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 $\$5000$ 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.
If 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).
A car worth $\$25,000$ depreciates at $15\%$ per year. After 3 years: $$ V = 25000(1-0.15)^3 = 15,438.28 $$
In 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.
