3.12: Chapter Review and Glossary

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Chapter Review

Percents

A percentage is a number that tells us how much a part is out of 100 and can also be written as a decimal or a fraction. Percentages describe a relative portion of a whole. It is important to consider the base when using a percent number.

Simple Interest

Simple interest is calculated by multiplying the principle, the rate, and the time of an investment where the interest is not reinvested. \[I=P r t \nonumber\]

Compound Interest

Compound interest is used for an investment where any interest we earn is automatically added to the balance a fixed number of times per year. \[A=P\left(1+\dfrac{r}{n}\right)^{n t} \nonumber\]

Annuities

An annuity is an account where a regular deposit (or a regular withdrawals) of some fixed amount are made in the account at equal intervals of time. The deposit (withdrawal) intervals match the compounding intervals. Typical savings annuities are 401k or IRA accounts, and typical payout annuities are retirement accounts used as income.

Savings annuity: \(A =\dfrac{d\left[\left(1+\dfrac{r}{n}\right)^{n t}-1\right]}{\left(\dfrac{r}{n}\right)}\nonumber\) Payout annuity: \(A=\dfrac{d\left[1-\left(1+\dfrac{r}{n}\right)^{- nt}\right]}{\left(\dfrac{r}{n}\right)} \nonumber\)

Loans

Conventional loans (also called amortized loans or installment loans), include for example payday loans, auto and student loans, and home mortgages. Loans are payout annuities in reverse.

Loan amount: \(A=\dfrac{d\left[1-\left(1+\dfrac{r}{n}\right)^{-n t}\right]}{\left(\dfrac{r}{n}\right)} \nonumber \) payment: \(d=\dfrac{A\left(\dfrac{r}{n}\right)}{\left[1-\left(1+\dfrac{r}{n}\right)^{- nt}\right]} \nonumber\)

Which equation to use?

If you're letting the money sit in the account with nothing but interest changing the balance, then you're looking at a compound interest problem. The exception would be bonds and other investments where the interest is not reinvested; in those cases you’re looking at simple interest.

If you're making regular payments or withdrawals, then the problem is an annuity problem.

Glossary:

annual percentage rate (APR)	the total interest that will be paid in the year
principal	the amount invested or borrowed
effective rate	the annual simple interest rate that would be equivalent to the annual compound interest accrued for the stated rate and number of compounding periods
savings annuity	a sequence of payments of some fixed amount are made into a savings account at equal intervals of time
payout annuity	a regular withdrawal from a starting amount at equal intervals of time
amortization	the process of paying off a loan with equal payments over time