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Transcript of © 2003 The McGraw-Hill Companies, Inc. All rights reserved. Introduction to Valuation: The Time...
![Page 1: © 2003 The McGraw-Hill Companies, Inc. All rights reserved. Introduction to Valuation: The Time Value of Money (Calculators) Chapter Five.](https://reader036.fdocuments.net/reader036/viewer/2022062417/5519f8775503462e378b45fa/html5/thumbnails/1.jpg)
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
Introduction to Valuation: The Time
Value of Money
(Calculators)
Chapter
Five
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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 a spreadsheet to solve time value of money problems
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Chapter Outline
• Future Value and Compounding
• Present Value and Discounting
• More on Present and Future Values
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Basic Definitions
• 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
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Future Value and Compounding
• Single Period
• Multiple Periods
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Investing for a single period
• If you invest $X today at an interest rate of r, you will have $X + $X(r) = $X(1 + r) in one period.
• Example: $100 at 10% interest gives $100(1.1) = $110
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Investing for more than one period
• Simple Interest: no interest is paid on previous interest
• Compounding: Reinvesting the interest, we earn interest on interest, i.e., compounding
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Examples
• Two periods: FV = $X(1 + r)(1 + r) = $X(1 + r)2
• $100 at 10% for 2 periods = $100(1.1)(1.1) = $100(1.1)2=$121
• In general, for t periods, FV = $X(1 + r)t where (1 + r)t is the future value interest factor, FVIF(r,t)
• Example: $100 at 10% for 10 periods gives $100(1.1)10 = $259.37
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Example
• Assume you just started a new job and your current annual salary is $25,000. Suppose the rate of inflation is about 4% annually for the next 40 years, and you receive annual cost‑of‑living increases tied to the inflation rate. What will your ending salary be?
• How much will the final salary will be should you receive average raises of 5% annually. – The difference is striking: 25,000(1.05)40 = $176,000; or
approximately $56,000 in additional purchasing power in that year alone!
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Present Value and Discounting
• Single Period
• Multiple Period
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The Single‑Period Case
• Given r, what amount today (Present Value or PV) will produce a given future amount?
• Remember that FV = $X(1 + r).
• Rearrange and solve for $X, which is the present value. Therefore,
• PV = FV / (1 + r).
• Example– $110 in 1 period with an interest rate of 10% has a
PV = 110 / (1.1) = $100
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Discounting
• Process of finding the present value.
• Present Value ‑ One Period Example– Determine the present value of $110 and $121 if
the amounts are received in one year and two years, respectively, and the interest rate is 10%.
– $100 = $110 (1 / 1.1) = 110 (.9091)– $100 = $121 (1 / 1.12) = 121(.8264)
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Present Values for Multiple Periods
• PV = FV [1 / (1 + r)t] – where [1 / (1 + r)t] is the discount factor, or the
present value interest factor, PVIF(r,t)
• Example: If you have $259.37 in 10 periods and the interest rate was 10%, how much did you deposit initially?– PV = 259.37 [1/(1.1)10] = 259.37(.3855) = $100
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Discounted Cash Flow (DCF)
• Example: effect of compounding over long periods.– Vincent Van Gogh's "Sunflowers" was sold at auction in
1987 for approximately $36 million. It had been sold in 1889 for $125. At what discount rate is $125 the present value of $36 million, given a 98‑year time span?
– 125 = 36,000,000 [1 / (1 + r)98]
– (36,000,000 / 125)1/98 ‑ 1 = r = .13685 = 13.685%
• If your great‑grandfather had purchased the painting in 1889 and your family sold it for $36 million, the average annually compounded rate of return on the $125 investment was ____?
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Patterns
• Present value decreases as interest rates increase. • Future values increase as the interest rate increases.• Start with a present value of $100 and computing the
future value under different interest rate scenarios.– Future Value of $100 at 10% for 5 years = 100(1.1)5 =
$161.05
– Future Value of $100 at 12% for 5 years = 100(1.12)5 = 176.23
– Future Value of $100 at 14% for 5 years = 100(1.14)5 = 192.54
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More on Present and Future Values
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Present versus Future Value
• Present Value factors are reciprocals of Future Value factors:– PVIF(r,t) = 1 / (1 + r)t and FVIF(r,t) = (1 +
r)t
– Example: FVIF(10%,4) = 1.14 = 1.464– PVIF(10%,4) = 1 / 1.14 = .683
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More on Time Values
• There are four variables in the basic time value equation. If we know three of the four, we can always solve for the fourth.
• Determining the Discount Rate– Start with the basic time value of money equation and
rearrange to solve for r:
– FV = PV(1 + r)t
– r = (FV / PV)1/t ‑ 1
• Example: What interest rate makes a PV of $100 become a FV of $150 in 6 periods?– r = (150 / 100)1/6 ‑ 1 = 7%
– or PV = ‑100; FV = 150; N = 6; CPT I/Y = 7%
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Finding the Number of Periods and Rule of 72
• Finding the Number of Periods– FV = PV(1 + r)t ‑ rearrange and solve for t. – t = ln(FV / PV) / ln(1 + r)
• Example: How many periods before $100 today grows to $150 at 7%? – t = ln(150 / 100) / ln(1.07) = 6 periods
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Rule of 72
• Time to double your money, (FV / PV) = 2.00 is approximately (72 / r%) periods.
• The rate needed to double your money is approximately (72/t)%.
• Example: To double your money at 10% takes approximately (72/10) = 7.2 periods.
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Future Values
• Suppose you invest $1000 for one year at 5% per year. What is the future value in one year?– Interest = 1000(.05) = 50– Value in one year = principal + interest = 1000 +
50 = 1050– Future Value (FV) = 1000(1 + .05) = 1050
• Suppose you leave the money in for another year. How much will you have two years from now?– FV = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50
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Future Values: General Formula
• FV = PV(1 + r)t
– FV = future value– PV = present value– r = period interest rate, expressed as a decimal– T = number of periods
• Future value interest factor = (1 + r)t
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Effects of Compounding
• Simple interest
• Compound interest
• Consider the previous example– FV with simple interest = 1000 + 50 + 50 = 1100– FV with compound interest = 1102.50– The extra 2.50 comes from the interest of .05(50)
= 2.50 earned on the first interest payment
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Calculator Keys
• Texas Instruments BA-II Plus– FV = future value– PV = present value– I/Y = period interest rate
• P/Y must equal 1 for the I/Y to be the period rate• Interest is entered as a percent, not a decimal
– N = number of periods– Remember to clear the registers (CLR TVM) after
each problem– Other calculators are similar in format
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Future Values – Example 2
• Suppose you invest the $1000 from the previous example for 5 years. How much would you have?– 5 N– 5 I/Y– 1000 PV– CPT FV = -1276.28
• The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1250, for a difference of $26.28.)
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Future Values – Example 3
• Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?– 200 N
– 5.5 I/Y
– 10 PV
– CPT FV = -447,189.84
• What is the effect of compounding?– Simple interest = 10 + 200(10)(.055) = 210.55
– Compounding added $446,979.29 to the value of the investment
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Future Value as a General Growth Formula
• Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?– 5 N– 15 I/Y– 3,000,000 PV– CPT FV = -6,034,072 units (remember the sign
convention)
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Quick Quiz – Part I
• What is the difference between simple interest and compound interest?
• Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years.– How much would you have at the end of 15 years
using compound interest?– How much would you have using simple interest?
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Present Values
• How much do I have to invest today to have some amount in the future?– FV = PV(1 + r)t
– Rearrange to solve for PV = FV / (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.
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Present Value – One Period Example
• Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?
• PV = 10,000 / (1.07)1 = 9345.79• Calculator
– 1 N– 7 I/Y– 10,000 FV– CPT PV = -9345.79
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Present Values – Example 2
• You want to begin saving for you daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?– N = 17– I/Y = 8– FV = 150,000– CPT PV = -40,540.34 (remember the sign
convention)
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Present Values – Example 3
• Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?– N = 10– I/Y = 7– FV = 19,671.51– CPT PV = -10,000
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Present Value – Important Relationship I
• For a given interest rate – the longer the time period, the lower the present value– What is the present value of $500 to be received in
5 years? 10 years? The discount rate is 10%– 5 years: N = 5; I/Y = 10; FV = 500
CPT PV = -310.46– 10 years: N = 10; I/Y = 10; FV = 500
CPT PV = -192.77
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Present Value – Important Relationship II
• For a given time period – the higher the interest rate, the smaller the present value– What is the present value of $500 received in 5
years if the interest rate is 10%? 15%?• Rate = 10%: N = 5; I/Y = 10; FV = 500
CPT PV = -310.46
• Rate = 15%; N = 5; I/Y = 15; FV = 500CPT PV = -248.58
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Quick Quiz – Part II
• What is the relationship between present value and future value?
• Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today?
• If you could invest the money at 8%, would you have to invest more or less than at 6%? How much?
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The Basic PV Equation - Refresher
• PV = FV / (1 + r)t
• There are four parts to this equation– PV, FV, r and t– If we know any three, we can solve for the fourth
• If you are using a financial calculator, be sure and remember the sign convention or you will receive an error when solving for r or t
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Discount Rate
• Often we will want to know what the implied interest rate is in an investment
• Rearrange the basic PV equation and solve for r– FV = PV(1 + r)t
– r = (FV / PV)1/t – 1
• If you are using formulas, you will want to make use of both the yx and the 1/x keys
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Discount Rate – Example 1
• You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest?– r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%– Calculator – the sign convention matters!!!
• N = 5
• PV = -1000 (you pay 1000 today)
• FV = 1200 (you receive 1200 in 5 years)
• CPT I/Y = 3.714%
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Discount Rate – Example 2
• Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?– N = 6– PV = -10,000– FV = 20,000– CPT I/Y = 12.25%
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Discount Rate – Example 3
• Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5000 to invest. What interest rate must you earn to have the $75,000 when you need it?– N = 17– PV = -5000– FV = 75,000– CPT I/Y = 17.27%
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Quick Quiz – Part III
• What are some situations where you might want to compute the implied interest rate?
• Suppose you are offered the following investment choices:– You can invest $500 today and receive $600 in 5
years. The investment is considered low risk.– You can invest the $500 in a bank account paying
4%.– What is the implied interest rate for the first choice
and which investment should you choose?
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Finding the Number of Periods
• Start with basic equation and solve for t (remember you logs)– FV = PV(1 + r)t
– t = ln(FV / PV) / ln(1 + r)
• You can use the financial keys on the calculator as well, just remember the sign convention.
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Number of Periods – Example 1
• You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?– I/Y = 10– PV = -15,000– FV = 20,000– CPT N = 3.02 years
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Number of Periods – Example 2
• Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?
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Number of Periods – Example 2 Continued
• How much do you need to have in the future?– Down payment = .1(150,000) = 15,000
– Closing costs = .05(150,000 – 15,000) = 6,750
– Total needed = 15,000 + 6,750 = 21,750
• Compute the number of periods– PV = -15,000
– FV = 21,750
– I/Y = 7.5
– CPT N = 5.14 years
• Using the formula– t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years
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Quick Quiz – Part IV
• When might you want to compute the number of periods?
• Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money?
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Spreadsheet Example
• Use the following formulas for TVM calculations– FV(rate,nper,pmt,pv)– PV(rate,nper,pmt,fv)– RATE(nper,pmt,pv,fv)– NPER(rate,pmt,pv,fv)
• The formula icon is very useful when you can’t remember the exact formula
• Click on the Excel icon to open a spreadsheet containing four different examples.
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Work the Web Example
• Many financial calculators are available online
• Click on the web surfer to go to Cigna’s web site and work the following example:– You need $50,000 in 10 years. If you can earn 6%
interest, how much do you need to invest today?– You should get $27,920
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Table 5.4