Paper 3_web

Conditional value-at-risk estimation using
non-integer values of degrees of freedom in
Student’s t-distribution
Andriy Andreev
Swedish School of Economics, P.O. Box 479, FIN-00101, Helsinki, Finland
Antti Kanto
Helsinki School of Economics, P.O. Box 1210, FIN-00101, Helsinki, Finland
This paper provides an analytical formula for CVAR calculated using t-distributions with non-integer degrees of freedom. We generalize standard formulas,
calculated on the assumption of normal log-returns without compromising on
the difficulty of the calculation procedure involved. We also extend the results
of Heikkinen and Kanto (2002) to show the impact of kurtosis on values of
CVAR. The results are summarized in a closed-form formula that can, with
little effort, be used by risk managers in the evaluation risk exposures for a
family of heavy-tailed distributions.
1 Introduction
Risk managers and regulators need measures for the evaluation of risk. There are
two common measures of risk: value-at-risk (VAR) and conditional value-at-risk
(CVAR) (Jorion, 2000; Artzner et al, 1999). The former is the lower bound that is
reached with given probability, usually 95%, 97.5%, 99% or 99.9%. The latter
gives the expected loss assuming that the lower bound is reached.
The simplest way to proceed is to assume that returns are normally distributed.
However, in practice this assumption seldom holds because the tails are heavier
than in the normal case. One possible alternative is Student’s t-distribution
(Student, 1908), which has slightly heavy tails. In his seminal article, Student
considered distributions with integer degrees of freedom, but mathematically this
is not necessary. In this paper, therefore, we allow degrees of freedom to be nonintegers. A nice property of this class of distributions is that kurtosis and degrees
of freedom have a simple relationship. Thus, degrees of freedom can easily be
estimated using the method of moments. In practice, the kurtosis is often larger
than six, leading to non-integer degrees of freedom of between four and five.
The critical values of Student’s t-distribution with integer values are often
reported in standard textbooks. Recently, Heikkinen and Kanto (2002) reported
them with several non-integer values. In this paper we present a compact formula
for CVAR with non-integer degrees of freedom. It shows that if the data are
The detailed comments of the Editor-in-Chief are gratefully acknowledged.
55
56
Andriy Andreev and Antti Kanto
heavy-tailed, ie, have large kurtosis, the CVAR calculated using the normal
assumption may differ significantly from the t-distributed one.
Section 2 provides proofs and formulas. In Section 3 we illustrate the findings
with examples. Section 4 concludes.
2 Calculation of VAR and CVAR for Student’s t-distributions
The literature on VAR (see, eg, Alexander, 1998; Jorion, 2000) often assumes
logarithmic returns to be normally distributed, implying that excess kurtosis is
zero. This condition is too restrictive and does not find empirical support.
Standard statistical tests suggest that most financial time series have heavy tails.
Student’s t-distributions allow for the modeling of non-zero excess kurtosis.
Furthermore, for applications in multivariate quantitative risk management, the
most useful copula is the t-copula (see, eg, Dematra and McNeil, 2004). Extreme
value theory (EVT) estimates the distribution of the maximum (see, eg, McNeil,
Frey and Embrechts, 2004) and completes the evaluation of risk exposure using
an algorithmic procedure. Our results suggest a closed-form formula to determine
risk exposure in a one-dimensional case.
The density of a non-central Student-t distribution has the following form
(Abramowitz and Stegun, 1971):
( ) 1 + ( x − µ ) 
f ( x) =


βν 
Γ ( ) πβ ν 
Γ
ν +1
2
2
ν
2
−
1+ ν
2
(1)
where µ is a location parameter, β is a dispersion parameter and ν is a shape
parameter, or degrees of freedom. The standard t-distribution assumes that µ = 0,
β = 1 and that ν is an integer. We follow Heikkinen and Kanto (2002) by assuming non-integer degrees of freedom and applying the method of moments to
estimate the parameters.
The calculation of CVAR for normal random variables amounts basically to
hazard rate evaluation at a significance level quantile, ie,
CVAR n = − s 2
f (q)
F (−q)
(2)
By construction, CVAR is always smaller than VAR when both are taken as negative numbers. A simple check shows that the classical CVARn formula for normal
random variables produces misleading results if applied directly to t-distributions, ie, one can easily find quantiles of the t-distribution for which Equation (2)
gives a number larger than VARt .
We calculate analytically the correction term that fixes the classical formula
q
for t-distributions. Since ∫ – f (x)dx = F(– q), the CVAR of the t-distribution (see
Equation (1)) can be written as follows:
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CVAR estimation using non-integer values of degrees of freedom in Student’s t-distribution
( )  x
= ∫ x f ( x )d x = ∫ x
1 + 
Γ ( ) πβ ν  β ν 
q
F ( − q ) CVAR t
−∞
q
Γ
−∞
ν
2
ν +1
2
2
−
1+ ν
2
dx
Let ν > 1. Straightforward integration by substituting y = x 2 ⁄ βν gives
q
βν 
q2 
∫− ∞ x f ( x ) d x = − ν − 1 1 + β ν f ( q )
Furthermore,
CVAR t = −
βν 
q2  f (q)
1+ 

ν − 1  β ν F ( − q )
The second central moment of the t-distribution can be estimated as m2 =
βν ⁄ (ν – 2). Assuming that ν > 2, we get the moment estimator β = ((ν – 2) ⁄ ν)m2 ,
yielding
(
CVAR t = − (1 − w ) m 2 + wq 2
) F (−q)
f (q)
where w = 1 ⁄ (ν – 1).
Let ν > 4. Simple calculation yields w = kur(6 + 3kur), since ν = 4 + 6 ⁄ kur,
and
where kur stands for excess kurtosis. Using moment estimators s 2, kur
(6 + 3kur
), we finally obtain the estimator
= kur
w
(
= − (1 − wˆ ) s 2 + wq
ˆ 2
CVAR
t
) F (−q)
f (q)
(3)
which is a weighted sum of sampled variance and squared quantile.
grows with
Figure 1 demonstrates that the impact of the quantile weight w
excess kurtosis but that it is bounded by 1 ⁄ 3. One can see that a kurtosis value of
40 is already large enough to use Equation (3) in the limiting form
= −  2 s 2 + 1 q 2 f ( q )
CVAR
t
3
3  F ( − q )

(4)
Another limiting case arises when excess kurtosis is zero. Equation (3) can be
rewritten as a function of kurtosis:
 
= −  2 kur + 6 s 2 + kur q 2 f ( q )
CVAR
t
 F (−q)
 6 + 3 kur
6 + 3 kur

(5)
= 0 suggests that the quantile weight has no effect. The result is a classiand kur
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57
Andriy Andreev and Antti Kanto
0.6
Volatility weight (1 – W )
0.4
Quantile weight (W )
0.2
Weights
0.8
1.0
FIGURE 1 The weights as a function of kurtosis in Equation (3).
0.0
58
0
20
40
60
80
100
Excess kurtosis
cal CVARn formula, Equation (2), ie, hazard evaluated at the appropriate quantile
scaled by volatility.
Equations (3) and (5) are consistent estimates. They—play the same role as the
) ÷
formula for the calculation of value-at-risk VARt = tαν βˆ , where βˆ = ((3 + kur
2
)) × s (see Heikkinen and Kanto, 2002).
(3 + 2kur
3 Simulation results
The results of numerical integration for non-integer degrees of freedom and
correcting coefficient values for the calculation of VARt have been summarized
in Table 2 of Heikkinen and Kanto (2002). The effect of kurtosis on VARt at
different probability levels is presented in Figure 1 of the same paper.
Our first objective is to report the effect of excess kurtosis on CVARt at different probability levels and to compare these findings with results one would obtain
by assuming normal log-returns. Equation (3) is the basis for the results shown in
Figure 2.
Heikkinen and Kanto (2002) found that the effect of kurtosis on the value of
VARt is almost insignificant at the 97.5% significance level, with a somewhat
lesser effect at lower levels and an increasing effect at higher levels. In contrast to
these findings, higher kurtosis increases CVARt at all levels, with a mild exception for the 90% significance level, when CVARt decreases slightly for small
values of kurtosis. This tendency of CVARt to increase manifests itself more
strongly as the significance level approaches 100%. At levels above 97.5%, the
increase becomes very marked in comparison to VARt calculations.
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CVAR estimation using non-integer values of degrees of freedom in Student’s t-distribution
FIGURE 2 The effect of excess kurtosis at different confidence levels on CVARt ,
s = 1.
4
0
2
CVAR
6
8
0.999
0.995
0.990
0.975
0.950
0.900
0
10
20
30
40
50
Excess kurtosis
Figure 2 makes clear the differences between CVARt and CVARn. Since the
excess kurtosis is zero for normal random variables, the corresponding values of
CVARn are points on the graph. They are obviously ordered inasmuch as higher
points correspond to higher significance levels. By definition, as excess kurtosis
tends to zero Student’s t-distribution tends to a normal distribution. Surprisingly,
one can see that CVARt < CVARn at the 90% level. This effect is mild but
present. As the significance level is increased the plots come more into line with
intuition: the effect of excess kurtosis is to increase the risk value.
Figures 3a and 3b scale the effect of excess kurtosis on (C)VARt (C)VARn.
Surprisingly, the intuitive hypothesis that higher kurtosis implies higher (C)VARt
fails for both ratios, with the more pronounced effect observed for VARt VARn.
Figure 3c presents a contrast by plotting CVARt VARt , indicating the importance
of the choice of risk measure. The ratio is strictly larger than one for all confidence levels. Unlike Figures 3a and 3b, there is no clear ordering in the graphs.
This phenomenon should be examined further, especially when excess kurtosis is
less than 10.
All the curves in Figures 3a and 3b are ordered as functions of p-values: a larger
p-value corresponds to a larger value of (C)VARt (C)VARn. This observation
makes analysis simple. Another important observation is that CVARt CVARn >
VARt VARn for all p-values.
The most interesting question is what happens when the ratio equals one, ie,
when (C)VARt = (C)VARn. The answers are different for CVAR and VAR. In line
with Heikkinen and Kanto (2002), Figure 3b suggests p = 0.975 to be the level
Volume 7/Number 2, Winter 2005
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59
Andriy Andreev and Antti Kanto
10
20
30
40
Excess kurtosis
50
2.0
1.5
CVAR t ⁄ VAR t
0.5
0.5
0
0.999
0.995
0.990
0.975
0.950
0.900
1.0
1.5
VAR t ⁄ VAR n
2.0
0.999
0.995
0.990
0.975
0.950
0.900
1.0
1.5
1.0
CVAR t ⁄ CVAR n
2.0
0.999
0.995
0.990
0.975
0.950
0.900
2.5
2.5
2.5
FIGURE 3 The effect of kurtosis on ratio of (C)VARt to (C)VARn.
0.5
60
0
10
20
30
40
Excess kurtosis
50
0
10
20
30
40
50
Excess kurtosis
at which VARt VARn for all values of kurtosis. Alternatively, VARt > VARn for
p > 0.975, while VARt < VARn for p < 0.975.
Somewhat surprisingly, a similar effect is observed for the ratio CVARt CVARn
(see Figure 3a). This is not as straightforward as for VARt VARn: the threshold
level depends on the value of kurtosis and belongs to the p-value interval
[0.9, 0.95]. CVARt CVARn takes values on both sides of one for the same
p-level. For instance, at the 90% level, CVARt CVARn < 1 for all levels, but, if
p = 0.95, CVARt CVARn > 1 for kur > 1, while CVARt CVARn = 0.99 for
kur = 1.
Finally, we demonstrate our findings using the same Nokia stock that was used
by Heikkinen and Kanto (2002) to illustrate the effect of excess kurtosis on the
calculation of VAR. The stock has been followed for four and a half years, during
which it shows large fluctuations on a monthly scale. We apply Equation (5) with
s = 20% and kur = 10, indicating a Student’s t-distribution with 4.6 degrees of
freedom.
The results are summarized in Table 1. These indicate that there is a rapidly
growing difference in the value of CVAR at higher p-values, in contrast to the
much smoother behavior of the VAR estimates.
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CVAR estimation using non-integer values of degrees of freedom in Student’s t-distribution
TABLE 1 Monthly estimates of VAR and CVAR (%) for Nokia stock.
VAR (normal)
VAR (Student, kur = 10)
CVAR (normal)
CVAR (Student, kur = 10)
95%
97.5%
36.2
34.0
45.4
50.7
43.1
43.6
51.4
66.4
Confidence level
99%
99.5%
51.2
57.8
58.6
90.1
56.7
69.8
63.6
110.7
99.9%
68.0
103.5
74.1
169.2
4 Concluding remarks
We have presented a simple closed-form formula for the calculation of conditional
value-at-risk (CVAR) using Student’s t-distribution. Equation (3) is a weighted
average of the estimated variance and the square of the allowed critical point. It
contains a classical formula for the calculation of CVAR for normal distributions
as a partial case. Since financial data usually have heavy tails, it will be of interest
to practitioners. Assuming finite kurtosis, the weights are easy to estimate from
the data, so Equation (3) provides a quick and useful tool for risk management.
REFERENCES
Abramowitz, M., and Stegun, I. A. (1971). Handbook of mathematical functions with formulas,
graphs and mathematical tables. Dover Publications, Inc., New York.
Alexander, C. (ed.) (1998). Risk management and analysis: Volume 1: Measuring and modelling financial risk. John Wiley & Sons, New York.
Artzner, P., Delbaen, F., Eber, J., and Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9, 203–28.
Dematra, S., and McNeil, A. J. (2004). The t copula and related copulas. Preprint, ETHZ
Zurich. Available from www.math.ethz.ch/~mcneil.
Heikkinen, V-P., and Kanto, A. (2002). Value-at-risk estimation using non-integer degrees of
freedom of Student’s distribution. Journal of Risk 4(4), 77–84.
Jorion, J. (2000). Value at risk. The new benchmark for managing financial risk. McGraw-Hill,
New York.
McNeil, A. J., Frey, R., and Embrechts, P. (2004). Quantitative risk management: concepts,
techniques and tools. Book manuscript, to be published.
Student (W. S. Gosset) (1908). The probable error of a mean. Biometrika VI(1), 1–25.
Volume 7/Number 2, Winter 2005
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