Measures of financial risks and market crashes
The problem of particular importance in financial risk management is forecasting the magnitude of a market crash. We address this problem using statistical inference on heavy–tailed distributions. Our approach involves accurate estimates of the tail index, extreme quantiles, and the mean excess func...
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| citation_txt | Measures of financial risks and market crashes / S.Y.Novak // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 182-193. — Бібліогр.: 24 назв.— англ. |
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| description | The problem of particular importance in financial risk management is forecasting the magnitude of a market crash. We address this problem using statistical inference on heavy–tailed distributions. Our approach involves accurate estimates of the tail index, extreme quantiles, and the mean excess function. We apply our approach to real financial data, and argue that the September 2001 crash had two components: one (systematic) could be predicted, while another (non–systematic) was due to the shock of the event. We present empirical evidence that the degree of tail heaviness can change considerably as one switches to less frequent data. This fact has important implications to the problem of estimating financial risks.
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.182-193
S.Y.NOVAK
MEASURES OF FINANCIAL RISKS AND
MARKET CRASHES
The problem of particular importance in financial risk management is
forecasting the magnitude of a market crash. We address this prob-
lem using statistical inference on heavy–tailed distributions. Our
approach involves accurate estimates of the tail index, extreme quan-
tiles, and the mean excess function. We apply our approach to
real financial data, and argue that the September 2001 crash had
two components: one (systematic) could be predicted, while another
(non–systematic) was due to the shock of the event. We present em-
pirical evidence that the degree of tail heaviness can change consid-
erably as one switches to less frequent data. This fact has important
implications to the problem of estimating financial risks.
1. Introduction
The classical mean–variance portfolio theory utilizes the idea of balanc-
ing the expected return vs. the risk represented by the standard deviation.
The use of the standard deviation as a measures of risk is justified if portfolio
returns are light–tailed. In reality, financial data often appear heavy–tailed.
The evidence of heavy tails was accounted as early as in 1960s (see Mandel-
brot (1963) and Fama & Roll (1968)). This fact is nowadays the subject of
textbooks, see, e.g., Luenberger [12], p. 302, and Embrechts et al. (1997),
p. 404–405. The feature is particularly common to “frequent” data, e.g.,
daily log–returns of stock prices, while log–returns of less frequent data can
exhibits lighter tails, well in line with the central limit theorem.
If data is heavy–tailed, then the standard deviation is no longer respon-
sible for extreme movements of portfolio returns even if the portfolio is
optimal in the sense of the mean–variance theory, and can hardly be con-
sidered a proper measure of risk. Recall that if data is heavy–tailed then
a single sample element, e.g., the loss over one particular day, can make
Invited lecture.
2000 Mathematics Subject Classifications 62G32.
Key words and phrases heavy–tailed distribution, Value–at–Risk, Expected Shortfall.
182
MEASURES OF FINANCIAL RISKS 183
a major contribution to the total loss over a considerable period of time.
For instance, on “black Monday” 19.10.1987 the S&P500 index fell by 20.5%
erasing all the index had gained since March 1986. Sometimes data exhibits
such heavy tails that the variance is likely to be infinite. Value–at–Risk and
Expected Shortfall appear more suitable measures of risk.
Value–at–Risk (VaR) was popularized as a measure of financial risks by
JP Morgan in 1990s. It indicates how much money a bank should put aside
in order to offset the risk of an unfavorable market movement. If one deals
with daily data, then 1%–VaR shows how far the value of a portfolio can fall
once in approximately 100 days. Another measure of risk, Expected Short-
fall (ES), presents the average loss given there is a fall beyond VaR. Many
banks routinely calculate VaR in order to monitor the current exposure of
their portfolios to market risks. For instance, Goldman Sachs uses 5%–VaR;
Citigroup, Credit Suisse First Boston, Deutsche Bank, JP Morgan Chase
and Morgan Stanley use 1%–VaR.
The important practical question is how to evaluate VaR and ES. If data
followed the normal distribution, then VaR would be a constant times the
standard deviation. In reality, daily and weekly rates of return of many
stocks and stock indexes appear heavy–tailed, ruling out the assumption of
normality.
The problem of reliable estimation of VaR and ES in the presence of
heavy tails is demanding, it remained open for a long while. A reliable
procedure of practical estimation of VaR and ES from (possibly dependent)
heavy–tailed data was introduced in Novak (2002). The approach was tested
on samples of simulated data where the true values of the tail index, VaR,
and ES are known. An application of the procedure to the problem of
predicting the magnitude of the “black Monday” crash in Novak & Beirlant
(2006) confirmed the accuracy of the approach in comparison with those of
McNeil (1998) and Matthys & Beirlant (2001).
While the market crash in October 1987 was purely “market driven”,
the collapse of S&P500 after September the 11th was obviously triggered by
the tragic event in New York. We argue in this paper that the 2001 crash
had two components: one (systematic) could be predicted using the method
presented below, while another (non–systematic) was due to the shock of
the event.
We show also that the degree of heaviness of the data tails can change
considerably as one switches to less frequent data. This fact has important
implications in risk management: it means that the textbook rule for the
evaluation of VaR over a period of time using daily VaR estimates multiplied
by the square root of the number of days (cf. Jorion (2001), Ho & Lee (2004),
p. 533, Tapiero (2004), p. 313) can be misleading.
The next two sections are devoted to the problems of statistical inference
on dependent heavy–tailed data. We introduce a new estimator of the tail
184 S.Y.NOVAK
index and establish its consistency. In section 4 we apply our approach
to real data and show what a forecast of the magnitude of possible losses
one could make on the eve of September the 11th. We present empirical
evidence that more frequent financial data has considerably heavier tails.
This fact contradicts to the theory, and has important implications to the
problem of risk evaluation.
2. Heavy tails
Which tails should be considered heavy? Following Resnick (1997), we
say that the distribution has a heavy tail if
F (x) := IP(X ≤ x) = L(x)|x|−α (α > 0), (1)
where the (unknown) function L is slowly varying: lim
x→−∞L(xt)/L(x) = 1
for all t > 0. Here X stands for a sample element, say, daily log–return of
a stock price. The definition is given for the left tail; it can be obviously
reformulated for the right tail:
IP(X > x) = L(x)x−1/a (a > 0), (1∗)
where the (unknown) function L is slowly varying at infinity.
The number α in (1) or (1∗) is called the tail index. It is the main
characteristic describing the tail behavior. The smaller is α, the heavier
is the tail as well as the likelihood of extreme movements. Thus, the tail
index appears a proper measure of the “degree of heaviness” of the tail of
a distribution.
Distributions that obey (1) form a non–parametric (semi–parametric)
family of probability laws. It includes Cauchy, Student’s, Pareto and ARCH
distributions, among others. The non–parametric setup is the advantage of
our approach since there are doubts that parametric models accurately de-
scribe real financial data (cf. Capobianco (2002), Section 4.3). The problem
with the parametric approach is that we never know if the unknown distrib-
ution belongs to a chosen parametric family (hypothesis H0); “parametric”
papers usually do not tell readers what authors’ findings are worth if the
unknown distribution does not come from a chosen parametric family. An-
other problem with the parametric approach is the lack of robustness. For
instance, McNeil (1997, Section 4.6) indicates that parametric tail index and
VaR estimators can change by 20% if one sample element is removed. The
advantage of the non–parametric setup is that a chosen class of distributions
is so rich that the problem of testing H0 does not arise.
There is a number of procedures to check if the tail is heavy. One
possibility is to use a qq–plot (quantile–quantile–plot). Recall that the
quantile F−1 is the inverse to the distribution function F :
F−1(y) = inf{t : F (t) ≥ y} (y ∈ IR).
MEASURES OF FINANCIAL RISKS 185
One can plot empirical quantiles versus quantiles of a given distribution
function Fo. If the line is approximately linear, then tails of F and Fo are
close up to a linear transform, cf. Embrechts et al. (1997), p. 292–293, and
Mikosch (2004), p. 90–91. Another procedure is based on the mean excess
function, see Embrechts et al. (1997), p. 355, Novak (2002) and Mikosch
(2004). For instance, the analysis in Matthys & Beirlant (2001) confirms
that daily log–returns of the S&P500 index are heavy–tailed.
Up to a sign, VaR is the extreme quantile:
m%–VaR = −F−1(m/100).
For instance, if 1%–VaR equals 0.04, then in 1% worst cases X can fall
below –0.04. ES is the average fall beyond VaR: ES = IE{y − X|X ≤ y},
where y = −VaR.
Below we prefer to deal with positive numbers; in particular, we speak
about “20.5% fall” of the S&P500 index on the “black Monday” instead of
“–20.5% rate of return”, and switch from X to −X, where X stands for
a sample element. We assume (1∗); q–VaR (denoted in the sequel by yq) is
now defined as the upper quantile of level q (the inverse of F̄ = 1− F ):
yq = F̄−1(q); (2)
ES is now the average excess over the VaR: if y =VaR, then ES is
E(y) = IE{X − y|X > y} .
It is well defined if the tail index α > 1. Novak (2002) points out that
E(y) ∼ y/(α − 1) as y → ∞ if (1∗) holds. One can accurately estimate
ES if consistent estimators of α and VaR are available and α is not close
to 1 (the difficulty of ES estimation when α is close to 1 is pointed out
by Yamai & Yoshiba (2002)). The sum VaR+ES = IE{X|X > yq} is the
average value of X given X exceeds q–VaR; it is a “typical” value of a
non–typical (extreme) observation.
When one estimates quantiles of level q with q bounded away from 0
and 1, the empirical quantile estimator (the inverse of the empirical dis-
tribution function) is the most natural choice. Simulation study in Novak
(2002) indicates that the empirical quantile estimator works poorly when
one estimates extreme quantiles (quantiles of level q ≤ 0.05). This observa-
tion is in line with the theory: if q is “small” (close to 0), then the empirical
inference is based on very few (if not none) elements of a sample. By con-
trast, our approach is based on observations of “moderate magnitude” and
hence is much more robust.
186 S.Y.NOVAK
3. Inference on heavy tails
It is widely accepted that financial data is typically dependent. One can
measure dependence using, for instance, mixing (week–dependence) coeffi-
cients ϕ and ρ (the definition of mixing coefficients can be found, e.g., in
Embrechts et al. (1997) and Novak (2005)). The fact that past data has
little effect on today’s price movements can be formalised by the equation
lim
k→0
ϕ(k) = 0 (3)
or ∑
k≥1
k−1ρ(k) < ∞ . (4)
The latter holds, e.g., if ρ(k) � (ln k)−c (c > 1). Note that ρ(k) ≤ ϕ1/2(k).
We assume in the sequel that observations {Xi} form a stationary se-
quence obeying (4).
Note that conditions (3) and (4) are among the weakest in the literature
on dependent random variables. In many particular parametric models,
including GARCH process, ϕ(·) decays exponentially fast, see Basrak et
al. (2002). Thus, papers assuming GARCH model implicitly require ϕ(·)
to have the exponential rate of decay. Recall that data has “long memory”
if ϕ(k) decays not faster than k−d for some d > 0. Hence GARCH model
is not applicable to “long memory” data. The evidence of the long memory
phenomenon for daily log–returns of S&P500 and some other financial data
is provided by Ding et al. (1993).
Any quantity of interest in the tail appears a function of the tail index.
Given the sample X1, ..., Xn, we want to estimate the tail index, q–VaR (2),
where q = q(n) is allowed to tend to 0 as the sample size n grows, and the
corresponding ES (denoted by E(yq)).
All known estimators of VaR and ES in the tail seem to involve an esti-
mator of the tail index. The problem of tail index estimation was addressed
by many authors, see the survey paper by Novak (2005). Goldie & Smith
(1987) have introduced the ratio estimator (RE)
an := an(x) =
n∑
i=1
ln(Xi/x)1I{Xi > x}
/
Nn(x) (5)
of index a = 1/α (equivalently, 1/an is the ratio estimator of the tail
index). Here n is the sample size, x = xn is the chosen threshold, and
Nn(x) =
∑n
i=1 1I{Xi > x}. Novak (2002, 2005) argues that RE has ad-
vantages over Hill’s and some other tail index estimators. Resnick (1997),
p.1839, and Embrechts et al. (1997), p. 406, point out drawbacks of Hill’s
and related estimators.
MEASURES OF FINANCIAL RISKS 187
The statistic
an,m := an,m(x) =
n∑
i=1
lnm(Xi/x)1I{Xi > x}
/
(Nn(x)m!), (6)
where m ∈ IN, can be called a generalised ratio estimator. It is the sample
analog of IE{lnm(X/x)|X > x}/m!. In fact, formula (6) generates a list of
tail index estimators. For instance, ãn = an,2/an is a consistent estimator
of index a if (7) holds.
Theorem. Denote pn = IP(X > xn), and suppose that
pn → 0 , npn →∞ (7)
as n→∞. Then a1/m
n,m is a consistent estimator of index a.
Assumption (7) means that the threshold xn is neither “too small” nor
“too large”. It guarantees the consistency of typical estimators implement-
ing the so–called peak–over-threshold approach, see Novak (2005).
The ratio estimator is a function of the threshold. It is a typical situa-
tion in non–parametric statistics that an estimator is not initially a single
number but a function of a nuisance parameter. The important practical
question is how to choose the nuisance parameter and produce the final
estimate as a number.
The procedure of practical estimation of index a = 1/α: choose the
interval [x−; x+] formed by a significant number of sample points, where the
function a1/m
n,m (·) is stable, and take the average value â = mean{a1/m
n,m (x) :
x ∈ [x−; x+]} of the generalised ratio estimator over that interval; then x̂n
is a level such that a1/m
n,m (x̂n) = â. Similarly we estimate VaR and ES.
The feature of this procedure is that it yields almost one and the same
result despite possibly individual choice of the interval [x−; x+]: since we
take the average over an interval formed by a significant number of sample
points, the variability with the choice of end–points is virtually eliminated.
Theorems establishing consistency of tail index, VaR and ES estimators (see
Novak (2002, 2005)) form a theoretical background to this procedure.
We use in the next section the following estimators of VaR and ES
(Novak (2002)):
ŷq(xn) = (Nn/qn)an xn , En(xn) = ŷqan/(1− an) . (8)
The possible versions of estimators (8) are
ỹq(xn) = (Nn/qn)â xn , y∗
q = ŷq(x̂n) , E∗
n = y∗
q â/(1− â) . (9)
Results of the simulation study in Novak (2002) indicate an acceptable
degree of the accuracy of estimators ŷq, ỹq and y∗
q . The comparison of
188 S.Y.NOVAK
three different approaches in Novak (2005) and Novak & Beirlant (2006)
suggests that (8), (9) are probably the best currently available estimator of
VaR.
A number of authors suggested to use the bootstrap approach in order
to choose the nuisance parameter when using Hill’s estimator aH
n (k) of in-
dex a, see, e.g., Danielsson et al. (2001) and Matthys et al. (2004). One
problem with the bootstrap is that it is designed for samples of independent
random variables. Thus, the application of the bootstrap approach to the
obviously dependent S&P500 data would not be justified. Another prob-
lem with the bootstrap is that it does not eliminate the nuisance parameter
but replaces it with new ones, cf. Danielsson et al. (2001). The optimal
value of the nuisance parameter k in Matthys et al. (2004) aims to mini-
mize the asymptotic mean squared error (AMSE) of
√
k (aH
n (k)− a), while
Drees (2003) minimizes the asymptotic variance only (ignoring the asymp-
totic bias). The problem is that the limiting distribution of
√
k (aH
n (k)− a)
is determined by the rate of growth of k = k(n), which we never know in
practical situations. Moreover, for a particular rate of growth of k, the
limiting distribution of
√
k (aH
n (k)− a) can be normal N (0; 1) or can have
infinite expectation; in both cases the optimal k cannot be determined by
minimizing AMSE. For some other rates of k the limiting distribution of√
k (aH
n (k)− a) is not normal at all.
4. September the 11
th
Forecasting the scale of possible extreme movements of stock prices and
indexes is one of the major tasks of a risk manager. A particular question of
this kind was raised in McNeil (1998): having the “historical” data available
on the eve of the “black Monday”, was it possible to predict the magnitude
of the crash? The comparison of few approaches to this problem was given
in Novak (2005) and Novak & Beirlant (2006); the procedure presented in
Section 3 came up a clear winner.
We have analyzed daily and weekly log–returns of the S&P500 index
over the period from 01.11.1987 till 10.09.2001 (3500 trading days, or 700
weeks). Figure 1 presents the plot of the ratio estimator for daily negative
log–returns of the S&P500 index. The plot looks stable in the interval [1.6;4]
which is formed by 163 points. The average value of an(x) as x∈ [1.6; 4]
is â = 0.294. Hence the estimate of the tail index for daily data, αd, is
3.4. We have estimated also 1%–VaR at 2.6% and the corresponding ES at
1.1%; the worst possible daily fall of the log–return of the S&P500 in 100
days was likely to be around VaR+ES=3.7%.
Comparing the tail index estimate with that for the period 01.01.1960
– 16.10.1987 (see Novak (2005), Novak and Beirlant (2006)), we conclude
that the left tail of daily log–returns of the S&P500 index became heavier
MEASURES OF FINANCIAL RISKS 189
Tail index estimation (daily)
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Threshold (%)
R
a
ti
o
e
st
im
a
to
r
Tail index estimation (weekly)
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8 9 10
Threshold (%)
R
a
ti
o
e
st
im
a
to
r
Figure 1: Tail index estimation for daily (left) and weekly (right) negative log–
returns of S&P500 index over the period from 01.11.1987 till 10.09.2001. Daily
data: the tail index αd is estimated at 3.4. Weekly data: the tail index αw is
estimated at 4.5.
after the “black Monday” crash. Recall that the heavier the tail, the higher
is the chance of an extreme movement.
In order to evaluate the worst possible weekly fall of the log–return of
the S&P500 index after the “black Monday”, we put q = 1/700. We have
estimated the tail index, αw, at 4.5 (see Figure 1), the q–VaR at 8.3% (see
Figure 2) and the corresponding ES at 2.4%. Hence the worst possible fall
of the weekly log–return of the S&P500 index in 700 weeks after the “black
Monday” was likely to be around 8.3% + 2.4% = 10.7%.
The trade at New York Stock Exchange (NYSE) reopened on 17.09.2001.
The S&P500 index finished the day at 1039, or 5% lower than its 1092.5
closing level on 10.09.2001. On 21.09.2001 the log–return of the index was
12.3% below its 10.09.2001 level. The fall during that extended week was
larger than 10.7% predicted by our method. The possible explanation is
that the market crash after September the 11th had two components: one
(systematic) was determined by the historical data, while another (non–
systematic) was due to the effect of the tragic event.
Recall that the tail index of the daily data, αd, was estimated at 3.4
while the tail index of the weekly data, αw, was estimated at 4.5. Thus,
we have empirical evidence that financial data over different time horizons
may have distinct tail indexes. This observation seems to be new, it has
important implications to the problem of VaR estimation. It is known, see,
190 S.Y.NOVAK
VaR estimation (weekly, q=1/700)
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6 7 8 9
Threshold (%)
Q
u
a
n
ti
le
e
st
im
a
to
r
VaR estimation (weekly, q=1/700)
0
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Threshold (%)
Q
u
a
n
ti
le
e
st
im
a
to
r
(t
il
d
e)
Figure 2: VaR estimation for weekly negative log–returns of S&P500 index over
the period 01.11.1987 – 10.09.2001: ŷq (left) and ỹq (right). The estimate of
q–VaR is 8.3%, ES is estimated at 2.4%.
e.g., Embrechts et al. (1997), that for large enough x,
F̄k(x) := IP(X1 + ... + Xk > x) ∼ kF̄ (x) (10)
(at least in the case of independent observations) if k is fixed and the
distribution of Xi obeys (1∗); in other words, the sum X1 + ... + Xk obeys
(1∗) with the same tail index.
Let Xi denote a daily log–return. Then a weekly log–return is a sum
of five consecutive daily log–returns. Since Value–at–Risk over a k–days
period, F̄−1
k , is the inverse to F̄k, (1∗) and (10) yield
F̄−1
k (q) ∼ k1/αF̄−1(q) (11)
as q → 0. Let q–VaRk denote q–VaR over a k–days period. Then (11)
means
q−VaRk ≈ q−VaR1k
1/α . (12)
This simple relation, if it was true for real data, would mean that one could
only need to evaluate, say, daily VaR and then use (12) in order to derive
VaR over any desired time horizon. The latter is very attractive as sample
sizes can get small when one deals with less frequent data.
In the case of normally distributed data, VaRk is a constant times σk,
where σk is the standard deviation of a sum of k daily log–returns. Since
σk = σ1
√
k , many textbooks, e.g., Jorion (2001), Ho & Lee (2004), Tapiero
(2004), recommend the formula
VaRk ≈ VaR1
√
k . (13)
MEASURES OF FINANCIAL RISKS 191
Relation (12) means that one has to replace
√
k in (13) with k1/α if data
is heavy–tailed.
The empirical evidence that the tail index of daily log–returns, αd, is
essentially different from the tail index of weekly log–returns, αw, forces us
to conclude that even formula (12) is not automatically applicable, and one
should estimate VaRk and ESk separately for different values of k. Indeed,
we have evaluated 1%–VaR5 at 5.3%, while an application of (12) would
give us 2.6%·51/αd = 4.2%.
Appendix
Proof of Theorem 1. Denote
a∗
m ≡ a∗
m(x) = IE{lnm(X/x)|X > x}/m! , 1Ii ≡ 1Ii(x) = 1I{Xi > x} ,
where m ≥ 1, and let
Yi,m = (lnm(Xi/x)− a∗
mm!) 1Ii .
Property (1∗) yields a∗
m ∼ am as x→∞, cf. formula (7) in Novak (2002).
Using Chebyshev’s inequality, we derive
IP(an,m − a∗
m > ε) = IP
(∑n
i=1
Yi,m > εm!
∑n
i=1
1Ii
)
= IP
(∑n
i=1
(
Yi,m − εm!1̄Ii
)
> εm!npn
)
≤ (εm!npn)−2 var
(∑n
i=1
(
Yi,m − εm!1̄Ii
))
for any ε > 0, where 1̄Ii = 1Ii − pn. By Theorem 1.1 in Utev (1989), there
exists a constant cρ, depending only on ρ(·), such that
varNn ≤ cρnpn , var
(∑n
i=1
Yi,m
)
≤ cρnvarY1,m ≤ cρ,mnpn .
Hence var
(∑n
i=1
(
Yi,m − εm!1̄Ii
))
≤ Cρ,mnpn, and IP(an,m − a∗
m > ε) → 0
as n →∞. Similarly one checks that IP(an,m − a∗
m < −ε) → 0 as n →∞.
The result follows.
Acknowledgement. The author is grateful to the organizers of the In-
ternational conference “MODERN STOCHASTICS: THEORY AND AP-
PLICATIONS” for their kind hospitality, and to the referee for helpful re-
marks.
192 S.Y.NOVAK
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|
| id | nasplib_isofts_kiev_ua-123456789-4488 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T17:23:04Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Novak, S.Y 2009-11-19T10:20:43Z 2009-11-19T10:20:43Z 2007 Measures of financial risks and market crashes / S.Y.Novak // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 182-193. — Бібліогр.: 24 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4488 The problem of particular importance in financial risk management is forecasting the magnitude of a market crash. We address this problem using statistical inference on heavy–tailed distributions. Our approach involves accurate estimates of the tail index, extreme quantiles, and the mean excess function. We apply our approach to real financial data, and argue that the September 2001 crash had two components: one (systematic) could be predicted, while another (non–systematic) was due to the shock of the event. We present empirical evidence that the degree of tail heaviness can change considerably as one switches to less frequent data. This fact has important implications to the problem of estimating financial risks. en Інститут математики НАН України Measures of financial risks and market crashes Article published earlier |
| spellingShingle | Measures of financial risks and market crashes Novak, S.Y |
| title | Measures of financial risks and market crashes |
| title_full | Measures of financial risks and market crashes |
| title_fullStr | Measures of financial risks and market crashes |
| title_full_unstemmed | Measures of financial risks and market crashes |
| title_short | Measures of financial risks and market crashes |
| title_sort | measures of financial risks and market crashes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4488 |
| work_keys_str_mv | AT novaksy measuresoffinancialrisksandmarketcrashes |