Estimating Volatility for a Two-Outcome Investment Return

Summary

The document asks how a quoted volatility estimate is obtained for a fair coin-flip investment with two possible outcomes: a gain of 50% or a loss of about one-third. It also references the approximation that geometric average return is close to arithmetic average return minus half the squared volatility, and asks for a derivation of that relation. The response explains that volatility in finance is measured as the standard deviation and gives an approximate average return based on the two outcomes.

The reply notes that the reported volatility is rounded and presents a shortcut based on the distances between the average and either outcome. This is a compact explanation of the numerical estimate, not a derivation of the geometric-return approximation requested by the question. It does not discuss assumptions behind the volatility-drag formula or its limits, so readers should treat the cited relationship as an approximation rather than a general identity.

Key ideas

• Financial volatility is measured as the standard deviation of returns.
• For a two-outcome coin-flip example, the average return is calculated from the midpoint of the possible outcomes.
• The response attributes the quoted volatility to rounding and relates it to the spread between outcomes.
• The document asks about the volatility-drag approximation but does not derive it.

Tags

# How to calculate the variance of this coin flip?


# How to calculate the variance of this coin flip?












I am reading the article “Shannon’s Demon & How Returns Can Be Created Out of Thin Air” by Richmond Quantitative Advisors (2021).

The main premise is a fair coin flip. If heads, you gain 50%. If tails, you lose 33.3%. They present the following equation

```
Average Arithmetic Return - Volatility Drag ≈ Geometric Average Return
```

where Volatility Drag = Volatility^2 ÷ 2.

They calculate the volatility as 42%. How was this deduced? Additionally, where could one find a resource that explains the mathematical derivation of both equations?

## Answer by AKdemy (score 2, accepted)

https://quant.stackexchange.com/a/76415

Volatility is computed as the standard deviation (SD) in finance.

- Average return (avg) is (150-66.6)/2 = 108.3



They just rounded it. Simplified, you can also just get the difference between max gain and average (150-avg) = 41.7

Or the difference between avg and min, (108.3 - 66.7) = 41.6

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