FINM7405 Chap.2 Volatility, Correlation and Tail Risk
Volatility, Correlation and Tail Risk
Volatility measures return dispersion, but financial risk changes through time and concentrates in the tails. This chapter moves from sample variance to conditional volatility, showing how ARCH reacts to squared shocks and GARCH carries both new information and lagged variance. It then combines asset risks through covariance and explains why crisis correlation can erode diversification.
Skewness, kurtosis, fat tails and Extreme Value Theory extend the analysis beyond symmetric normal dispersion so that a risk estimate addresses the losses most relevant to capital and solvency.
The calculation sequence matters. Variance keeps squared units, volatility returns to the original return scale, and covariance combines asset volatilities with correlation.
A GARCH update adds a constant, a weighted squared innovation and lagged conditional variance. Because the innovation is squared, both a large gain and a large loss raise the forecast. A high persistence sum means the effect of a shock decays slowly.
Portfolio volatility introduces dependence.
Lower correlation can reduce the total below a weighted average of stand-alone volatilities, but that benefit is conditional on the estimate. During crises, assets exposed to the same funding or macro shock can move together. The final layer examines distribution shape: negative skew locates the longer loss tail, while excess kurtosis warns that a normal model assigns too little frequency to extremes.
Volatility, dependence and tail shape are three distinct questions.
What this chapter covers
- 01
Variance and standard deviation
- 02
Historical and implied volatility
- 03
ARCH and GARCH conditional variance
- 04
Correlation and portfolio diversification
- 05
Skewness, kurtosis and fat tails
Worked example · free
Updating a GARCH variance forecast
This five-point weighting organises the GARCH update and is not a published university assessment scheme.
- +1Square the shock: (-3.00)^2 = 9.00.
- +1Weight new information: 0.08 x 9.00 = 0.72.
- +1Carry memory: 0.90 x 1.00 = 0.90.
- +1Add the floor: 0.05 + 0.72 + 0.90 = 1.67.
- +1Take the square root: volatility is approximately 1.292%.
Key terms
- Conditional Variance
- A variance forecast that changes with current information and recent shocks.
- Volatility Clustering
- The tendency for large return movements to follow large movements and calm periods to follow calm periods.
- Correlation
- A standardised measure of linear co-movement between two returns.
- Kurtosis
- A distributional property associated with tail weight and the frequency of extreme observations.
Volatility, Correlation and Tail Risk FAQ
Why does GARCH remember a shock?
The new squared shock enters through alpha, while beta carries the previous conditional variance into the new forecast. When alpha plus beta is close to one, the effect fades slowly rather than disappearing after one calm day.
How does correlation affect portfolio volatility?
Correlation enters the covariance cross-term. Lower correlation reduces portfolio variance for given weights and volatilities, while correlation near one removes much of the diversification benefit. Stressed correlation should be examined because dependence can rise during crises.
Why is volatility alone incomplete?
Standard deviation treats positive and negative moves symmetrically and says little about tail shape. Negative skew, excess kurtosis and fat-tail analysis matter when the decision focuses on large losses rather than ordinary dispersion.
Exam move
Keep a unit ledger for every calculation. Compute GARCH in variance units, then take a square root only when volatility is requested. Reproduce the calm, shock and decay path so you can explain persistence without relying on a graph. For portfolio risk, write both stand-alone variance terms and the covariance cross-term, then test the result against the weighted average volatility.
Change correlation while holding weights fixed to see exactly where diversification enters. Build a three-column comparison of historical, implied and conditional volatility, focusing on the information each uses and the decision each supports. Pair every number with a limitation: parameter stability for GARCH, correlation stability for diversification, and distribution shape for tail risk.
When an answer mentions a normal model, ask whether negative skew or excess kurtosis makes the left tail more dangerous than the symmetric estimate suggests. Finish by recomputing a two-asset portfolio, interpreting the reduction, and rewriting the conclusion for a stressed correlation state. Build a small diagnostic sheet with raw returns, squared returns, conditional variance and conditional volatility in separate columns.
This makes it harder to mix percent, decimal and variance units. After each GARCH update, compare the new variance with the long-run level and explain whether mean reversion should push the next forecast up or down. For correlation problems, recompute the portfolio under correlations of one, zero and a stressed positive value; these boundary cases reveal impossible answers immediately.
Then ask whether the covariance matrix is internally consistent and whether the weights sum to the intended net or gross exposure. Treat annualisation as an assumption rather than a formatting step, especially when clustering or serial dependence is visible.
For tail interpretation, sketch how negative skew changes loss asymmetry and how excess kurtosis raises the frequency of large moves without necessarily changing ordinary-day variance. Compare a calm-window estimate with a shocked-window estimate and explain the governance consequence for limits. Before finishing, translate the statistic into a decision: position size, capital, hedge urgency or scenario design.
A number without that bridge is measurement, not risk management.
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