6.5: Classification of Finance Problems
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
In this section, you will learn to:
 Reexamine the types of financial problems and classify them.
 Reexamine the vocabulary words used in describing financial calculations
Before you get started, take this prerequisite quiz.
1. Describe the differences between solving \(50=x^9\) vs solving \(50=9^x\).
 Click here to check your answer

If the exponent is known, you'll need to take that root of each side of the equation. If the exponent is unknown, you'll need to take the logarithm of each side of the equation.
If you missed this problem, review Section 5.1. (Note that this will open in a new window.)
2. Describe the differences between simple interest vs compound interest.
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Simple interest is calculated once, based on the agreed timeframe of the loan. Compound interest is calculated multiple times and added to the account balance before it is calculated again.
If you missed this problem, review Section 6.2. (Note that this will open in a new window.)
3. Describe the differences between periodically compounded interest vs continuously compounded interest.
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Periodically compounded interest is calculated at regular time intervals, such as every quarter, month, day, etc. The \(n\)value tells how many times that time interval occurs in a year. Continuously compounded interest is calculated at an infinitely small time interval, using \(e\) instead of an \(n\)value.
If you missed this problem, review Section 6.3. (Note that this will open in a new window.)
4. Describe the differences between a lumpsum vs an annuity.
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In a lumpsum, money is added to the account once. The only thing that changes the account balance after that is the interest. In an annuity, money is added (and\or removed) from the account at regular time intervals.
If you missed this problem, review Section 6.3. (Note that this will open in a new window.)
5. Describe the differences between the future value of an annuity vs the present value of an annuity. (Be careful because either formula can be used to find what should be done now (in the present) and what would be the result in the future!)
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The future value formula is used when interest and regular payments both add to the account balance. The present value formula is used when interest and regular payments work in opposite directions, one increasing the balance and one decreasing the balance.
If you missed this problem, review Section 6.3. (Note that this will open in a new window.)
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 lumpsum 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 lumpsum, only a single deposit is made, while in an annuity numerous deposits are made at equal spaced time intervals. To help interpret the vocabulary used in the problems, we include a glossary at the end of this section.
Students often confuse the present value of an annuity with the future value of an annuity. Don't get too caught up on the words, as either formula can be used to find what should be done now (in the present) and what would be the result in the future.
 The future value formula will be used in the context of investments, where the regular deposits and interest both work together to add to the account balance.
 The present value formula will be used in the context of loans or bank withdraws, where the regular payments and interest work against each other. In a loan, the monthly payments lower the account balance while the interest payments raise the account balance.
CLASSIFICATION OF PROBLEMS AND EQUATIONS
We now list eight problems that form a basis for all finance problems. Read each question and determine which formula would be used to find the missing information.
Example \(\PageIndex{1}\)
Dillon takes out a simple interest loan for $5,000 at 10.5% interest with an agreement to pay it back in 10 months. What is the total amount he will repay at the end of the 10 months?
Classification: Since the loan is a simple interest loan and we are looking for the total amount he will repay, we will use the Accumulated Value of Simple Interest Formula.
Equation: \[\mathrm{A}=5,000(1+0.105(\frac{10}{12}))\]
Example \(\PageIndex{2}\)
If $2,000 is invested at 7% compounded quarterly, what will the final amount be in 5 years?
Classification: Since the $2000 is invested once, this is a lumpsum. The interest is compounded each quarter, so this will use the Accumulated Value of Periodic Compounding Formula.
Equation: \[\mathrm{A}=\$ 2000(1+.07 / 4)^{20}\]
Example \(\PageIndex{3}\)
If $2,000 is invested at 7% compounded continuously, what will the final amount be in 5 years?
Classification: Since the $2000 is invested once, this is a lumpsum. The interest is compounded continuously, so this will use the Accumulated Value of Continuous Compounding Formula.
Equation: \[\mathrm{A}=\$ 2000e^{0.07*5}\]
Example \(\PageIndex{4}\)
How much should be invested at 8% compounded yearly, for the final amount to be $5,000 in five years?
Classification: Since this question is asking for a onetime investment, this is a lumpsum. The interest is compounded each year, so this will use the Accumulated Value of Periodic Compounding Formula. Note that we are given the accumulated value at the end of the 5 years and are looking for the principal.
Equation: \[\$ 5,000=\mathrm{P}(1+.08)^{5}\]
Example \(\PageIndex{5}\)
If $200 is invested each month at 8.5% compounded monthly, what will the final amount be in 4 years?
Classification: Since the money is invested each month, this is an example of an annuity. This is an investment where the monthly payments and the interest both add to the value, so this will use the Future Value of an Annuity Formula.
Equation: \[\mathrm{A}=\frac{\$ 200\left[(1+.085 / 12)^{12*4}1\right]}{.085 / 12}\]
Example \(\PageIndex{6}\)
How much should be invested each month at 9% for it to accumulate to $8,000 in three years?
Classification: Since the money is invested each month, this is an example of an annuity. This is an investment where the monthly payments and the interest both add to the value, so this will use the Future Value of an Annuity Formula. Note that we are given the accumulated value at the end of the 3 years and are looking for the monthly payment.
Equation: \[\$ 8,000=\frac{m\left[(1+.09 / 12)^{36}1\right]}{.09 / 12}\]
Example \(\PageIndex{7}\)
Keith wants to buy a new car worth $30,000 and needs to apply for an auto loan. The dealership is offering him financing at 9.25% compounded monthly over the next 5 years. How much will each monthly payment be?
Classification: Since the money will be paid each month, this is an example of an annuity. This is a loan where the monthly payments lower the account balance and the interest raises the account balance, so this will use the Present Value of an Annuity Formula.
Equation: \[\$ 30,000(1+.0925/12)^{12*5}=\frac{m \left[(1+.0925/12)^{12*5}1\right]}{.0925/12}\]
Example \(\PageIndex{8}\)
Maria is saving money for her retirement. The retirement fund earns 8% interest compounded monthly, and she plans to withdraw $1,500 each month for 25 years after her retirement. How much does she need in this fund before she retires?
Classification: Since the money will be withdrawn each month, this is an example of an annuity. This is a fund where the monthly withdraws lower the account balance and the interest raises the account balance, so this will use the Present Value of an Annuity Formula.
Equation: \[\mathrm{P}(1+.08/12)^{12*25}=\frac{\$1,500\left[(1+.08/12)^{12*25}1\right]}{.08/12}\]
GLOSSARY: VOCABULARY AND SYMBOLS USED IN FINANCIAL CALCULATIONS
As we’ve seen in these examples, it’s important to read the problems carefully to correctly identify the situation. It is essential to understand to vocabulary for financial problems. Many of the vocabulary words used are listed in the glossary below for easy reference.
\(t\) 
Term 
Time period for a loan or investment. In this book \(t\) is represented in years and should be converted into years when it is stated in months or other units. 
\(\mathrm{P}\) 
Principal 
Principal is the amount of money borrowed in a loan. If a sum of money is invested for a period of time, the sum invested at the start is the Principal. 
\(\mathrm{P}\) 
Present Value 
Value of money at the beginning of the time period. 
\(\mathrm{A}\) 
Accumulated Value Future Value 
Value of money at the end of the time period 
\(D\) 
Discount 
In loans involving simple interest, a discount occurs if the interest is deducted from the loan amount at the beginning of the loan period, rather than being repaid at the end of the loan period. 
\(m\) 
Periodic Payment 
The amount of a constant periodic payment that occurs at regular intervals during the time period under consideration (examples: periodic payments made to repay a loan, regular periodic payments into a bank account as savings, regular periodic payment to a retired person as an annuity,) 
\(n\) 
Number of payment periods and compounding periods per year 
In this book, when we consider periodic payments, we will always have the compounding period be the same as the payment period. 
\(nt\) 
Number of periods 
\(nt\) = (number of periods per year)\(\times\)(number of years) \(nt\) gives the total number of payment and compounding periods In some situations we will calculate \(nt\) as the multiplication shown above. In other situations the problem may state \(nt\), such as a problem describing an investment of 18 months duration compounded monthly. In this example: \(nt\) = 18 months and \(n\) = 12; then \(t\) = 1.5 years but \(t\) is not stated explicitly in the problem. The TI84+ calculators built in TVM solver uses \(N = nt\). 
\(r\) 
Annual interest rate Nominal rate 
The stated annual interest rate. This is stated as a percent but converted to decimal form when using financial calculation formulas. If a bank account pays 3% interest compounded quarterly, then 3% is the nominal rate, and it is included in the financial formulas as \(r\) = 0.03 
\(r/n\) 
Interest rate per compounding period 
If a bank account pays 3% interest compounded quarterly, then \(r/n\) = 0.03/4 = 0. 075, corresponding to a rate of 0.75% per quarter. Some Finite Math books use the symbol \(i\) to represent \(r/n\) 
\(r_{EFF}\) 
Effective Rate Effective Annual Interest Rate APY Annual Percentage Yield APR Annual Percentage Rate 
The effective rate is the interest rate compounded annually that would give the same interest rate as the compounded rate stated for the investment. The effective rate provides a uniform way for investors or borrowers to compare different interest rates with different compounding periods. 
\(I\) 
Interest 
Money paid by a borrower for the use of money borrowed as a loan. Money earned over time when depositing money into a savings account, certificate of deposit, or money market account. When a person deposits money in a bank account, the person depositing the funds is essentially temporarily lending the money to the bank and the bank pays interest to the depositor. 
Sinking Fund 
A fund set up by making payments over a period of time into a savings or investment account in order to save to fund a future purchase. Businesses use sinking funds to save for a future purchase of equipment at the end of the savings period by making periodic installment payments into a sinking fund. 

Annuity 
An annuity is a stream of periodic payments. In this book it refers to a stream of constant periodic payments made at the end of each compounding period for a specific amount of time. In common use the term annuity generally refers to a constant stream of periodic payments received by a person as retirement income, such as from a pension. Annuity payments in general may be made at the end of each payment period (ordinary annuity) or at the start of each period (annuity due). The compounding periods and payment periods do not need to be equal, but in this textbook we only consider situations when these periods are equal. 

Lump Sum 
A single sum of money paid or deposited at one time, rather than being spread out over time. An example is lottery winnings if the recipient chooses to receive a single “lump sum” onetime payment, instead of periodic payments over a period of time or as. Use of the word lump sum indicates that this is a one time transaction and is not a stream of periodic payments. 

Loan 
An amount of money that is borrowed with the understanding that the borrower needs to repay the loan to the lender in the future by the end of a period of time that is called the term of the loan. The repayment is most often accomplished through periodic payments until the loan has been completely repaid over the term of the loan. However there are also loans that can be repaid as a single sum at the end of the term of the loan, with interest paid either periodically over the term or in a lump sum at the end of the loan or as a discount at the start of the loan. 