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CAGR Explained Step by Step

If you've watched a stock double in three years, tripled in five, or seen a startup valued at $1 billion after starting at $10 million, you've probably seen growth described as a percentage: "30% annual growth" or "20% CAGR." But what does that number actually mean? How do you calculate it, and why is the formula different from simple division?

CAGR—Compound Annual Growth Rate—is the annual percentage rate at which an investment grows from a starting value to an ending value over a specific time period. Unlike simple average growth (which ignores compounding), CAGR accounts for the fact that each year's growth builds on the previous year's total. This article walks you through the formula, shows you how to calculate it for real investments, and reveals why CAGR matters for comparison and planning.

Quick definition: CAGR is the compound annual growth rate—the single annual percentage rate that, if applied each year, transforms a starting value into an ending value over a given time period.

Key Takeaways​

  • CAGR is calculated using the formula: CAGR = (Ending Value / Beginning Value)^(1 / years) – 1
  • CAGR accounts for compounding and is more accurate than simple average growth
  • It allows you to compare the annual performance of investments with different time horizons
  • Volatility is hidden in CAGR; a smooth 10% annual return and a volatile path to the same ending value both show 10% CAGR
  • CAGR is useful for stocks, funds, company revenue, and any metric that compounds over time

The CAGR Formula and Why It Works​

The compound interest formula we explored earlier was:

A = P(1 + r)^t

where A is the ending value, P is the starting value, r is the annual rate, and t is time in years.

Solving for r (the annual rate) gives us CAGR:

A = P(1 + r)^t
A/P = (1 + r)^t
(A/P)^(1/t) = 1 + r
CAGR = (A/P)^(1/t) – 1

In words: divide the ending value by the beginning value, take the t-th root (where t is the number of years), and subtract 1. The result is CAGR as a decimal; multiply by 100 to express as a percentage.

Let's verify this makes sense. If an investment starts at $100 and ends at $200 over 10 years:

CAGR = (200 / 100)^(1/10) – 1
CAGR = (2)^(0.1) – 1
CAGR = 1.0718 – 1
CAGR ≈ 0.0718 or 7.18%

Check: Does $100 at 7.18% annual growth for 10 years reach $200?

A = 100 × (1.0718)^10
A = 100 × 2.00
A = $200 ✓

Yes. The formula is self-consistent. A 7.18% annual compound rate transforms $100 into $200 in 10 years.

Example 1: A Simple Stock Investment​

You bought a stock for $50 per share on January 1, 2020. On December 31, 2024 (5 years later), it trades at $95 per share. What's the CAGR?

Setup:

  • Beginning Value (P) = $50
  • Ending Value (A) = $95
  • Time (t) = 5 years

Calculation:

CAGR = (95 / 50)^(1/5) – 1
CAGR = (1.9)^(0.2) – 1
CAGR = 1.1384 – 1
CAGR ≈ 0.1384 or 13.84%

Interpretation: Your stock grew at an average compound rate of 13.84% per year. If it had grown exactly 13.84% annually (no volatility), the share price would follow: Year 0: $50 → Year 1: $56.92 → Year 2: $64.85 → Year 3: $73.84 → Year 4: $84.12 → Year 5: $95.76 (approximately $95).

Example 2: A Diversified Fund with Multiple Periods​

Your mutual fund had these values:

  • January 1, 2020: $25,000
  • December 31, 2024: $48,750

What's the CAGR over this 5-year period?

Setup:

  • P = $25,000
  • A = $48,750
  • t = 5

Calculation:

CAGR = (48,750 / 25,000)^(1/5) – 1
CAGR = (1.95)^(0.2) – 1
CAGR = 1.1361 – 1
CAGR ≈ 0.1361 or 13.61%

Interpretation: Your fund grew at a 13.61% compound annual rate, nearly identical to the stock example. Despite different starting values ($50 vs. $25,000), the growth rate is comparable—this is why CAGR is so useful for comparison.

Example 3: Revenue Growth for a Private Company​

A startup had:

  • 2019 Revenue: $500,000
  • 2024 Revenue: $8,000,000

What's the compound annual growth rate from 2019 to 2024 (5 years)?

Setup:

  • Beginning Value = $500,000
  • Ending Value = $8,000,000
  • Time = 5 years (2019 to 2024)

Calculation:

CAGR = (8,000,000 / 500,000)^(1/5) – 1
CAGR = (16)^(0.2) – 1
CAGR = 2.0112 – 1
CAGR ≈ 1.0112 or 101.12%

Interpretation: This startup's revenue grew at a 101% compound annual rate. That means each year, revenue roughly doubled (100% = 2x). This is not uncommon for successful early-stage ventures. Let's verify: $500k → $1.01M → $2.04M → $4.11M → $8.29M ≈ $8M. Yes.

Understanding the T-th Root​

The key operation in CAGR is taking the t-th root. When t = 5, you're taking the 5th root, written as ^(1/5) or ^0.2. This extracts the annual growth rate from the total compound growth.

Think of it this way: if you grow 95% in total over 5 years, your annual growth rate isn't 95% ÷ 5 = 19% (that's the simple average). Instead, you find the rate that, compounded 5 times, reaches 95% total growth.

(1 + r)^5 = 1.95
1 + r = 1.95^(1/5)
1 + r = 1.1384
r ≈ 13.84%

Why? Because 13.84% × 5 = 69.2%, not 95%. But 13.84% compounded 5 times gives: 1.1384^5 ≈ 1.95, which corresponds to 95% total growth. The math accounts for the compounding effect.

CAGR with Volatility: Why the Path Doesn't Matter​

Here's a crucial insight: CAGR only cares about the starting and ending values. The path in between is invisible.

Scenario A (Smooth): Investment grows 13.84% every year for 5 years.

  • Year 0: $100.00
  • Year 1: $113.84
  • Year 2: $129.63
  • Year 3: $147.59
  • Year 4: $168.02
  • Year 5: $191.30
  • CAGR: 13.84%

Scenario B (Volatile): Investment is chaotic but ends at the same place.

  • Year 0: $100.00
  • Year 1: $85.00 (15% loss)
  • Year 2: $120.00 (41% gain)
  • Year 3: $95.00 (21% loss)
  • Year 4: $200.00 (111% gain)
  • Year 5: $191.30 (4.3% loss)
  • CAGR: 13.84%

Both show 13.84% CAGR because both start at $100 and end at $191.30. But the investor in Scenario B lived through wild swings, while Scenario A offered a calm ride. This is why CAGR is incomplete: it doesn't tell you about risk, volatility, or the emotional journey. It's a summary metric, useful for comparing returns but blind to the road taken.

When Time Matters: Partial Years and Non-Calendar Periods​

CAGR works with any time period, not just whole years. If you held an investment for 3.5 years, use t = 3.5. If you measured performance from March 15, 2020 to August 20, 2025, calculate the precise time difference in years.

Example: An investment grew from $10,000 to $13,000 over 2.5 years.

CAGR = (13,000 / 10,000)^(1/2.5) – 1
CAGR = (1.3)^(0.4) – 1
CAGR = 1.1066 – 1
CAGR ≈ 0.1066 or 10.66%

The investment's compound annual growth rate is 10.66%, even though the holding period isn't a round number of years.

CAGR vs. Simple Average Growth: Why They Differ​

A common mistake is calculating CAGR as a simple average:

Wrong approach: If an investment grows 10%, 15%, 8%, 20%, and 12% over 5 years, don't just average those percentages:

(10 + 15 + 8 + 20 + 12) / 5 = 13%

This gives 13%, but that's not CAGR. The correct approach is to compound year by year and then back out the equivalent annual rate.

Correct approach:

  • Start: $1,000
  • After Year 1 (+10%): $1,100
  • After Year 2 (+15%): $1,100 × 1.15 = $1,265
  • After Year 3 (+8%): $1,265 × 1.08 = $1,366.20
  • After Year 4 (+20%): $1,366.20 × 1.20 = $1,639.44
  • After Year 5 (+12%): $1,639.44 × 1.12 = $1,836.18

Now calculate CAGR:

CAGR = (1,836.18 / 1,000)^(1/5) – 1
CAGR = (1.8362)^0.2 – 1
CAGR = 1.1313 – 1
CAGR ≈ 0.1313 or 13.13%

CAGR is 13.13%, not 13%—close, but different. The difference is small in this example but grows when volatility increases.

Visualization of CAGR​

Each year, the value is multiplied by (1 + CAGR). Over t years, it's multiplied (1 + CAGR)^t times total. CAGR is the annual multiplication factor that, applied repeatedly, reaches the ending value.

Real-World CAGR Examples​

Example 1: Amazon Stock Performance (2014–2019)

  • January 2014: $398 per share
  • December 2019: $1,681 per share
  • Time: approximately 6 years
CAGR = (1,681 / 398)^(1/6) – 1
CAGR = (4.223)^(0.1667) – 1
CAGR = 1.2887 – 1
CAGR ≈ 0.2887 or 28.87%

Amazon's CAGR from 2014–2019 was approximately 29%. This is high but not unusual for mega-cap tech stocks during a strong market period.

Example 2: S&P 500 Long-Term Performance According to Federal Reserve Economic Data (FRED) and academic sources, the S&P 500 has historically returned approximately 10% annually over the long term (1950–2024). This figure is already a CAGR, and it's the benchmark most financial advisors use for equity returns. The SEC requires fund managers to disclose CAGR performance, making it the standard metric for comparing investment vehicles.

Example 3: Bond Fund Performance A bond fund shows:

  • End of 2019: $100 (value per share)
  • End of 2024: $108 (value per share)
  • Time: 5 years
CAGR = (108 / 100)^(1/5) – 1
CAGR = (1.08)^(0.2) – 1
CAGR = 1.01554 – 1
CAGR ≈ 0.01554 or 1.554%

A 1.55% CAGR reflects typical bond returns in a low-interest-rate environment. Over 5 years, $100,000 grows to $108,100—modest but steady, with lower volatility than stocks.

Common Mistakes with CAGR​

Mistake 1: Assuming CAGR Predicts Future Returns CAGR is backward-looking. A stock with 20% CAGR over the past 5 years might crash tomorrow. Past CAGR is useful context, but it's not a forecast. Always distinguish between historical CAGR (what happened) and expected CAGR (what you think will happen).

Mistake 2: Ignoring the Time Period 10% CAGR over 1 year means something different from 10% CAGR over 20 years (the longer period is more impressive because it's sustained). Always specify the time frame.

Mistake 3: Comparing CAGRs Across Different Time Periods Without Thought A fund with 12% CAGR over 2 years and another with 10% CAGR over 10 years aren't directly comparable. The 2-year period might have been a bull market; the 10-year includes cycles. Use multiple time periods to assess consistency.

Mistake 4: Forgetting About Taxes and Fees CAGR on a pre-tax, pre-fee basis can be misleading. A 15% CAGR after 30% in taxes might be more like an 10.5% CAGR to your pocket. Always ask: "Is this gross or net?" and "Does it include fees?"

Mistake 5: Using CAGR for Highly Volatile Short Periods CAGR is most meaningful over periods of several years or more. A stock that's down 30%, then up 60% over 2 years shows a CAGR of 10.5%, but calling that 10.5% "annual growth" is misleading because it's the result of swings. For short, volatile periods, consider volatility and drawdown alongside CAGR.

FAQ​

Q: How do I calculate CAGR on a spreadsheet? A: In Excel or Google Sheets, use: =(ending_value/beginning_value)^(1/years)-1 and format as a percentage. For example, =(95/50)^(1/5)-1 returns 0.1384, which is 13.84% when formatted as a percentage.

Q: Can CAGR be negative? A: Yes. If an investment declines from $100 to $60 over 5 years, CAGR = (60/100)^(1/5) – 1 = 0.6^0.2 – 1 ≈ -0.0920 or -9.20%. The investment is losing value at an average of 9.2% per year.

Q: What's the difference between CAGR and IRR? A: CAGR applies to a single investment with a beginning and ending value. IRR (internal rate of return) applies to a series of cash flows at different times. We'll explore IRR in detail in the next article.

Q: Is CAGR the same as average return? A: No. Average return ignores compounding; CAGR accounts for it. If you have annual returns of 10%, 15%, and 8%, the average is 11%, but CAGR is different (about 10.99% in this case). CAGR is always more accurate for comparing compound growth.

Q: How do I annualize returns if I only have a 3-month holding period? A: Use CAGR with t = 0.25 (3 months = 1/4 year). For example, if $10,000 became $10,500 in 3 months, CAGR = (10,500/10,000)^(1/0.25) – 1 ≈ 0.205 or 20.5%. That's the annualized rate.

Q: Does CAGR account for dividends or withdrawals? A: No, unless you reinvest them and include the reinvested amounts in your ending value. If you track total return including dividends reinvested, CAGR works perfectly. If you withdrew dividends, those withdrawals reduce your ending value, and CAGR reflects that.

Summary​

CAGR is the compound annual growth rate—the single annual percentage that, if applied each year, transforms a starting value into an ending value. The formula, CAGR = (Ending Value / Beginning Value)^(1 / years) – 1, is straightforward but powerful. It enables fair comparison of investments with different time horizons and starting values. CAGR accounts for compounding, unlike simple average growth, making it the correct metric for understanding long-term investment performance. However, CAGR has a critical limitation: it reveals nothing about volatility, drawdowns, or the path taken. A smooth 10% annual return and a chaotic path to the same ending show identical CAGR. For this reason, CAGR is best used alongside risk metrics and always interpreted with the full time period and context in mind. Master CAGR, and you have a tool for evaluating any long-term investment—whether it's a stock, a fund, a business, or your own portfolio.

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CAGR vs Average Return