Chapter 3: Math of Finance
Which Formula to Use?
Now that we have surveyed the basic kinds of finance calculations that are used, it may not always be obvious which one to use when you are given a problem to solve. Here are some hints on deciding which equation to use, based on the wording of the problem.
Loans
The easiest types of problems to identify are loans. Loan problems almost always include words like loan, amortize (the fancy word for loans), finance (i.e. a car), or mortgage (a home loan). Look for words like monthly or annual payment.
The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.
Loans Formula
[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]
- P0 is the balance in the account at the beginning (the principal, or amount of the loan).
- d is your loan payment (your monthly payment, annual payment, etc)
- r is the annual interest rate in decimal form.
- k is the number of compounding periods in one year.
- N is the length of the loan, in years.
Interest-Bearing Accounts
Accounts that gain interest fall into two main categories. The first is on where you put money in an account once and let it sit, the other is where you make regular payments or withdrawals from the account as in a retirement account.
Interest
- 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. Look for words like compounded, or APY. Compound interest assumes that you put money in the account once and let it sit there earning interest.
COMPOUND INTEREST
[latex]P_{N}=P_{0}\left(1+\frac{r}{k}\right)^{Nk}[/latex]
- PN is the balance in the account after N years.
- P0 is the starting balance of the account (also called initial deposit, or principal)
- r is the annual interest rate in decimal form
- k is the number of compounding periods in one year
- If the compounding is done annually (once a year), k = 1.
- If the compounding is done quarterly, k = 4.
- If the compounding is done monthly, k = 12.
- If the compounding is done daily, k = 365.
- The exception would be bonds and other investments where the interest is not reinvested; in those cases you’re looking at simple interest.
SIMPLE INTEREST OVER TIME
[latex]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex]
- I is the interest
- A is the end amount: principal plus interest
- [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
- r is the interest rate in decimal form
- t is time
The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.
Annuities
- If you’re putting money into the account on a regular basis (monthly/annually/quarterly) then you’re looking at a basic annuity problem. Basic annuities are when you are saving money. Usually in