# 0.8 Mathematics of finance  (Page 9/9)

 Page 9 / 9

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 $\left(\text{outstanding balance}\right)\left(\text{the monthly interest rate}\right)=\left(\text{500}\right)\left(\text{.}\text{12}/\text{12}\right)=5$ 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 $\text{500}-\text{81}\text{.}\text{27}=\text{418}\text{.}\text{73}$ .

Similarly, the second month, the outstanding balance is $418.73, and the monthly interest on the outstanding balance is $\left(\text{418}\text{.}\text{73}\right)\left(\text{.}\text{12}/\text{12}\right)=4\text{.}\text{19}$ . 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 $\text{418}\text{.}\text{73}-\text{82}\text{.}\text{08}=\text{336}\text{.}\text{65}$ . The process continues in the table below.

 Payment # Payment Interest Debt Payment Balance 1 $86.27$5 $81.27$418.73 2 $86.27$4.19 $82.08$336.65 3 $86.27$3.37 $82.90$253.75 4 $86.27$2.54 $83.73$170.02 5 $86.27$1.70 $84.57$85.45 6 $86.27$0.85 $85.42$0.03

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 $\text{15},\text{000}{\left(1+\text{.}\text{09}/\text{12}\right)}^{\text{60}}$ , and

the future value of Mr. Credit's annuity was $\frac{x\left[{\left(1+\text{.}\text{09}/\text{12}\right)}^{\text{60}}-1\right]}{\text{.}\text{09}/\text{12}}$ .

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 $y{\left(1+\text{.}\text{09}/\text{12}\right)}^{\text{60}}$ , and

the future value of Mr. Credit's annuity is $\frac{\text{311}\text{.}\text{38}\left[{\left(1+\text{.}\text{09}/\text{12}\right)}^{\text{60}}-1\right]}{\text{.}\text{09}/\text{12}}$ .

Setting them equal we get,

$\begin{array}{l}y{\left(1+\text{.}\text{09}/\text{12}\right)}^{\text{60}}=\frac{\text{311}\text{.}\text{38}\left[{\left(1+\text{.}\text{09}/\text{12}\right)}^{\text{60}}-1\right]}{\text{.}\text{09}/\text{12}}\\ y\left(1\text{.}\text{5657}\right)=\left(\text{311}\text{.}\text{38}\right)\left(\text{75}\text{.}\text{4241}\right)\\ y\left(1\text{.}\text{5657}\right)=\text{23},\text{485}\text{.}\text{57}\\ y=\text{15},\text{000}\text{.}\text{04}\end{array}$

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.

## Classification of problems and equations for solutions

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. Equation: $\text{FV}=\text{2000}{\left(1+\text{.}\text{07}/4\right)}^{\text{20}}$ . 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.

Equation: $\text{PV}{\left(1+\text{.}\text{08}\right)}^{5}=5,\text{000}$

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. Equation: $\text{FV}=\frac{\text{200}\left[{\left(1+\text{.}\text{085}/\text{12}\right)}^{\text{48}}-1\right]}{\text{.}\text{085}/\text{12}}$ How much should be invested each month at 9% for it to accumulate to$8,000 in three years?

Classification: Sinking Fund Payment

Equation: $\frac{m\left[{\left(1+\text{.}\text{09}/\text{12}\right)}^{\text{36}}-1\right]}{\text{.}\text{09}/\text{12}}=8,\text{000}$

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. Equation: $\text{PV}{\left(1+\text{.}\text{076}/\text{12}\right)}^{\text{120}}=\frac{\text{2000}\left[{\left(1+\text{.}\text{076}/\text{12}\right)}^{\text{120}}-1\right]}{\text{.}\text{076}/\text{12}}$ 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.

Equation: $\frac{m\left[{\left(1+\text{.}\text{082}/\text{12}\right)}^{\text{60}}-1\right]}{\text{.}\text{082}/\text{12}}=\text{25},\text{000}{\left(1+\text{.}\text{082}/\text{12}\right)}^{\text{60}}$ .

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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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