Geometric Drift Index Calculator

Calculate the geometric drift index — the volatility-adjusted compounded growth rate (mean return minus half the variance) of a geometric Brownian motion process — along with volatility drag and projected values.

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years
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Results

Calculated
Geometric Drift Index
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Annualized compounded growth rate (μ − σ²/2)
Volatility Drag
—
Growth lost to variance (σ²/2)
Projected Value (Typical Path)
—
Median outcome after N years using the geometric drift
Projected Value (Expected Path)
—
Mean outcome after N years before volatility drag

How to use this calculator

Enter the expected annual return (μ), the annual volatility (σ), a time horizon in years, and a starting value, then click Calculate. The calculator returns the geometric drift index — the compounded growth rate after adjusting for volatility — plus the volatility drag and two projected values. Click Clear to reset all fields and start a new calculation.

The formula

For a process that follows geometric Brownian motion (the standard model behind log-normal growth, used for things like stock-price and portfolio-value modeling), the arithmetic mean return μ and the compounded, or geometric, growth rate are not the same number. The geometric drift index is:

Geometric drift = μ − σ²/2

where μ is the expected annual return (as a decimal) and σ is the annual volatility, or standard deviation of returns (as a decimal). The term σ²/2 is called volatility drag: it is the amount by which compounding pulls the typical, or median, growth path below the simple average return. Projected values follow from continuous compounding: the typical path after t years is Starting Value × e^((μ − σ²/2) × t), while the expected (mean) path is Starting Value × e^(μ × t).

Understanding the inputs

Expected Annual Return (μ) is the arithmetic average return you expect per year, entered as a percentage. Annual Volatility (σ) is the standard deviation of those annual returns — a measure of how much they swing around the average — also entered as a percentage. Time Horizon is how many years to project forward, and Starting Value is the dollar amount at time zero.

Interpreting the results

The Geometric Drift Index is the headline number: it is the rate at which a typical (median) outcome actually compounds once volatility is taken into account, and it is always less than or equal to μ. Volatility Drag shows exactly how much return volatility costs you — the higher the volatility, the bigger the gap. The two projected values let you compare the typical outcome against the average outcome: the expected (mean) path is pulled up by rare large gains, while the typical (median) path better reflects what most single outcomes actually look like.

Frequently Asked Questions

What is the geometric drift index?
The geometric drift index is the annualized growth rate of a geometric Brownian motion process after adjusting for volatility, computed as the mean return minus half the variance (μ − σ²/2). It is the rate at which the typical, or median, path of the process actually compounds, as opposed to the arithmetic average return.
Why is the geometric drift lower than the average return?
Compounding is multiplicative, so a loss and an equal-sized gain do not cancel out: a 50% drop needs a 100% gain to recover. This asymmetry pulls the compounded (geometric) growth rate below the simple arithmetic mean, by an amount equal to half the variance. This gap is often called volatility drag.
How do I use the geometric drift index to project a value?
Multiply the starting value by e raised to the geometric drift times the number of years to get the typical (median) projected value. Multiplying by e raised to the arithmetic mean return times years instead gives the expected (mean) value, which is always at or above the typical path once volatility is positive.
What are the limitations of this model?
The formula assumes returns follow a geometric Brownian motion: a constant annualized mean return and volatility, and log-normally distributed outcomes. Real-world returns can have fat tails, changing volatility, and jumps that this simplified model does not capture, so treat the result as an estimate rather than a guarantee.