Financial Management Spring 2012

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1 3-1 Financial Management Spring 2012 Week 4 How to Calculate Present Values III 4-1

2 3-2 Topics Covered More Shortcuts Growing Perpetuities and Annuities How Interest Is Paid and Quoted 4-2

3 Example 3-3 Problem: Your parents have made you an offer you can t refuse. They re planning to give you part of your inheritance early. They ve given you a choice. They ll pay you $10,000 per year for each of the next seven years (beginning today) or they ll give you their 2007 BMW M6 Convertible, which you can sell for $61,000 (guaranteed) today. If you can earn 7% annually on your investments, which should you choose?

4 PV Annuities Due 3-7 Annuity due: a level of stream of payments starting immediately(선불연금) Go back to the lottery prize and the first payment of the 25 yearly payments was made immediately PV(Annuity due)=(1+r) PV(Annuity) $152.6 mil*(1+r)= $152.6 mil*(1+.059) An annuity due is worth (1+r) times the value of an ordinary annuity! 4-7

5 FV Annuity Short Cut 3-8 Future Value of an Annuity The future value of an asset that pays a fixed sum each year for a specified number of years How to calculate FV of an annuity? 1) calculate PV 2) multiply by (1+r) t FV of annuity C 1 r r t 1 4-8

6 FV Annuity Short Cut 3-9 Example What is the future value of $20,000 paid at the end of each of the following 5 years, assuming your investment returns 8% per year? FV 20,000 $117,

7 Example 3-10 Problem: Adam is 25 years old, and he has decided it is time to plan seriously for his retirement. At the end of each year until he is 45 he will save $10,000 in a retirement account. At that time, he will stop paying into the account, though he does not plan to retire until he is 65. If the account earns 10% per year, how much will Adam have saved at age 65?

8 3-15 Growing Perpetuities and Annuities Constant growth perpetuities a stream of cash flows that grow at a constant rate, g C n =C 1 (1+g) n-1 PVof growing perpetuity C 1 1 r C 1 (1 (1 r) g) 2 C 1 (1 g) (1 r) 3 2 C1 /(1 r) 1 g 1 1 r r C 1 g 4-15

9 Constant Growth Perpetuity 3-16 NOTE: This formula can be used to value a perpetuity at any point in time! PV 0 r C 1 g PV t C t 1 r g 4-16

10 Constant Growth Perpetuity 3-17 Example What is the present value of $1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and a constant growth rate of 4%? PV $ billion 4-17

11 Growing Annuities 3-18 A three-year stream of cash flows that grows at the rate g is equal to the difference between two growing perpetuities. Year: Present value 1. Growing perpetuity A $1 2. Growing perpetuity B 3. Growing 3-yr annuity (1-2) $1

12 Compound Interest and PV 3-19 Simple vs. Compound interest In the simple interest care, the interest is paid only on the initial investment In contrast, in the compound interest case, each interest payment is reinvested to earn more interest in subsequent periods! Under compound interest our wealth grows at a constant rate! In when quoting an annual interest rate, companies should use an annual percentage rate, or APR 4-19

13 Compound Interest 3-20 i ii iii iv v Periods Interest Value Annually per per APR after compounded year period (i x ii) one year interest rate 1 6% 6% % = = = = =

14 Simple and Compound Interest 3-21 The value of a $100 investment earning 10% annually.

15 FV of $1 Compound Interest % Simple 10% Compound Number of Years

16 Effective Interest Rates 3-25 Effective Annual Interest Rate - Interest rate that is annualized using compound interest. Annual Percentage Rate - Interest rate that is annualized using simple interest. 4-25

17 Effective Interest Rates 3-26 Example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 4-26

18 Effective Interest Rates 3-27 Example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? EAR = (1+.01) EAR = (1+.01) = r -1 =.1268 or 12.68% APR =.01x 12 =.12 or 12.00% 4-27

19 Compound Interest 3-28 Example Suppose you are offered an automobile loan at an APR of 6% per year. What does that mean, and what is the true rate of interest, given monthly payments? Assume $10,000 loan amount.

20 Compound Interest 3-29 Example - continued Suppose you are offered an automobile loan at an APR of 6% per year. What does that mean, and what is the true rate of interest, given monthly payments? Assume $10,000 loan amount. Loan payment EAR 10,000(1.005) 10, % 12

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