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.
Example
- 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
Hint
- 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.
Example
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:
- $V$ is final value
- $P$ is initial value
- $r$ is rate of depreciation
- $n$ is number of years
Common Mistake
- 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).
Example
- A car worth $\text{\$}25,000$ depreciates at $15\%$ per year.
- After 3 years: $$ V = 25000(1-0.15)^3 = 15,438.28 $$
Note
In real-world applications, depreciation rates can vary based on:
- Type of asset
- Industry standards
- Tax regulations
- Company policies
Exam technique
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
Hint
When solving financial problems, always round your final answer to two decimal places, as this represents currency accurately.
