Power Law vs CAGR: Two Ways to Model Bitcoin

Why simple compound-return math underestimates emerging assets — and what to use instead

The Core Problem

Most retirement calculators use the same math for Bitcoin that they use for the S&P 500: pick a constant annual return, compound it forward, done. That math assumes the asset’s price behavior is roughly random walk with a predictable mean. For mature markets, fine. For a still-emerging asset class with strong network effects, it gets the shape wrong — usually too pessimistic in long-horizon projections.

CAGR Model (Standard)

Compound Annual Growth Rate. The same percentage gain every year, applied multiplicatively. Standard for stocks, bonds, real estate. Easy to compute, easy to defend in a Monte Carlo.

P(t) = P₀ × (1 + r)t

Assumes percentage growth is constant over time
Best for: mature, liquid, mean-reverting assets
Behavior: exponential curve, accelerating
Risk model: well-suited to volatility-of-returns framing

Power Law Model

Price grows as a power of time. Developed for Bitcoin by analysts including Bjarne Hagen, Plan B variants, and Burger Crypto. Fits BTC’s first 16 years better than CAGR. Growth rate decreases over time but stays positive.

P(t) = A × tn

Assumes annualized return slows as market matures
Best for: emerging assets with network effects (early internet, early electricity, BTC)
Behavior: steep early, gradually flattening
Risk model: captures S-curve adoption better

How they diverge over 30 years

Starting at $100,000 BTC value. CAGR @ 12% real. Power law calibrated to BTC’s actual 2012-2026 trajectory.

Year

CAGR 12%

$100k

$176k

$310k

$547k

$965k

$1.7M

$3.0M

Power Law

$100k

$245k

$465k

$735k

$1.05M

$1.42M

$1.85M

Illustrative. Power law fitted with n≈2.0 to BTC’s price record per Hagen formulation. Past performance does not guarantee future results.

Notice what changes at the long end

CAGR’s exponential keeps accelerating — at year 30 it’s bigger. Power law starts faster (more realistic for early Bitcoin years) but its growth slows over time, ending lower at year 30. Different models, different long-horizon shapes. The choice matters.

When to use which

Use CAGR when:

Modeling traditional assets (stocks, bonds, REITs, real estate). Your time horizon is short (< 10 years). You want conservative, defensible math for a retirement plan. Volatility-of-returns analysis matters more than long-horizon shape.

Use Power Law when:

Modeling Bitcoin specifically over 10+ years. You’re testing how an emerging asset class with adoption dynamics behaves through its S-curve. You want to compare against actual BTC price history rather than assume it’ll behave like the S&P 500.

Best practice: run both. If retirement target works in CAGR but breaks in Power Law (or vice versa), you’ve found a sensitivity worth understanding.

Five takeaways

CAGR and Power Law tell different long-horizon stories. Same starting price, same general direction, different shape.
Power Law fits BTC’s historical trajectory better than CAGR over its full life — 2012-2026 actual prices match a power curve, not an exponential.
Power Law’s growth slows over time. If you assume BTC’s adoption follows an S-curve (like every prior network), this is the more honest projection.
CAGR is more conservative at short horizons but more aggressive at long horizons. The crossover point matters for retirement planning.
Run both in your FIRE calculation. If the plan only works in one model, you don’t have a plan — you have a hope.

Try both models in the Multi-Asset FIRE Calculator — the Bitcoin asset offers both a CAGR input and a Power Law preset