Measuring volatility

Two analysts measure the volatility of the same equity index fund on the same day. One reports 14.1% a year, the other 24.0%. Neither has made a mistake. They just used different stretches of history.

You met volatilityThe standard deviation of an asset's returns: how widely they typically swing around their average. in L1.1.08: Returns and risk: a first look as the size of a typical swing, and in L1.5.06: Equity returns and volatility as the standard deviation of returns, scaled to a year. Every risk measure in this module starts from it. Here we estimate it properly from data, and look at the choices that change the answer.

Why n − 1 and not n? The mean was worked out from the same returns, so the returns sit a little closer to it than to the true, unknown average. Dividing by n − 1 makes up for that. The result is the sample standard deviationThe standard deviation estimated from data, dividing the sum of squared deviations by n − 1., and it is the convention in this module whenever a standard deviation is estimated from data. With 10 days it makes the answer 5.4% larger than dividing by n; with 500 days, only 0.10%.

Over all 500 days, the equity index fund's daily volatility is 1.10% and the bond fund's is 0.32%.

Scaling with the square root of time

Volatility is usually quoted per year. If each day's return is independent of the others and has the same volatility, the variances of the days add up. Over h days the variance is h times the daily one, so the volatility is $\sqrt{h}$ times the daily one. We use 252 trading days a year:

• Equity index fund: 1.10% × √252 = 17.4% a year, or 3.47% over 10 days.
• Bond fund: 0.32% × √252 = 5.1% a year.

The square-root-of-time ruleScaling a volatility from one horizon to another by the square root of the ratio of the horizons. is the standard convention, but it rests on assumptions that real markets break:

• Volatility clusters. Calm days follow calm days and stormy days follow stormy days (volatility clusteringThe tendency of calm days to follow calm days and stormy days to follow stormy days.). Today's regime says more about the next 10 days than the long-run average does.
• Returns are not independent. If moves tend to continue, they pile up and the long-run spread is wider than √h suggests. If they tend to reverse, it is narrower.
• Positions change. Over a long horizon a portfolio is traded and hedged, so the thing whose volatility you scaled no longer exists.
• Tails behave differently. The rule scales the standard deviation, not the shape of the extremes. That matters for value at risk (L2.5.06: Value at risk: the idea).

How much history? The window

The windowThe stretch of past data used for an estimate, such as the last 60 or 250 trading days. is how many past days go into the estimate. Here is the equity index fund, measured on day 500 with five different windows:

The 120-day window covers the turbulent spell from about day 381 to day 440, so it gives 24.0%. The 20- and 60-day windows see only the calm weeks since, and give 14.1% and 13.2%. That is the hook's puzzle. A short window reacts quickly to a change in conditions but jumps around; a long window is steady but slow, and can remember a storm long after it has passed.

A popular middle way gives every day a weight that fades with age. In the exponentially weighted moving averageAn exponentially weighted moving average: each older squared return gets a smaller weight, set by a decay factor λ. (EWMA), yesterday's squared return gets weight 1 − λ, the day before (1 − λ)λ, and so on. RiskMetrics used λ = 0.94 for daily data. On day 500 that gives the equity fund an annualised volatility of 12.8%: closer to the calm recent weeks, but still remembering the storm.

Write $r_t$ for the return on day $t$, $n$ for the number of days and $\bar r$ for their mean, $\bar r = \frac{1}{n}\sum_{t=1}^{n} r_t$. The sample variance is new notation, so let's build it one piece at a time.

The square-root-of-time rule, on a few lines. Over $h$ days the return is (approximately, for small daily moves) the sum of the daily returns. If the days are independent with the same variance $\sigma^2$:

The first line is where independence is used: without it, covariance terms between days appear (you will meet covariance in L2.5.03: Correlation and diversification). The sum of daily returns is exact for log returns and a close approximation for simple returns over a few days (L1.5.06: Equity returns and volatility).

Start with the equity index fund and a 60-day window. Slide the window up to 120 days and watch the estimate jump as the turbulent spell comes in. Then try 500, and switch to the bond fund. In the lower chart, notice how much a 20-day window swings compared with a 250-day one.

Estimate the equity index fund's volatility from its last 10 days (days 491 to 500), then compare with the full 500 days. Returns are shown in per cent, so the squared deviations are in per cent squared. Try each step before revealing it.

Supervisors care about the data window because a model fed only calm data underestimates risk. When the Basel Committee first allowed banks to use their own value-at-risk models for market risk capital, in the 1996 Market Risk Amendment, it required at least one year of historical data, updated at least every three months. After the 2008 crisis, the 2009 revisions (known as Basel 2.5) added a stressed VaR, calibrated to a 12-month period of significant market stress, so that capital could not fall just because markets had been quiet. The detail is in M4.3: Market risk capital (FRTB) and CVA, part of Level 4: Regulation, Capital & Accounting.

• Daily volatility is estimated by the sample standard deviation of daily returns: square the deviations from the mean, add them, divide by n − 1 and take the square root.
• Annualise with the square root of time: σ_annual = σ_daily × √252. Over h days, multiply by √h, never by h.
• The square-root rule assumes independent days with constant volatility. Volatility clustering, trends, mean reversion and changing positions all break it.
• The window matters: short windows react fast but are noisy; long windows are steady but slow. Always state the window.
• EWMA weights recent days more heavily, a middle way between the two.

Volatility

The standard deviation of an asset's returns: how widely they typically swing around their average.

Sample standard deviation

The standard deviation estimated from data, dividing the sum of squared deviations by n − 1.

Annualise

Restate a figure for a shorter period, such as a day, as its equivalent for a year.

Square-root-of-time rule

Scaling a volatility from one horizon to another by the square root of the ratio of the horizons.

Window

The stretch of past data used for an estimate, such as the last 60 or 250 trading days.

Volatility clustering

The tendency of calm days to follow calm days and stormy days to follow stormy days.

EWMA

An exponentially weighted moving average: each older squared return gets a smaller weight, set by a decay factor λ.