Mathematics
Compound Interest
Compound interest refers to the interest calculated on the initial principal and also on the accumulated interest from previous periods. It is a powerful concept in finance and investment, as it allows for exponential growth of wealth over time. The formula for compound interest takes into account the compounding periods, interest rate, and time, making it a fundamental concept in financial mathematics.
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10 Key excerpts on "Compound Interest"
eBook - PDF- Akihito Asano(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
Two more mathematical notions, however, will need to be introduced eventually. 3.4 Compound Interest Most of us borrow money when we purchase a car, a house, education, etc. Most businesses borrow money when they purchase new equipment or just to keep up the daily operation of their businesses. We call the amount of money we borrow the (original) principal. When people borrow money, they have to agree to repay this amount – usually plus some extra – in the future. This extra amount is called the interest. These terms are also used when we invest money and earn interest, and our story indeed involves investing money. Consider the following situation. Suppose you have invested $100 at an interest rate of 10 per cent per annum. Suppose also that interest is compounded annually, i.e. the interest earned by the principal is reinvested so that it, too, earns interest. How much will you obtain by the end of the tenth year? After one year, the value of investment will be the original principal ($100), plus the interest on the principal ($100 × 0.1): 100 + 100 × 0.1 = $110. So the interest for the second year is earned for $110, not just $100. At the end of the second year, the value of investment will be the principal at the end of the first year ($110), plus the interest on it ($110 × 0.1): 110 + 110 × 0.1 = $121. It means that the principal increases each year by 10 per cent. The $121 represents the original principal, plus all accrued interest, and is called the compound amount. The difference between the compound amount and the original principal is called the 66 Financial mathematics Compound Interest. In this example, the Compound Interest (at the end of the second year) is $121 − $100 = $21. In general, the compound amount S t of the principal P at the end of t years at the rate of i (expressed as a fraction or decimal) compounded annually can be expressed as follows. (1) After the first year: S 1 = P + P i = P (1 + i ).
eBook - PDFFinite Mathematics
Models and Applications
- Carla C. Morris, Robert M. Stark(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
2 Mathematics of Finance 2.1 Simple and Compound Interest 47 Simple Interest 47 Example 2.1.1 Simple Interest 48 Example 2.1.2 A Savings Account 48 Compound Interest 49 Example 2.1.3 Compound Interest 49 Example 2.1.4 Effects of Compounding Periods 50 Example 2.1.5 Compounding Continuously 51 Effective Rate of Interest 51 Example 2.1.6 Effective Rate of Interest 51 Example 2.1.7 Annual Percentage Rate 52 Present Worth 53 Example 2.1.8 Present Worth 53 Example 2.1.9 Real Estate Bargains 53 2.2 Ordinary Annuity 55 Example 2.2.1 Future Value Calculation 56 Example 2.2.2 An Annuity Accumulation 56 Example 2.2.3 An Annuity Payout 57 Example 2.2.4 Present Value of an Annuity 58 2.3 Amortization 59 Example 2.3.1 Amortization Schedule 60 Example 2.3.2 Amortization 61 Example 2.3.3 Mortgage Payment 61 2.4 Arithmetic and Geometric Sequences 63 Example 2.4.1 Arithmetic Sum 64 Example 2.4.2 Geometric Series 64 Example 2.4.3 Compound Interest 65 Example 2.4.4 Benjamin Franklin’s Bequest 65 Fibonacci Numbers 66 Example 2.4.5 Fibonacci Sequence 66 Historical Notes 69 2.1 SIMPLE AND Compound Interest Simple Interest You have heard the refrain: “A dollar today is not the same as a dollar tomorrow.” Used in many contexts, here we mean that money earns more money! Finite Mathematics: Models and Applications, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc. Companion Website: http://www.wiley.com/go/morris/finitemathematics 47 48 MATHEMATICS OF FINANCE Interest is monies paid or received for the use of money. When you borrow money, your promise to pay interest is the inducement for lenders to part with their money. In your savings or money market accounts, you are the lender, and your accounts are credited with interest. The money borrowed is the principal. The interest rate is expressed in dollars per dollar per unit time. Interest rates are often expressed in decimal format.
eBook - PDFMathematics NQF4 SB
TVET FIRST
- M Van Rensburg, I Mapaling M Trollope(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
385 Module 14 14.1.2 Compound growth Compound Interest is interest payable or earned on both the principal amount and the interest earned or accrued on the principal amount so far. The closing amount at the end of each year becomes the principal amount at the beginning of the next year. The formula to calculate Compound Interest is A = P (1 + i ) n , where: • A is the total amount payable or available at the end of the loan or investment period. • P is the initial or principal amount you borrow or invest. • i = r ____ 100 where r is the interest rate (%) per annum. • n is the term in years. Note: We use the Compound Interest formula in the form A = P (1 + i ) n to calculate the accumulated amount (A). To calculate P, i or n , we can manipulate the formula as follows: • To calculate the principal amount : P = A _____ (1 + i ) n • To calculate the interest rate : i = n √ __ A __ P − 1 • To calculate the term : n = log A __ P _______ log (1 + i ) (Not for examination purposes) To calculate I, the amount of Compound Interest , we can simply use the formula I = A − P. Example 14.6 An amount of R10 000 is invested for five years. Interest is earned at 12% per year, compounded annually. Calculate the value of the investment after five years. Solution A = P (1 + i ) n = P ( 1 + r ____ 100 ) n = 10 000 (1 + 12 ____ 100 ) 5 = R17 623, 42 386 Module 14 Remember: In Level 3 you learned how to solve problems related to the compound growth formula when interest is compounded annually, half-yearly, monthly and daily. The formula for calculating Compound Interest is: A = P(1 + i __ m ) m × n where m is the number of times that the growth is compounded annually. The formulae will therefore be: • A = P [ 1 + i _____ 2 ] 2 n if interest is added half-yearly. • A = P [ 1 + i ______ 12 ] 12 n if interest is added monthly. • A = P [ 1 + i _______ 365 ] 365 n if interest is added daily. Example 14.7 Amos deposits R10 000 into a savings account.
eBook - PDF- Robert Zipf(Author)
- 2003(Publication Date)
- Academic Press(Publisher)
You should be able to use tables to approximate the solutions to Compound Interest problems. The equations and mathematical concepts in this unit are the foundation for all our work in this book. It is absolutely essential that you understand the equations and the mathematical thinking behind them before you move on to the rest of the book. You simply will not be able to follow the book if you do not understand these equations or the mathematics. WHAT IS Compound Interest? Suppose you have $100, and you put it in the bank for 1 year. The bank promises to pay you 8% per year interest. At the end of 1 year, you will have You can also compute the total amount on deposit after 1 year as follows: Equation 3.1 In the preceding case, i = .08 in decimal form, and S = 100, so we have Equation 3.2 If the total amount at the end of 1 year is left on deposit for the next year, the total time is 2 years, and the equation becomes S S S S 2 2 1 08 1 08 1 08 1 1664 100 1 1664 116 64 = ( )( ) = ( ) = ( ) = ( ) = . . . . . . S 1 100 1 8 100 1 08 108 = + ( ) = ( ) = . . Total amount after 1 year since = = + = + = + = ( ) = + ( ) S S I S Sit S Si t S i 1 1 1 Interest $100 .08 1 $8 Principal still on deposit $100 Total amount on deposit $108 = = ¥ ¥ = ¥ ¥ = = = I S i t 20 Compound Interest Here is another way to figure the total: On deposit at the end of year 1: $108.00 This process of the interest paid earning interest on its own is called “com-pounding,” and the result is called “Compound Interest.” The $.64 earned during the second year on the interest paid during the first year is called “interest on interest,” and is the result of compounding. If simple interest were used, the result would be only 116. The $.64 doesn’t look like much. Actually, over a period of years, interest on interest can amount to a large sum. In fact, given enough time, there is no limit on how much it can amount to.
eBook - PDF- Richard Aufmann, Joanne Lockwood, Richard Nation, Daniel K. Clegg(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
In general, an increase in the num- ber of compounding periods results in an increase in the interest earned by an account. In the example above, the formulas I = Prt and A = P + I were used to show the amount of interest added to the account each quarter. The formula A = P( 1 + rt ) can be used to calculate A at the end of each quarter. For example, the amount in the account at the end of the first quarter is A = P( 1 + rt ) A = 1000 F 1 + 0.05 S 3 12 DG A = 1000( 1.0125) A = 1012.50 This amount, $1012.50, is the same as the amount calculated on the preceding page using the formula A = P + I to find the amount at the end of the first quarter. The formula A = P( 1 + rt ) is used in Example 1. EXAMPLE 1 Calculate Future Value You deposit $500 in an account earning 6% interest, compounded semiannually. How much is in the account at the end of 1 year? Solution The interest is compounded every 6 months. Calculate the amount in the account after the first 6 months. t = 6 12 . A = P( 1 + rt ) A = 500 F 1 + 0.06 S 6 12 DG A = 515 Calculate the amount in the account after the second 6 months. A = P( 1 + rt ) A = 515 F 1 + 0.06 S 6 12 DG A = 530.45 The total amount in the account at the end of 1 year is $530.45. CHECK YOUR PROGRESS 1 You deposit $2000 in an account earning 4% interest, compounded monthly. How much is in the account at the end of 6 months? Solution See page S38. ◀ In calculations that involve Compound Interest, the sum of the principal and the inter- est that has been added to it is called the compound amount. In Example 1, the com- pound amount is $530.45. The calculations necessary to determine Compound Interest and compound amounts can be simplified by using a formula. Consider an amount P deposited into an account paying an annual interest rate r , compounded annually. The interest earned during the first year is I = Prt I = Pr( 1) I = Pr POINT OF INTEREST The top seven sources of financial news for professionals in the field of finance are listed below.
eBook - PDF- Steve Slavin, Tere Stouffer(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Sim- ilarly, when you deposit money in a savings account or another savings vehicle, the bank pays you interest because it wants to give you an incentive to keep your money there. The first section in this chapter discusses simple interest; the second section explains Compound Interest, the third discusses the concept of present value, and the final section discusses basic consumer credit. 9.1 Simple Interest What is simple interest? Suppose you put $100 in the bank for 1 year, and the bank pays 4% simple interest. At the end of the year, you have $104—the $100 you started with (called the principal), and the $4 simple interest you earned dur- ing the year. The total amount (in this case, $104) is called the maturity value. There is another type of interest—called Compound Interest—that is dis- cussed later in this chapter. 9.1.1 Solving for the Unknown The formula for finding simple interest is or In the simple interest formula, ▲ Principal is the original investment or borrowed amount, represented by p. ▲ Rate (or interest rate) is the percentage charged to lend or borrow money, represented by r. ▲ Time is the length of the investment or loan, expressed in years, months, or days, represented by t. Simple interest is the amount paid or received, calculated on the principal. Maturity value is the value of the investment after a period of time, calculated using the following formula: MV p I I p r t Interest Principal Rate Time 9.1.1 SOLVING FOR THE UNKNOWN 151 For example, how much interest is earned if $1,000 is loaned for 2 years at 6%? To figure this out, you use the following equation: As another example, how much interest is earned if $100 is invested at 8% for 6 months? (Remember to calculate 6 months as half a year, or 0.5 year.) You figure this out as follows: Use the simple interest formula to solve the following loan problems: 1. Paul Adams borrowed $2,000 for 1 year, at a rate of 7%.
eBook - PDFUnderstanding the Mathematics of Personal Finance
An Introduction to Financial Literacy
- Lawrence N. Dworsky(Author)
- 2009(Publication Date)
- Wiley(Publisher)
The interest column therefore shows $0, and the balance (how much you owe) shows the principal, $5,000.00. 1 The term “10% loan,” unless expressly explained otherwise, is synonymous with a “10% APR loan.” 28 Chapter 2 Compound Interest The first interest calculation is performed on July 14, 2008. Since this is the first such calculation, column 1 shows this as compounding interval #1. Five percent APR compounded annually is just 5% interest per calculation. This interest is therefore Interest = ( )( ) = $ , . . $ . . 5 000 00 0 05 250 00 The balance, that is, how much you owe, is now the previous balance ($5,000.00) plus this interest ($250.00), which is $5,250.00. July 14, 2009, is the date of the second Compound Interest calculation. Here’s where Compound Interest is very different from simple interest. The interest calcula- tion is based not on the principal ($5,000.00) but on the balance at the time of the calculation, in this case $5,250.00: Interest = ( )( ) = $ , . . $ . . 5 250 00 0 05 262 50 Notice that this interest value is higher than the first interest value ($262.50 vs. $250.00), and that the balance grows to $5,612.50. I’ll repeat this calculation in detail one more time. On July 14, 2010, the third Compound Interest calculation is Interest = ( )( ) = $ , . . $ . , 5 512 50 0 05 275 63 and the new balance is $5,788.13. Each time the calculation is performed, the interest is larger.
eBook - PDFFundamentals of Financial Instruments
An Introduction to Stocks, Bonds, Foreign Exchange, and Derivatives
- Sunil K. Parameswaran(Author)
- 2022(Publication Date)
- Wiley(Publisher)
Compound Interest Let us take the case of an investment of $ P that has been made for N measurement periods. However, we will assume this time that interest is compounded at the end of every year. Notice, we are assuming that the interest conversion period is equal to the measurement period, namely a year. In other words, the quoted rate is equal to the effective rate. In this case, an original investment of $ P will become $ P (1 + r ) dollars after one period. The difference as compared to the earlier case, however, is that during the second period the entire amount will earn interest and consequently the balance at the end of two periods will be P (1 + r ) 2 . Extending the logic the balance after N periods will be P (1 + r ) N . Once again you should note that N need not be an integer. Thus, the following observations are valid if interest is paid on a compound inter-est basis. ▪ Every time interest is earned it is automatically reinvested at the same rate for the next conversion period. ▪ Interest is paid on the accumulated value at the start of the conversion period and not on the original principal. ▪ The interest earned every period will not be a constant but will steadily increase. EXAMPLE 2.4 Assume that Katherine Mitchell has deposited $25,000 with Continental Bank for four years, and that the bank pays 8% interest per annum compounded annually. The initial investment of $25,000 will become 25 , 000 × 1.08 = $27 , 000 ( continued ) 52 MATHEMATICS OF FINANCE ( continued ) after one year. The difference in this case, as opposed to the earlier example (2.2) where simple interest was considered, is that the entire accumulated value of $27,000 will earn interest during the second year.
eBook - PDFFinancial Rules for New College Grads
Invest before Paying Off Debt—and Other Tips Your Professors Didn't Teach You
- Michael C. Taylor(Author)
- 2018(Publication Date)
- Praeger(Publisher)
It’s not magic or wizardry, it’s just pretty simple math. As I said you don’t have to do this math yourself to benefit from the les- sons of this book. But optimistically speaking, if you did master this, I think you would find yourself incentivized to do the right things with savings and investments. The next chapter takes the same algebra formula and simply reverses it. The reversal allows us to figure out how money in the future can be valued as money today. Appendix In this Appendix, I assume you have already read or understand Chap- ter 4 on Compound Interest on kittens, snowballs, pine forests, retirement sav- ings, inflation, and government debt. This Appendix offers a practical illustration of how investors might use Compound Interest math when it comes to bond and stock calculation. I also return to feral cats, and then the extraordinary potential profitabil- ity of credit card debt, for banks. More Frequent Compounding than Annual Compounding Now, are you ready to become a next-level, compound-interest-calculating Jedi? Because in the real world of financial calculations, we need to introduce one more easy little step to achieve better precision with our numbers. In many situations, money does not compound annually, but rather more than once a year. When you look at the real world, you realize compounding occurs more frequently than annually, like: • Bonds—usually 2 times a year • Dividend stocks—usually 4 times a year • Feral kittens (with their 60-day gestation period)—6 times a year, or • Credit card debt—12 times per year On Compound Interest 33 To calculate more-often-than-annual compounding, we have to make sure the yield or annual growth number (Y) gets properly divided by the number of periods per year (which I’ll call “p”). The Compound Interest formula I’ve been using could now be written with that extra tweak as FV = PV ∗ (1 + Y/p)^N.
eBook - PDFMathematical Interest Theory
Third Edition
- Leslie Jane Federer Vaaler, Shinko Kojima Harper, James W. Daniel(Authors)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
That is, they say they will Compound Interest constantly, and this is the limiting case of compounding more and more frequently. Calculus is the mathematics of limits, so it takes calculus to analyze to what annual effective rate of interest this continuous compounding is equivalent. In fact, it is equivalent to an annual effective interest rate of i = e . 05 − 1 ≈ 5 . 1271% . More generally, A nominal interest rate of δ convertible continuously is equivalent to an annual effective rate of i = e δ − 1. When these equivalent rates govern the growth of money, a ( t ) = (1 + i ) t = e δt and v = 1 1+ i = e − δ . The constant δ is called the force of interest . Alternatively, we may start with an annual effective interest rate and look for an equivalent force of interest (nominal rate convertible continuously). An annual effective interest rate of i is equivalent to a nominal rate of interest δ convertible continuously where δ = ln (1 + i ). EXAMPLE 1.13.1 Problem: Swift Bank promises 3.5% interest compounded continuously. If Ken deposits $3,000 in Swift Bank, what will his balance be four years later? Solution At Swift Bank, δ = . 035, a ( t ) = e δt = e . 035 t , and Ken’s balance after four years is $3 , 000 e ( . 035)4 ≈ $3 , 450 . 82 . 1.14 QUOTED RATES FOR TREASURY BILLS Practical applications of simple interest and simple discount may be limited, but they do exist. The rates for short-term loans to the United States federal government and the Government of Canada are quoted using simple discount and simple interest, respectively. When an individual or a small business needs to borrow money, it often turns to a commercial bank or a savings and loan institution. However, if 62 Chapter 1 The growth of money a country’s government needs a loan, the amount it needs is often much too large for one or even several banks to lend. Instead, a government borrows from numerous entities, each of which can designate an acceptable loan amount.
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