Interest and the Cost of Borrowing

With simple interest, interest is calculated on the original principal only. You earn (or owe) the same number of dollars every year, so the amount grows in a straight line, a linear relation with initial value $P$ and rate of change $Pr$ dollars per year. Because $r$ is a rate per year, $t$ must be in years: 6 months is $t=0.5$ and 18 months is $t=1.5$. For example, $2,500 at 4% simple interest earns $2500\times 0.04=100$ dollars each year, so 300 dollars in 3 years.

Simple Interest (Grade 9 Destreamed, MTH1W)

With compound interest, the interest earned at the end of each period is added to the balance, and from then on it earns interest too. Each period the balance is multiplied by $1+i$, where $i$ is the interest rate per compounding period. After $n$ periods the amount is $A=P{(1+i)}^n$. If interest is compounded once a year (annually), $i$ is the yearly rate and $n$ is the number of years. If it is compounded monthly, divide the yearly rate by 12 and count months: 6% per year compounded monthly for 2 years gives $i=0.06÷12=0.005$ and $n=2\times 12=24$. Compounding is the same multiplying pattern as appreciation in the last lesson. Round money to the nearest cent at the end of the calculation.

For the first year, simple interest and interest compounded annually give the same amount, since there is no earlier interest to earn interest yet. After that, compound interest pulls ahead, and the gap grows faster and faster. That is great news for savings left alone for a long time, and bad news for a debt left unpaid.

Compound Interest (Grade 9 Destreamed, MTH1W)

• A higher interest rate means more interest every year.
• A longer time means more years of interest, and with compound interest each year's interest is larger than the last.
• A bigger down payment (money paid up front) means you borrow less, and interest is charged on a smaller principal.
• More frequent compounding at the same yearly rate (monthly instead of yearly) adds slightly more interest.
• Paying a loan off sooner, or paying more than the minimum, cuts the time interest has to build up.

Here is the down payment effect in a simplified model. A car costs $12,000. A loan charges 7% simple interest per year for 4 years. With no down payment you borrow all 12,000 dollars and pay $12000\times 0.07\times 4=3360$ dollars of interest, a total cost of 15,360 dollars. With a 3,000 dollar down payment you borrow 9,000 dollars and pay $9000\times 0.07\times 4=2520$ dollars of interest, a total cost of $3000+9000+2520=14520$ dollars, which is 840 dollars less. (Most real car loans are repaid monthly and charge interest only on the balance still owed, so the dollar amounts are lower, but a bigger down payment, a lower rate and a shorter term all lower the cost in the same way.)

Every credit card purchase is a short loan. In Canada, banks must give you an interest-free grace period of at least 21 days on new purchases, but only if you pay your full balance by the due date (Financial Consumer Agency of Canada). Pay even slightly less and interest is charged, often at about 20% per year; cash advances get no grace period at all: interest is charged from the day you take the cash. For example, at 19.99% per year, one month's interest on a $400 balance is about $400\times \frac{0.1999}{12}\approx 6.66$ dollars. Leave $500 unpaid for a whole year with no payments, compounding monthly, and it grows to about $500{\left(1+\frac{0.1999}{12}\right)}^{12}\approx 609.64$ dollars: almost 110 dollars of interest for nothing new. (Card issuers usually work out interest day by day, so a real statement can differ by a few cents.)