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Inside Collection: Applied Finite Mathematics
Summary: This chapter covers principles of finance. After completing this chapter students should be able to: solve financial problems that involve simple interest; solve problems involving compound interest; find the future value of an annuity; find the amount of payments to a sinking fund; find the present value of an annuity; and find an installment payment on a loan.
In this chapter, you will learn to:
In this section, you will learn to:
It costs to borrow money. The rent one pays for the use of money is called the interest. The amount of money that is being borrowed or loaned is called the principal or present value. Simple interest is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period he keeps the money. Although the interest rate is often specified for a year, it may be specified for a week, a month, or a quarter, etc. The credit card companies often list their charges as monthly rates, sometimes it is as high as 1.5% a month.
If an amount
The total amount
or
Where interest rate
Ursula borrows $600 for 5 months at a simple interest rate of 15% per year. Find the interest, and the total amount she is obligated to pay?
Jose deposited $2500 in an account that pays 6% simple interest. How much money will he have at the end of 3 years?
Darnel owes a total of $3060 which includes 12% interest for the three years he borrowed the money. How much did he originally borrow?
A Visa credit card company charges a 1.5% finance charge each month on the unpaid balance. If Martha owes $2350 and has not paid her bill for three months, how much does she owe?
Banks often deduct the simple interest from the loan amount at the time that the loan is made. When this happens, we say the loan has been discounted. The interest that is deducted is called the discount, and the actual amount that is given to the borrower is called the proceeds. The amount the borrower is obligated to repay is called the maturity value.
If an amount
The proceeds
or
Where interest rate
Francisco borrows $1200 for 10 months at a simple interest rate of 15% per year. Determine the discount and the proceeds.
If Francisco wants to receive $1200 for 10 months at a simple interest rate of 15% per year, what amount of loan should he apply for?
In this section you will learn to:
In the Section 2, we did problems involving simple interest. Simple interest is charged when the lending period is short and often less than a year. When the money is loaned or borrowed for a longer time period, the interest is paid (or charged) not only on the principal, but also on the past interest, and we say the interest is compounded.
Suppose we deposit $200 in an account that pays 8% interest. At the end of one year, we will have
Now suppose we put this amount, $216, in the same account. After another year, we will have
So an initial deposit of $200 has accumulated to $233.28 in two years. Further note that had it been simple interest, this amount would have accumulated to only $232. The reason the amount is slightly higher is because the interest ($16) we earned the first year, was put back into the account. And this $16 amount itself earned for one year an interest of
Now suppose we leave this amount, $233.28, in the bank for another year, the final amount will be
Now let us look at the mathematical part of this problem so that we can devise an easier way to solve these problems.
After one year, we had
After two years, we had
But
After three years, we get
Which can be written as
Suppose we are asked to find the total amount at the end of 5 years, we will get
We summarize as follows:
Banks often compound interest more than one time a year. Consider a bank that pays 8% interest but compounds it four times a year, or quarterly. This means that every quarter the bank will pay an interest equal to one-fourth of 8%, or 2%.
Now if we deposit $200 in the bank, after one quarter we will have
After two quarters, we will have
After one year, we will have
After three years, we will have
Therefore, if we invest a lump-sum amount of
If $3500 is invested at 9% compounded monthly, what will the future value be in four years?
How much should be invested in an account paying 9% compounded daily for it to accumulate to $5,000 in five years?
If a bank pays 7.2% interest compounded monthly, what is the effective interest rate?
Interest can be compounded yearly, semiannually, quarterly, monthly, daily, hourly, minutely, and even every second. But what do we mean when we say the interest is compounded continuously, and how do we compute such amounts. When interest is compounded "infinitely many times", we say that the interest is compounded continuously. Our next objective is to derive a formula to solve such problems, and at the same time put things in proper perspective.
Suppose we put $1 in an account that pays 100% interest. If the interest is compounded once a year, the total amount after one year will be
If the interest is compounded semiannually, in one year we will have
If the interest is compounded quarterly, in one year we will have
We show the results as follows:
| Frequency of compounding | Formula | Total amount |
| Annually |
|
$2 |
| Semiannually |
|
$2.25 |
| Quarterly |
|
$2.44140625 |
| Monthly |
|
$2.61303529 |
| Daily |
|
$2.71456748 |
| Hourly |
|
$2.71812699 |
| Every second |
|
$2.71827922 |
| Continuously |
|
$2.718281828... |
We have noticed that the $1 we invested does not grow without bound. It starts to stabilize to an irrational number 2.718281828... given the name "
In mathematics, we say that as
Therefore, it is natural that the number
Therefore, it follows that if we invest
If $3500 is invested at 9% compounded continuously, what will the future value be in four years?
Next we learn a common-sense rule to be able to readily estimate answers to some finance as well as real-life problems. We consider the following problem.
If an amount is invested at 7% compounded continuously, what is the effective interest rate?
If an amount is invested at 7%, estimate how long will it take to double.
By doing a few similar calculations we can construct a table like the one below.
| Annual interest rate | 1% | 2% | 3% | 4% | 5% | 6% | 7% | 8% | 9% | 10% |
| Number of years to double money | 70 | 35 | 23 | 18 | 14 | 12 | 10 | 9 | 8 | 7 |
The pattern in the table introduces us to the law of 70.
It is a good idea to familiarize yourself with the law of 70, as it can help you to estimate many problems mentally.
If the world population doubles every 35 years, what is the growth rate?
We summarize the concepts learned in this chapter in the following table:
If an amount
If a bank pays an interest rate
If an amount
The law of 70 states that
The number of years to double money = 70 ÷ interest rate
In this section, you will learn to:
In Section 2 and Section 4, we did problems where an amount of money was deposited lump sum in an account and was left there for the entire time period. Now we will do problems where timely payments are made in an account. When a sequence of payments of some fixed amount are made in an account at equal intervals of time, we call that an annuity. And this is the subject of this section.
To develop a formula to find the value of an annuity, we will need to recall the formula for the sum of a geometric series.
A geometric series is of the form:
The following are some examples of geometric series.
In a geometric series, each subsequent term is obtained by multiplying the preceding term by a number, called the common ratio. And a geometric series is completely determined by knowing its first term, the common ratio, and the number of terms.
In the example,
In your algebra class, you developed a formula for finding the sum of a geometric series. The formula states that the sum of a geometric series is
We will use this formula to find the value of an annuity.
Consider the following example.
If at the end of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years?
When the payments are made at the end of each period rather than at the beginning, we call it an ordinary annuity.
If a payment of
Tanya deposits $300 at the end of each quarter in her savings account. If the account earns 5.75%, how much money will she have in 4 years?
Robert needs $5000 in three years. How much should he deposit each month in an account that pays 8% in order to achieve his goal?
A business needs $450,000 in five years. How much should be deposited each quarter in a sinking fund that earns 9% to have this amount in five years?
If the payment is made at the beginning of each period, rather than at the end, we call it an annuity due. The formula for the annuity due can be derived in a similar manner. Reconsider Example 15, with the change that the deposits be made at the beginning of each month.
If at the beginning of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years?
So, in the case of an annuity due, to find the future value, we increase the number of periods
Most of the problems we are going to do in this chapter involve ordinary annuity, therefore, we will down play the significance of the last formula. We mentioned the last formula only for completeness.
Finally, it is the author's wish that the student learn the concepts in a way that he or she will not have to memorize every formula. It is for this reason formulas are kept at a minimum. But before we conclude this section we will once again mention one single equation that will help us find the future value, as well as the sinking fund payment.
If a payment of
In this section, you will learn to:
In Section 4, we learned to find the future value of a lump sum, and in Section 6, we learned to find the future value of an annuity. With these two concepts in hand, we will now learn to amortize a loan, and to find the present value of an annuity.
Let us consider the following problem.
Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept?
We now consider another problem that involves the same logic.
Find the monthly payment for a car costing $15,000 if the loan is amortized over five years at an interest rate of 9%.
If a payment of
where the amount
We have already developed the tools to solve most finance problems. Now we use these tools to solve some application problems.
One of the most common problems deals with finding the balance owed at a given time during the life of a loan. Suppose a person buys a house and amortizes the loan over 30 years, but decides to sell the house a few years later. At the time of the sale, he is obligated to pay off his lender, therefore, he needs to know the balance he owes. Since most long term loans are paid off prematurely, we are often confronted with this problem. Let us consider an example.
Mr. Jackson bought his house in 1975, and financed the loan for 30 years at an interest rate of 9.8%. His monthly payment was $1260. In 1995, Mr. Jackson decided to pay off the loan. Find the balance of the loan he still owes.
The Orange computer company needs to raise money to expand. It issues a 10-year $1,000 bond that pays $30 every six months. If the current market interest rate is 7%, what is the fair market value of the bond?
An amount of $500 is borrowed for 6 months at a rate of 12%. Make an amortization schedule showing the monthly payment, the monthly interest on the outstanding balance, the portion of the payment contributing toward reducing the debt, and the outstanding balance.
Most of the other applications in this section's problem set are reasonably straight forward, and can be solved by taking a little extra care in interpreting them. And remember, there is often more than one way to solve a problem.
We'd like to remind the reader that the hardest part of solving a finance problem is determining the category it falls into. So in this section, we will emphasize the classification of problems rather than finding the actual solution.
We suggest that the student read each problem carefully and look for the word or words that may give clues to the kind of problem that is presented. For instance, students often fail to distinguish a lump-sum problem from an annuity. Since the payments are made each period, an annuity problem contains words such as each, every, per etc.. One should also be aware that in the case of a lump-sum, only a single deposit is made, while in an annuity numerous deposits are made at equal spaced time intervals.
Students often confuse the present value with the future value. For example, if a car costs $15,000, then this is its present value. Surely, you cannot convince the dealer to accept $15,000 in some future time, say, in five years. Recall how we found the installment payment for that car. We assumed that two people, Mr. Cash and Mr. Credit, were buying two identical cars both costing $15, 000 each. To settle the argument that both people should pay exactly the same amount, we put Mr. Cash's cash of $15,000 in the bank as a lump-sum and Mr. Credit's monthly payments of
the future value of Mr. Cash's lump-sum was
the future value of Mr. Credit's annuity was
To solve the problem, we set the two expressions equal and solve for
The present value of an annuity is found in exactly the same way. For example, suppose Mr. Credit is told that he can buy a particular car for $311.38 a month for five years, and Mr. Cash wants to know how much he needs to pay. We are finding the present value of the annuity of $311.38 per month, which is the same as finding the price of the car. This time our unknown quantity is the price of the car. Now suppose the price of the car is
the future value of Mr. Cash's lump-sum is
the future value of Mr. Credit's annuity is
Setting them equal we get,
We now list six problems that form a basis for all finance problems. Further, we classify these problems and give an equation for the solution.
If $2,000 is invested at 7% compounded quarterly, what will the final amount be in 5 years?
How much should be invested at 8% compounded yearly, for the final amount to be $5,000 in five years?
If $200 is invested each month at 8.5% compounded monthly, what will the final amount be in 4 years?
How much should be invested each month at 9% for it to accumulate to $8,000 in three years?
Keith has won a lottery paying him $2,000 per month for the next 10 years. He'd rather have the entire sum now. If the interest rate is 7.6%, how much should he receive?
Mr. A has just donated $25,000 to his alma mater. Mr. B would like to donate an equivalent amount, but would like to pay by monthly payments over a five year period. If the interest rate is 8.2%, determine the size of the monthly payment?
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Last edited by Collins Williamson on Jul 14, 2011 5:37 pm -0500.
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