Compound Interest Formula: A Complete Guide with Worked Examples
The formula A = P(1 + r/n)nt explained — variables defined, compounding frequencies compared, continuous compounding, and how to solve for rate or time.
The Compound Interest Formula
Compound interest is interest calculated on both the original principal and the accumulated interest from previous periods. Unlike simple interest — which only ever charges on the principal — compound interest generates "interest on interest," causing balances to grow exponentially over time.
Variable definitions:
A — final amount (principal + interest)P — principal (starting amount)r — annual interest rate as a decimaln — compounding periods per yeart — time in yearsnt — total number of compounding periodsTo find the interest earned only, subtract the principal: Interest = A − P.
Compounding Frequency: How n Affects Your Return
The compounding frequency (n) determines how often interest is added to the balance. More frequent compounding means interest starts earning interest sooner, producing a marginally higher final amount at the same nominal rate.
| Frequency | n value | £5,000 at 6% for 10 years | Extra vs annual |
|---|---|---|---|
| Annually | 1 | £8,954 | — |
| Semi-annually | 2 | £9,030 | +£76 |
| Quarterly | 4 | £9,070 | +£116 |
| Monthly | 12 | £9,097 | +£143 |
| Daily | 365 | £9,110 | +£156 |
The difference between monthly and daily compounding is small (£13 over 10 years). Most of the compounding benefit comes simply from switching from annual to quarterly compounding.
Step-by-Step Worked Example
£5,000 invested at 6% annual interest, compounded monthly, for 10 years.
Identify variables: P = 5,000 · r = 0.06 · n = 12 · t = 10
Step 1: Monthly rate → r/n = 0.06 ÷ 12 = 0.005
Step 2: Total periods → nt = 12 × 10 = 120
Step 3: Compounding factor → (1 + 0.005)120 = (1.005)120 = 1.8194
Step 4: Final amount → A = 5,000 × 1.8194 = £9,097
Interest earned: £9,097 − £5,000 = £4,097
Simple Interest vs Compound Interest
Simple interest is charged only on the original principal: I = P × r × t. It grows linearly. Compound interest grows exponentially because each period's interest is added to the base before the next calculation.
Interest earned: £3,000
Interest earned: £4,097
The additional £1,097 produced by compounding is "interest on interest" — the cumulative effect of 120 monthly periods in which the growing balance itself earned a return.
Continuous Compounding
As the compounding frequency approaches infinity, the formula converges to a limit described by Euler's number (e ≈ 2.71828). This is called continuous compounding and represents the theoretical maximum return for a given nominal rate and time.
rt = 0.06 × 10 = 0.6
e0.6 = 1.8221
A = 5,000 × 1.8221 = £9,111
Only £14 more than monthly compounding — the theoretical maximum adds very little over monthly in practice.
Solving for Other Variables
The compound interest formula can be rearranged to solve for any unknown variable, not just the final amount.
Finding the present value (P): How much do you need to invest today to reach a target amount?
Finding the time (t): How long until you reach a target amount?
Finding the rate (r): Requires numerical methods (iteration) for exact values, but the Rule of 72 gives a quick estimate — see the FAQ below. For precise rate calculations from known P, A, n, and t, use the Average Return (CAGR) calculator.
Use the free compound interest calculator — enter principal, rate, period, and compounding frequency to see your final balance and interest earned instantly.
Open Compound Interest Calculator →Frequently Asked Questions
What is the Rule of 72?
Divide 72 by the annual interest rate to estimate how many years it takes to double your money. At 6%, 72 ÷ 6 = 12 years. At 9%, 72 ÷ 9 = 8 years. It is a quick mental shortcut that works best for rates between 3% and 12% with annual compounding.
How often does compound interest compound?
It depends on the account or loan. UK savings accounts typically compound daily or monthly. Mortgages compound monthly. Some bonds compound annually. The more frequently interest compounds, the higher the effective annual rate (EAR or AER) compared to the quoted nominal rate.
What is the difference between APR and APY?
APR (Annual Percentage Rate) is the nominal rate without intra-year compounding. APY (Annual Percentage Yield) — called AER in the UK — is the effective annual rate after compounding. A 6% APR compounded monthly gives an AER of (1.005)12 − 1 = 6.168%. Always compare AERs when choosing savings accounts.
How does compound interest hurt you on debt?
On debt, compounding works against you. Unpaid interest is added to the principal, and future interest is then charged on that larger balance. Credit cards typically compound daily. Paying only the minimum on a large balance can mean most of each payment goes to interest rather than reducing what you owe.
Evidence & Methodology
How This Page Is Grounded
Method
Derives A = P(1 + r/n)^(nt) and extends it to regular contributions and different compounding frequencies.
Important limitation: Examples assume a constant nominal rate and exclude tax, fees, inflation and market variation.
Primary Sources
- Compound Interest Calculator Investor.gov — U.S. Securities and Exchange Commission
- Compound interest Investor.gov — U.S. Securities and Exchange Commission
Quality Checks
Formula examples are checked independently and against the compound-interest calculator.
See how sources are selected and corrections are handled in our Editorial & Calculation Methodology, and which automated checks this page has to pass in How We Test.