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# Loans

In the last section, you learned about payout annuities. In this section, you will learn about conventional loans (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front. One great thing about loans is that they use exactly the same formula as a payout annuity. To see why, imagine that you had $10,000 invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests$10,000 in you. Since you’re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.

Loans Formula [latex-display]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex-display] 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
Like before, the compounding frequency is not always explicitly given, but is determined by how often you make payments.
When do you use this 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. Compound interest: One deposit Annuity: Many deposits. Payout Annuity: Many withdrawals Loans: Many payments