Think Value

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How to Assess a Stock’s Risk Based on Historical Data

We know that in finance, “risk” usually refers to volatility, i.e., uncertainty regarding changes. Here, I want to highlight a few important points.

The volatility (σ) measure is very commonly used in practical finance when we:

• assess the riskiness of an asset,
• construct a stock portfolio, or
• even calculate an option’s value.

It is therefore important to understand clearly what this measure captures and what it does not:

1. This measure is not Beta (β).
   • Beta measures relative risk, describing the correlation between a stock’s price and the market.
   • Volatility (σ) measures absolute risk (standard deviation).
2. This is not the volatility of stock prices themselves, but the volatility of returns (i.e., percentage changes in stock prices).
3. This is not simple arithmetic return volatility, but log-normal volatility (based on continuously compounded returns).

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Let’s look at an example of a non-dividend-paying stock. Below is a table showing the stock’s price changes over 21 trading days:

The third column shows the stock’s price relative to the previous day (price change).

The fourth column shows the logarithm of that change, to convert returns to a continuously compounded basis.

The formula used to calculate volatility is:

In our example, the daily volatility is 1.21%.

To convert daily volatility to annual volatility, we account for the number of trading days and use the formula:

For our example, the annual volatility is 19.3%.

We can also estimate the standard error (uncertainty) of this volatility estimate using:

In our example, the standard error is 3.05%.

Excel File: Estimating Volatility

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P.S.
These calculations can be easily adapted for dividend-paying stocks by including the dividend amount in the return calculation on ex-dividend days.

P.P.S.
In practice, professionals usually use the number of trading days, not calendar days, when calculating annual volatility, because volatility on non-trading days is effectively zero. The same principle applies to options pricing: when we talk about an option’s time to maturity in years, we mean trading-year equivalents.

Adapted from: Options, Futures & Other Derivatives, John C. Hull

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