Modeling drawdowns and drawups in financial markets - Risk.net

Modeling drawdowns and drawups in 

financial markets 

Beatriz Vaz de Melo Mendes 

Statistics Department/IM-COPPEAD, Federal University at Rio de Janeiro, RJ, Brazil 

Vinicius Ratton Brandi 

COPPEAD, Federal University at Rio de Janeiro, RJ, Brazil 

Periods of turbulence are often characterized by observed consecutive drops 

in prices, called “drawdowns”. In such cases, static, one-period measures of 

risk insufficiently describe downside risk. In this paper, we assess risk by formally 

modeling these random strings of negative returns. We use long-tailed 

distributions from extreme value theory to model the severity and duration 

of the drawdowns. This leads to the concept of drawdown-at-risk (DAR) and 

conditional DAR. The proposed distributions adjust properly to extreme values 

previously found to be outliers. The paper illustrates the importance of these 

concepts with stock market indices. 

1 Introduction 

Periods of turbulence are often characterized by observed consecutive drops in 

prices, called “drawdowns”. According to Mandelbrot (1963), for daily changes of 

stock prices with infinite second moment, the total price change over the span of 

the data is usually concentrated in a few hectic runs of trading days. As noticed by 

traders and investors, in that case, static, one-period measures of risk quite often 

fail to fully describe downside risk. 1 

Many return distributions have infinite second moment, and theoretical 

research in this area has led to the concept of distributions with fractional dimensions. 

2 The study of multifractal trading time was first introduced by Mandelbrot 

(1972) in the context of modeling phenomena showing intermittent turbulence. 

His work was motivated by the observation that the time interval between successive 

transactions, which follows variations in the trading volume, varies greatly, 

suggesting that trading time is related to volume. 

These ideas were pursued by Clark (1973), who argues that there may exist a 

time-dependent subordinated process explaining the turbulent cascades in stock 

The authors gratefully acknowledge Brazilian financial support from CNPq-PRONEX, 

FAPERJ, and COPPEAD research grants. We are also very grateful to the Editor-in-Chief 

Philippe Jorion and an anonymous referee for helpful comments and suggestions. 

1 Examples are standard deviation, value-at-risk (VAR), shortfall. 

2 See Ghashghaie et al (1996), Arneodo et al (1998), and Focardi and Fabozzi (2003) and 

references therein, for a review of related concepts such as fat-tailed, power law, stable distributions 

and scaling. 

Volume 6/Number 3, Spring 2004 53 

URL: www.thejournalofrisk.com

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. 

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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

56 


FIGURE 2 Densities of the MGPD with ξ = 0.6 and γ = 0.5, 1.0, 1.5, and the 

stretched exponential for γ = 0.8, 1.0, 1.2, respectively. 

0.0 0.1 0.2 0.3 0.4 

MGPD 

0.5 

1.0 

1.5 

0 2 4 6 8 10 12 14 

Both distributions possess scale equal to 3. 

0 2 4 6 8 10 12 14 

for y ≥ 0, where ψ is the scale parameter and γ > 0 is a tuning constant. When 

γ =1 we are back to the unit exponential distribution. The class of sub-exponential 

distributions is generated by letting γ < 1. Distributions with γ > 1 are called 

super-exponentials. The right-hand side of Figure 2 shows the stretched exponential 

densities for γ = 0.8, 1, 1.2, and scale ψ = 3. 

Johansen and Sornette (2001) fitted model (1) to several data sets, including 

indexes, commodities and currencies. They found that, typically, this distribution 

gives a good fit to the bulk of the data but underestimates between one and 

10 extreme observations. Their conclusion raises the question of whether these 

extreme observations should actually be considered outliers, or whether a better 

fit could be found for all observations. In fact, the authors considered this issue 

and commented that the success in using the stretched exponential for fitting 

drawdowns does not mean that a better parametrization does not exist. Thus, it 

seems natural to look for a flexible distribution from the extreme value theory 

suitable for modeling long-tailed data. 

In the 1990s we saw a large amount of research applying techniques from 

the extreme value theory in risk management. Examples include Longin (1996), 

Danielsson and de Vries (1997), McNeil and Frey (2002), Embrechts, Resnick, 

and Samorodnitsky (1998), among others. Extreme value theory is mostly concerned 

with the study of the behavior of extremes from a strictly stationary 

stochastic process {X i }, i = 1, 2,…, where all X i has a common distribution F X . A 

main result is the Fisher–Tippett theorem (Fisher and Tippett, 1928), which gives 

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0.0 0.1 0.2 0.3 0.4 

Stretched exponential 

0.8 

1.0 

1.2


the limit distribution of the normalized maximum M n , M n = max 1 ≤ i ≤ n X i . This 

non-degenerated limit distribution H ξ depends on a shape parameter ξ and must 

be one of the three types of extreme value distributions. In this case, we say that 

F X belongs to the maximum domain of attraction of an extreme value distribution, 

4 and write X ∈MDA(H ξ ) for some ξ. 

According to the Balkema–de Haan–Pickands theorem (Balkema and de 

Haan, 1974; Pickands, 1975), X ∈MDA(H ξ ) if and only if there exists a measurable 

positive function a(·) such that, for 1 + ξx > 0 

lim 

u↑xFX FX ( u + xa( u) 

⎧( 

1+ 

ξ x) 

−1 

ξ 

⎪ , if ξ ↑ 0 

= ⎨ 

F( u) 

e− x 

⎩⎪ , if ξ=0 

where, for any distribution function F, xF is the right endpoint of the support of F, 

and F¯ (x) = 1 – F(x). 

Let u be a high threshold, a value close to xFX . The result above suggests 

an approximation for the distribution of the extreme tail of X. Provided that 

X ∈ MDA(Hξ ), the conditional distribution of the excess beyond u given that 

X > u, normalized by a scaling factor a(u), is approximated by the generalized 

Pareto distribution (GPD), which has distribution function 

⎧ y ξ 

⎪1 

− ( 1+ 

ξ ) −1 

, if ξ ↑ 0 

Pξ( y) 

= ⎨ 

− − y 

⎩⎪ 1 e , if ξ=0 

for y ≥ 0 if ξ ≥ 0, and 0 ≤ y ≤ –1⁄ ξ if ξ < 0. The scale family is generated when y is 

replaced by y ⁄ ψ. Note that ξ = 0 corresponds to the unit exponential distribution. 

These results hold whenever the observations X 1 ,…, X n are independent and 

identically distributed (i.i.d.). For series presenting weak long-range dependence, 

Leadbetter, Lindgren, and Rootzén (1983) show that the convergence theorems 

still hold. However, the convergence may be slower than in the i.i.d. case. 5 

Even though the GPD seems to be a good candidate to model long-tailed 

observations, we go further and propose the use of a more powerful model, the 

modified generalized Pareto distribution. This distribution seems suitable for 

modeling data aggregated according to some subordinated process, as explained 

in Anderson and Dancy (1992). The subordinator causing this temporary dependency 

could be either a measurable external variable or some subjective fact. 6 We 

propose fitting the MGPD to the severity of the drawups and (absolute value of) 

drawdowns. The MGPD has distribution function given by 

4 See details in Embrechts, Klüppelberg, and Mikosch (1997). 

5 As noted by the referee, the drawdowns and drawups may show weak short range dependence. 

We do not consider this issue here and leave it open for a future work. 

6 For more details on the two-dimensional point process related to the extremal behavior of 

{X j } see Mori (1977) and Hsing (1987). 

Volume 6/Number 3, Spring 2004 URL: www.thejournalofrisk.com 

(2)

58 


( ) ↑ 

⎧ yγ 

−1 

ξ 

⎪ 

1− 1+ 

ξ , if ξ 0 

ψ 

G( γ, ξ, ψ; 

y) 

= ⎨ 

yγ 

⎪ − 

ψ 

⎩⎪ 

1− 

e , if ξ=0 

where γ > 0, and ψ > 0 is a scale parameter. We observe that the GPD (2) may be 

obtained from (3) by letting γ = 1, and that the stretched exponential corresponds 

to ξ = 0. As pointed out by the referee, fitting the MGPD to Y is equivalent to 

taking a Box–Cox transformation of Y, and modeling Y γ using equation (2). We 

chose to keep the random variable Y, the drawdown or drawup, and to fit the large 

family MGPD, which allowed us to pursue standard log-likelihood nested statistical 

tests. 

The MGPD density function is given by 

−1− ξ 

⎧ 

⎪ 

g( γ, ξ, ψ; 

y) 

( ξψ yγ ξ ) ψ γ yγ 

= 

1+ −1 −1 −1, 

if ξ ↑ 0 

⎨ 

e− ψ−1y γ 

ψ−1γ γ −1 

⎩⎪ 

y ξ 

( ) , if =0 

The left side of Figure 2 illustrates the flexibility of the MGPD density according 

to the shape ξ, the scale ψ, and the “modifying” γ parameters. In this plot 

ξ = 0.6, ψ = 3, and γ = 0.5, 1.0, 1.5. When γ < 1 the densities are strictly decreasing 

with heavier tails; the GPD is recovered by setting γ = 1; γ > 1 induces concavity. 

The MGPD can accommodate several distribution shapes and may present fatter 

tails than the stretched exponential (at the right-hand side). 

Summarizing, the following three constrained models are obtained from the 

full model MGPD(γ, ξ, ψ): (1) The MGPD(γ, 0, ψ), or the stretched exponential 

(SE); (2) The MGPD(1, ξ, ψ), or the GPD; (3) The MGPD(1, 0, ψ), or the unit 

exponential distribution. 

2.2 Estimation and tests 

We fit by maximum likelihood the full model MGPD(γ, ξ, ψ) and the three constrained 

models to the samples y 1 , y 2 ,…, y n , containing n drawups or absolute 

value of drawdowns. Other estimation methods may be found in the literature, 

such as the method of moments (see Embrechts, Klüppelberg, and Mikosch, 1997, 

or Bayesian methods – Reiss and Thomas, 1999). We chose to estimate by maximum 

likelihood because these estimates possess good (asymptotic) properties and 

allow the use of well-known statistical tests. 

The standard likelihood ratio test is used to discriminate between the nested 

models. Standard errors of the estimates may be obtained from the inverse of the 

observed information matrix. Confidence intervals based on simple bootstrap 

techniques are also easily obtained. 

Goodness of fits may be checked graphically by means of QQ-plots. We plot 

the quantiles from the fitted models against the sorted data. We formally test the 

goodness of fit using the Kolmogorov test (Bickel and Doksum, 1977) whose test 

statistic is D n = max i max{i⁄n – F 0 (y (i) ), F 0 (y (i) ) – (i–1) ⁄n}, where y (1) ≤ y (2) ≤ … ≤ 


(3) 

(4)


y (n) are the ordered observations of Y, n is the sample size, and F 0 is the distribution 

of Y under the null hypothesis. Large values of the test statistic leads one to 

reject the null hypothesis and to conclude that Y is not well modeled by F 0 . 

3 Drawdowns and drawups from stock markets indexes 

We now fit the full and constrained MGPD models to drawdowns and drawups 

from a set of nine financial series, which include some of those previously used by 

Johansen and Sornette (2001). Results are compared with the stretched exponential 

fit. The nine stock market indexes used are: the Dow Jones Industrial Average 

(US); the S&P500 (US); the Nasdaq Composite (US); the Paris stock exchange, 

CAC40 (France); the Hong Kong stock exchange, Hang Seng (Hong Kong); the 

London stock exchange, FTSE 100 (UK); the Tokyo stock exchange, Nikkei 225 

(Japan); the São Paulo stock exchange, IBOVESPA (Brazil); and the Korea stock 

exchange, KOSPI 200. 

Table 1 presents some statistics of all indexes. This table contains, for each 

series, the sample size N of daily closing prices, the period covered (month/year), 

the sample size nD of drawdowns, the three smallest drawdowns (%), the sample 

size nU of drawups, and the three largest drawups (%). 

The observed frequencies (%) of the duration D (in number of days) of drawdowns 

and drawups for all indexes are given in Table 2. 

We first look for (a)symmetries in the behavior of consecutive losses and gains, 

by examining the numbers in Tables 1 and 2. For the Dow Jones and the S&P500, 

the drawdowns and drawups seem to possess similar duration distribution, but 

different magnitudes (drawdowns possess longer tails). For the Nasdaq and the 

FTSE 100, the sequences of consecutive losses are shorter than the sequences 

of consecutive gains, although the cumulative gains tend to be less extreme than 

the cumulative losses. The Nikkei 225 is quite symmetric, in the sense that its 

drawdowns and drawups seem to possess similar severity and duration distribu- 

TABLE 1 Simple statistics of all indexes. 

Index N Period nD 3 smallest DD (%) nU 3 largest DU (%) 

Dow Jones 15099 11/40–06/00 3471 –30.7 –13.3 –12.4 3472 12.3 12.5 16.6 

S&P500 15060 11/40–06/00 3465 –28.5 –13.7 –12.4 3464 2.30 14.9 16.8 

Nasdaq 7378 02/71–06/00 1492 –25.3 –24.5 –17.2 1492 14.2 15.9 16.0 

CAC40 2625 01/90–06/00 637 –11.3 –10.4 –10.3 635 10.9 16.3 18.1 

Hang Seng 5041 02/80–06/00 1180 –41.7 –26.4 –25.6 1182 19.1 19.1 19.9 

FTSE 100 4095 04/84–06/00 1003 –23.3 –13.4 –9.00 1002 8.30 8.80 10.3 

Nikkei 225 6788 01/73–06/00 1630 –17.8 –16.9 –15.6 1631 13.8 14.6 16.2 

IBOVESPA 1852 07/94–12/01 446 –34.6 –31.2 –31.0 447 27.6 45.1 51.7 

KOSPI 200 3479 01/90–06/02 796 –19.2 –18.7 –18.6 797 24.6 24.9 26.1 

The sample size N of daily closing prices; the period covered (month/year); the sample size nD of drawdowns; 

the three smallest drawdowns (%); the sample size nU of drawups; the three largest drawups (%). 

DD and DU denote, respectively, the drawdown and the drawup. 


60 


TABLE 2 Observed frequencies (%) of the duration D (in number of days) for the drawdowns and drawups from all indexes. 

Drawdowns 

Number of days 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 ≥15 

Dow Jones 47.0 26.4 13.6 6.42 3.77 1.56 0.78 0.23 0.03 0.06 0.09 0.06 0.00 0.00 0.00 

S&P500 46.4 26.7 14.3 6.21 3.39 1.69 0.71 0.27 0.18 0.06 0.06 0.03 0.00 0.00 0.00 

Nasdaq 46.4 25.9 13.7 5.90 4.09 1.88 1.07 0.67 0.20 0.20 0.00 0.00 0.00 0.00 0.07 

CAC40 48.8 24.2 15.5 6.30 3.30 1.30 0.30 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

Hang Seng 48.9 25.2 13.7 6.53 2.37 2.03 0.76 0.17 0.08 0.17 0.08 0.00 0.00 0.00 0.00 

FTSE 100 51.7 23.4 13.9 6.38 2.79 1.40 0.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

Nikkei 225 49.0 24.5 14.4 6.93 3.07 1.10 0.61 0.25 0.12 0.06 0.00 0.00 0.00 0.00 0.00 

IBOVESPA 48.9 27.4 12.8 5.38 2.91 1.35 0.45 0.45 0.45 0.00 0.00 0.00 0.00 0.00 0.00 

KOSPI 200 40.9 29.0 12.7 6.66 5.65 2.89 1.13 0.25 0.38 0.25 0.00 0.00 0.13 0.00 0.00 

Drawups 

Number of days 


1 2 3 4 5 6 7 8 9 10 11 12 13 14 ≥15 

Dow Jones 42.0 26.2 16.0 7.31 4.08 2.34 0.94 0.54 0.31 0.14 0.06 0.03 0.03 0.00 0.00 

S&P500 40.3 25.9 16.3 8.10 4.37 2.32 1.29 0.65 0.32 0.18 0.09 0.12 0.00 0.00 0.00 

Nasdaq 35.8 21.0 16.0 10.1 5.97 3.82 2.88 1.81 0.74 0.54 0.6 0.40 0.07 0.13 0.14 

CAC40 44.7 26.3 14.3 5.51 5.83 2.05 0.47 0.31 0.31 0.00 0.00 0.00 0.16 0.00 0.00 

Hang Seng 44.5 24.5 13.0 7.90 4.60 2.20 1.61 0.93 0.42 0.08 0.17 0.00 0.00 0.00 0.00 

FTSE 100 44.1 25.7 14.6 6.99 4.79 1.80 1.00 0.70 0.20 0.00 0.10 0.00 0.00 0.00 0.00 

Nikkei 225 47.3 24.6 11.3 7.85 3.86 1.90 1.35 1.04 0.31 0.18 0.12 0.00 0.00 0.00 0.06 

IBOVESPA 45.9 24.6 14.1 7.61 3.80 1.79 1.12 0.00 0.45 0.45 0.00 0.22 0.00 0.00 0.00 

KOSPI 200 47.2 25.2 13.9 7.53 2.89 1.25 1.13 0.38 0.13 0.25 0.13 0.00 0.00 0.00 0.00

TABLE 3 Parameters of maximum likelihood estimates for the two models fitted to the severity, Y, of the drawdowns and drawups. 

Drawdowns Drawups 

MGPD Stretched exp. MGPD Stretched exp. 

Index ˆ� � ˆ � ˆ LL � ˆ ˆ� LL ˆ� � ˆ � ˆ LL � ˆ ˆ� LL 

Dow Jones 1.01 0.10 1.11 –4158 1.20 0.95 –5591 1.13 0.11 1.36 –4604 1.42 1.06 –5284 

S&P500 1.08 0.21 1.06 –4087 1.22 0.96 –5356 1.13 0.10 1.41 –4580 1.46 1.06 –5134 

Nasdaq 0.96 0.23 1.11 –2039 1.36 0.84 –2571 0.98 0.10 1.61 –2379 1.76 0.93 –2441 

CAC40 1.06 0.00 1.91 –1009 1.91 1.06 –1009 1.07 0.00 2.24 –1091 2.24 1.07 –1091 

Hang Seng 1.12 0.32 2.01 –2197 2.32 0.91 –1930 0.98 0.00 2.75 –2415 2.75 0.98 –2415 

FTSE 100 1.13 0.17 1.26 –1304 1.38 1.01 –1550 1.10 0.00 1.73 –1454 1.73 1.10 –1454 

Nikkei 225 1.03 0.20 1.26 –2320 1.48 0.91 –2683 1.09 0.20 1.46 –2467 1.66 0.97 –2591 

IBOVESPA 1.11 0.32 3.06 –1012 3.40 0.90 –1130 1.24 0.35 4.65 –1093 3.40 0.92 –1199 

KOSPI 200 1.00 0.00 2.92 –1650 2.92 1.00 –1650 1.01 0.19 2.55 –1688 2.96 0.90 –1318 


Volume 6/Number 3, Spring 2004 URL: www.thejournalofrisk.com 

LL denotes log-likelihood. 

TABLE 4 Empirical and MGPD DAR � , for � = 0.05, 0.01, and 0.001, for three indexes. 

� = 0.05 � = 0.01 � = 0.001 

Nasdaq Hang Seng IBOVESPA Nasdaq Hang Seng IBOVESPA Nasdaq Hang Seng IBOVESPA 

Empirical –5.23 –8.05 –13.20 –10.01 –14.47 –24.07 –20.90 –26.26 –33.09 

MGPD –5.12 –7.85 –11.74 –10.01 –15.14 –22.85 –21.40 –33.12 –50.62

62 


tions. For the IBOVESPA, the strings of consecutive gains are longer than those 

of consecutive losses and may result in extreme cumulative gains. 

The longest duration observed correspond to 19 consecutive gains of the 

Nasdaq. Nasdaq also showed the smallest frequency of gains of duration one day, 

and the highest frequency of consecutive losses and gains lasting more than 15 

days. 

In Table 3 we present results from the MGPD and the stretched exponential fits. 

This table gives the parameters maximum likelihood estimates and the log-likelihood 

value for each model. For the sake of brevity, we do not report the results 

from the statistical tests. The estimates of ξ and γ suggest that, for most indexes, 

drawdowns possess heavier tails than drawups. 7 An exception is the IBOVESPA, 

whose drawups possess the heavier tail among all indexes, as indicated by the 

estimate of the shape parameter, ˆ ξ = 0.35. 

As noted by Johansen and Sornette (2001), extreme (larger than 10%) drawdowns 

and drawups from the three US indexes, showed a strong departure from 

the stretched exponential. The full MGPD model provided a better fit for them and 

the statistical tests strongly rejected the stretched exponential. 

Figure 3 shows QQ-plots based on the Nasdaq drawdowns and drawups fits. In 

this figure the reference distribution is the MGPD for the two upper plots, and the 

stretched exponential for the two lower plots. At the left-hand side, we can see that 

the five large drawdowns outliers observed on the stretched exponential fit do not 

seem to be so atypical under the MGPD model. The MGPD fitted the moderately 

large drawups well and pointed out a single outlier (right-hand side). We note that 

this largest observation had not shown up as an outlier under the stretched exponential 

model (bottom right). This phenomenon is well known in robustness as the 

masking effect (Rousseeuw and Leroy, 1987). 

For the drawdowns from the Dow Jones, the stretched exponential fit detected 

one large and five moderately large outliers. The MGPD also found the large 

outlier, but fitted the remaining data points well. For the drawups from the Dow 

Jones the stretched exponential provided a nice fit only for drawups with size up 

to 6%, whereas the MGPD gave a good fit to all data, spotting only the largest 

observation. 

In the case of the S&P500, the stretched exponential provided a poor fit for 

drawdowns smaller than –8% and for drawups greater than 8%. The MGPD 

found one extreme and one large drawdown outlier and gave a good fit to drawups 

up to 14%, detecting just two moderately large outliers. 

Johansen and Sornette (2001) found that drawdowns from CAC40 are almost 

perfect exponential. Our results shown in Table 3 confirm this statement. The 

QQ-plots indicated two drawups outliers. 

The Hang Seng index is another nice example that improvements may be 

achieved by using the MGPD distribution. The stretched exponential fit on the 

7 It seems there might be a connection between the value of the modifying parameter γ and 

the duration of a drawdown. This requires a more comprehensive investigation. 



FIGURE 3 QQ-plots of the fitted MGPD (upper row) and the stretched exponential 

(lower row) of the drawdowns (left) and drawups (right) in the case of the 

Nasdaq index. 

MGPD fit 

-30 -20 -10 -5 0 

Stretched exponential fit 

-30 -20 -10 -5 0 

-30 -25 -20 -15 -10 -5 0 

MGPD fit 

0 5 10 15 20 

0 5 10 15 20 



0 5 10 15 20 

-30 -25 -20 -15 -10 -5 0 

0 5 10 15 20 


drawdowns resulted in one extreme and at least four large outliers. As illustrated 

in Figure 4, the MGPD fitted the whole drawdowns in the data set well. With 

respect to drawups, the stretched exponential misfitted only the largest drawup. 

In the case of the FTSE 100, improvements when using the MGPD were 

observed on the drawdowns. For example, the larger drawdown of value –23.3% 

was estimated by the stretched exponential as approximately –10%, and by the 

MGPD as –14%. Even though in both models the two largest observations are misfitted, 

it is greater the adherence of the whole data set to the MGPD. The formal 

test strongly favored the MGPD when compared with the stretched exponential. 

For the Nikkei 225, and for both the drawdowns and drawups data, the 

stretched exponential fit found five outliers, whereas the MGPD was able to fit 

all data well except for a single, moderately large outlier. The IBOVESPA index 

provided an illustration of another situation where we could find a better fit for 

both distributions of drawdowns and drawups. The stretched exponential detected 

two large drawups outliers which do not seem atypical under the MGPD model, 

as we can see in Figure 5. 

For the drawups from the Korea Stock Price index KOSPI 200, we could again 


64 



(lower row) of the drawdowns (left) and drawups (right) in the case of the Hang 

Seng index. 

MGPD fit 


MGPD fit 




observe the masking effect. The largest observation had, under the stretched 

exponential, an almost perfect fit, at the cost of a poor fit for drawups between 

13% and 24%. The MGPD fitted the whole data set well, spotting a moderate 

(24.9%) and an extreme outlier (26%). 

In summary, for many data sets the stretched exponential provided a poor fit 

for a few extreme observations. According to Johansen and Sornette (2001), these 

atypical observations could belong to a different regime and might be considered 

outliers. However, for most of these data sets the MGPD was able to provide a 

good fit for most of the extreme points as well as the bulk of the data. Despite its 

improvement, there is still a small percentage of points with poor adherence to 

the MGPD. We wonder if such points could be an effect of long-lasting sequences 

of drops (rises), or an effect of spurious correlation. Under either assumption they 

could be considered outliers. 

To complete our analysis, Figure 6 illustrates the different perception of risk 

provided by the drawdowns. This figure shows in the x-axis the exceedance probabilities 

(0.002, 0.004, 0.006, 0.008, 0.010, 0.020, 0.030, 0.040) of observing 




(lower row) of the drawdowns (left) and drawups (right) in the case of the 

IBOVESPA index. 

MGPD fit 


MGPD fit 




drawdowns and drawups whose severities are given in the y-axis. We note that a 

very large cumulative loss (say, –24%), which would often correspond to a crash, 

possesses higher probability of occurrence for the Hang Seng (0.005) and the 

IBOVESPA (0.01). This is also true for cumulative gains. On the other hand, the 

small exceedance probability of 1% would correspond to a drawdown (drawup) 

of severity just –7% (9%) for the CAC40. In addition, we note that the smaller the 

exceedance probability, the greater the difference between the markets. 

The distribution of drawdowns may serve as the basis for many applications in 

finance. For example, its empirical version was used by Chekhlov et al (2000) to 

obtain optimal portfolios with drawdowns constraints. In that paper, the authors 

computed empirical estimates of the drawdown-at-risk with level α, the DAR α . 

Here, we estimate parametrically the DAR α using the (1 – α)% quantile of the 

MGPD. Note that Figure 6 illustrates the parametric DAR α for several exceedance 

probabilities α. 

Table 4 shows the empirical and the MGPD based DAR α , for α= 0.05, 0.01, and 

0.001, and in the cases of the Nasdaq, Hang Seng, and IBOVESPA. We observe 


66 


FIGURE 6 The exceedance probability from 0.002 to 0.040 of observing drawdowns 

(left) and drawups (right) for all indexes. 

-40 -30 -20 -10 

Nasdaq 

CAC40 

Hang Seng 

FTSE 

Nikkei 

IBOVESPA 

0.0 0.01 0.02 0.03 0.04 

0.0 0.01 0.02 0.03 0.04 

that for a risk of 0.05 the empirical estimates over-estimate the DAR, whereas for 

the smaller probability 0.001, they underestimate the DAR. 8 

Chekhlov et al (2000) also proposed the use of the conditional drawdown-atrisk, 

CDAR α , to obtain optimal portfolios. This risk measure is similar in spirit to 

the expected shortfall. 9 Chekhlov et al (2000) proposed estimating the threshold 

DAR α empirically, and then computing the CDAR α as the mean of the drawdowns 

exceeding the threshold. 

Here, we obtain a parametric estimate of the CDAR α by fitting the MGPD 

and its sub-models to the extreme tail of the drawdowns – ie, to the observations 

exceeding the model based DAR α . To illustrate, we fix α = 0.05 and compute both 

estimates in the case of the IBOVESPA. 

To estimate empirically, we just compute the mean of the 23 drawdowns more 

extreme than –13.20%, the empirical DAR 0.05 . We found the empirical CDAR 0.05 

of IBOVESPA to be –18.91%. 

To obtain the parametric CDAR 0.05 , we first compute the 0.95 quantile of the 

MGPD(γ = 1.11, ξ = 0.32,ψ = 3.06) distribution, fitted to the drawdowns from the 

IBOVESPA, obtaining the parametric DAR 0.05 as the value –11.74%. Then, we 


10 20 30 40 

Nasdaq 

CAC40 

Hang Seng 

FTSE 

Nikkei 

IBOVESPA 

8 This kind of behavior is also observed when one compares the VAR computed using, for 

example, Gaussian Garch models and extreme value methodologies. 

9 The expected shortfall is defined as the expected excess loss, given that the VARα (valueat-risk 

with level α) has occurred.


collect the 26 observations more extreme than –11.74% and fit the MGPD to the 

26 excesses. The best fit was the MGPD(γ = 1, ξ = 0.00,ψ = 6.46). The expected 

value of this distribution plus the threshold –11.74% yield the final model-based 

CDAR 0.05 estimate of –18.20%. 

As noted by Chekhlov et al (2000), solutions from the optimization procedure 

based on empirical estimation possess high variability, since they are often based 

on few observations. Using the more accurate estimated values, one may expect 

to obtain more reliable optimal portfolios based on the minimization of the drawdown-at-risk. 

We conclude this section by noting that confidence intervals for the parameter 

estimates, as well as for the DAR α and the CDAR α , can be easily obtained using 

bootstrap techniques. They will provide, for example, for the DAR α and any α an 

interval at which this quantity will lie according to some confidence level. 

4 Conclusions 

This paper addresses concerns expressed by traders and investors with respect to 

the ability of fixed time-scale risk measures to fully characterize risk. Investment 

risk is perceived differently during periods of consecutive losses, and for many 

financial series, the total price change over the range of the data is usually concentrated 

in a few of these strings. The drawdown is a new variable defined over an 

elastic time scale, and it measures directly the cumulative loss that an investment 

can incur. 

We estimated by maximum likelihood the distribution of the severity of 

drawdowns and drawups from nine stock market indexes, including the three 

major US indexes and two emerging markets indexes. We showed that the modified 

generalized Pareto distribution (MGPD) and its sub-models adjust properly 

to the majority of values previously found to be outliers. Theorems from the 

extreme value theory supported the use of the MGPD for modeling drawdowns 

and drawups. Formal statistical tests were carried out to test goodness of fit, confirmed 

by graphical analysis. 

The results were compared to the stretched exponential fit. For many data sets, 

the stretched exponential provided a poor fit for a few extreme observations. In 

some cases we could also observe the masking effect phenomenon, when the (hidden) 

largest outlier does not show up at the cost of a poor fit for moderately large 

observations. 

For the Nasdaq and the FTSE 100 we noted that, although the cumulative gains 

tend to be less extreme than the cumulative losses, the sequence of consecutive 

losses are shorter than the sequences of consecutive gains. The Dow Jones and 

the S&P500 present drawdowns and drawups with similar duration distributions. 

However, their drawdowns distributions possess longer tails. The Nikkei 225 

presents drawdowns and drawups of similar severity and duration distributions. 

Consecutive gains from IBOVESPA last longer than consecutive losses and a 

few may result in extreme cumulative gains. The MGPD parameter estimates 

indicated that, for most indexes, drawdowns possess heavier tails than drawups, 


68 


an exception being the IBOVESPA. Overall, the method presented here provides 

useful insights into the measurement of risk. 

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