Back to Course
Simple and Compound Interest
Simple Interest: Linear Growth
Simple Interest is the most straightforward way to calculate earnings on money. The interest is computed only on the original principal and never on accumulated interest, producing steady, predictable growth.
$$SI = \frac{P \times R \times T}{100}$$
Simple interest is the product of principal, rate, and time divided by 100.
$SI$=Simple interest earned (or charged)(currency)
$P$=Principal — the original amount of money deposited or borrowed(currency)
$R$=Annual [rate of interest] (in percentage)(% per year)
$T$=[Time period] for which the money is invested or borrowed(years)
$T$ is in months
→Convert to years: $T = \frac{\text{months}}{12}$
$T$ is in days
→Convert to years: $T = \frac{\text{days}}{365}$
Flat Growth: Interest each year is the same: $\frac{P \times R}{100}$. After $T$ years, total interest is simply $T$ times the yearly interest.
Total Amount: The amount at the end of $T$ years is $A = P + SI$.
Directly Proportional: SI is directly proportional to $P$, $R$, and $T$ individually — doubling any one of them doubles the interest.
The amount under simple interest combines the original principal with the total interest earned. This formula gives the final value you receive (or owe) after $T$ years.
$$A = P\left(1 + \frac{RT}{100}\right)$$
The amount grows linearly with time — it's a straight-line function of $T$.
$A$=Total amount after $T$ years (principal + interest)(currency)
$1 + \frac{RT}{100}$=Growth factor — the multiplier applied to the principal(unitless)
$T = 1$
→$A = P(1 + \frac{R}{100})$ — the one-year multiplier
Linear Relationship: Plotting $A$ vs $T$ gives a straight line with slope $\frac{PR}{100}$ and y-intercept $P$.
Finding Principal from Amount: Rearrange to $P = \frac{A \times 100}{100 + RT}$ — useful when you know the final amount and need the original investment.
Compound Interest: Exponential Growth
Compound Interest is calculated on both the original principal and any previously earned interest. This reinvestment of interest creates a snowball effect — the money grows faster each year because the base keeps getting larger.
$$A = P\left(1 + \frac{R}{n \times 100}\right)^{nT}$$
The amount after compounding depends on how frequently interest is added to the principal.
$A$=Total amount after $T$ years(currency)
$P$=Original principal(currency)
$R$=Annual [rate of interest] (percentage)(% per year)
$n$=[Compounding frequency] — number of times interest is compounded per year(per year)
$T$=Number of years(years)
$n = 1$ (annual)
→$A = P(1 + \frac{R}{100})^T$ — interest added once per year
$n = 2$ (semi-annual)
→$A = P(1 + \frac{R}{200})^{2T}$ — rate halved, periods doubled
$n = 4$ (quarterly)
→$A = P(1 + \frac{R}{400})^{4T}$ — rate quartered, periods quadrupled
$n = 12$ (monthly)
→$A = P(1 + \frac{R}{1200})^{12T}$ — rate divided by 12, periods multiplied by 12
Exponential Nature: CI follows $A = P(1 + r)^t$ where $r = \frac{R}{100}$ — this is an exponential function, not linear like SI.
Compounding Effect: The more frequently you compound, the larger the final amount — but with diminishing returns. Annual → semi-annual gives a noticeable jump; monthly → daily gives very little extra.
CI Calculation Shortcut: $CI = A - P$. Always compute $A$ first, then subtract $P$.
Greater Than SI: Compound interest is always greater than or equal to simple interest for the same $P$, $R$, and $T$ (equal only when $T \leq 1$).
CI for Non-Integer Years
When the time period includes a fraction of a year (e.g., 2.5 years), split the calculation into a whole number of compounding periods plus a simple interest portion for the remaining fraction.
$$A = P\left(1 + \frac{R}{100}\right)^t \times \left(1 + \frac{R \times f}{100}\right)$$
Compound the full years, then apply simple interest for the fractional part.
$t$=Whole number of years (integer part of $T$)(years)
$f$=Fractional part of a year ($f = T - t$)(years)
$T$ is already an integer
→This reduces to the standard compound interest formula — no fractional adjustment needed
Mixed Approach: For 3 years 4 months with annual compounding: compound for 3 full years, then add SI for $\frac{4}{12} = \frac{1}{3}$ year on the accumulated amount.
Why Not Just Use $T$ Directly?: The exponent $T$ in $(1 + R/100)^T$ only produces clean results when $T$ is an integer. For fractional exponents, this mixed approach is the conventional method.
Effective Annual Rate
The effective interest rate is the rate that, when compounded annually, would produce the same final amount as a nominal rate compounded more frequently. It reveals the true cost of borrowing or the true return on investment.
$$R_{\text{eff}} = \left(1 + \frac{R}{n \times 100}\right)^n - 1$$
Converts any compounding frequency into an equivalent annual rate for fair comparison.
$R_{\text{eff}}$=Effective annual rate (as a decimal, multiply by 100 for percentage)(decimal)
$R$=Nominal (stated) annual rate in percentage(% per year)
$n$=Number of compounding periods per year(per year)
$n = 1$
→$R_{\text{eff}} = R$ — nominal and effective rates are identical
$n \to \infty$ (continuous)
→$R_{\text{eff}} = e^{R/100} - 1$ — the maximum possible effective rate
Always Greater Than Nominal: For $n > 1$, $R_{\text{eff}} > R$. The more frequent the compounding, the higher the effective rate.
Comparison Tool: Use this to compare two investments with different compounding frequencies — the one with the higher effective rate is always better, regardless of how the nominal rate is stated.
Practical Example: 12% compounded quarterly gives $R_{\text{eff}} = (1 + 0.03)^4 - 1 = 0.1255 = 12.55\%$. You earn more than the stated 12%.
Simple vs Compound Interest: The Difference
The gap between compound interest and simple interest grows wider with time. In the first year they are equal, but from year two onward, compound interest pulls ahead because it earns interest on previously accumulated interest.
$$CI - SI = P\left(1 + \frac{R}{100}\right)^T - P\left(1 + \frac{RT}{100}\right)$$
The difference is the compound amount minus the simple amount, both starting from the same principal.
$CI - SI$=Excess of compound interest over simple interest(currency)
Year 1: No Difference: When $T = 1$, $CI = SI = \frac{PRT}{100}$. The gap only appears from year 2 onward.
Accelerating Gap: The difference grows faster each year because CI is exponential while SI is linear.
Shortcut for 2 Years: $CI - SI = P\left(\frac{R}{100}\right)^2$ — the difference equals the interest on one year's SI.
Shortcut for 3 Years: $CI - SI = P\left(\frac{R}{100}\right)^2 \times \left(3 + \frac{R}{100}\right)$.
For a two-year period, there is a clean shortcut to find the difference between compound and simple interest without computing each separately.
$$CI - SI = P\left(\frac{R}{100}\right)^2$$
The 2-year difference equals the principal multiplied by the square of the rate fraction.
$\frac{R}{100}$=Rate expressed as a decimal(decimal)
Difference and rate are given
→Find principal: $P = \frac{\text{Difference}}{(R/100)^2}$
Why It Works: In year 2, CI earns interest on year 1's interest too. The extra amount is exactly $\frac{R}{100}$ of year 1's SI = $\frac{R}{100} \times \frac{PR}{100} = P(R/100)^2$.
Inverse Problem: If $CI - SI = 80$ and $R = 10\%$, then $P = \frac{80}{0.01} = 8000$.
Installments and the Time Value of Money
An installment is a fixed periodic payment made to repay a loan or purchase over time. Because of the time value of money, each installment is worth less in present-value terms than the final amount — money received sooner is worth more than money received later.
$$\text{Present Value} = \frac{X}{\left(1 + \frac{R}{100}\right)^t}$$
Discounts a future payment back to its value today at the given interest rate.
$X$=Future payment (installment amount)(currency)
$R$=Interest rate per period(% per period)
$t$=Number of periods from now until the payment is made(periods)
Equal Installments Principle: If $n$ equal annual installments of $X$ repay a loan of $P$ at rate $R\%$, then $P = \frac{X}{(1+R/100)} + \frac{X}{(1+R/100)^2} + \dots + \frac{X}{(1+R/100)^n}$.
Each Installment's Time Value: The first installment (paid after 1 year) is discounted by 1 period, the second by 2 periods, and so on.
Total Repayment: Total money paid = $n \times X$, which is always greater than $P$ because of interest.