31 March 2016
- 6 minute read

Each curve increases by 1/n when a shock occurs for a single market in a sector containing n markets (or for the global total, when data for n markets are available). Totals across all sectors are shown in the top chart using 33-day and five-year volatilities. Blue boxes show mean shocks per market over each ten-year period across all sectors using 33-day volatility. The curve for each sector in the bottom chart starts when daily data from the first market becomes available and is calculated using 33-day volatility. See Appendix for more.

The increase on 3 December was particularly extreme against the background of recent returns.

Returns are divided by the previously estimated 33-day volatility. The three largest negative and positive returns are labelled.

Each curve increases by one when a shock occurs. A shock is defined as a volatility-adjusted return large enough to occur only once in a thousand years for normally distributed returns. For comparison, we also show (dashed) the total for simulated returns with a student’s t distribution having 7 degrees of freedom (using 33-day volatility).

* Daily excess returns calculated using total asset returns and US treasury-bill rate. Other markets’ excess returns are calculated from back-adjusted futures returns, generally using the front contract (closest to expiry) and rolling to the next a few days or weeks before expiry.

[1] For example: Gavin Jackson, Why market volatility is growing more intense, Financial Times, 14 September 2015; D. Sornette and S. von der Becke, Crashes and high-frequency trading, UK government Foresight report August 2011.

[2] We might have expected a event to occur once in a thousand years for normal data, since allowing for 262 business days in a year, we find , where is the normal cumulative distribution function. But if we estimate the volatility using a 33-day moving average the frequency of recorded shocks is a little higher than we expect because of fluctuations in the estimated volatility. For a five-year volatility the frequency is closer to the expected one.

[3] For financial assets such as stock indices, the futures returns are very close to the excess total returns from holding the asset (“excess” meaning over the risk-free rate, and “total” meaning including dividends for stocks and coupon payments for bonds). We therefore use futures returns back to the start of futures trading, and the excess returns from holding the index assets before this date.

[4] In fact, a t-distribution with 5 to 10 degrees of freedom is a better model of market returns, as shown in Figure 4.