The Value at Risk (VaR) concept emerged from the rubble in the aftermath of the market crash of 1987. Likewise, the current financial crisis has stimulated the development of new approaches to better measure and control our risk exposure.
This development is critical. New complex financial products are drawing traders and investors away from plain assets. Risk evaluation tools must evolve with these changing investing practices.
Last month, we looked at the shift from one-sided risk estimation to evaluating algorithms based on different principles. In addition, we discussed the extension of risk management from after-the-fact assessment to active utilization at the portfolio composition and structuring stage. We looked at four indicators that could help fill these roles: index delta, asymmetry coefficient, loss probability and traditional VaR. (For detailed explanations and descriptive equations, see the previous article.)
• Index delta. This gauge expresses the risk of a complex portfolio containing options on different underlying assets.
• Asymmetry coefficient. This indicator expresses the skewness of the payoff function of the option portfolio.
• Loss probability. A portfolio can yield a loss. The probability of this event can be estimated by Monte Carlo simulation.
• Value at Risk. VaR is an estimate of a maximum loss, which will not be exceeded with a given probability.
In practical application, the tools used must be unique and independent (not correlated). Each should supplement the information contained in the other indicators, not duplicate it. In this issue, we will look at these four measures with that goal in mind.
CORRELATION ANALYSIS
If different risk indicators are unique and do not duplicate each other, portfolios created using those risk indicators should be uncorrelated.
To test the interrelationship of the four risk indicators, we used prices of options and their underlying assets from January 2003 until August 2009. For each expiration date, we created 60 series of portfolios (each series comprised of 1,000 portfolios) corresponding to different time intervals left until the expiration. The most distant series was set up 60 days before the expiration, the next one 59 days before, and so on, right up to the last series. Thus, each number-of-days-to-expiration was represented by 1,000 portfolios, which gives 60,000 portfolios for each expiration day.
Each portfolio consisted of 10 short straddles for underlying assets randomly chosen from the S&P 500 index. Each straddle was constructed using the strike closest to the current stock price. The quantity of options corresponding to each underlying was determined as 1,000,000/x, where x is the price of the stock.
Within each series, values of the four risk indicators were calculated for each of the 1,000 portfolios. Subsequently, the best portfolio was selected for each indicator (thereby, four portfolios were chosen from every series). The returns of the selected portfolios were recorded on the expiration date. The returns were expressed as profit or loss normalized by the investment amount and by the time spent in a position.
As expected, the returns of the portfolios selected on the basis of the four risk indicators were interrelated to a certain extent (see “Interdependence of risk indicators,” right). However, apart from a single exception, the correlations turned out to be relatively low. In five of the six cases, the determination coefficient (squared correlation coefficient) was within the range of 0.3 to 0.4. This implies that information contained in our risk gauges is duplicated only by 30%-40%. Hence, introduction of an additional indicator to the risk evaluation system based on a single criterion can enrich this system with about 60%-70% of new information.
The exception was represented by one pair of indicators, VaR and loss probability. Using these measures together would not add sufficient volume of new information to the risk-management system.
CORRELATION FACTORS
The degree of correlation between risk indicators varies depending on certain factors. Two possibilities were analyzed: the time interval from the moment of portfolio creation until the expiration date and the market volatility at the moment of portfolio creation.
Within one trading day, the interrelationship between risk indicators can be measured by the variance of returns of the four portfolios selected by these indicators on that day. The higher the variance, the lower the correlation of risk indicators. If the indicators are perfectly correlated, each of them chooses the same portfolio; in this case the variance is zero.
“Time effect” (page 47) shows the inverse non-linear relationship between variance and the time interval left until the expiration. It means that close to the expiration risk indicators are weakly correlated and, hence, all of them (or, at least, some of them) carry a considerable load of additional information. On the other hand, the variance of returns of the portfolios created long before the expiration is rather low, which means that risk indicators are strongly interrelated during this period.
Market volatility also influences the interdependence of risk indicators, though this effect is stronger shortly before expiration. Thus, for portfolios created two days before expiration, the correlation coefficient between variance and volatility was r = 0.62 for historical volatility and r = 0.68 for implied volatility (see “Volatility effect,” page 47). Yet, for portfolios created 60 days before the expiration correlation coefficients were much lower (r = 0.24 and r = 0.28, correspondingly).
FURTHER DEVELOPMENTS
The risk indicators discussed in this series represent the type of evaluation tools that can be used to develop a multi-criteria risk management system. These indicators by no means exhaust the list of potential opportunities for creating additional risk forecasting algorithms.
At the same time, risk management systems should not involve too many indicators. This would overburden calculations and lower the efficiency of evaluation procedures. While considering new indicators, their uniqueness should be thoroughly measured. All potential candidates must provide new information not contained in other gauges. Two of the four indicators analyzed in this article duplicated each other — VaR and loss probability. Hence, we should use only one of them in a multi-criteria risk management system.
Our investigation shows that the multi-criteria approach to risk evaluation may be most appropriate when portfolios are formed close to expiration date and when the market volatility is relatively high. Under these conditions, evaluation tools are less correlated. Further research should reveal additional factors influencing the correlation of risk indicators and determine the necessity for additional gauges.
Sergey Izraylevich has a Ph.D. from Hebrew University of Jerusalem. He is chief investment officer of Horton SARL and chairman of High Technology Inc. Vadim Tsudikman is a director at Horton and develops trading systems based on genetic optimization algorithms. Contact the authors at izraylevich@htinvest.ru
