Structural breaks

1. Structural breaks

We'll now investigate how structural changes affect our measurement and estimation of risk in real data.

2. Risk and distribution

Up to now our main tool for risk mitigation is Modern Portfolio Theory, while our main measurement tools are Value at Risk and Conditional Value at Risk. We have identified risk with dispersion or volatility of risk factors, giving us statistical variance as a quantitative measure. In turn, this provides a connection between risk and an underlying statistical distribution that risk factors are assumed to follow.

3. Stationarity

This connection also assumes that the underlying distribution is the same over the time the risk factor data is collected. In other words, the loss (or returns) distribution is stationary. But as we saw in Chapter 1, the efficient frontier was different depending upon which part of the global financial crisis period was studied. So it may be the case that the distribution of losses is not stationary. This poses a problem for our estimation of risk, because all three estimation procedures we discussed in the last video (historical, parametric, and Monte Carlo) assume stationary distributions.

4. Structural breaks

If we knew that the distribution was changing over time, and we knew when distribution changes occur, then we could estimate our risk measures only during sub-periods when the distribution didn't change. In other words, distributions would be stationary only within certain sub-periods of the data. We first need to know when the data suggest that an underlying distribution is changing. This is the known as a "structural break" point in the data. You can think of a structural break as a change in a “trend”.

5. Example: China's population growth

As an example of a structural break, let's examine the population growth rate of China from 1950 to 2019. We can see that the growth is relatively linear over this period,

6. Example: China's population growth

but around 1990 the growth rate slows down. This indicates that there may be a structural break around 1990. In other words, the underlying distribution of net population gains (births minus deaths) changed, causing there to be fewer net gains from 1990 onwards. Some reasons for such a change could be changes in government policy, changes in living standards, and other micro- and macro-level factors.

7. The Chow test

A visual examination is useful to identify structural breaks. But there are also powerful statistical tools that can be applied to data. One tool is the “Chow Test”, which asks if the data support a structural break at a given break point of time, for a given linear risk factor model. Its null hypothesis is no structural break. In the test, three ordinary least squares regressions are performed: one for the entire period, and two for before and after the break point. The sum-of-squared residuals are then collected, and the Chow test statistic is created. The statistic is distributed as an "F"-distribution statistic.

8. The Chow test in Python

Let's use the Chow test to see if there was a structural break somewhere around 1990 in China's population growth. We'll use a very simple "factor model" where log-population is regressed against year. First we perform an OLS regression over the 1950 - 2019 period, regressing the natural logarithm of population against year and an intercept. This gives our 'baseline' regression, assuming no structural break. The sum-of-squared residuals from the regression are stored.

9. The Chow test in Python

Then we break the time period into two sub-periods: 1950 - 1989 and 1990 - 2019, and run regressions for each sub-period. We then store the sum of squared residuals for both regressions.

10. The Chow test in Python

Finally, we compute the Chow test statistic using the three sum of squared residuals and the degrees of freedom of the test. The Chow test statistic should be distributed as an "F" distribution. The "F" distribution has two degrees of freedom: the first (used in the numerator) is the number of regression parameters, here equal to 2: the intercept and slope coefficient. The second (used in the denominator), is the total number of data points, 70, minus twice the first degree of freedom, or 70 minus 4, which is 66. The computed test statistic is statistically different from zero at the 99-point-9% confidence level. We can reject the null hypothesis that there was no structural break in the data.

11. Let's practice!

Now you’ll apply the Chow test to identify a possible structural break in the investment bank portfolio data during the global financial crisis.

This exercise is part of the course

Quantitative Risk Management in Python

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Risk management begins with an understanding of risk and return. We’ll recap how risk and return are related to each other, identify risk factors, and use them to re-acquaint ourselves with Modern Portfolio Theory applied to the global financial crisis of 2007-2008.

Exercise 1: Welcome!Exercise 2: Portfolio returns during the crisis Exercise 3: Asset covariance and portfolio volatility Exercise 4: Risk factors and the financial crisis Exercise 5: Frequency resampling primer Exercise 6: Visualizing risk factor correlation Exercise 7: Least-squares factor model Exercise 8: Modern portfolio theory Exercise 9: Practice with PyPortfolioOpt: returns Exercise 10: Practice with PyPortfolioOpt: covariance Exercise 11: Breaking down the financial crisis Exercise 12: The efficient frontier and the financial crisis

Now it’s time to expand your portfolio optimization toolkit with risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). To do this you will use specialized Python libraries including pandas, scipy, and pypfopt. You’ll also learn how to mitigate risk exposure using the Black-Scholes model to hedge an options portfolio.

Exercise 1: Measuring Risk Exercise 2: VaR for the Normal distribution Exercise 3: Comparing CVaR and VaR Exercise 4: Which risk measure is "better"?Exercise 5: Risk exposure and loss Exercise 6: What's your risk appetite?Exercise 7: VaR and risk exposure Exercise 8: CVaR and risk exposure Exercise 9: Risk management using VaR & CVaR Exercise 10: VaR from a fitted distribution Exercise 11: Minimizing CVaR Exercise 12: CVaR risk management and the crisis Exercise 13: Portfolio hedging: offsetting risk Exercise 14: Black-Scholes options pricing Exercise 15: Options pricing and the underlying asset Exercise 16: Using options for hedging

In this chapter, you’ll estimate risk measures using parametric estimation and historical real-world data. You'll then discover how Monte Carlo simulation can help you predict uncertainty. Lastly, you’ll learn how the global financial crisis signaled that randomness itself was changing, by understanding structural breaks and how to identify them.

Exercise 1: Parametric Estimation Exercise 2: Parameter estimation: Normal Exercise 3: Parameter estimation: Skewed Normal Exercise 4: Historical and Monte Carlo Simulation Exercise 5: Historical Simulation Exercise 6: Monte Carlo Simulation Exercise 7: Structural breaks

Current Exercise

Exercise 8: Crisis structural break: I Exercise 9: Crisis structural break: II Exercise 10: Crisis structural break: III Exercise 11: Volatility and extreme values Exercise 12: Volatility and structural breaks Exercise 13: Extreme values and backtesting

It's time to explore more general risk management tools. These advanced techniques are pivotal when attempting to understand extreme events, such as losses incurred during the financial crisis, and complicated loss distributions which may defy traditional estimation techniques. You’ll also discover how neural networks can be implemented to approximate loss distributions and conduct real-time portfolio optimization.

Exercise 1: Extreme value theory Exercise 2: Block maxima Exercise 3: Extreme events during the crisis Exercise 4: GEV risk estimation Exercise 5: Kernel density estimation Exercise 6: KDE of a loss distribution Exercise 7: Which distribution?Exercise 8: CVaR and loss cover selection Exercise 9: Neural network risk management Exercise 10: Single layer neural networks Exercise 11: Asset price prediction Exercise 12: Real-time risk management Exercise 13: Wrap-up and Future Steps