In example 1, the amount John borrows ($500) is called the principal. The basic time unit used for the computation of the interest rate (one year) is called the interest period. The interest rate (6% per year) is always given as a percentage per interest period.
The general formulas governing computation of simple interest are straightforward.
Rules for Simple Interest
If one borrows a principal P at a rate r per interest period for a total of t interest periods, then the total amount of simple interest I is given by
The total amount A that must be repaid is
Although the rate r is always expressed as a percentage per interest period, when computing with it, remember to convert the percentage to a number (by dividing by 100).
If one is borrowing money, the principal P is sometimes called the present value of the loan, and the total amount A to be repaid is called the future value.
Notice that in example 1 the interest is added onto the principal, and the entire amount is paid back when the loan period (three years in example 1) expires. Add-on interest is the amount of simple interest that is added to the principal.
Compound Interest
Look again at example 1. Sue loans John $500 for three years, and John has the use of the entire $500 for all three years. An important feature of simple interest is that the recipient has the use of all the money loaned for the full period of the loan.
(Compound interest continued from p. 56)
Now let’s take a look at the money in the Alma Steadman Trust in the story. If she deposits the original $2 million at 6% compounded annually, it is the same as loaning the money to the bank. At the end of every year, the money that was deposited at the start of the year, plus the interest that money has earned, is returned by the bank to Alma, and she immediately redeposits the new amount for another year.
Notice that, after one year, the balance on 12/31/04 is $2,000,000 × 1.06. After two years, the balance on 12/31/05 is $2,000,000 × 1.062. After three years, the balance on 12/31/06 is $2,000,000 × 1.063. Finally, after 10 years, the balance on 12/31/13 is $2,000,000 × 1.0610. So $2,000,000 is the original principal, 1.06 is 1 plus the annual interest rate, and 10 is the number of years that the deposit has been earning interest. This shows that the future value A of a principal P deposit at an annual interest rate r for N years is given by the formula
This equation is extremely important. It involves four quantities: the principal P, the future value A, the annual interest rate r, and the number of years N that the money has been earning interest. If any three of these quantities are known, it is possible to solve for the fourth quantity in terms of the three missing quantities. For example, the present value P as a function of A, r, and N is given by
It is often important to compute the present value of an amount that will be needed in the future.
Example 2: Jose’s parents decide to give him a car when he graduates from college in four years. If they estimate the cost of the car as $10,000, and a bank is paying 6% compounded annually, how much must they deposit now to be able to buy the car when Jose graduates?
Solution: We must find the present value P of an amount whose future value A = $10,000, when money is compounded at an annual rate r = 0.06 for four years. So
P = $10,000/1.064 = $7,920.94
We can check that this is correct by seeing that $7,920.94 deposited for four years at 6% compounded annually yields a future value of $10,000. ■
OTHER COMPOUNDING PERIODS
When money is compounded annually, the interest is computed and added on at the end of the year, and the new total is used as the principal for the next year. Other frequently used compounding periods are semiannual compounding (twice a year), quarterly compounding (four times a year), monthly compounding (12 times a year), and daily compounding (360 times a year). The fact that a banking year is only 360 days is a reminder of how difficult computation was B.C. (before calculators) because it is much easier to work with semiannual, quarterly, and monthly compounding when the year is 360 days rather than 365.
We could simply use formula [6.3] to compute the future value of any principal if we are given the interest rate r per compounding period. Then N would be the number of compounding periods.
Example 3: Meredith deposits $3,000 in a bank that pays a quarterly compounding rate of 1.5%. What is the amount in her account at the end of three years?
Solution: Since there are four quarters in a year, there will be 12 quarters in three years. Using formula [6.3],
A = $3,000 × 1.01512 = $3,586.85 ■
Most of the time, however, a loan does not specify the interest rate r per compounding period but rather the annual compounding rate and the number of times per year that the loan is compounded. In example 3, the bank would say that it paid 6% compounded quarterly. The quarterly compounding rate is determined by dividing the annual compounding rate of 6% by 4, the number of compounding periods in a year.
This situation leads to the following modification of formula [6.3]. If a principal P is borrowed for N years at an annual rate r that is compounded t times a year, then the future value A is given by
A = P(1 + r/t)Nt
Example 4: Suppose that $8,000 is deposited in a high-yield corporate bond for five years at 8% annually. Compute the amount in the account if compounding is done (a) annually, (b) semiannually, (c) quarterly, (d) monthly, and (e) daily.
Solution:
(a) A = $8,000 × 1.085 = $11,754.62
(b) A = $8,000 × 1.0410 = $11,841.95
(c) A = $8,000 × 1.0220 = $11,887.58
(d) A = $8,000 × 1.006666760 = $11,918.79
(e) A = $8,000 × 1.00022221800 = $11,933.59
While (a), (b), and possibly even (c) can be done without a calculator that can handle exponentials, (d) and (e) are simply too much work unless such a calculator is available. ■
Example 4 makes two things clear. The first is that the more frequently money is compounded at the same annual rate, the greater the future value. The second is that either financial calculators, or calculators with an exponentiation key, are indispensable for computing when the compounding period is monthly or daily. Before the invention of electronic computers, the word “computer” did not refer to a machine but to an individual who was employed to do these sorts of calculations every day! A “computer” was a job description, not something that could be bought at an electronic supply store.
AMORTIZATION AND BUYING ON INSTALLMENT
New cars cost in the tens of thousands of dollars, and houses cost in the hundreds of thousands of dollars—and most of us simply don’t have that much money on hand. Fortunately, there are companies whose business is to loan money to be repaid over a period of time—typically four or five years for cars, fifteen to thirty years for houses. The typical way that this is done is for the purchaser to immediately pay a fraction of the purchase price, known as the down payment, and pay the rest off periodically (usually monthly, but sometimes biweekly) in installments. Each installment payment is for the same amount.
Before diving into the formulas, let’s see how this works for the first few payments in a typical case. Suppose you have your heart set on a new car that costs $20,000. The dealer asks for 10% down, so you write a check for $2,000. That leaves $18,000 to be financed, and the company supplying that $18,000 does so at a rate of 6%. You are then informed that you will make monthly payments of $422.73, and the first payment will be made a month from now.
The arithmetic of how this works is pretty straightforward. The moment you take possession of the car, you owe $18,000 to the credit company. At 6% annually, that’s 0.5% a month. So, after the first month, you have borrowed $18,000 and the interest that has accrued is 0.5% of $18,000, or $90. Consequently, you now actually owe the company $18,090. When you make your first payment of $422.73, the amount you owe, called the balance, is reduced to $18,090 − $422.73 = $17,667.27.
When you made that first payment of $422.73, $90 went to pa
y off the interest, and the remainder, $422.73 − $90 = $332.73 was used to reduce the balance. That $332.73 is usually called the principal.
Another month goes by. You still have to pay 0.5% monthly on the balance of $17,667.27, but that’s only $88.34. So, when you make that second payment of $422.73, the principal is $422.73 − $88.34 = $334.39; this reduces the balance to $17,667.27 − $334.39 = $17,332.88. And so it goes; each month the interest is less, the principal is larger, and the 48th payment reduces the debt to zero. The company now sends you the pink slip, and the car is all yours.
This method of paying off a loan is called amortization. There are many spreadsheets available online; you can either use them on a website or download them. Typically, a spreadsheet for the above loan will look something like this; because there are 48 separate payments, only the first three and the last three are shown.
Initial Balance = $18,000 Installment Payment = $422.73
As you can see, you still owe the company a nickel, and it’s not going to let you get away with that! They’ll ask you to make a last payment of $422.78 rather than $422.73. After all, if they let a million borrowers walk away with a nickel each, that’s $50,000.
HOW INSTALLMENT PAYMENTS ARE COMPUTED
The formula for finding the amount of an installment payment comes from a clever bit of algebra.
Suppose that k is a constant, N is a positive integer, and we want to compute the sum
Multiply both sides by k.
Subtract equation [6.6] from equation [6.5].
There’s a lot of cancellation on the right side; what remains is
(1 − k)S = (k − kN+1)
So
The basic principle used in figuring out your installment payments is that the sum of the present values of all the installment payments has to equal the initial balance. Suppose that the interest rate per period is given by i (in the example of the auto loan above, i = 0.5% = 0.005 when expressed as a decimal). Then by formula [6.4], if each installment payment is P, the present value of the nth payment, the one made after n payment periods, is P/(1 + i)n. The sum of the present values of installment payments 1 through N, which must be the current balance B, is therefore
B = P/(1 + i) + P/(1 + i)2 + … + P/(1 + i)N
= P(k + k2 + … + kN), where k = 1/(1 + i)
We’ll spare you the tiresome details of substituting equation [6.7] into this, solving for P, and simplifying. The result is that
If you haul out a calculator with an exponentiation key and let B = 18,000 and i = 0.005 in the above formula, you will see that P = 422.73.
Maybe formula [6.8] doesn’t get the PR that Einstein’s E = mc2 does, but formula [6.8] has a lot more impact because almost everyone will buy something on installment at some time in his or her life.
APPENDIX 7
SET THEORY IN “ANIMAL PASSIONS”
Set theory is a branch of mathematics that is very different from arithmetic or algebra. The fundamental questions of arithmetic and algebra deal with computation—what rules govern it, how best to perform it, and where to use it.
Although there are computational aspects to set theory, the primary use of set theory is to provide a framework in which to phrase and study a variety of important mathematical topics. It is impossible to conceive of mathematics without arithmetic and algebra, but almost all the branches of mathematics that use set theory were in existence long before the invention of set theory and were surviving quite nicely without it.
Nonetheless, once set theory had been invented, it was immediately put to use in a variety of situations.
SETS AND INCLUSION
The first order of business in constructing a new branch of mathematics is to define the objects we shall consider. A set is defined to be a collection of things. Of course, this raises the obvious question: what is a “thing”? Well, we can’t define it, but we know what “things” are—a triangle, Abraham Lincoln, and the number 7 are all examples of things.
A set S can be specified by listing its things. For example, S = {1, 2, 3, 4} is the set consisting of the whole numbers 1, 2, 3, and 4. This method of listing the elements (“element” is a more impressive-sounding synonym for “thing”) of a set works well if the set has only a few elements, but for sets with many elements, it is inefficient and is replaced by the “set-builder” method. To describe the set of all whole numbers between 1 and 1,000, one uses set-builder notation. When set-builder notation is used, the “entrance requirements” for membership in the set are described by using letters as variables.
An example of describing a set using set-builder notation is to write the set S of all whole numbers between 1 and 1,000 as S = {x : x is a whole number between 1 and 1,000}. This is read, “S is the set of all x such that x is a whole number between 1 and 1,000.” The symbol x is sometimes called a dummy variable because the particular symbol x plays no part in the actual set. We could have used any other symbol to denote the same set, such as
S = {y : y is a whole number between 1 and 1,000}
or
S = {☺ : ☺ is a whole number between 1 and 1,000}
Given a specific thing, which we shall denote by t, and a set S, we can ask whether or not t belongs to the set S. We write t ∈ S to indicate that thing t belongs to set S, and t ∉ S to indicate that thing t does not belong to S. Note that drawing a diagonal line through the symbol ∈ negates it. This is a standard mathematical convention, which we are already familiar with from arithmetic: 2 + 2 ≠ 3 means that 2 plus 2 does not equal 3. The same convention is followed with public information signs: A picture of a cigarette with a diagonal line through it means “no smoking.” It is customary to use capital letters, such as A, B, and C, to denote sets and lowercase letters, such as a, b, and c, to denote elements of sets.
Example 1: Formulate the sentence, “George Washington was a President of the United States who did not play baseball,” using sets. Use set-builder notation, and ∈ and ∉ symbols.
Solution:
Let A = {x : x was a President of the United States}
Let B = {x : x played baseball}
George Washington ∈ A, George Washington ∉ B. ■
Once a type of mathematical object has been defined (sets in our case), there are many questions that can be asked. Can we compare two such objects? How can we combine them? Of course, you are familiar with these questions for numbers, but these questions are among those typically asked when new mathematical objects are first studied.
Two sets A and B are equal if they contain the same elements. If A = {1, 2, 3, 4} and B = {3, 1, 4, 2}, these sets are equal (which we write A = B) because they contain the same things, even though they were listed in different orders. In this sense, a set is like a lunchbox because only the contents of the lunchbox matter—the order in which they were placed in the lunchbox, or the order in which they were removed from the lunchbox, is unimportant.
Subsets
We say that A is a subset of B if every element of A is an element of B. If A = {1, 2, 3} and B = {1, 2, 3, 4}, then A is a subset of B, written A ⊆ B. If A is a subset of B, we sometimes (but not often) say that B is a superset of A, written B ⊇ A. If A is not a subset of B, there must be some element of A that is not a member of B. Informally, if A ⊆ B, we say that A is contained in B, and B contains A.
Example 2: Let A = {p, q, r}. Is p a subset of A? Is {p} a subset of A? Is {r, q, p} a subset of A?
Solution: No, p is not a subset of A; it is an element belonging to A. {p} is the subset of A consisting of the single element p. Every element of {r, q, p} belongs to A, so it is a subset of A (it is, in fact, A itself, so A is a subset of A). ■
The subset relation between sets is in some ways similar to the “less than or equal to” (≤) relation between numbers. Both include equality as a possibility: We know that 8 ≤ 8, and also A ⊆ A (every element of the set A on the left side of the ⊆ symbol is certainly an element of the same set A on the right side of the ⊆ symbol). Both symbols, ≤ and
⊆, have horizontal bars under them to allow for the possibility that the object on the left of the symbol may be equal to the object on the right. This parallel among symbols can be extended to eliminate the possibility of equality by removing the horizontal bar: the symbol 3 < 5 means that 3 is less than 5, and A ⊂ B means that A is a subset of B but is not equal to B (sometimes we say that A is a proper subset of B).
Similarities between ≤ and ⊆
Reflexivity: a ≤ a and A ⊆ A
Antisymmetry: If a ≤ b and b ≤ a, then a = b
If A ⊆ B and B ⊆ A, then A = B
Transitivity: If a ≤ b and b ≤ c, then a ≤ c
If A ⊆ B and B ⊆ C, then A ⊆ C
The first property, reflexivity, is more an observation than something that is generally useful, but the last two properties are quite useful. The second property, antisymmetry, is often used to show that two sets are equal by showing that each is a subset of the other.
Universal Sets and Complements
Set theory provides a useful framework for discussing certain types of problems. However, often the items under discussion during these problems are limited: We might be discussing positive numbers, or presidents of the United States, or poker hands. In such situations, it is useful to have a universal set available, which may be loosely regarded as the “smallest” set containing all the items under discussion. If we are discussing positive numbers, for instance, we don’t want to worry about presidents or poker hands, and so it would be sensible to define U as a universal set consisting of all possible positive numbers. It is then understood that all sets mentioned will be subsets of this universal set.
Unlike most of the concepts in mathematics, the concept of universal set is a little nebulous. For instance, if A = {Iowa, Illinois}, and B = {Illinois, Indiana}, possible universal sets are states or places beginning with the letter I.
L.A. Math: Romance, Crime, and Mathematics in the City of Angels Page 19