Growth Rate & CAGR: Formulas and Annualized Returns

Year

Invested

Interest

Balance

112,000

5,000

117,000

124,000

10,850

134,850

136,000

17,593

153,593

148,000

25,272

173,272

160,000

33,936

193,936

172,000

43,633

215,633

184,000

54,414

238,414

196,000

66,335

262,335

208,000

79,452

287,452

Y10

220,000

93,824

313,824

Key Takeaways

CAGR formula: (end value ÷ start value)^(1/n) − 1, where n = years

Example: 100k → 200k in 5 years: 2^(1/5) − 1 = 14.87%/year — the smooth rate that compounds to the same result

Arithmetic mean trap: +50% then −50% averages 0% but you lost 25%; CAGR (−13.4%) tells the truth

Rule of 72: 72 ÷ rate ≈ years to double; 12% growth doubles in ~6 years

Core answer: Growth rate = (new − old) ÷ old × 100%. CAGR (compound annual growth rate) = (end ÷ start)^(1/years) − 1: revenue growing ¥1M → ¥2M in 5 years has CAGR = 2^(1/5) − 1 = 14.87%/year — NOT (100% ÷ 5 = 20%). Average growth and CAGR diverge wildly in volatile series; CAGR is the only honest multi-year rate. Double-check with the Rule of 72: 72 ÷ 14.87 ≈ 4.8 years to double. ✓

The three rates people confuse

Metric	Formula	Use	Trap
YoY growth	(this − last) ÷ last	single-period	base effects
Average growth	mean of YoY rates	rough feel	ignores compounding
CAGR	(end/start)^(1/n) − 1	multi-year truth	hides volatility

Why average ≠ CAGR. +100% then −50%: average +25%, actual 0% (¥100 → ¥200 → ¥100). CAGR handles it correctly: (100/100)^(1/2) − 1 = 0%.

Worked examples

Example 1 — Investment check. Fund: ¥100k → ¥161k in 5 years. CAGR = (1.61)^(0.2) − 1 = 10%/yr. The brochure said "61% in 5 years" — true and useless; 10%/yr is comparable against alternatives.

Example 2 — Business planning. Revenue ¥5M, target ¥20M in 4 years: needed CAGR = 4^(1/4) − 1 = 41.4%/yr. Sanity-check: can the market, sales team, and delivery each sustain ~41%? If not, extend the timeline or shrink the target — before promising it to investors.

Example 3 — Negative and recovery. Portfolio ¥1M → ¥600k in 2 years: CAGR = (0.6)^(0.5) − 1 = −22.5%/yr. Recovery to ¥1M from here needs +66.7% total, i.e. CAGR +29.1% over the next 2 years — asymmetric math after losses.

Example 4 — Smooth-looking lie. A company grows +80%, +10%, −30%, +5% (¥1M → ¥1.17M over 4 years): CAGR = 4.0%/yr — respectable-looking, but the path included a near-death year. CAGR says nothing about risk; pair it with max drawdown.

Common mistakes and myths

Dividing total growth by years — 100% over 5 years is 14.9%/yr, not 20%/yr; compounding is not linear.
CAGR over cherry-picked dates — start at a crash bottom, end at a bubble top: any number you want. Disclose your endpoints.
Using CAGR for lumpy cash flows — money in and out mid-period needs XIRR, not CAGR.
Comparing CAGRs across different risk — 15% from an index fund and 15% from a leveraged scheme are different animals; risk-adjust before ranking.
Forgetting inflation — 8% nominal CAGR with 3% inflation = ~4.9% real; long-term plans belong in real terms.

Frequently Asked Questions

How do I calculate CAGR?

CAGR = (ending value ÷ beginning value)^(1 ÷ years) − 1. Revenue grew from ¥2M to ¥5M over 4 years: (5÷2)^(1/4) − 1 = 2.5^0.25 − 1 = 25.7%. It answers: "what steady annual rate would compound to the same endpoint?"

Why is CAGR better than average annual growth?

Average growth uses arithmetic mean, which ignores compounding and is fooled by volatility. Returns of +60% and −30% average +15%, but your money went 1.0 → 1.6 → 1.12: CAGR is only +5.8%. CAGR is the geometric mean — it reflects what actually happened to your money.

What is the Rule of 72?

Years to double ≈ 72 ÷ annual rate (%). At 6%: ~12 years; at 12%: ~6 years; at 24%: ~3 years. Reverse it too: doubling in 5 years needs 72÷5 ≈ 14.4%/year. It works because ln2 ≈ 0.693 and holds well for rates between 2% and 30%.

What are CAGR's limitations?

It only sees the start and end points — a smooth 15% CAGR could hide a −40% crash in year 2. It also says nothing about cash flows in between (deposits/withdrawals distort it; use IRR/XIRR for those). Always pair CAGR with max drawdown or volatility for the full risk picture.

How do I compare investments over different time spans?

Annualize everything with CAGR. Fund A: +80% over 5 years → 12.5% CAGR. Fund B: +40% over 2 years → 18.3% CAGR. B performed better annually despite the smaller total gain. Comparing raw totals across different horizons is meaningless — different compounding periods hide the true rate.

Can CAGR be negative or mislead with short periods?

Yes to both. Negative CAGR means the value shrank. And over 1-year or partial-year periods, CAGR equals the simple return — annualizing a hot 3-month streak (+10% → 46% annualized) projects unrealistic growth. Use CAGR for 2+ year spans; for sub-year analysis use absolute returns.