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.
- Let's deposit $\text{\$}5000$ 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 $\text{\$}(0.05\times5000)=\text{\$}250$.
- Meanwhile, account B's balance gets multiplied by $1.05$ every year.
- After 50 years, account A will have a balance of $$\text{\$}(5000+50(0.05\times5000))=\text{\$}17500$$
- After 50 years, account B will have a balance of $$\text{\$}5000(1.05)^{50}\approx \text{\$}57337$$
- 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:
- $A$ is final amount
- $P$ is principal (initial investment)
- $r$ is interest rate (as a decimal)
- $n$ is 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)
