Value-at-Risk scaling for long-term risk estimation
L. Spadafora1,2
M. Dubrovich1
1
2
M. Terraneo1
UniCredit S.p.A.
Faculty of Mathematical, Physical and Natural Sciences, Università Cattolica del Sacro Cuore, Brescia
XVI Workshop on Quantitative Finance,
Parma, January 29-30, 2015
The views and opinions expressed in this presentation are those of the author and
do not necessarily represent official policy or position of UniCredit S.p.A.
Main Reference:
L. Spadafora, M. Dubrovich and M. Terraneo,Value-at-Risk time scaling for long-term risk
estimation, arXiv:1408.2462, 2014.
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Outline
1
Introduction and Motivation
2
Value-at-Risk Scaling
3
Modelling P&L Distributions
4
Time Scaling
5
VaR scaling on a real portfolio
6
Summary and Conclusions
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Introduction and Motivation
Introduction: Value-at-Risk vs Economic Capital
Regulatory Capital: 99% Value-at-Risk at a short time-horizon (1 day)
Economic capital (EC): capital required to face losses within a 1-year time-horizon
at a more conservative percentile (we refer to 99.93%)
Possible estimation approaches for EC:
1
1
Scenario generation (for the risk factors) and portfolio revaluation to obtain a 1-year
profit-and-loss (P&L) distribution
2
Extension of the short-term market-risk measures to longer time-horizons/higher
percentiles
The first approach has the following drawbacks:
Assumptions are needed to generate scenarios at 1 year (both with historical
simulation and with Monte-Carlo methods...)
No rebalancing of portfolio (i.e. unrealistic assumption of freezing positions for 1 year)
2
The second approach bypasses such difficulties:
Assumes hedging/rebalancing of the portfolio
Relies on models already approved and used into day-to-day activities
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Introduction and Motivation
Motivation: Economic Capital as a scaled Value-at-Risk
Main idea: follow the second approach and develop a scaling mechanism for
computing efficiently Economic Capital out of Regulatory Value-at-Risk measures
Model short-term P&L using iid RVs distributed according to some benchmark PDFs
Apply convolution theorem to subsequent time-steps and interpret scaling in light of
the Central Limit Theorem (CLT), to derive conditions needed for normal convergence
Main results: generalized VaR-scaling methodology to be used for calculating EC, in
dependence of the short-term PDF’s properties:
If the √
P&L distribution has exponential decay, VaR-scaling can be correctly inferred
using T -rule, even if starting distribution is not Normal
√
With power-law decay, T -rule can be applied naively only if tails are not too fat.
Otherwise the long-term P&L distribution needs to be determined explicitly, and EC
can be significantly larger than what would have been inferred under Normal
assumptions
Theoretical results are integrated by a numerical simulation performed on a test equity
trading portfolio.
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Value-at-Risk Scaling
The VaR-scaling approach
Given x(t) P&L over time-horizon t (e.g. 1 day) and its PDF p(x(t)), VaR at
confidence level (CL) 1 − α (e.g. 99%) is defined by:
Z VaR(α,t)
1−α=
p(x(t))dx(t)
(1)
−∞
General VaR-scaling approach: find h(·) such that, given α2 6= α and T 6= t
VaR(α2 , T ) = h(VaR(α, t))
(2)
For EC, i.e. VaR at CL 1 − α2 = 99.93% √
over horizon T = 1y, commonly done
assuming normality of PDF and applying T -rule:
VaRN (99.93%, 1y) =
√
Φ−1 (0.01%)
250 N −1
VaRN (99%, 1d)
ΦN (1%)
(3)
where ΦN denotes CDF of N(ormal) distribution
We propose a generalization:
1
2
3
Fit short-term P&L distribution and choose PDF with best explanatory power
Calculate long-term P&L distribution (analytically or numerically), given chosen PDF
Compute EC as the desired extreme-percentile
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Modelling P&L Distributions
Modelling real-world P&L distributions: candidates
Which theoretical PDF class better fits empirical P&L data?
Benchmark the basic normal assumption using leptokurtic distributions:
1
Normal distribution (N):
pN (x; µ, σ, T ) = √
2
(x − µT )2
exp −
2σ 2 T
2πσ 2 T
1
Student’s t-distribution (ST, power-law decay):
"
#− ν+1
2
)
( x−µ
)2
Γ( ν+1
2
σ
pST (x; µ, σ, ν) = √
1+
σ νπΓ( ν2 )
ν
3
(4)
(5)
Variance-Gamma distribution (VG, exponential decay):

θ(x−µT )
 r|x−µT |
2σ 2
pVG (x;µ,σ,k,θ,T )= 2e T

√
2σ 2 +θ 2
σ πk k Γ( T )
k
k
√
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T −1


r
k
2
2σ 2 +θ 2

 |x−µT |

k
KT 1 


−
σ2
k
2
Value-at-Risk scaling for long-term risk estimation
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(6)
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Modelling P&L Distributions
Fit performances over time
Test 1: fitting performance on a 250-day P&L strip (each P&L distribution made by
N = 500 obs.)
Test 2: fitting performance on a single P&L distribution with N = 8000 obs.
100
100
10-1
10-2
0.025
CDF
CDF
10-1
0.020
0.015
0.010
Return
Actual Data
N
ST
VG
0.005 0.000
10-2
10-3
10-4
0.12
0.10
0.08
0.06 0.04
Return
0.02 0.00
0.02
N performs much worse than ST and VG in explaining empirical P&L data
VG and ST: comparable performances when N = 500
Raising the number of observations clarifies which PDF better fits the P&L dataset:
in the IBM example the winner is ST
Takeaway: though a challenging task, the determination of the PDF to fit P&L data
is crucial to implement any efficient VaR-scaling methodology
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Time Scaling
Convolution and the CLT (1)
Long-term PDF can be calculated (analytically or numerically) by convoluting the
short-term PDF p(xk (t)):
Xk (t) RV (with values xk (t)) describing P&L over horizon t at time (day) k
P&L over the long horizon T = nt is given by:
P&L0→T = x1 (t) + x2 (t) + ... + xn (t) =
n
X
xk (t)
(7)
k=1
The PDF of the sum of two independent (as we assume the P&Ls) RVs is given by:
Z +∞
p(y ) =
p(y − x1 (t))p(x1 (t))dx1 (t)
(8)
−∞
where RV Y = X1 + X2 with values y = x1 + x2 .
Apply n times to the short-term PDF to obtain long-term PDF
What about our benchmark distributions?
Normal: well-known
√
T -rule
VG: analytic expression (see Eq. (6))
ST: numerical convolution (we apply Eqs. (7) and (8) with FFT algorithm)
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Time Scaling
Convolution and the CLT (2)
Is it possible to obtain an asymptotic behaviour? (n → ∞) Yes! Use CLT!
Given RV X distributed as pD (x; ·), with E (X ) = µ∆t and Var(X ) = σ 2 ∆t, the
n-times convoluted distribution satisfies (for all finite α and β):
(
) Z
n
β
X
(x−µn∆t)2
1
−
√
lim
P(α <
xi < β) =
e 2σ2 n∆t
(9)
n→+∞
2πσ 2 n∆t
α
i=1
The above holds for n → ∞. For finite n it is understood that convergence takes
place only in the central region of the PDF, which needs to be quantified somehow
(see next slides)
Therefore, we have the crucial result:
If percentile xα (considered for VaR estimation) falls into central region of PDF in the
sense of CLT after n = T /∆t convolutions, the normal approximation holds:
√
VaRD (α, T ) ' Φ−1
(10)
N (α; µT , σ T ) = VaRN (α, T )
Otherwise (convergence not achieved) long-term P&L distribution to be computed by
explicit convolution (n times)
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Time Scaling
The Normal limit: ST distribution
Which conditions to define the central region of the ST PDF ”Normal”?
We propose a quantitative method following Bouchaud et al.1
1
2
3
4
Define critical value x ∗ beyond which the two PDFs become substantially different
Intuitively take x ∗ as the point where the two PDFs intersect
After some math we find that, as expected, region where CLT holds enlarges slowly:
√ p
(11)
x ∗ = σ ν T log(T )
Using Eq. (11) we estimate the percentile at which convergence condition is satisfied
after exactly 1 year, as a function of ν:
(ν + 1)Γ ν+1
√ p
1
2
P(σ ν T log(T ) < x < +∞) =
(12)
√
ν−2
πΓ ν2
2
T
(log(T ))ν/2
Imposing P = 0.07% and T = 250 days = 1 year, we obtain ν ∗ = 3.41
Using the above-defined criterion we have a discrimination:
convergence regime (ν > ν ∗ ): the ST distribution becomes sufficiently ”normal” for
our purposes2
non-convergence regime (ν < ν ∗ ) where the ST distribution cannot be approximated
with a normal. Accordingly, the lower ν, the fatter the tails.
1
J. P. Bouchaud and M. Potters,Theory of Financial Risks - From Statistical Physics to Risk Management,
Cambridge University Press, 1998
2
Recall that, for ν → ∞ the ST is a Normal.
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Time Scaling
The Normal limit: VG distribution
0.05
|FVG(x)−FN (x)| / FN (x)
0.04
0.03
0.02
0.01
0.00
5
4
3
x/σ
2
1
0
In the VG case, convergence takes place in much quicker way, due to exponential
decay
We present a proof just by numerical example: after convolving the VG PDF a
number of times, we compare its CDF with the target Normal CDF
Already for n = 50 iterations, the relative deviation at 4σ (corresponding to
PN (x < 4σ) ' 0.006%) is smaller than 2%
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VaR scaling on a real portfolio
The methodology test: setup
To assess our VaR-scaling methodology we built a test equity trading portfolio
composed by 10 FTSE stocks and ATM european calls to achieve ∆-hedging:
representative of real portfolios, convex and asymmetric.
1
Perform a (1-day) historical simulation to infer the short-term P&L distribution
2
Fit the P&L distribution with the benchmarks (N, VG and ST)
VaR-scaling calculation:
3
Normal VaR: through application of CLT, the 1-year P&L distribution is Normal with
µ(T ) = µ(t)T = 0
2
(by assumption)
2
σ (T ) = σ (t)T
where µ(s) and σ 2 (s) are mean and variance of the PDF over horizon s
Convoluted VaR: given the short-term fitted PDF pD (x; ·), convolve it n = 250 times
to extract the long-term PDF:
If pD is VG, the long-term PDF is given by Eq. (6)3
If pD is ST, the long-term PDF can be estimated numerically by explicitly convolving pD .
4
3
Repeat steps (1-3) for 10000 different (random) portfolio weight combinations to
derive the statistical properties w.r.t. asset allocation
As mentioned before, in our case it always reaches convergence to the normal limit.
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VaR scaling on a real portfolio
600
5
500
4
400
VaRST / VaRN
Number of Observations (ν)
The methodology test: outcomes
300
200
2
1
100
0
3
2
3
4
5
ν
6
7
8
0
2.0
2.5
3.0
ν
3.5
4.0
As in the single-stock case, VG and ST provide comparable goodness-of-fit; again, N
yields the worst performance
In the VG case, Normal approximation always holds (as expected)
The majority (∼ 70%) of fitted ν values for the ST case lies below the critical value
ν ∗ = 3.41: the assumption of normal convergence is often unsafe
When ν > ν ∗ , ST has reached Normal convergence and VaRST /VaRN ' 1
When ν < ν ∗ , the scaled ST-VaR is greater than the scaled N-VaR, and, in the ν → 2
limit, VaRST /VaRN ∼ 4: the assumption of normal convergence underestimates risk!
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Summary and Conclusions
Summary and Conclusions (1)
Derived a generalized VaR-scaling methodology for calculating Economic Capital
(i.e. 1-year 99.93% VaR)
Chosen as benchmarks for explaining empirical (daily) P&L data were Normal,
Student’s t- (leptokurtic power-law) and Variance-Gamma (leptokurtic exponential)
distributions
Defined long-term P&L distribution by means of convolution and explored its
asymptotic properties using the Central Limit Theorem (CLT)
Theoretical results are a range of possible VaR-scaling approaches depending on
PDF chosen as best fit
given confidence level and
given time horizon
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Summary and Conclusions
Summary and Conclusions (2)
Main discriminant: reaching of Normal convergence (in the sense of CLT) by the
chosen PDF
If assuming exponential decay (Variance-Gamma case) CLT can be safely applied for
the typical time-horizons and percentiles
If assuming power-law decay (Student’s t- case), CLT can be applied only if number of
degrees of freedom ν exceeds a critical value ν ∗ depending on chosen percentile and
time horizon
Outcome of methodology test by portfolio simulation:
In the VG case, Normal convergence is always reached and the
scaling VaR (even if the short-term distribution is not Normal)
√
T -rule is safe for
In the ST case, Normal convergence
√ is often not achieved. In this case, CLT cannot be
applied. The naive usage of the T -rule in the non-convergence regime (ν < ν ∗ , the
most likely in our simulation) can lead to severe underestimation of the risk measure
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