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Credit rating. Convertible 9. Coupe Sedan 2. Black 5. Blue 4. This fee is called interest. In this chapter, you will learn how to calculate interest using simple interest. Although most financial transactions use compound interest introduced in chapter 2 , simple interest is still used in many short-term transactions. Many of the concepts introduced in this chapter will be used throughout the rest of this book and are applicable to compound interest. The chapter starts off with some fundamental relationships for calculating interest and how to determine the future, or accumulated, value of a single sum of money invested today.
You will also learn how to calculate the time between dates. Section 1. This "present value of a future cash flow" is one of the fundamental calculations underlying the mathematics of finance. This section also introduces equations of value, which allow you to accumulate or discount a series of financial transactions and are used to solve many problems in financial mathematics. The chapter ends with section 1. Simple Interest In any financial transaction, there are two parties involved: an investor, who is lending money to someone, and a debtor, who is borrowing money from the investor.
The debtor must pay back the money originally borrowed, and also the fee charged for the use of the money, called interest. From the investor's point of view, interest is income from invested capital. The capital originally invested in an interest transaction is called the principal. The sum of the principal and interest due is called the amount or accumulated value. Any interest transaction can be described by the rate of interest, which is the ratio of the interest earned in one time unit on the principal.
In early times, the principal lent and the interest paid might be tangible goods e. Now, they are most commonly in the form of money. The practice of charging interest is as old as the earliest written records of humanity.
Four thousand years ago, the laws of Babylon referred to interest payments on debts. At simple interest, the interest is computed on the original principal during the whole time, or term of the loan, at the stated annual rate of interest. We shall use the following notation: P - the principal, or the present value of S, or the discounted value of S, or the proceeds. We can display the relationship between S and P on a time diagram.
When the time is given in days, there are two different varieties of simple interest in use: number of days 1. Exact interest, where i.
The general practice in Canada is to use exact interest, whereas the general practice in the United States and in international business transactions is to use ordinary interest also referred to as the Banker's Rule.
In this textbook, exact interest is used all the time unless specified otherwise. When the time is given by two dates we calculate the exact number of days between the two dates from a table listing the serial number of each day of the year see the table on the inside back cover. The exact time is obtained as the difference between serial numbers of the given dates. In leap years years divisible by 4 an extra day is added to February.
When calculating the number of days between two dates, the most common practice in Canada is to count the starting date, but not the ending date. The reason is that financial institutions calculate interest each day on the closing balance of a loan or savings account. On the day a loan is taken out or a deposit is made, there is a non-zero balance at the end of that day, whereas on the day a loan is paid off or the deposit is fully withdrawn, there is a zero balance at the end of that day.
However, it is easier to assume the opposite when using the table on the inside back cover. That is, unless otherwise stated, you should assume that interest is not calculated on the starting date, but is calculated on the ending date. That way, in order to determine the number of days between two dates, all you have to do is subtract the two Aluminum Boats Kelowna Net values you find from the table on the inside back cover. Example 1 will illustrate this. Since a date is given when the loan was actually taken out, we must use days.
Seven months after April 7 is November 7. Using the table on the inside back cover, we find that April 7 is day 97 and November 7 is day Notice that ordinary interest is always greater than the exact interest and thus it brings increased revenue to the lender.
What was the annual interest rate? On August 5, the interest rate changes to 4. How much is in the fund on October 23? Demand Loans On a demand loan, the lender may demand full or partial payment of the loan at any time and the borrower may repay all of the loan or any part at any time without notice and without interest penalty.
Interest on demand loans is based on the unpaid balance and is usually payable monthly The interest rate on demand loans is not usually fixed but fluctuates with market conditions. Interest on the loan, calculated on the unpaid balance, is charged to her account on the 1st of each month. T h e rate was changed to Calculate the interest payments required and the total interest paid. Invoice Cash Discounts To encourage prompt payments of invoices many manufacturers and wholesalers offer cash discounts for payments in advance of the final due date.
Otherwise, the full amount must be paid not later than thirty days from the date of the invoice. A merchant may consider borrowing the money to pay the invoice in time to receive the cash discount. Assuming that the loan would be repaid on the day the invoice is due, the interest the merchant should be willing to pay on the loan should not exceed the cash discount.
W h a t is the highest simple interest rate at which he can afford to borrow money in order to take advantage of the discount? H e would borrow this money on day 30 and repay it on day the day the original invoice is due resulting in a day loan. T h e highest simple interest rate at which the merchant can afford to borrow money is T h i s is a break-even rate. Exercise 1. What simple interest rate is implied? What amount must he repay in 7 months?
Determine die amount of interest earned. T h e bank assured Jacob that his investment would be adequate to cover the purchase. Determine the minimum simple interest rate that Jacob's money must be earning. Interest is paid at the end of each quarter March 31, June 30, September 30, December 31 and at the time of the last payment.
Interest is calculated at the. Mustafa repaid the loan with the following payments:. Interest on the loan is charged to the company's current account on the 11 th of each month. The company repaid the loan with the following payments:.
At what interest rate could you afford to borrow money to take advantage of this discount? What is the highest simple interest rate at which he can afford to borrow money in order to take advantage of the discount? If the store were to borrow the money to pay the bill in 10 days, what is the highest interest rate at which the store could afford to borrow?
Optical actually save by using the cash discount? W e can display the relationship between P and S on a time diagram. W h e n we calculate P from S, we call P the present value of S or the discounted value of 5. For a given interest rate r, the difference S - P has two interpretations. T h e interest I on P which when added to P ffives S. T h e discount D on S which when subtracted from S gives P. What is the simple discount?
Eight months after September 15 day is May 15 day Promissory Notes A promissory note is a written promise by a debtor, called the maker of the note, to pay to, or to the order of, the creditor, called the payee of the note, a sum of money, with or without interest, on a specified date.
The following is an example of an interest-bearing note. The term of the note is 60 days. The due date is 60 days after September 1, , that is October 31, By a maturity value of a note we shall understand the value of the note at the maturity date. If Mr. B chooses to pay on October 31, he will pay interest for 60 days, not 63 days.
However, no legal action can be taken against him until the expiry of the three days of grace. Thus, Mr. B would be within his legal rights to repay as late as November 3, and in that case he would pay interest for 63 days. A promissory note is similar to a short-term bank loan, but it differs in two important ways: 1. The lender does not have to be a financial institution. A promissory note may be sold one or several times before its maturity.
The resulting value is called the proceeds and is paid to die seller. The proceeds are determined by formula 3. The procedure for discounting of promissory notes can be summarized in 2 steps: Step 1 Calculate the maturity value, S, of the note. The maturity value of a noninterest-bearing note is die face value of die note.
The maturity value of an interest-bearing note is the accumulated value of the face value of the note at the maturity date. Step 2 Calculate the proceeds, P, by discounting the maturity value, S, at a specified simple interest rate from the maturity date back to the date of sale. In Canada, "three days of grace" are allowed for the payment of a promissory note.
This means that the payment becomes due on the third day after the due date of the note. If interest is being charged, the three days of grace must be added to the time stated in the note. If the third day of grace falls on a holiday, payment will become due the next day that is itself not a holiday. If the time is stated in months, these must be calendar months and not months of 30 days. For example, 2 months after July 5 is September 5.
Thus the legal due date is September 8, and the number of days to be used in calculating the interest if any will be 65 days. In the month in which the payment falls due, there may be no corresponding date to that from which the time is computed.
In such a case, the last day of the month is taken as the corresponding date. For example, two months after December 31 is February 28 or February 29 in leap years and March 3 would be the legal due date. T h e promissory note described in the beginning of this section is sold by Mr. A receive for the note? A realize on his investment, when he sells the note on October 1, ? W e arrange the dated values on a time diagram below.
Solution b. T h e rate of interest Mr. A will realize is I W h a Starcraft Center Console Aluminum Boats Network t should be the face value of this note to give him the exact amount needed to pay cash for the goods? Treasury Bills or T-bills Treasury bills are popular short-term and low-risk securities issued by the Federal Government of Canada every other Tuesday with maturities of 13, 26, or 52 weeks 91, , or days. T h e y are basically promissory notes issued by the government to meet their short-term financing needs.
T h e face value of a T-bill is the amount the government guarantees it will pay on the maturity date. T h e r e is n o interest rate stated on a T-bill. Instead, to determine the purchase price of a T-bill, you need to discount the face value to the date of sale at an interest rate that is determined by market conditions. W h a t price is paid? W h a t rate of interest is assumed? A day note promises to pay Ms. Chiu receive? Chiu realize on her investment?
How much does the bank pay for the note? What is the investor's profit? What is the bank's profit on this investment when the note matures? W h a t are the proceeds? What should be the face value of this note to give the retailer the exact amount needed to pay cash for the goods? T h e note is due in days with interest at 6.
An investor bought a day treasury bill to yield 3. What price did tlie T-bill sell for? Determine the proceeds if the supplier sells the 9.
T h e note is due in days with interest at 7. W h a t yield rate did the other investor wish to earn? W h a t rate of return did the dealer end up earning?
T h e dealer sold it to. Equations of Value All financial decisions must take into account the basic idea that m o n e y has time value. In a financial transaction involving money due on different dates, every sum of money should have an attached date, the date on which it falls due. T h a t is, the mathematics of finance deals with dated values. T h i s is one of the most important facts in the mathematics of finance.
W h i c h one would you choose? N o t e that the three dated values are not equivalent at some other rate of interest. In general, we compare dated values by the. T h e following time diagram illustrates dated values equivalent to a given dated value X. Based on the time diagram above we can state the following simple rules: 1. W h e n we move money forward in time, we accumulate, i. W h e n we move money backward in time, we discount, i. Let us arrange the data on a time diagram below.
T h e sum of a set of dated values, due on different dates, has n o meaning. W e must take into account the time value of money, which means we have to replace all the dated values by equivalent dated values, due on the same date. T h e sum of the equivalent values is called the dated value of the set. T h e lender pay off these two debts with a single payment. W e calculate equivalent dated values of both obligations at the three different times and arrange them in the table below.
Equation of Value One of the most important problems in the mathematics of finance is the replacing of a given set of payments by an equivalent set. We say that two sets of payments are equivalent at a given simple interest rate if the dated values of the sets, on any common date, are equal. An equation stating that the dated values, on a common date, of two sets of payments are equal is called an equation of value or an equation of equivalence.
The date used is called the focal date or the comparison date or the valuation date. A very effective way to solve many problems in mathematics of finance is to use the equation of value. The procedure is carried out in the following steps: Step 1 Make a good time diagram showing the dated values of original debts on one side of the time line and the dated values of replacement payments on the other side. A good time diagram is of great help in the analysis and solution of problems. Step 2 Select a focal date and bring all the dated values to the focal date using the specified interest rate.
Step 3. We arrange the djrted values on a time diagram. W e arrange the dated values X, Y, and Z on a time diagram, listing for each date the day n u m b e r from the table on die inside back cover. In mathematics, an equivalence relationship must satisfy the so-called property of transitivity, that is, if X i s equivalent to F a n d F i s equivalent to Z, then X is equivalent to Z. Example 4 illustrates that the definition of equivalence of dated values at simple interest r does n o t satisfy the property of transitivity.
As a result, the solutions to the problems of equations of value at simple interest do depend on the selection of the focal date. T h e following Example 5 illustrates that in problems involving equations of value at simple interest the answer will vary slightly with the location of the focal date. It is therefore important that the parties involved in the financial transaction agree on the location of the focal date.
She is to repay the debt with 3 equal payments, the first at the end of 3 months, the second at the end of 6 months, and the third at the end of 9 months. Determine the size of the payments. P u t the focal date a at the present time; b at the end of 9 months.
Arrange all the dated values on a time diagram. These two debts are to be replaced with a single payment due in 8 months. Determine the value of the single payment if money is worth 6. Since the two original debts are due with interest, we first must determine the maturity value of each of them. In each case determine the equivalent payments for each of the debts. What if the focal date is today?
T h e last payment is to be on January 2, Adams has two options available in repaying a loan. What is the answer if the focal date is 3 months hence and the options are equivalent? A will pay Mr. Determine Xii the comparison date is 2 years hence i. He is to repay the debt with 4 equal payments, one at the end of each 3-month period for 1 year.
Determine the size of the payments given a focal date a at the present time; b at the end of 1 year. Determine X using all four transaction dates as possible focal dates. Partial Payments Financial obligations are sometimes liquidated by a series of partial payments during the term of obligation. T h e n it is necessary to determine the balance due on the final due date. T h e r e are two common ways to allow interest credit on short-term transactions. T h e interest on the unpaid balance of the debt is computed each time a partial payment is made.
If the payment is greater than the interest due, the difference is used to reduce the debt. If the payment is less tfian the interest due, it is held without interest until other partial payments are made whose sum exceeds the interest due at the time of the last of these partial payments.
This point is illustrated in Example 3. T h e balance due on the final date is the outstanding balance after the last partial payment carried to the final due date. W h a t is Aluminum Boats Gibsons Stores the balance due on August 15 using the declining balance method?
W e arrange all dated values on a time diagram. Instead of having one comparison date we have 4 comparison dates. T h e calculations are given below.
The above calculations also may be carried out in a shorter form, as shown below. Method 2 also known as Merchant's Rule The entire debt and each partial payment earn interest to the final settlement date. The balance due on the final due date is simply the difference between the accumulated value of the debt and the accumulated value of the partial payments. We arrange all dated values on a time diagram. We can write an equation of value with August 15 as the focal date.
Note: T h e methods result in two different concluding payments. It is important that the two parties to a business transaction agree on die mediod to be used. Common business practice is die Declining Balance Method.
W h a t is the balance due on December 15 using the Declining Balance Method? T h e Merchant's Rule? What is the balance due on April 18, , by the Declining Balance Method? By the Merchant's Rule? Determine the balance due in one year using the Declining Balance Method.
What is the balance due on the final statement date by the Declining Balance A-lethod? Simple Discount at a Discount Rate T h e calculation of present or discounted value over durations of less than one year is sometimes based on a simple discount rate. T h e annual simple discount rate d is the ratio of the discount D for the year to the amount 5" on which the discount is given. T h e simple discount D on an amount 5 for t years at the discount rate d is calculated by means of the formula:.
T h e factor 1 - dt in formula 5 is called a discount factor at a simple discount rate d. Simple discount is not very common. However, it should not be ignored.
Some financial institutions offer what are referred to as discounted loans. For these types of loans, the interest charge is based on the final amount, S, rather.
The lender calculates the interest which is called discount D, using formula 4 and Aluminum Boats Gibsons Bc 104 deducts this amount from S. The difference, P, is the amount the borrower actually receives, even though the actual loan amount is considered to be S. The borrower pays back S on the due date. For this reason the interest charge on discounted loans is sometimes referred to as interest in advance, as the interest is paid up front, at the time of the loan, instead of the more common practice of paying interest on the final due date.
In theory, we can also accumulate a sum of money using simple discount, although it is not commonly done in practice. From formula 5 , we can express S in terms of P, d, and t and obtain. The factor 1 - dt l in formula 6 is called an accumulation factor at a simple discount rate d. Formula 6 can be used to calculate the maturity value of a loan for a specified proceeds. We can conclude that a given simple discount rate results in a larger money return to a lender than the same simple interest rate.
In general, we can calculate equivalent rates of interest or discount by using the following definition: Two rates are equivalent if they have the same effect on money over the same period of time. The above definition can be used to determine simple or compound equivalent rates of interest or discount. Notice that for simple rates of interest and discount, the equivalent rates are dependent on the period of time.
Determine Brown's purchase price. We arrange the dated values on a time diagram. Determine the discount and the proceeds. W h a t simple discount rate can the purchaser expect to earn?
For what amount should he make the note? Determine the amount of interest paid on the first note and the face value of the second note. If the actual amount of money that Mr. If this is a discounted loan, what rate of simple discount is assumed? Note: The above definition of equivalence of dated values at a simple interest rate r does not satisfy the property of transitivity.
The lack of transitivity leads to different answers when different comparison dates are used for equations of value. Equivalent rates: Two rates are equivalent i: they have the same effect on money over the same period of time. Note: At simple rates of interest and discount, the equivalent rates are dependent on the period of time.
Review Exercises 1. What rate of simple interest will the tax discounter earn? She was given two options. Today Mr. Determine the size of the equal payments using a the end of 6 months as a focal date. Ninety days after date, I promise t o pay t o the order of J.
Signed J. Prasad receive? Prasad realize on his investment? Lau pays off the loan as due in exactly 90 days? On the maturity date, the maker of the note paid the interest in full and gave a. Determine the interest paid on the first note and the face value of the second note. In how many days should the final payment be made?
H e is to repay the debt with 3 equal payments, the first at the end of 3 months, the second at the end of 7 months, and the third at the end of 12 months. Determine the difference between payments resulting from the selection of the focal date at the present time and at the end of 12 months.
How much does the finance company pay for the note? Learning Objectives In chapter 1, we learned how to perform financial calculations using simple interest. Beginning with this chapter, the rest of this textbook is devoted to financial calculations using compound interest. Tbegins with the concept of interest earned in a given period being added to your principal and thereafter earning interest.
Simple interest reappears in this section. In many loans and investments, the interest rate changes over time and section 2. And finally, section 2. Fundamental Compound Interest Formula If the interest due is added to the principal at the end of each interest period and thereafter earns interest, the interest is said to be compounded. The sum of the original principal and total interest is called the compound amount or accumulated value. The difference between the accumulated value and the original principal is called the compound interest.
The time between two successive. This time unit need not be a year. Most of you will already be familiar with situations where interest is "payable quarterly," or "compounded semi-annually" or "convertible monthly. At the End of. In this section we shall develop quicker methods for calculating the compound interest.
Continuous compounding that uses compounding intervals shorter than one day is covered in section 2. N o t e 2: T h e rate i equals � and is always used in the compound interest calculation. Let P represent the principal at the beginning of the first interest period and i the interest rate per compounding period. We shall calculate the accumulated values at the ends of the successive interest periods for n periods. At the end of the 1st period: the interest due the accumulated value.
Continuing in this manner for n periods the accumulated value S at the end of n periods is given by the fundamental compound interest formula.
The process of calculating S from P is called accumulation. This illustrates the power of compound interest over a long period of time. It is assumed that students will be using pocket calculators equipped with the functions yx and log x to solve the problems in Mathematics of Finance. In the examples in this textbook, we have used all digits of the factors provided by a pocket calculator and rounded off to the nearest cent only in the final answer.
The table and graph below show the effect of time and rate on the growth of money at compound interest. Using a Computer Spreadsheet A computer spreadsheet is tailor made for many of the financial calculations that will be studied in this textbook. In this text, we will present Excel spreadsheets. We will illustrate the formulas needed to perform the calculations of Example 2 on a spreadsheet. The entries in an Excel spreadsheet are summarized below. In A2 to A9, type 5, 10, 15, 20, 25, 30, 35, and In B2, we need to type the formula that will accumulate the initial investment given in B l l to the end of 5 years at the given interest rate.
Remember that the given interest rate needs to be divided by 12 to obtain die monthly rate and the exponent must be multiplied by 12 to give the term in months. A copy of the resulting spreadsheet is given below. A Years. How much interest will be earned: a during the first year? When using the fundamental compound interest formula 7 in calculations, do not round off the value of i. Use all digits provided by your calculator. How much has been accumulated in the fund a at the end of 33 months; b at the end of six and a half years?
To calculate n, we observe that 33 months is equal to 2. But m represents the number of interest periods per year which is 4 , not the number of months in the interest period. How much is in the account on December 31 if a the interest is compounded daily?
Note how the account with interest compounded daily earns more interest during the year than the account with interest calculated daily. Problems 1 to 8 make use of the following table. In each case find the accumulated value and the compound interest earned. How much interest will she earn during the first year if.
Equivalent Compound Interest Rates T h e yearly nominal rate is meaningless until we specify the frequency of conversion m. Frequency of Conversion. At the same nominal rate the accumulated value depends on the frequency of conversion; it increases with the increased frequency of conversion.
For a given nominal ratejm compounded m times per year, we define the corresponding annual effective rate to be that rate j which, if compounded annually, will produce the same amount of interest per year. Calculating equivalent interest rates and, in particular, the annual effective rate is very useful when you need to compare investments that have interest rates with different compounding periods. The following example illustrates this point.
You wish to invest a sum of money for a number of years and have narrowed your choices to the following three interest rates: Investment A: j 2 - On the surface it looks like investment A is best, as it has the highest nominal rate of interest. However, it pays interest only twice a year. What about investment B?
It pays interest 12 times a year, but it has the lowest nominal rate of interest. To choose which investment is best we need to calculate the equivalent rate of interest, compounded with the same frequency, for each investment. Investment A:. It turns out that investment C has the highest annual effective rate and should be the investment that is chosen.
In Chapter 1 we defined equivalent rates as the rates that have the same effect on money over the same period of time. See p. Applying the general definition specifically for compound interest rates we say: Two nominal compound interest rates are equivalent if they yield the same accumulated values at the end of one year, and hence at the end of any number of years. Let r be the unknown simple interest rate. W h e n calculating unknown yearly rates of interest, carry all decimal digits and then round off your final answer to the nearest hundredth of a percent.
Exercise 2. Which option yields the higher annual effective rate of interest? Which rate gives the best and the worst rate of return on your investment? What is the minimum frequency of compounding for bank B in order that the rate at bank B be at least as attractive as that at bank A? Part B 1. Determine what is the current best and worst interest rate available on 5-year guaranteed investment certificates by checking three different financial institutions.
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