Financial Applications: Compound Interest and Annual Depreciation
Compound Interest
Simple interest is interest that provides a fixed amount each period - for example, 5% of the original amount you put in. Compound interest, meanwhile, increases each period by a percentage of the current amount.
This doesn't sound like a big difference, but exponential growth is a powerful thing, and a bank account with compound interest can end up significantly richer than one with simple interest.
ExampleLet's deposit $\$5000$ into two accounts, both with 5% annual interest. Account $A$ has simple interest, and account $B$ has compound interest.
Account A's balance increases every year by $\$(0.05\times5000)=\$250$.
Meanwhile, account B's balance gets multiplied by $1.05$ every year.
After 50 years, account A will have a balance of $\$(5000+50(0.05\times5000))=\$17500$.
After 50 years, account B will have a balance of $\$5000(1.05)^{50}\approx\$57337$.
You can see here how compound interest is a lot more profitable than simple interest!
The Basic Formula
The compound interest formula is: $$ A = P(1 + r)^n $$ Where:
- $A$ = Final amount
- $P$ = Principal (initial investment)
- $r$ = Interest rate (as a decimal)
- $n$ = Number of compounding periods
When converting interest rates to decimals, divide the percentage by 100. For example, 5% becomes 0.05
This might look familiar. This is because this is really just a geometric series where the common ratio is $1 + r$.
For example, if the compound interest rate is $5\%$, the common ratio the balance increases by each year is $1.05$.
Different Compounding Periods
Interest can be compounded at different frequencies:
- Annually (once per year)
- Semi-annually (twice per year)
- Quarterly (four times per year)
- Monthly (twelve times per year)
- Daily (365 times per year)
