AP Business with Personal Finance · Lesson 1

Saving, interest and growth

Know Know it

Saving means setting money aside now for later. People save for short-term goals (a phone in six months), long-term goals (college or a first car) and emergencies. An emergency fund, often three to six months of essential expenses, pays for surprises without borrowing. Every choice to save has an opportunity cost: the next-best use of that money, given up. Paying yourself first, saving a set amount as soon as you are paid, often by automatic transfer, makes saving a habit instead of whatever is left over.

A good savings goal is specific and has a deadline, so you can work out the plan. To find the monthly amount, take the goal, subtract what you already have, and divide by the number of months: monthly saving = (goal - amount saved) / months.

Simple interest is paid only on the principal: interest = principal x rate x time. Compound interest is also paid on interest already earned: balance = principal x (1+r)n(1 + r)^n, where rr is the rate per period and nn is the number of periods. When interest compounds more than once a year, divide the yearly rate by the number of periods in a year and count every period. Over many years, compounding makes a large difference. The Rule of 72 gives a quick estimate: at a yearly rate of rr percent, money doubles in about 72/r72 / r years.

The annual percentage yield (APY) is the yearly return once compounding is counted; more frequent compounding makes it slightly higher than the stated rate. Inflation raises prices, so if savings earn less than the inflation rate, their buying power shrinks. A rough guide: real return ≈ interest rate - inflation rate.

Where to keep savings depends on liquidity (how fast you can reach the money), safety and return. A checking account is for spending and usually pays little or no interest. A savings account or high-yield savings account pays more and keeps money easy to reach. A certificate of deposit pays a fixed rate for a set term but charges a penalty for early withdrawal. Deposits at an FDIC-insured bank, or an NCUA-insured credit union, are protected up to 250,000 dollars per depositor, per institution, for each ownership category. Stocks, bonds and mutual funds are not deposits, so they are not covered, even when a bank sells them.

Calculator On the exam

A four-function calculator is allowed on both sections of the exam. Powers such as 1.0431.04^3 are just repeated multiplication: key in 1.04 x 1.04 x 1.04. Write each step down so a reader can follow your work, and round money to the cent only at the end.

Trap Compound interest is not rate times years

With compounding, you cannot just multiply the rate by the number of years. After the first period, each period’s interest is figured on a bigger balance, so over more than one period the total is more than simple interest at the same rate. Multiply by (1 + rate) once for each period instead.

Worked example Easy

Rosa deposits 1,800 dollars in a fictional account that pays 3 percent simple interest per year. She makes no other deposits or withdrawals. How much interest does she earn in 4 years?

  1. 54 dollars
  2. 216 dollarsAnswer
  3. 225.92 dollars
  4. 2,016 dollars

How to solve it

  1. Simple interest = principal x rate x time.
  2. 1,800 x 0.03 = 54 dollars a year.
  3. 54 x 4 = 216 dollars of interest.

Why each choice is right or wrong

  • A. Incorrect. 54 dollars is the interest for one year, 1,800 x 0.03; she earns it for 4 years.
  • B. Correct. Simple interest = 1,800 x 0.03 x 4 = 216 dollars.
  • C. Incorrect. 225.92 dollars is what compound interest would give; the account pays simple interest.
  • D. Incorrect. 2,016 dollars is her whole balance after 4 years, 1,800 + 216, not the interest alone.

Explain Explain it

Task verbs: calculate, compare

For a compound-interest question, write the formula with the numbers in it before you multiply, and say what each number is: the principal, the rate per period and the number of periods.

To compare two ways of saving, put them on the same basis, such as the balance after the same number of years, then say which is larger and by how much.

Worked example Medium

Ben deposits 2,500 dollars in a fictional account that pays 4 percent interest compounded once a year. If he makes no other deposits or withdrawals, what is his balance after 3 years?

  1. 2,600.00 dollars
  2. 2,800.00 dollars
  3. 2,812.16 dollarsAnswer
  4. 312.16 dollars

How to solve it

  1. Balance = principal x (1 + r) to the power n, with r = 0.04 and n = 3.
  2. 1.04 x 1.04 x 1.04 = 1.124864.
  3. 2,500 x 1.124864 = 2,812.16 dollars.

Why each choice is right or wrong

  • A. Incorrect. 2,600.00 dollars is the balance after only one year, 2,500 x 1.04.
  • B. Incorrect. 2,800.00 dollars uses simple interest, 2,500 + 2,500 x 0.04 x 3, and ignores interest earned on interest.
  • C. Correct. 2,500 x 1.04 x 1.04 x 1.04 = 2,812.16 dollars.
  • D. Incorrect. 312.16 dollars is the interest earned over 3 years, not the balance.

Calculate Your turn

Mei wants 2,400 dollars for a fictional study-abroad trip in 18 months. She has already saved 600 dollars. Ignoring interest, calculate how much she must save each month to reach her goal on time. Then name one habit that would make it easier for her to save that amount every month.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • Subtracts what Mei has already saved from her goal
  • Divides by 18 months to get 100 dollars a month
  • Names a habit that helps her save regularly

Model answer

Mei still needs 2,400 - 600 = 1,800 dollars. Spread over 18 months, that is 1,800 / 18 = 100 dollars a month. To make it easier, she could pay herself first: set up an automatic transfer of 100 dollars into savings on each payday, before she spends anything.

Apply Apply it

Practice this topic. AP Practice questions on exactly what this lesson covers, with an explanation for every answer.