What is Compound Interest?
Compound interest is the interest calculated on the initial principal balance plus all the accumulated interest from previous periods. Unlike simple interest, which only yields returns on the baseline deposit, compounding allows your wealth to grow exponentially rather than linearly.Albert Einstein is often famously (if apocryphally) quoted as calling compound interest the "eighth wonder of the world: he who understands it, earns it; he who doesn't, pays it."---The Mathematical Formula
The standard formula for compound interest on an initial deposit is:A = P \left(1 + \frac{r}{n}\right)^{nt}
Where:
* A = Future value of the investment / loan
* P = Initial principal balance
* r = Annual interest rate (expressed as a decimal, e.g. 7% = 0.07)
* n = Number of compounding periods per year (e.g. 12 for monthly, 4 for quarterly, 365 for daily)
* t = Number of years the money is invested or borrowedWhen recurring monthly deposits ($PMT$) are added, the total accumulated future value expands to:A = P \left(1 + \frac{r}{n}\right)^{nt} + PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}
---The Impact of Compounding Frequencies
The more frequently interest compounds, the greater the effective annual yield (APY). Let us compare how a $10,000 deposit grows at 8% annual interest over 10 years under different frequencies:| Compounding Frequency | Periods/Year ($n$) | Final Balance ($A$) | Total Interest Earned | | :--- | :--- | :--- | :--- | | Annually | 1 | $21,589.25 | $11,589.25 | | Quarterly | 4 | $22,080.40 | $12,080.40 | | Monthly | 12 | $22,196.40 | $12,196.40 | | Daily | 365 | $22,253.46 | $12,253.46 |Notice that daily compounding yields $664.21 more in pure interest compared to annual compounding on the exact same starting amount.---Step-by-Step Practical Calculation Example
Suppose you invest $5,000 today at an 8% annual return, compounded monthly for 5 years.1. Identify variables: $P = 5000$, $r = 0.08$, $n = 12$, $t = 5$. 2. Calculate periodic rate: $r / n = 0.08 / 12 \approx 0.0066667$. 3. Calculate total periods: $n \times t = 12 \times 5 = 60$. 4. Compute base factor: $(1 + 0.0066667)^{60} \approx 1.4898457$. 5. Multiply by principal: $5000 \times 1.4898457 = \mathbf{\$7,449.23}$.Your total compound interest earned is $2,449.23.---The Rule of 72 Shortcut
When you want to know how long it will take your investment to double without using a calculator, use the Rule of 72:\text{Years to Double} \approx \frac{72}{\text{Annual Interest Rate (\%)}}
* At 6% return: $72 / 6 = 12$ years to double.
* At 8% return: $72 / 8 = 9$ years to double.
* At 12% return: $72 / 12 = 6$ years to double.