How to Calculate CAGR and Annualized Returns

What CAGR and annualized returns mean

Retail traders often see the terms "CAGR" and "annualized return" used interchangeably. Both measure performance on a per-year basis, but there are nuances.

• Compound Annual Growth Rate (CAGR) is the constant annual growth rate that takes an investment from its beginning value to its ending value over a specified period, assuming profits are reinvested.
• Annualized return can refer more generally to expressing any multi-period return as a per-year rate and may account for irregular cash flows when using time-weighted or money-weighted methods.

This article focuses on the standard compound annual growth rate formula and how to perform basic annualized return calculation for buy-and-hold scenarios. For portfolios with deposits/withdrawals, we'll point to the right methods (TWRR and IRR) and how they differ.

The compound annual growth rate formula

The simple, widely used CAGR formula is:

CAGR = (Ending Value / Beginning Value)^(1 / n) - 1

Where:

• Ending Value = value at the end of the period
• Beginning Value = value at the start of the period
• n = number of years (can be fractional for months/days)

This yields a single annual rate that, if applied each year, would compound the beginning value to the ending value.

Worked example: CAGR step-by-step

Suppose you bought shares for $8,000 on 1 Jan 2018 and the holding was worth $13,240 on 1 Jan 2023. No dividends or cash flows (simple buy-and-hold).

1. Beginning Value = $8,000
2. Ending Value = $13,240
3. Period n = 2023 - 2018 = 5 years

Apply the formula:

CAGR = (13,240 / 8,000)^(1 / 5) - 1 CAGR = (1.655)^(0.2) - 1 CAGR ≈ 1.105 - 1 = 0.105 ≈ 10.5% per year

Interpretation: A 10.5% annual compounded rate would turn $8,000 into $13,240 in 5 years.

Annualized return calculation for non-whole years

If your holding period is not a whole number of years, convert days or months into fractional years. For example, for a holding of 2 years and 3 months (2.25 years):

CAGR = (Ending / Beginning)^(1 / 2.25) - 1

Always use consistent time units when computing n. For daily precision you can use n = days / 365 (or 252 for trading days, but this is less common for CAGR which measures calendar years).

When cash flows exist: TWRR and IRR

CAGR assumes no intermediate cash flows. If you deposit or withdraw funds during the period, use:

• Time-Weighted Rate of Return (TWRR): removes the impact of timing of investor cash flows; best for comparing fund/manager performance.
• Internal Rate of Return (IRR) or Money-Weighted Return: solves for a discount rate where the net present value of cash flows (including final value) equals zero; reflects investor's actual dollar experience.

IRR is computed by solving for r in the equation sum(CF_t / (1+r)^t) = 0, which typically requires a spreadsheet or financial calculator.

Concrete worked example with a cash flow (IRR)

You invest $5,000 today (t=0), add $2,000 after 1 year (t=1), and end up with $9,000 at the end of year 2 (t=2). Find the annualized money-weighted return (IRR).

Cash flows: CF0 = -5,000 (outflow), CF1 = -2,000, CF2 = +9,000.

Solve for r in: -5,000/(1+r)^0 - 2,000/(1+r)^1 + 9,000/(1+r)^2 = 0

This doesn't have an algebraic closed form; use Excel's IRR or a numerical method. Using a spreadsheet returns r ≈ 12.34% per year.

Interpretation: Your actual annualized return, taking the extra $2,000 contribution into account, is about 12.34%.

Common pitfalls and tips

• Do not use CAGR if there are significant intermediate cash flows — use IRR or TWRR.
• Beware of calendar vs. trading days when converting short periods to years. For CAGR, prefer calendar days (365) unless you have a specific reason to use trading days.
• CAGR hides volatility. Two investments with the same CAGR can have very different risk profiles and drawdowns.
• When reporting performance, always disclose the period and whether the returns include dividends, fees, or taxes.

Quick formulas and cheat-sheet

• CAGR: (End / Start)^(1/n) - 1
• n in years = (end_date - start_date) in days / 365
• IRR: solve sum(CF_t / (1+r)^t) = 0 (use Excel's IRR or XIRR for irregular dates)
• TWRR: link sub-period returns multiplicatively and convert to annual form

Example comparison: CAGR vs simple average

Suppose yearly returns are +40%, -20%, +10% over 3 years.

• Arithmetic average = (40% - 20% + 10%) / 3 = 10%
• CAGR = (1.40 0.80 1.10)^(1/3) - 1 = (1.232)^(1/3) - 1 ≈ 7.2%

The CAGR (7.2%) is lower because it accounts for compounding and volatility.

Use practice to learn

Calculating CAGR and annualized returns correctly is a basic but essential skill for retail traders. Practice with spreadsheets, work through examples with and without cash flows, and always verify assumptions about dividends and fees. For risk-free practice of ideas using real market data, you can try AIYUG's free paper-trading race at https://aiyug.trading/race.

No part of this article is financial advice or a promise of future performance. These are mechanics and educational examples only.