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)
For non-annual compounding, we modify our formula to: $$ A = P(1 + \frac{r}{k})^{kn} $$ Where $k$ is the number of times interest is compounded per year.
ExampleIf you invest $1000 at 6% annual interest compounded monthly for 2 years:
- $P = 1000$
- $r = 0.06$
- $k = 12$ (monthly)
- $n = 2$ (years)
$A = 1000(1 + \frac{0.06}{12})^{24} = 1127.49$
Annual Depreciation
Depreciation is the opposite of compound interest – i.e. when the value of an asset decreases by a fixed percentage each year. This usually happens to physical assets that degrade over time, such as cars or real estate.
Declining Balance Depreciation
This method assumes the asset loses a fixed percentage of its value each year.
Formula: $$ V = P(1-r)^n $$ Where:
- $V$ = Final value
- $P$ = Initial value
- $r$ = Rate of depreciation
- $n$ = Number of years
Don't confuse the declining balance formula with the compound interest formula! While they look similar, one shows growth (compound interest) and the other shows reduction (depreciation).
ExampleA car worth $\$25,000$ depreciates at $15\%$ per year. After 3 years: $$ V = 25000(1-0.15)^3 = 15,438.28 $$
NoteIn real-world applications, depreciation rates can vary based on:
- Type of asset
- Industry standards
- Tax regulations
- Company policies
Problem-Solving Strategy
- Identify whether you're dealing with growth (compound interest) or reduction (depreciation)
- Determine the time period and frequency of compounding/depreciation
- Gather all given values and match them to the appropriate formula
- Pay attention to units and decimal places
- Check if your answer makes logical sense
When solving financial problems, always round your final answer to two decimal places, as this represents currency accurately.
