Compound Interest

Compound interest applies growth to a changing balance, creating exponential saving and borrowing models.

Cash-flow timing determines how financial quantities accumulate. Dated value markers sit above a timeline and grow across successive comparison dates. The caption beneath the axis requires every cash flow to be moved to one common date before totals are compared.

Compound interest is interest on the current balance, including previously earned or charged interest.

The compound-growth model

For principal $P$, nominal annual rate $r$, $n$ compounding periods per year, and time $t$ years,

A=P\left(1+\frac rn\right)^{nt}.

The interest is $I=A-P$. The periodic growth factor is $1+r/n$, and the number of periods is $nt$.

Why compounding is exponential

The balance is multiplied by the same factor each period. If $1000$ grows by $5\%$ annually, balances begin $1000$, $1050$, $1102.50$, and $1157.63$. The dollar increase grows because the rate applies to a larger balance.

Simple interest adds a constant amount; compound interest multiplies by a constant factor.

Reading the formula structurally

Separate the calculation into four questions:

What is the starting amount $P$?
What decimal rate applies each period, $r/n$?
What is the periodic multiplier, $1+r/n$?
How many periods occur, $nt$?

This structure transfers to population growth, inflation, depreciation, and repeated percent change.

Solving for time

When time is unknown, logarithms isolate the exponent. For annual compounding,

$$ A=P(1+r)^t $$

gives

t=\frac{\log(A/P)}{\log(1+r)}.

Before using the formula, decide whether the result should be rounded up to a complete payment or compounding period.

Common mistakes

Using percent form directly. $4\%$ becomes $0.04$.

Dividing the exponent by $n$. Frequency increases periods: use $nt$.

Forgetting to divide rate by $n$. Each period receives $r/n$.

Subtracting $r$ for growth. Growth factor is $1+r/n$; depreciation uses subtraction.

Calling the final amount “interest.” Interest is $A-P$.

Quick self-check

What is the compounding period?
Did I use decimal rate per period and total number of periods?
Is this growth or depreciation?
Does the question ask for balance, interest, or time?
Have I delayed rounding until the end?