1. Interest Rate. Three components of interest: Principal Interest rate Investment horizon (Time)

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Stanley Harvey
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1 Key Concepts The future value of an investment made today The present value of cash to be received at some future date The return on an investment The number

1 1 Key Concepts The future value of an investment made today The present value of cash to be received at some future date The return on an investment The number of periods that equates a present value and a future value given an interest rate Effect of compounding more than once a year Quoted rate vs. effective rate 2 1

2 1. Interest Rate Three components of interest: Principal Interest rate Investment horizon (Time) 3 ex) $1,000 to invest (principal) at 10% annual interest. What will be the value of investment in two years? Simple Interest Year one, interest = $1, = $100. total value of investment = $ $100 1 = $1,100 Year two, interest = $1, = $100. total value of investment = $ $100 2 = $1,200 Compound Interest Year one, interest =.10 $1,000 = $100. total value of investment = $ = $1,100 Year two, interest =.10 x $1,100 = $110. total value of investment = $1000 (1.10) 2 = $1,

3 Compounding The total value of an investment of some principal amount at an (compound) interest rate, r, after t years equals to: Value of invstmt = principal (1+interest rate) t interest on interest 5 Example: $100 invested at 10% for five years 6 3

4 2. Present Value and Future Value Future Value the amount to which an investment grows after earning interest. Present Value the current value of future cash flows discounted at the appropriate interest rate. Interest rate exchange rate between the present value and the future value 7 Future Value 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 (FVIF) = (1 + r) t 8 4

5 Future Value Interest Factors (FVIF) 9 Future Value Example Suppose you invest $1,000 for one year at 5% per year. What is the future value in two years? FV 2 = 1000(1.05)(1.05) = 1000(1.05) 2 = 1,

11 Discounting Discounting: the process to find the present value of a future cash flow (discounted at the appropriate interest rate). Example) Suppose you need $133.

6 11 Discounting Discounting: the process to find the present value of a future cash flow (discounted at the appropriate interest rate). Example) Suppose you need $ in three years, and you know the fair market interest is 10% on your money. How much do you have to invest today to reach your goal? How much now? $ in 3 yrs. 12 6

7 Present Value Formula PV = FV = future value PV = present value r = discount rate, expressed as a decimal t = number of periods Present value interest factor (PVIF) = 13 Figure: Present Value, Discount Rate, and Time: As the length of time until payment grows, present values decline. As shown in the above figure, present values tend to become smaller as the time horizon grows. Also, for given length of time, 14 the higher the discount rate is, the lower is the present value. 7

8 Present Value 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 = 15 Example: Present Value Example Mr. Smart, who works for a securities firm, has just been selected by his employer to start his MBA education in one year with financial support from the company. The tuition and fees of MBA education is expected to be $50,000 in the first year and $52,000 in the second year. Assuming that all expenses occur at the beginning of each year, calculate the present value of the support he will get from his employer. The current deposit and borrowing rates are 2% and 4% respectively. If the deposit rate is applied, PV = If the borrowing rate is applied, PV = 16 8

9 3. Return on an Investment 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 the above formula, you will want to make use of both the y x and the 1/x keys in your calculator. 17 Discount Rate Example You are looking at an investment that will pay $1,200 in 5 years if you invest $1000 today. What is the implied rate of interest? r = 18 9

10 In 1934, the first edition of a book described by many as the bible of financial statement analysis was published. Security Analysis has proven so popular among financial analysts that it has never been out of print. According to an item in The Wall Street Journal, a copy of the first edition was sold by a rare book dealer in 1996 for $7,500. The original price of the first edition was $3.37. What is the annually compounded rate of increase in the value of the book? Future value = $7,500 Present value = $3.37 t = = 62 years 19 r = 20 10

11 4. Finding the Number of Years Start with the TVM equation and solve for t (remember you use natural log) FV = PV(1 + r) t t = ln(fv / PV) / ln(1 + r) 21 Example: Finding the number of years 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? Interest Rate = 10% PV = -$15,000 FV = $20,000 T = 22 11

Classroom Practice You find an analyst who projects the stock price for Samsung Electronics will increase 10% per year for the foreseeable future.

12 Classroom Practice You find an analyst who projects the stock price for Samsung Electronics will increase 10% per year for the foreseeable future. Based on the most recent stock price, if the projection holds true, when will the stock price reach 4 Mil. Won? When will it reach 5 Mil. Won? 23 Classroom Practice The Most Expensive Painting in the World 24 12

13 Picasso s Boy with a Pipe sells for $104 million Sale beat auction record set by Van Gogh's Portrait of Doctor Gachet Updated: 10:36 a.m. ET May 6, 2004 NEW YORK - Pablo Picasso s 1905 painting Boy with a Pipe sold for $104 million Wednesday at Sotheby s, shattering the record for an auctioned painting. The total includes the auction price of $93 million plus the auction house s commission of about $11 million. This is the finest work in public hands that was for sale, Sotheby s senior vice president David Norman said. John H. Whitney, the late husband of the seller of the painting, Betsey Whitney from New York, bought it at $30,000 in Classroom Practice The Most Expensive Painting in the World What is the growth rate of Mr. Whitney s investment? FV = $93M + $11M = $104M PV = $30K T = = 54 years r = (104M/30K) (1/54) 1 = or 16.29% 26 13

14 5. The Effect of Compounding more than once a year Until now we have assumed that we pay interest once a year (i.e., compounding once a year). What if the compounding interval is less than a year, or we compound more than once in a year? Is paying 10% a year equal to paying 5% every six months? 27 Suppose the annual quote rate of an investment is 10%. Let s see how much our investment of $1 grows as we change the compounding interval. (1) 10% annual compounding for 1 year is: $1 (1.10) = $1.10 (2) 10% semiannual compounding: $1 (1.05) x (1.05) = $1 x (1.05) 2 = $ (3) 10% Quarterly compounding: $1 (1.025) 4 = $ (4) 10% Monthly compounding: $1 (1.0083) 12 = $

15 The future value of an investment at annual rate r compounded m times a year equals: 29 Compounding Interval Times Compounding Actual annual rate you earn Annual % Quarterly % Monthly % Weekly % Daily % By Hour 8, % By Minute 525, % By Second 31,536, % By Mili-second 3,153,600, % 1/ (continuous) % 30 15

16 Ex-President Chun, Doo-Hwan, Enjoyed a Billion Won Profit by Refusing to Pay Fines 전두환씨측이 10 일추징금완납계획을발표했지만과연그가내야할추징액이올바르게산정됐는지에대한의구심이사라지지않고있다. 이날전씨측이내겠다고밝힌금액은 1672 억원이다. 16 년전인 1997 년확정판결을통해부과받은추징금총액 2205 억원가운데그동안단계적으로추징당한금액을제외한금액이다. 이금액에는국가가그동안전씨로부터추징금을납부받지못해발생한기회비용은제외돼있다. 16 년전받아야했던 1672 억원을현재의가치로다시산정해서받는방안에대한고려가빠져있는것이다. 통계청물가동향과에따르면 1997 년라면의소비자물가지수는 이었다. 지난해이지수는 로 2.09 배정도뛰었다. 이물가상승률을추징액에적용하면전씨측은 1672 억원보다 1815 억원이많은 3487 억원을내야맞다. 여기에이자까지더하면전씨는이보다 2~3 배더많은돈을내야한다. According to the article. Verdict in 1997: Billion Won should be paid by Chun Sept. 2013: Chun announces that he will pay off Billion Won billion 53.3 billion (had been collected so far) = billion What about the opportunity cost? CPI in 1997 = CPI in 2013 = (2.09 times higher now) Hence, the actual value of billion in 1997 is equivalent to billion 2.09 = billion in 2013 Add foregone interests billion = billion 32 16

17 Let s find out what the fair amount is: Rate on long-term deposits (Sept., 1997 ~ July 2013): 5.31% 1,672 ( /4) 4 16 = 3, Rate on long-term Korea Treasury bonds (Sept ~ Aug. 2013): 5.90% 1,672 ( /2) 2 16 = 4, Quoted Rate vs. Effective Rate Quoted or Stated (Annual) Interest Rate: The annual rate expressed in terms of the interest payment made each period. Effective Annual Rate: The interest rate expressed as if it were compounded once per year

18 In general, an investment of $1 at a stated interest rate r per annum compounded m times a year amounts to : 35 EAR Questions Q. Depending on the issuer, a typical credit card agreement quotes an interest rate of 25 to 29% APR on the revolving balance. Monthly payments are required. What is the actual interest rate you pay on such a credit card? At APR 25%: EAR = At APR 29%: EAR = 36 18

19 Extreme Case of Compounding: Continuous Compounding What about compounding infinitely ( ) during the year or compounding continuously? Continuous compounding where e = Continuous Compounding Example What is the effective annual rate of 10% interest rate compounded continuously? EAR = 38 19

Time Value of Money Question: Will time travel ever be possible? 39 VS.

It can t move people in time but it can move our money Economic value moves

20 Time Value of Money Question: Will time travel ever be possible? 39 VS. In essence, finance technology is a time machine we have built ourselves. It can t move people in time but it can move our money Economic value moves forwards and backwards in time Any time traveler could engage in arbitrage across calendar time. arbitrage opportunity will be eliminated only if the nominal rate of interest is zero at all times. time travelers will drive the nominal 40 rate of interest to zero 20

Chapter 5. Finance 300 David Moore

Chapter 5. Finance 300 David Moore Chapter 5 Finance 300 David Moore Time and Money This chapter is the first chapter on the most important skill in this course: how to move money through time. Timing is everything. The simple techniques