Modeling drawdowns and drawups in financial markets - Risk.net

More documents

Recommendations

Info

54 Beatriz Vaz de Melo Mendes and Vinicius Ratton Brandi FIGURE 1 Drawdowns and drawups during a period of 50 days for the CAC40. 4300 4400 4500 4600 4700 0 10 20 30 40 50 Drawdowns are indicated by sequences of downward empty triangles, and drawups by sequences of upward filled triangles. markets. 3 Two examples of such common factors are the transaction volume or the number of trades (see also Geman and Ané, 1996). Dependent subordinated processes may induce sequences of successive drops when measured with a regular time scale. Nowadays, however, there is a growing concern whether fixed physical time would be the most appropriate basis for collecting data for tracking stock price moves and tendencies. Recently, the literature has provided alternative ways for collecting and analyzing financial data, which aim to reveal some characteristics (so far intangible) that are not captured by series of returns computed daily or monthly. Müller et al (1995), Dacorogna et al (1996), and Guillaume et al (1995) have used the concept of elastic time, which suggests the expansion of periods of large volatility and the contraction of periods of low volatility. This data manipulation would “normalize” the magnitude of returns relatively to the duration of some stress period and thus provide a better understanding of the importance of such events. Levitt (1998) presented an alternative time scale produced by dynamic sampling techniques, which was used to improve the performance of technical analysis and trading systems. Mandelbrot (1972), Clark (1973), and Johansen and Sornette (2001) argue that the drawdown, being a direct measurement of the cumulative loss that can affect an investment, may be a more suitable random variable for capturing the dynamics and price flow of an investment. For example, Figure 1 shows sequences of drops and rises in the price of the 3 Strong local dependence is observed during these periods of turbulence. However, there may also exist (spurious) short-range, positive autocorrelation in financial series. It may be caused by the lack of synchrony among closing prices of stocks composing a stock market index; or by the low liquidity of some of these components; or by the information asymmetry caused by inside information. URL: www.thejournalofrisk.com Journal of Risk
Modeling drawdowns and drawups in financial markets 55 Paris stock exchange index, CAC40 (France). A drawdown is defined as the price change between the last filled triangle in a sequence of rises and the last empty triangle in the following sequence of drops. Let P t represent the price of an index or a portfolio at day t, and r t = ln(P t ⁄P t –1 ) be the log return. Suppose that P t –1 is a local maximum, ie, P t–2 ≤ P t –1 > P t , and consider a sequence of d + 1 price drops, ie, P t –1 > P t > P t +1 > … > P t + d . This results in a sequence of d negative log returns, r t , r t +1 ,…, r t + d . The drawdown Y, with duration D equal to d days, is defined as Y = ln(P t + d ⁄P t –1 ) = r t + d + r t + d –1 + … + r t . The same concept applies to the drawups. Both quantities – the severity Y and the duration D of a drawdown – are not collected over a fixed time scale, and their probability distributions may provide a new insight on the risk profile of an investment. For example, for a client of a financial institution with or without expertise in the field, the drawdown is a very impressive indicator of what is occurring to his wealth. He may decide to invest elsewhere purely on the basis of the sizes and durations of his investments’ drawdowns. In this paper, we use parametric models from the extreme value theory to characterize the distribution of the severity of drawdowns and drawups. We illustrate these concepts using nine stock market indices. We show that the modified generalized Pareto distribution (MGPD), and its nested models, adjust properly to the data, giving a good fit to most of those observations found to be “atypical” under the stretched exponential fit used by Johansen and Sornette (2001). The MGPD is a flexible distribution that encompasses as a sub-model the pure exponential, which, for many data sets, models adequately the distribution of drops under independence. In Section 2, we present the statistical models and methods used for inference. Theorems from extreme value theory supporting the use of the MGPD are briefly reviewed. In Section 3, we fit the stretched exponential and the MGPD to the drawdowns and drawups of nine stock markets indexes. We compute an accurate estimate of the drawdown-at-risk (DAR α ) which, like the unconditional VAR α , is basically the (1 – α)-quantile of the fitted distribution. Results from extreme value theory allow us to estimate the distribution of the conditional drawdown-atrisk. Finally, in Section 4, we summarize the results and present our conclusions. 2 Models and statistical inference for drawdowns 2.1 Statistical models Let Y represent the drawup or the absolute value of the drawdown. Johansen and Sornette (2001) suggested approximating the finite sample distribution of Y using a stretched exponential (actually, the well-known Weibull distribution), which has cumulative distribution function y N ( y) = − exp − (1) ⎧ γ ⎫ 1 ⎨ ⎬ ⎩ ψ ⎭ Volume 6/Number 3, Spring 2004 URL: www.thejournalofrisk.com
Page 1: Modeling drawdowns and drawups in f
Page 5 and 6: Modeling drawdowns and drawups in f
Page 9 and 10: TABLE 3 Parameters of maximum likel
Page 17: Modeling drawdowns and drawups in f

Modeling drawdowns and drawups in financial markets - Risk.net

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?