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Chapter 4

Chapter 4. Introduction to Valuation: The Time Value of Money. Overview. Future Value and Compounding Present Value and Discounting More on Present and Future Values. The Time Value of Money Concept.

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Chapter 4

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  1. Chapter 4 Introduction to Valuation: The Time Value of Money

  2. Overview • Future Value and Compounding • Present Value and Discounting • More on Present and Future Values

  3. The Time Value of Money Concept • We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs • The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner • Present Value – earlier money on a time line • Future Value – later money on a time line • Interest rate – “exchange rate” between earlier money and later money • Discount rate • Cost of capital • Opportunity cost of capital • Required return

  4. Future Value for a Single Payment • Notice that 1. $110.00 = $100 X (1 + 0.10) 2. $121.00 = $110 X (1 + 0.10) = $100 X 1.10 X 1.10 = $100 X (1.10)2 3. $133.10 = $121 X (1 + 0.10) = $100 X 1.10 X 1.10 X 1.10 = $100 X (1.10)3 • In general, the future value, FVt, of $1 invested today at r% for t periods is • FVt = $1 x (1 + r)t or FVt = PV x (1 + r)t • The expression (1 + r)t is the future value interest factor. • This is the process of compounding – interest earns interest

  5. Example 1 • Q. Deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest? • A. Multiply the $5,000 by the future value interest factor: • $5,000 X (1 + r)t = $5,000 X (1.12)6 • = $5,000 X 1.9738227 • = $9,869.11 • At 12%, the simple interest is 0.12 X $5000 = $600 per year. • After 6 years, this is 6 X $600 = $3,600 • The difference between compound and simple interest is thus $4,869.11 – $3,600 = $1,269.11

  6. Future Value of $100 at 10 Percent

  7. Future Value, Time, and Interest Rate

  8. Using TI BA II PLUS

  9. Some Important Notes on TI BA II PLUS • We will first use keys located at 3rd row from top of your calculator • To clear previous entries you can use • 2ND CLR TVM • I prefer entering “0” for irrelevant variables • For most cases, PV should be entered as a negative number • Always keep in mind that “N” is number of periods not necessarily years. • Compounding per Year (C/Y) can be modified by changing 2ND P/Y. This determines how frequently interest is earned. • Therefore we will use C/Y and P/Y interchangeably. • There are situations in which you would have to distinguish between the two that we will see in the next chapter • Amount of interest would be based on periodic rate • Periodic Rate = Annual Rate / Number of Periods in a Year • Always make sure P/Y is consistent with the problem at hand – this means you need to check its setting every time. • P/Y also help us determine number of periods • N = P/Y X Number of Years

  10. Example 2Interest on Interest Illustration • Q. You have just won a $1 million jackpot in the state lottery. • You can buy a ten year certificate of deposit which pays 6% compounded annually. • Alternatively, you can give the $1 million to your brother-in-law, who promises to pay you 6% simple interest annually over the ten-year period. • Which alternative will provide you with more money at the end of ten years? • Answer: • The future value of the CD is $1 million x (1.06)10 = $1,790,847.70. • The future value of the investment with your brother-in-law, on the other hand, is $1 million + [$1 million X (0.06) X (10)] = $1,600,000. • Thus, compounding (or the payment of “interest on interest”), results in incremental wealth of nearly $191,000.

  11. Present Value of a Single Payment • How much do I have to invest today to have some amount in the future? • FVt = PV x (1 + r)t • Rearrange to solve for PV = FVt / (1 + r)t • When we talk about discounting, we mean finding the present value of some future amount. • When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

  12. Example 3 • Want to be a millionaire? No problem! Suppose you are currently 21 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate $1 million by the time you reach age 65? • First define the variables: • FV = $1 million, r = 10 percent and t = 65 - 21 = 44 years • PV = $1 million/(1.10)44 = $15,091.

  13. Example 4 • Q. Suppose you need $20,000 in three years to pay your college tuition. If you can earn 8% on your money, how much do you need today? • A. Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the unknown present amount? • PV = $20,000/(1.08)3 = $15,876.64

  14. Present Value, Time, and Interest Rate

  15. Example 5Finding the Rate • Suppose you deposit $5,000 today in an account paying r percent per year. If you are promised to get $10,000 in 10 years, what rate of return are you being offered? • Set this up as present value equation: • FV = $10,000, PV = $ 5,000, t = 10 years, r = ? • PV = FVt / (1 + r)t • $5,000 = $10,000 / (1 + r)10 • Now solve for r: • (1 + r)10 = $10,000 / $5,000 = 2 • (1 + r)10 = 2 then r = (2)1/10 - 1 = 0.0718 = 7.18%

  16. Example 6Finding the Number of Periods • Suppose you need $2,000 to buy a new stereo for your car. If you have $500 to invest at 14% compounded annually, how long will you have to wait to buy the stereo? • Set this up as future value equation: • FV = $2,000, PV = $ 500, r = 14%, t = ? • FVt = PV x (1 + r)t • $2,000 = $500 x (1 + 0.14)t • Now solve for t: • (1 + 0.14)t = $2,000 / $500 = 4 • (1 + 0.14)t = 4 and t = ln (4) / ln (1.14) = 10.58 years

  17. A Note on Non-Annual Compounding – Continued • Remember that FVt = PV0 x (1 + r)t • When dealing with non annual interest and payment you should make two adjustment • Interest rate should be periodic • Number of periods should reflect the total number of periods given number of years and frequency of interest earnings in a year • Formulations should be adjusted as follows • If “m” is the number of compounding in a year then where you see “r” divide it by “m” and multiply “t” by “m.” • This assumes that “t” is the number of years • FVt = PV0 x (1 + r/m)t X m • Excel adjustment are same as formulations that can be made in a function dialog box or data cells

  18. A Note on Non-Annual Compounding • FVt = PV0 x (1 + r/m)t X m • Entering the above equation into TI BA II PLUS • Method 1: • Keep P/Y = 1 (so is C/Y = 1) • N = t X m (Number of periods) • I/Y = r/m (Periodic interest rate) • Rest of the information is entered as before • Method 2: • Change P/Y = m (so is C/Y = m) • N = t X m (Number of periods) • I/Y = r (Annual interest rate) • Rest of the information is entered as before • Note that P/Y has to be checked for consistency every time

  19. Example: Spreadsheet Strategies • Use the following formulas for TVM calculations • FV(rate,nper,pmt,pv,type) • PV(rate,nper,pmt,fv,type) • RATE(nper,pmt,pv,fv,type) • NPER(rate,pmt,pv,fv,type) • The default setting for “type” is “0” which sugests end-of-period payments (pmt). It should be set to “1” for beginning-of-period payments. • See Excel file for the solution of examples.

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