •