Simple Interest, Compound Interest and Depreciation

In short: Simple interest grows by the same amount each period, $I=Prn$, while compound interest grows by the same percentage, $A=P(1+r{)}^n$. Depreciation is the same idea in reverse: an item loses value by a fixed percentage each year, $S=V_0(1-r{)}^n$.

Starting point: Years 9 and 10

Simple interest

Simple interest is paid only on the original amount, the principal $P$.

$$I=Prn$$

$r$ is the rate per period as a decimal and $n$ is the number of periods.

Compound interest

With compound interest, each period's interest is added to the balance, and the next period's interest is calculated on the new balance.

$$A=P(1+r{)}^n$$

$A$ is the final amount. The interest earned is $A-P$.

Compounding periods

If interest is compounded more than once a year, divide the yearly rate and multiply the years.

• Monthly: $r=\frac{\text{annual rate}}{12}$, $n=12\times \text{years}$.
• Quarterly: divide by 4, multiply by 4.

8% per year compounded monthly for 5 years: $r=\frac{0.08}{12}$ and $n=60$.

Depreciation

Cars, phones and machines lose value. Declining balance depreciation takes a fixed percentage off each year:

$$S=V_0(1-r{)}^n$$

$V_0$ is the starting value and $S$ is the salvage value after $n$ years.

Straight-line depreciation takes the same amount off each year, a percentage of the purchase price: $$S=V_0-Dn$$ where $D$ is the amount lost each year. A 30 000 dollar car losing 10% of its purchase price each year loses 3000 dollars a year, so after 4 years it is worth $30000-4\times 3000=18000$ dollars. Compare declining balance: $30000(0.9{)}^4\approx 19683$ dollars.

Quick summary

• Simple interest: $I=Prn$. Grows by the same amount each period.
• Compound interest: $A=P(1+r{)}^n$. Grows by the same percentage each period.
• For monthly compounding, divide the yearly rate by 12 and multiply the years by 12.
• Depreciation: $S=V_0(1-r{)}^n$.
• Use logs to find how long an investment takes to reach a target.

Quiz