6.6 Classification of Finance Problems

This section is the capstone of Chapter 6. By now, the algebra in finance problems is usually not the hardest part. The real challenge is deciding which kind of problem you are looking at. If you choose the wrong category, you can use the wrong formula even if your calculations are perfect. That is why this section focuses on classification first and computation second.

In this section, you will review the ideas of Chapter 6 in order to:
  1. re-examine the types of financial problems and classify them.
  2. re-examine the vocabulary words used in describing financial calculations.

Think of classification as a decision tree. Before you calculate anything, ask two questions: Is this a lump-sum or an annuity? and Am I looking for a present value or a future value? Those two questions usually point you to the right model.

Why Classification Matters

The hardest part of solving a finance problem is often determining the category it belongs to. That is why, in this section, we emphasize classification of problems rather than carrying every problem all the way to a numerical answer.

Read each problem carefully and look for words that reveal the type of situation. Students often confuse a lump-sum problem with an annuity problem. In an annuity, payments are made every period, so the wording often includes clues such as each, every, monthly, quarterly, or per. In a lump-sum problem, only one deposit or one payment is made.

Classification Clues

  • Lump-sum means one single deposit or one single payment.
  • Annuity means equal payments made at equally spaced time intervals.
  • Future value asks for how much the account will be worth later.
  • Present value asks for how much something is worth now.
  • Sinking fund payment asks for the regular deposit needed to reach a future goal.
  • Installment payment asks for the regular payment needed to repay a present cost or loan.

Lump-Sum or Annuity?

A lump-sum problem involves one amount deposited or paid one time. An annuity problem involves many equal payments made at regular intervals.

If you see wording such as “invested once,” “deposited today,” or “borrowed now,” think lump-sum. If you see wording such as “each month,” “every quarter,” or “per year,” think annuity.

Try It Now 6.6.1

Source: Applied Finite Math

Classify each situation as either a lump-sum or an annuity.

  1. A student deposits \$150 at the end of every month into a savings account.
  2. An investor places \$4,000 into a certificate of deposit today and leaves it there for 3 years.
Try It Now 6.6.1 Solution
  1. This is an annuity because the deposits happen repeatedly at regular monthly intervals.
  2. This is a lump-sum because there is one single deposit made today.

Present Value or Future Value?

Students also often confuse present value with future value. The present value is the value of money at the beginning of the time period. The future value is the value at the end of the time period.

If a car costs \$15,000 today, then \$15,000 is its present value. A dealer would not agree to accept that same \$15,000 five years from now with no adjustment, because money changes value over time when interest is involved.

A quick test is this: if the question asks for a value now, think present value. If it asks what something will grow to later, think future value.

Try It Now 6.6.2

Source: Applied Finite Math

Decide whether each question is asking for a present value or a future value.

  1. How much will \$8,000 grow to in 6 years at 5% compounded annually?
  2. How much do you need to invest today so that the account will be worth \$12,000 in 8 years?
Try It Now 6.6.2 Solution
  1. This asks for a future value, because we want the amount at the end of 6 years.
  2. This asks for a present value, because we want the amount that must be invested now.

Spotting the Clues in Problem Wording

When you classify a finance problem, focus on the wording before you focus on the formula. The vocabulary often tells you exactly what type of model you need.

A Quick Decision Process

  1. Ask whether there is one payment or many equal payments.
  2. Ask whether the problem is about a value now or a value later.
  3. If there are many payments and the goal is a future amount, decide whether you are finding the future value of an annuity or the sinking fund payment.
  4. If there are many payments and the comparison is to a value now, decide whether you are finding the present value of an annuity or the installment payment.

A Car-Purchase Comparison: Why Present Value and Future Value Get Confused

Recall how we found the installment payment for a car costing \$15,000. Imagine that two people, Mr. Cash and Mr. Credit, buy two identical cars that each cost \$15,000.

To make sure both people really pay equivalent amounts, we compare the future values of their accounts at the same time. At an interest rate of 9%:

the future value of Mr. Cash's lump-sum is

\[ 15000\left(1 + \frac{0.09}{12}\right)^{60} \]

and the future value of Mr. Credit's annuity is

\[ \frac{x\left[\left(1 + \frac{0.09}{12}\right)^{60} - 1\right]}{0.09 / 12}. \]

To solve the installment-payment problem, we set those expressions equal and solve for \(x\).

Now suppose Mr. Credit is told that he can buy a particular car for \$311.38 per month for five years, and Mr. Cash wants to know how much he must pay now. In this case, we are finding the present value of the annuity of \$311.38 per month. That present value is the same as the price of the car.

Let \(P\) be the price of the car. Then:

the future value of Mr. Cash's lump-sum is

\[ P\left(1 + \frac{0.09}{12}\right)^{60} \]

and the future value of Mr. Credit's annuity is

\[ \frac{311.38\left[\left(1 + \frac{0.09}{12}\right)^{60} - 1\right]}{0.09 / 12}. \]

Setting them equal gives

\[ P\left(1 + \frac{0.09}{12}\right)^{60} = \frac{311.38\left[\left(1 + \frac{0.09}{12}\right)^{60} - 1\right]}{0.09 / 12} \] \[ P(1.5657) = (311.38)(75.4241) \] \[ P(1.5657) = 23485.57 \] \[ P = 15000.04 \]

So the present value of that annuity is about \$15,000.04, which matches the car's price.

This comparison explains why finance problems can feel tricky. A stream of payments in the future can be equivalent to one lump sum now, but only after we account for interest. In consumer finance, that idea is what connects the sticker price of an item to the payment plan attached to it.

Try It Now 6.6.3

Source: Applied Finite Math

Suppose a store offers a computer for \$82 per month for 3 years, and you want to know the cash price today. Is this problem asking for the future value of an annuity, the present value of an annuity, or an installment payment?

Try It Now 6.6.3 Solution

It asks for the present value of an annuity. The monthly payments form an annuity, and the question asks what lump sum now is equivalent to that payment stream.

The Six Core Finance Problem Types

The six problems below form a foundation for nearly all finance problems in this chapter. For each one, we identify the classification and write the equation used for the solution.

Example 6.6.1

Source: Applied Finite Math

If \$2,000 is invested at 7% compounded quarterly, what will the final amount be in 5 years?

Example 6.6.1 Solution

This is the future (accumulated) value of a lump-sum because one amount is invested now and the question asks for the amount at the end.

\[ \mathrm{FV} = A = 2000\left(1 + \frac{0.07}{4}\right)^{20} \]
Example 6.6.2

Source: Applied Finite Math

How much should be invested at 8% compounded yearly for the final amount to be \$5,000 in 5 years?

Example 6.6.2 Solution

This is the present value of a lump-sum because the future amount is known and we want to know the single amount that must be invested now.

\[ \mathrm{PV}(1 + 0.08)^5 = 5000 \]
Example 6.6.3

Source: Applied Finite Math

If \$200 is invested each month at 8.5% compounded monthly, what will the final amount be in 4 years?

Example 6.6.3 Solution

This is the future (accumulated) value of an annuity because equal monthly deposits are being made, and we want the amount at the end of 4 years.

\[ \mathrm{FV} = A = \frac{200\left[\left(1 + \frac{0.085}{12}\right)^{48} - 1\right]}{0.085 / 12} \]
Example 6.6.4

Source: Applied Finite Math

How much should be invested each month at 9% for it to accumulate to \$8,000 in 3 years?

Example 6.6.4 Solution

This is a sinking fund payment problem because we know the target future amount and want to find the equal monthly deposit needed to reach it.

\[ \frac{m\left[\left(1 + \frac{0.09}{12}\right)^{36} - 1\right]}{0.09 / 12} = 8000 \]
Example 6.6.5

Source: Applied Finite Math

Keith has won a lottery paying him \$2,000 per month for the next 10 years. He would rather have the entire sum now. If the interest rate is 7.6%, how much should he receive?

Example 6.6.5 Solution

This is the present value of an annuity because the monthly lottery payments form an annuity, and the question asks for the equivalent lump sum now.

\[ \mathrm{PV}\left(1 + \frac{0.076}{12}\right)^{120} = \frac{2000\left[\left(1 + \frac{0.076}{12}\right)^{120} - 1\right]}{0.076 / 12} \]
Example 6.6.6

Source: Applied Finite Math

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 5-year period. If the interest rate is 8.2%, determine the size of the monthly payment.

Example 6.6.6 Solution

This is an installment payment problem because the present lump-sum amount is known, and we want the equal monthly payments that are financially equivalent.

\[ \frac{m\left[\left(1 + \frac{0.082}{12}\right)^{60} - 1\right]}{0.082 / 12} = 25000\left(1 + \frac{0.082}{12}\right)^{60} \]

The six categories naturally pair up. Lump-sum problems come in future value and present value forms. Annuity problems also come in future value and present value forms. Then there are two “payment-finding” problems: sinking fund payment for saving up, and installment payment for paying off.

Finance Classification Map

  • One amount now \(\rightarrow\) one amount later: lump-sum problem.
  • Many equal payments \(\rightarrow\) one amount later: annuity or sinking fund problem.
  • Many equal payments \(\leftrightarrow\) one amount now: present value of an annuity or installment payment problem.
  • If the future amount is unknown, you are often finding a future value.
  • If the payment is unknown, you are often finding a sinking fund payment or an installment payment.
Try It Now 6.6.4

Source: Applied Finite Math

Classify each problem.

  1. You want to know how much \$6,500 invested today will be worth in 12 years.
  2. You want to know how much to deposit every month so that the account reaches \$40,000 in 10 years.
  3. You want to know the cash value now of receiving \$900 every month for 6 years.
Try It Now 6.6.4 Solution
  1. Future value of a lump-sum: one deposit now, amount later.
  2. Sinking fund payment: regular deposits are unknown, target future amount is known.
  3. Present value of an annuity: a stream of monthly payments is being converted into one lump sum now.

Section 6.6 Problem Set: Classification of Finance Problems

Problem Set 6.6

Source: Applied Finite Math

Let the letters \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) represent the following:

\[ A = \text{FV of a lump-sum} \] \[ B = \text{PV of a lump-sum} \] \[ C = \text{FV of an annuity} \] \[ D = \text{Sinking fund payment} \] \[ E = \text{Installment payment} \] \[ F = \text{PV of an annuity} \]

Classify each by writing the appropriate letter in the box, and write an equation for the solution.

Problem 1. What monthly deposits made to an account paying 9% will grow to \$10,000 in 4 years?

Problem 2. An amount of \$4000 is invested at 6% compounded daily. What will the final amount be in 5 years?

Problem 3. David has won a lottery paying him \$10,000 per month for the next 20 years. He would rather have the whole amount in one lump sum now. If the current interest rate is 7%, how much money can he hope to get?

Problem 4. Each month Linda deposits \$250 in an account that pays 9%. How much money will she have in 4 years?

Problem 5. Find the monthly payment for a \$15,000 car if the loan is amortized over 4 years at a rate of 10%.

Problem 6. What lump-sum deposited in an account paying 7% compounded daily will grow to \$10,000 in 5 years?

Problem 7. What amount of quarterly payments will amount to \$250,000 in 5 years at a rate of 8%?

Problem 8. The Chang family bought their house 25 years ago. They had their loan financed for 30 years at an interest rate of 11% resulting in a payment of \$1350 a month. Find the balance of the loan.

Problem 9. A 10-year \$1000 bond pays \$35 every six months. If the current interest rate is 8%, find the present value of \$1000.

Problem 10. A 10-year \$1000 bond pays \$35 every six months. If the current interest rate is 8%, find the present value of the \$35 per six-month payments.

Problem Set Continuation

Source: Applied Finite Math

The source repeats the classification key before Questions 11-20; it is preserved here for continuity.

\[ A = \text{FV of a lump-sum} \] \[ B = \text{PV of a lump-sum} \] \[ C = \text{FV of an annuity} \] \[ D = \text{Sinking fund payment} \] \[ E = \text{Installment payment} \] \[ F = \text{PV of an annuity} \]

Problem 11. What lump-sum deposit made today is equal to 33 monthly deposits of \$500 if the interest rate is 8%?

Problem 12. What monthly deposits made to an account paying 10% will accumulate to \$10,000 in 6 years?

Problem 13. A department store charges a finance charge of 1.5% per month on the outstanding balance. If Ned charged \$400 three months ago and has not paid his bill, how much does he owe?

Problem 14. What will the value of \$300 monthly deposits be in 10 years if the account pays 12% compounded monthly?

Problem 15. What lump-sum deposited at 6% compounded daily will grow to \$2000 in 3 years?

Problem 16. A company buys an apartment complex for \$5,000,000 and amortizes the loan over 10 years. What is the yearly payment if the interest rate is 14%?

Problem 17. In 2002, a house in Rock City cost \$300,000. Real estate in Rock City has been increasing in value at the annual rate of 5.3%. Find the price of that house in 2016.

Problem 18. You determine that you can afford to pay \$400 per month for a car. What is the maximum price you can pay for a car if the interest rate is 11% and you want to repay the loan in 4 years?

Problem 19. A business needs \$350,000 in 5 years. How much lump-sum should be put aside in an account that pays 9% so that 5 years from now the company will have \$350,000?

Problem 20. A person wishes to have \$500,000 in a pension fund 20 years from now. How much should he deposit each month in an account paying 9% compounded monthly?

Glossary: Vocabulary and Symbols Used in Financial Calculations

Source: Applied Finite Math

As we have seen throughout this chapter, reading finance problems carefully is essential. The glossary below collects the vocabulary and symbols that appear often in financial calculations.

Symbol Term Meaning
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.
P Principal 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.
P Present Value The value of money at the beginning of the time period.
A Accumulated Value / Future Value The 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 include payments made to repay a loan, periodic deposits into savings, or periodic payments received as an annuity.
n Number of payment periods and compounding periods per year In this book, when we consider periodic payments, the compounding period is the same as the payment period. In general, these periods do not have to be the same, but the calculations are more complicated when they differ. In that case, formulas can be found in finance textbooks or online resources, and technology such as spreadsheets or financial calculators can be used.
nt Number of periods nt = (number of periods per year) \(\times\) (number of years). This gives the total number of payment and compounding periods. Sometimes we compute nt directly from that multiplication. In other situations, the problem may state nt implicitly. For example, an 18-month investment compounded monthly has nt = 18 and n = 12, so t = 1.5 years. On a TI-84+ calculator, the built-in TVM solver uses N = nt.
r Annual interest rate / Nominal rate The stated annual interest rate. It is usually given as a percent but converted to decimal form in formulas. For example, if a bank account pays 3% interest compounded quarterly, then 3% is the nominal rate and we use 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 = 0.0075, which corresponds to a rate of 0.75% per quarter. Some finite math books use the symbol i for r/n.
rEFF Effective Rate / Effective Annual Interest Rate / APY / APR The effective rate is the interest rate compounded annually that would produce the same result as the stated compounded rate. It gives investors and borrowers a uniform way to compare interest rates with different compounding periods.
I Interest Money paid by a borrower for the use of borrowed money, or money earned over time on deposits in savings accounts, certificates of deposit, or money market accounts. When a person deposits money in a bank, the depositor is essentially lending the money to the bank, and the bank pays interest in return.
Sinking Fund A fund created by making periodic payments into a savings or investment account in order to finance a future purchase. Businesses often use sinking funds to save for future equipment purchases.
Annuity An annuity is a stream of periodic payments. In this book, it refers to a stream of constant payments made at the end of each compounding period for a specific amount of time. In general use, the term often refers to retirement income. Payments may be made at the end of each period (ordinary annuity) or at the beginning of each period (annuity due). In this textbook, we consider only cases where payment periods and compounding periods are equal.
Lump Sum A single sum of money paid or deposited at one time rather than spread out over time. For example, lottery winnings may be taken as one lump-sum payment instead of a stream of payments. The phrase lump sum signals a one-time transaction rather than an annuity.
Loan An amount of money borrowed with the understanding that it must be repaid by the end of a period called the term of the loan. Repayment usually occurs through periodic payments, although some loans are repaid in a single sum at the end, with interest paid periodically, at the end, or as a discount at the beginning.