1. © 2003 The McGraw-Hill Companies, Inc. All rights reserved. Introduction to Valuation: The Time Value of Money (Formulas) Chapter Five

2. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.1 Key Concepts and Skills Be able to compute the future value of an investment made today Be able to compute the present value of cash to be received at some future date Be able to compute the return on an investment Be able to compute the number of periods that equates a present value and a future value given an interest rate Be able to use a financial calculator and/or a spreadsheet to solve time value of money problems

3. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.2 The Present Value Relationship Where PV = present value CF t = the cash flow occurring at time t r = the discount rate N = the number of periods

4. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.3 The Present Value Relationship and the discount rate, r What determines r? –The real rate compensates investors for deferring consumption when they lend their money. –A premium for expected inflation keeps investors from losing expected purchasing power of their money. –A premium for risk, induces investors to invest in risky projects that might otherwise not attract investment dollars.

5. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.4 The Present Value Relationship and the discount rate, r What is the present value of $1 received one year from now if the appropriate discount rate is 5% versus 10%? –PV = $1/(1.05) = $.952 –PV = $1/(1.10) = $.909 Why does the term “discount” make sense in this context? Suppose the discount rate were zero. What would that mean?

6. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.5 The Present Value Relationship and the timing of cash flows Timing refers to the time at which cash flows arrive. A time line is useful in forming a picture of the arrival process. 0123 What is the present value of $1 received one year from now versus two years from now if the appropriate discount rate is 10%? PV = $1/(1.10) 1 = $.909 PV = $1/(1.10) 2 = $.826

7. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.6 The Present Value Relationship and Cash Flow Patterns A Lump Sum An annuity An annuity and a lump sum $10 $1 $1 + $10

8. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.7 Future Values: General Formula FV = PV(1 + r) N –FV = future value –PV = present value –r = period interest rate, expressed as a decimal –N = number of periods Future value interest factor = (1 + r) t

9. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.8 Sample Problems You’re trying to save to buy a new $120,000 Ferrarri. You have $40,000 today that can be invested in the bank. The bank pays 5.5% annual interest on its accounts. How long will it be before you have enough to buy the new car? What does Excel say? If you believe your mutual fund can achieve an 11 percent annual rate of return and you want to buy the car in 10 years, how much must you invest today? What does Excel say?

10. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.9 Sample Problems (Continued) Your friend is celebrating her 35 th birthday today and wants to start saving for her retirement at age 65. She wants to be able to withdraw $80,000 from her savings account on each birthday for 15 years following retirement; the first withdrawal will be on her 66 th birthday. She expects to get 9% per year on her investments. She wants to make equal annual payments on each birthday into her account. –If she makes her first investment on her 36 th birthday, and continues to make deposits until her 65 th birthday, what amount must she invest annually to be able to make the desired withdrawals at retirement? What does Excel say?What does Excel say? –Suppose your friend has just inherited a large sum of money. Rather than making equal annual payments, she has decided to make one lump-sum payment on her 35 th birthday. What amount does she have to deposit? What does Excel say?What does Excel say? –Suppose your friend’s employer will contribute $1500 to the account every year. In addition, your friend also expects a $30,000 distribution from a family trust on her 55 th birthday which she will put into her retirement account. What amount must she deposit annually now to provide for retirement? What does Excel say?What does Excel say?

11. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.10 Sample Problems (Continued) An insurance company is offering a new policy to customers that is typically bought by a parent or grandparent for a child at the child’s birth. The purchaser of the policy makes the following six payments: –First birthday $750 –Second birthday$750 –Third birthday$850 –Fourth birthday$850 –Fifth birthday$950 –Sixth birthday $950 After the sixth birthday, no more payments are made. At age 65, the child receives $175,000. If the relevant interest rate is 10% for the first six years and 6% for all subsequent years, is the policy worth buying? What does Excel say? What does Excel say?

12. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 5.11 The Present Value Relationship A Practice Problem Assume you want to buy a home that costs $150,000. You will pay regular monthly payments for 10 years, and then pay a balloon payment of $50,000. The current interest rate is 8%. Create an amortization schedule. If interest payments are tax deductible and you are in the 35% tax bracket, what is the present value of the tax savings? What the present value of the cost to you for purchasing the home? What does Excel say?What does Excel say?