In this note it is argued that the estimation error in Value-at-Risk predictors gives rise to underestimation of portfolio risk. We propose a simple correction and find in an empirical illustration that it is economically relevant.Estimation Error; Finance; Garch; Prediction; Risk Management

Lönnbark, Carl

English

Research Papers in Economics

A Corrected Value-at-Risk Predictor￿
Carl L￿ onnbark
Department of Economics, Ume￿ a University
SE-901 87 Ume￿ a, Sweden
email: carl.lonnbark@econ.umu.se
March, 2008
Abstract
In this note it is argued that the estimation error in Value-at-Risk predictors
gives rise to underestimation of portfolio risk. We propose a simple correction
and ￿nd in an empirical illustration that it is economically relevant.
Key Words: Estimation Error, Finance, GARCH, Prediction, Risk Management.
JEL Classi￿cation: C32, C51, C53, G10.
Ume￿ a Economic Studies 734
￿The ￿nancial support from the Wallander-Hedelius Foundation is gratefully acknowledged.
Kurt Br￿ ann￿ as is thanked for many valuable comments and suggestions. The paper has bene￿tted
from discussions with David Granlund.1 Introduction
Value-at-Risk (V aR) has become a standard measure of market risk and it is widely
used by ￿nancial institutions and their regulators. The V aR is de￿ned as the max-
imum potential portfolio loss that will not be exceeded with a given probability,
or
Pr
￿
portfolio loss ￿ V aR
1￿￿￿
= ￿: (1)
The Basel Committee on Banking Supervision, which includes governors of the main
central banks, imposes on ￿nancial institutions such as banks and investment ￿rms
to meet capital requirements based on V aR. Accurate V aR estimates are therefore
crucially important, and V aR has already received much attention in the literature
(see Jorion (2007) for a survey). Although the literature dealing with di￿erent
modeling issues is large, surprisingly little is written about the uncertainty of V aR
predictors. Three studies attempting to quantify the uncertainty are Jorion (1996),
Christo￿ersen and Gon￿ calves (2005) and Chan, Deng, Peng, and Xia (2007).
The question a practitioner naturally poses is how uncertainty in the V aR a￿ects
risk management, i.e. does it in some way change what value to report. Tsay (2005,
ch. 7) points out that V aR should be computed using the predictive distribution
of returns and should take into account the parameter uncertainty in a properly
speci￿ed model. The uncertainty arises from two primary sources. The true data
generating process is not known, which gives rise to model risk, and the parameters
of the hypothesized model must be estimated, which gives rise to estimation risk.
The focus of this paper is on how to incorporate the estimation error in the
V aR predictor. In particular, we take a time series model and demonstrate that the
implied conventional plug-in V aR predictor does not satisfy eq. (1) asymptotically.
In fact, if V aR1￿￿ in eq. (1) is replaced by its predictor, a stochastic variable, the
corresponding probability is higher than ￿, i.e. the portfolio risk is underestimated.
This is of course an undesirable feature, but it is relatively straightforward to correct
the predictor to give the correct risk measure interpretation. We propose a corrected
V aR predictor that accounts for estimation risk. Schaller (2002) discusses along
similar lines and suggests an alternative approach. We emphasize that the correction
is due to the randomness of the V aR predictor and it is not due to conventional
bias, i.e. that the expected value of the V aR predictor is di￿erent from the true
1value. Two studies attempting to correct for conventional bias are Bao and Ullah
(2004) and Hartz, Mittnik, and Paolella (2006).
2 VaR and uncertainty
A general multivariate time series model with conditional mean and variance is
yt = ￿(￿1;￿t￿1) + H
1=2 (￿2;￿t￿1)"t; t = 1;:::;T; (2)
where ￿t￿1 is the information set at t ￿ 1, "t ￿ i:i:d:(0;I) and ￿ and H are,
respectively, vector and matrix valued functions. In the sequel we occasionally use
the shorthand notation ￿t = ￿(￿1;￿t￿1) and Ht= H(￿2;￿t￿1).
Models of this type are usually estimated by maximizing the log-likelihood func-
tion lnLT (￿) / ￿1
2
PT
t [lnjHtj + (yt ￿ ￿t)0H
￿1
t (yt ￿ ￿t)], i.e. assuming "t ￿
n:i:d:(0;I). Given some regularity conditions the estimator vector, ^ ￿ = (^ ￿
0
1;^ ￿
0
2)0, is
asymptotically, normally distributed with the true parameter vector, ￿0, as its mean
and covariance matrix ￿ = ￿[E (@2 lnLT (￿0)=@￿@￿
0)]
￿1.
For a portfolio of assets with returns generated by model (2), the one-period-
ahead conditional V aR, V aR
1￿￿
T+1, satis￿es Pr
￿
w0
TyT+1 ￿ ￿V aR
1￿￿
T+1j￿T
￿
= ￿, where
wT is a vector of portfolio weights that remains unchanged between T and T + 1.
Assuming normally distributed errors a predictor of the V aR is
[ V aR
1￿￿
T+1 = ￿w
0
T￿
￿
^ ￿1;￿T
￿
￿ ￿
￿1 (￿)
r
w0
TH
￿
^ ￿2;￿T
￿
wT: (3)
Since the parameters of the underlying model are estimated, [ V aR
1￿￿
T+1 is subject
to estimation error and is therefore random. It can be decomposed as
[ V aR
1￿￿
T+1 = V aR
1￿￿
T+1 + eT+1 + bT+1;
where eT+1 accounts for sampling variation and has zero mean and variance ￿2
V aR;T+1.
The ￿nite sample bias of the predictor is denoted bT+1. In a related study, Hansen
(2006) showed that asymptotically
p
TeT+1
d ￿! N(0; T￿2
V aR;T+1); where ￿2
V aR;T+1 =
2(@V aR
1￿￿
T+1=@￿
0)￿(@V aR
1￿￿
T+1=@￿). Due to eT+1 and bT+1, [ V aR
1￿￿
T+1 satis￿es
Pr
n
w
0
TyT+1 ￿ ￿[ V aR
1￿￿
T+1j￿T
o
= ￿
￿;
where ￿￿ need not equal ￿.
Now, introduce a correction term cT+1 such that Prfw0
TyT+1 ￿ ￿(V aR
1￿￿
T+1 +
eT+1 +bT+1 +cT+1)j￿Tg = ￿. Assume that yT+1 and eT+1 are independent1 and we
have that
Pr
8
<
:
w0
TyT+1 + eT+1 ￿ w0
T￿T+1 q
w0
THT+1wT + ￿2
V aR;T+1
￿
￿
￿
V aR
1￿￿
T+1 + bT+1 + cT+1
￿
￿ w0
T￿T+1 q
w0
THT+1wT + ￿2
V aR;T+1
j￿T
9
=
;
= ￿.
By the normality of eT+1 and yT+1 the correction can be obtained as
cT+1 = ￿￿
￿1 (￿)
hq
w0
THT+1wT + ￿2
V aR;T+1 ￿
p
w0
THT+1wT
i
￿ bT+1:
Due to bT+1, the correction may in small samples be either positive or negative.
Asymptotically, however, bT+1 is zero, cT+1 is positive, and then ￿￿ > ￿. Now,
add the estimator of the correction to the conventional predictor, [ V aR
1￿￿
T+1, and the
corrected V aR predictor becomes
\ CV aR
1￿￿
T+1 = ￿w
0
T ^ ￿T+1 ￿ ￿
￿1 (￿)
q
w0
T ^ H
1=2
T+1wT + ^ ￿
2
V aR;T+1:
3 A numerical illustration
To illustrate the properties of the correction we conduct a small simulation exper-
iment with data generated according to a GARCH(1;1) model: yt =
p
ht"t, with
ht = 40=252 + 0:1y2
t￿1 + 0:8ht￿1 and "t ￿ n:i:d:(0;1). We consider samples of sizes
250;500, 1000, and 2500 observations, ￿ = 0:01 and results are in each case based on
100 000 replications. The fractions of exceedences for the conventional (￿￿) and the
corrected V aR predictor (￿c) are computed and the hypotheses ￿￿ = ￿ and ￿c = ￿
are tested against the one-sided alternatives ￿￿ > ￿ and ￿c > ￿, respectively. Table
1In the simulation exercise in Section 3, the covariance between the yT+1 and the eT+1 series
was close to zero for all four sample sizes.
3Table 1: Statistics for the corrections in the numerical illustration. S.D. is the
standard deviation. The z’s are the test statistics for the hypotheses ￿￿ = 0:01 and
￿c = 0:01, respectively.
T Mean S.D. Skewness Kurtosis z￿￿ z￿c
250 0.061 0.063 5.007 60.236 3.909 -0.318
500 0.028 0.027 4.272 34.852 1.716 -0.222
1000 0.013 0.013 4.196 31.151 0.254 -0.636
2500 0.005 0.005 4.304 34.284 0.032 -0.254
1 gives some statistics for the estimated corrections and the z-statistics for the tests.
As expected the mean and the variance of the correction decreases with the
sample size. The hypothesis ￿￿ = ￿ is rejected at the 5% level for the sample sizes
250 and 500, while the hypothesis ￿c = ￿ is not rejected for any of the four sample
sizes. That is, the correction matters statistically for the two smaller sample sizes
and it appears that it does the job of taking the estimation error into account.
4 An empirical illustration
V aR corrections are obtained for the three major stock market indices: FTSE 100 of
UK, Nikkei 225 of Japan, and S&P 100 of USA. Five years of daily index data were
downloaded from DataStream and returns were calculated as yt = 100￿log(It=It￿1),
where It is the value of the index at t. The sample covers February 6, 2003 to
February 7, 2008, for a total of 1304 observations. We consider ￿ = 0:01 and the
predictor for V aR at t + 1 is based on observations t ￿ 653 to t, t = 654;:::;1304.
V aR0s are predicted for the ￿nal half of the sample, and are based on re-estimated
GARCH(1;1) models with constant means. Figure 1 gives the estimated corrections
in percentage points.
The corrections exhibit time variation and vary between 0:003 and 0:140 for the
FTSE 100 index, 0:003 and 0:069 for Nikkei 225 and 0:002 and 0:183 for S&P 100.
The average corrections are 0:016, 0:014 and 0:015. These small numbers must be
converted into monetary units to give a fair picture. For example, a correction of
0:05 for a portfolio with 100 billion dollars worth of assets is 50 million dollars on a
daily basis.
4Nikkei 225
sep-05 mar-06 sep-06 mar-07 sep-07 mar-08
0.00
0.05
0.10
0.15
0.20
FTSE 100
sep-05 mar-06 sep-06 mar-07 sep-07 mar-08
0.00
0.05
0.10
0.15
0.20
S&P 100
sep-05 mar-06 sep-06 mar-07 sep-07 mar-08
0.00
0.05
0.10
0.15
0.20
Figure 1: Corrections in percentage points.
5The few relatively large corrections are due to outliers and highlight the sensitiv-
ity of both the Quasi-Maximum Likelihood and the ￿2
V aR;T+1 estimators to extreme
observations.
5 Conclusion
This note argued that the estimation error in V aR predictors gives rise to underes-
timation of portfolio risk. We introduced an approach to correcting a predictor to
account for the estimation error, and in an empirical illustration we found that the
correction is of economic relevance. The proposed correction hinges on the normal-
ity of both the V aR estimator and the returns and does not apply directly to cases
with non-normally distributed returns. Adapting the proposed approach to other
distributions is in principle straightforward, though.
References
Bao, Y., Ullah, A., 2004. Bias of a Value-at-Risk estimator. Finance Research Letters
1, 241{249.
Chan, N. H., Deng, S.-J., Peng, L., Xia, Z., 2007. Interval estimation of value-at-risk
based on GARCH models with heavy-tailed innovations. Journal of Econometrics
137, 556{576.
Christo￿ersen, P., Gon￿ calves, S., 2005. Estimation risk in ￿nancial risk management.
Journal of Risk 7, 1{28.
Hansen, B. E., 2006. Interval forecasts and parameter uncertainty. Journal of Econo-
metrics 135, 377{398.
Hartz, C., Mittnik, S., Paolella, M., 2006. Accurate value-at-risk forecasting based
on the normal-GARCH model. Computational Statistics & Data Analysis 51,
2295{2312.
Jorion, P., 1996. Risk2: Measuring the Risk in Value at Risk. Financial Analysts
Journal 52, 47{56.
6Jorion, P., 2007. Value at Risk: The New Benchmark for Managing Financial Risk,
3rd Edition. McGraw-Hill, New York.
Schaller, P., 2002. Uncertainty of parameter estimates in VAR calculations. Bank
Austria Working Paper BA-RISK-20020419.
Tsay, R. S., 2005. Analysis of Financial Time Series, 2nd Edition. John Wiley &
Sons, Inc., Hoboken, New Jersey.
7

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