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Chapter 05 McGraw-Hill/Irwin Time Value of Money 2: Analyzing Annuity Cash Flows Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 1 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 Step 3 Add two sums together – FV of both is $690.66 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 0 $100 $116.82 $131.02 $0.00 1 2 $125 $150 3 $0 $125/(1.07) $150/ (1.07)2 $0 / (1.07)3 $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: 12 0.12 EAR 1 12 1 ... 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