In this chapter, you will learn to:
- Solve financial problems that involve simple interest.
- Solve problems involving compound interest.
- Find the future value of an annuity, and the amount of payments to a sinking fund.
- Find the present value of an annuity, and an installment payment on a loan.
Simple Interest and Discount
In this section, you will learn to:
- Find simple interest.
- Find present value.
- Find discounts and proceeds.
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 P is borrowed for a time t at an interest rate of r per time period, then the simple interest is given by
I = P ⋅ r ⋅ t
The total amount A also called the accumulated value or the future value is given by
A = P + I = P + Pr t
Where interest rate r is expressed in decimals.
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?
The interest is computed by multiplying the principal with the interest rate and the time.
The total amount is
Incidentally, the total amount can be computed directly as
Jose deposited $2500 in an account that pays 6% simple interest. How much money will he have at the end of 3 years?
The total amount or the future value is given by .
Darnel owes a total of $3060 which includes 12% interest for the three years he borrowed the money. How much did he originally borrow?
This time we are asked to compute the principal P.
So Darnel originally borrowed $2250.
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?
Before we attempt the problem, the reader should note that in this problem the rate of finance charge is given per month and not per year.
The total amount Martha owes is the previous unpaid balance plus the finance charge.
Once again, we can compute the amount directly by using the formula
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 M is borrowed for a time t at a discount rate of r per year, then the discount D is
D = M ⋅ r ⋅ t
The proceeds P, the actual amount the borrower gets, is given by
P = M − D
Where interest rate r is expressed in decimals.
Francisco borrows $1200 for 10 months at a simple interest rate of 15% per year. Determine the discount and the proceeds.
The discount D is the interest on the loan that the bank deducts from the loan amount.
Therefore, the bank deducts $150 from the maturity value of $1200, and gives Francisco $1050. Francisco is obligated to repay the bank $1200.
In this case, the discount D=$150, and the proceeds P=$1200−$150=$1050.
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 problem, we are given the proceeds P and are being asked to find the maturity value M.
We have P=$1200, r=.15, t=10/12 . We need to find M.
Therefore, Francisco should ask for a loan for $1371.43.
The bank will discount $171.43 and Francisco will receive $1200.
In this section you will learn to:
- Find the future value of a lump-sum.
- Find the present value of a lump-sum.
- Find the effective interest rate.
In the the section called “Simple Interest and Discount”, 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 $200+$200(.08)=$200(1+.08)=$216.
Now suppose we put this amount, $216, in the same account. After another year, we will have $216+$216(.08)=$216(1+.08)=$233.28.
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 $16(.08)=$1.28, thus resulting in the increase. So we have earned interest on the principal as well as on the past interest, and that is why we call it compound interest.
Now suppose we leave this amount, $233.28, in the bank for another year, the final amount will be $233.28+$233.28(.08)=$233.28(1+.08)=$251.94.
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 $216=$200(1+.08), therefore, the above expression becomes
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 or $204.
After two quarters, we will have or $208.08.
After one year, we will have or $216.49.
After three years, we will have or $253.65, etc.
Therefore, if we invest a lump-sum amount of P dollars at an interest rate r, compounded n times a year, then after t years the final amount is given by
If $3500 is invested at 9% compounded monthly, what will the future value be in four years?
Clearly an interest of .09/12 is paid every month for four years. This means that the interest is compounded 48 times over the four-year period. We get
How much should be invested in an account paying 9% compounded daily for it to accumulate to $5,000 in five years?
This time we know the future value, but we need to find the principal. Applying the formula , we get
For comparison purposes, the government requires the bank to state their interest rate in terms of effective yield or effective interest rate.
For example, if one bank advertises its rate as 7.2% compounded monthly, and another bank advertises its rate as 7.5%, how are we to find out which is better? Let us look at the next example.
If a bank pays 7.2% interest compounded monthly, what is the effective interest rate?
Suppose we deposit $100 in this bank and leave it for a year, we will get
Which means we earned an interest of $107.44−$100=$7.44
But this interest is for $100, therefore, the effective interest rate is 7.44%.
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 $1(1+1)=$2.
If the interest is compounded semiannually, in one year we will have $1(1+1/2)2=$2.25
If the interest is compounded quarterly, in one year we will have $1(1+1/4)4=$2.44, etc.
We show the results as follows:
|Frequency of compounding||Formula||Total amount|
|Annually||$ 1 ( 1 + 1 )||$2|
|Semiannually||$ 1 ( 1 + 1 / 2 ) 2||$2.25|
|Quarterly||$ 1 ( 1 + 1 / 4 ) 4||$2.44140625|
|Monthly||$ 1 ( 1 + 1 / 12 ) 12||$2.61303529|
|Daily||$ 1 ( 1 + 1 / 365 ) 365||$2.71456748|
|Hourly||$ 1 ( 1 + 1 / 8760 ) 8760||$2.71812699|
|Every second||$ 1 ( 1 + 1 / 525600 ) 525600||$2.71827922|
|Continuously||$ 1 ( 2 . 718281828 . . . )||$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 "e" after the great mathematician Euler.
In mathematics, we say that as n becomes infinitely large the expression equals e.
Therefore, it is natural that the number e play a part in continuous compounding. It can be shown that as n becomes infinitely large the expression .
Therefore, it follows that if we invest $P at an interest rate r per year, compounded continuously, after t years the final amount will be given by .
If $3500 is invested at 9% compounded continuously, what will the future value be in four years?
Using the formula for the continuous compounding, we get .
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?
Once again, if we put $1 in the bank at that rate for one year, and subtract that $1 from the final amount, we will get the interest rate in decimals.
If an amount is invested at 7%, estimate how long will it take to double.
Since we are estimating the answer, we really do not care how often the interest is compounded. Let us say the interest is compounded continuously. Then our problem becomes
We divide both sides by P
Now by substituting values by trial and error, we can estimate t to be about 10.
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.
- Definition: The Law of 70:
The number of years required to double money = 70 ÷ interest rate
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?
According to the law of 70,
Therefore, the world population grows at a rate of 2%.
We summarize the concepts learned in this chapter in the following table:
If an amount P is invested for t years at an interest rate r per year, compounded n times a year, then the future value is given by(9.30)
If a bank pays an interest rate r per year, compounded n times a year, then the effective interest rate is given by(9.31)
If an amount P is invested for t years at an interest rate r per year, compounded continuously, then the future value is given by(9.32)
The law of 70 states that
The number of years to double money = 70 ÷ interest rate
Annuities and Sinking Funds
In this section, you will learn to:
Find the future value of an annuity.
Find the amount of payments to a sinking fund.
In the section called “Simple Interest and Discount” and the section called “Compound Interest ”, 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, the first term of the series is a, the common ratio is r, and the number of terms are n.
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?
There are 60 deposits made in this account. The first payment stays in the account for 59 months, the second payment for 58 months, the third for 57 months, and so on.
The first payment of $500 will accumulate to an amount of $500(1+.08/12)59.
The second payment of $500 will accumulate to an amount of $500(1+.08/12)58.
The third payment will accumulate to $500(1+.08/12)57.
And so on.
The last payment is taken out the same time it is made, and will not earn any interest.
To find the total amount in five years, we need to add the accumulated value of these sixty payments.
In other words, we need to find the sum of the following series.
Written backwards, we have
This is a geometric series with a=500, r=(1+.08/12), and n=59. Therefore the sum is
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 m dollars is made in an account n times a year at an interest r, then the final amount A after t years is
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?
The future value of this annuity can be found using the above formula.
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?
If Robert saves x dollars per month, after three years he will have
But we'd like this amount to be $5,000. Therefore,
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?
Again, suppose that x dollars are deposited each quarter in the sinking fund. After five years, the future value of the fund should be $450,000. Which suggests the following relationship:
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 9.18, 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?
There are 60 deposits made in this account. The first payment stays in the account for 60 months, the second payment for 59 months, the third for 58 months, and so on.
The first payment of $500 will accumulate to an amount of $500(1+.08/12)60.
The second payment of $500 will accumulate to an amount of $500(1+.08/12)59.
The third payment will accumulate to $500(1+.08/12)58.
And so on.
The last payment is in the account for a month and accumulates to $500(1+.08/12)
To find the total amount in five years, we need to find the sum of the following series.
Written backwards, we have
If we add $500 to this series, and later subtract that $500, the value will not change. We get
Not considering the last term, we have a geometric series with a=$500, r=(1+.08/12), and n=60. Therefore the sum is
So, in the case of an annuity due, to find the future value, we increase the number of periods n by 1, and subtract one payment.
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 m dollars is made in an account n times a year at an interest r, then the future value A after t years is
Present Value of an Annuity and Installment Payment
In this section, you will learn to:
Find the present value of an annuity.
Find the amount of installment payment on a loan.
In the section called “Compound Interest ”, we learned to find the future value of a lump sum, and in the section called “Annuities and Sinking Funds”, 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?
This classic present value problem needs our complete attention because the rationalization we use to solve this problem will be used again in the problems to follow.
Consider for argument purposes that two people Mr. Cash, and Mr. Credit have won the same lottery of $1,000 per month for the next 20 years. Now, Mr. Credit is happy with his $1,000 monthly payment, but Mr. Cash wants to have the entire amount now. Our job is to determine how much Mr. Cash should get. We reason as follows: If Mr. Cash accepts x dollars, then the x dollars deposited at 8% for 20 years should yield the same amount as the $1,000 monthly payments for 20 years. In other words, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like the future values to equal.
Since Mr. Cash is receiving a lump sum of x dollars, its future value is given by the lump sum formula we studied in the section called “Compound Interest ”, and it is
Since Mr. Credit is receiving a sequence of payments, or an annuity, of $1,000 per month, its future value is given by the annuity formula we learned in the section called “Annuities and Sinking Funds”. This value is
The only way Mr. Cash will agree to the amount he receives is if these two future values are equal. So we set them equal and solve for the unknown.
The reader should also note that if Mr. Cash takes his lump sum of $119,554.36 and invests it at 8% compounded monthly, he will have $589,020.41 in 20 years.
We have just found the present value of an annuity of $1,000 each month for 20 years at 8%.
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%.
Again, consider the following scenario:
Two people, Mr. Cash and Mr. Credit, go to buy the same car that costs $15,000. Mr. Cash pays cash and drives away, but Mr. Credit wants to make monthly payments for five years. Our job is to determine the amount of the monthly payment. We reason as follows: If Mr. Credit pays x dollars per month, then the x dollar payment deposited each month at 9% for 5 years should yield the same amount as the $15,000 lump sum deposited for 5 years. Again, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like them to be the same.
Since Mr. Cash is paying a lump sum of $15,000, its future value is given by the lump sum formula, and it is
Mr. Credit wishes to make a sequence of payments, or an annuity, of x dollars per month, and its future value is given by the annuity formula, and this value is
We set the two future amounts equal and solve for the unknown.
Therefore, the monthly payment on the loan is $311.38 for five years.
If a payment of m dollars is made in an account n times a year at an interest r, then the present value P of the annuity after t years is
where the amount P is also the loan amount, and m the periodic payment.
Miscellaneous Application Problems
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 reader should note that the original amount of the loan is not mentioned in the problem. That is because we don't need to know that to find the balance.
As for the bank or lender is concerned, Mr. Jackson is obligated to pay $1260 each month for 10 more years. But since Mr. Jackson wants to pay it all off now, we need to find the present value of the $1260 payments. Using the formula we get,
The next application we discuss deals with bond problems. Whenever a business, and for that matter the U. S. government, needs to raise money it does it by selling bonds. A bond is a certificate of promise that states the terms of the agreement. Usually the businesses sells bonds for the face amount of $1,000 each for a period of 10 years. The person who buys the bond, the bondholder, pays $1,000 to buy the bond. The bondholder is promised two things: First that he will get his $1,000 back in ten years, and second that he will receive a fixed amount of interest every six months. As the market interest rates change, the price of the bond starts to fluctuate. The bonds are bought and sold in the market at their fair market value. The interest rate a bond pays is fixed, but if the market interest rate goes up, the value of the bond drops since the money invested in the bond can earn more elsewhere. When the value of the bond drops, we say it is trading at a discount. On the other hand, if the market interest rate drops, the value of the bond goes up, and it is trading at a premium.
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?
A bond certificate promises us two things – An amount of $1,000 to be paid in 10 years, and a semi-annual payment of $30 for ten years. Therefore, to find the fair market value of the bond, we need to find the present value of the lump sum of $1,000 we are to receive in 10 years, as well as, the present value of the $30 semi-annual payments for the 10 years.
The present value of the lump-sum $1,000 is
Note that since the interest is paid twice a year, the interest is compounded twice a year.
The present value of the $30 semi-annual payments is
The present value of the lump-sum $1,000=$502.57
The present value of the $30 semi-annual payments=$426.37
Therefore, the fair market value of the bond is $502.57+$426.37=$928.94
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.
The reader can verify that the monthly payment is $86.27.
The first month, the outstanding balance is $500, and therefore, the monthly interest on the outstanding balance is
This means, the first month, out of the $86.27 payment, $5 goes toward the interest and the remaining $81.27 toward the balance leaving a new balance of $500−$81.27=$418.73.
Similarly, the second month, the outstanding balance is $418.73, and the monthly interest on the outstanding balance is ($418.73)(.12/12)=$4.19. Again, out of the $86.27 payment, $4.19 goes toward the interest and the remaining $82.08 toward the balance leaving a new balance of $418.73−$82.08=$336.65. The process continues in the table below.
|Payment #||Payment||Interest||Debt Payment||Balance|
Note that the last balance of 3 cents is due to error in rounding off.
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.
Classification of Finance Problems
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 x dollars each as an annuity. Then we make sure that the future values of these two accounts are equal. As you remember, at an interest rate of 9%
the future value of Mr. Cash's lump-sum was $15,000(1+.09/12)60, and
the future value of Mr. Credit's annuity was .
To solve the problem, we set the two expressions equal and solve for x.
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 y, then
the future value of Mr. Cash's lump-sum is y(1+.09/12)60, and
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?
Classification: Future Value of a Lump-sum or FV of a lump-sum.
How much should be invested at 8% compounded yearly, for the final amount to be $5,000 in five years?
Classification: Present Value of a Lump-sum or PV of a lump-sum.
If $200 is invested each month at 8.5% compounded monthly, what will the final amount be in 4 years?
Classification: Future Value of an Annuity or FV of an annuity.
How much should be invested each month at 9% for it to accumulate to $8,000 in three years?
Classification: Sinking Fund Payment
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?
Classification: Present Value of an Annuity or PV of an annuity.
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?
Classification: Installment Payment.
- Definition: The Law of 70:
The number of years required to double money = 70 ÷ interest rate