1 Escuela Internacional de Ciencias Económicas y Administrativas RESEARCH SEMINAR: “FINANCIAL ECONOMETRICS” Lecture Notes Teacher: Javier Bonza VaR Estimation 1. Some useful definitions Value-at-Risk (VaR) uses two parameters, the time horizon and the confidence level, which are denoted by T and 1 – α, respectively. Given these, the VaR is a bound such that the loss over the horizon is less than this bound with probability equal to the confidence coefficient. For example, if the horizon is one week, the confidence coefficient is 99% (so α = 0.01), and the VaR is $5 million, then there is only a 1% chance of a loss exceeding $5 million over the next week. We sometimes use the notation VaR(α) or VaR(α, T) to indicate the dependence of VaR on α or on both α and the horizon T. Usually, VaR(α) is used with T being understood. If L is the loss over the holding period T, then VaR(α) is the α th upper quantile of L. Equivalently, if R = -L is the revenue, then VaR(α) is minus the α th quantile of R. For continuous loss distributions, VaR(α) solves (1) 2. Non parametric estimation of VaR Suppose that we want a confidence coefficient of 1 – α for the risk measures. Therefore, we estimate the α-quantile of the return distribution, which is the α -upper quantile of the loss distribution. In the nonparametric method, this quantile is estimated as the α-quantile of a sample of historic returns, which we will call (α). If S is the size of the current position, then the nonparametric estimate of VaR is
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Escuela Internacional de Ciencias Económicas y Administrativas
RESEARCH SEMINAR: “FINANCIAL ECONOMETRICS”
Lecture Notes
Teacher: Javier Bonza
VaR Estimation
1. Some useful definitions
Value-at-Risk (VaR) uses two parameters, the time horizon and the confidence level, which
are denoted by T and 1 – α, respectively. Given these, the VaR is a bound such that the loss
over the horizon is less than this bound with probability equal to the confidence coefficient.
For example, if the horizon is one week, the confidence coefficient is 99% (so α = 0.01), and
the VaR is $5 million, then there is only a 1% chance of a loss exceeding $5 million over the
next week. We sometimes use the notation VaR(α) or VaR(α, T) to indicate the dependence
of VaR on α or on both α and the horizon T. Usually, VaR(α) is used with T being understood.
If L is the loss over the holding period T, then VaR(α) is the α th upper quantile of L.
Equivalently, if R = -L is the revenue, then VaR(α) is minus the α th quantile of R. For
continuous loss distributions, VaR(α) solves
(1)
2. Non parametric estimation of VaR
Suppose that we want a confidence coefficient of 1 – α for the risk measures. Therefore, we
estimate the α-quantile of the return distribution, which is the α -upper quantile of the loss
distribution. In the nonparametric method, this quantile is estimated as the α-quantile of a
sample of historic returns, which we will call �̂�(α). If S is the size of the current position, then
the nonparametric estimate of VaR is
2
(2)
with the minus sign converting revenue (return times initial investment) to a loss. Here, the
superscript “np" means “nonparametrically estimated."
3. Parametric estimation of VaR
Parametric estimation of VaR has a number of advantages. For example, parametric
estimation allows the use of GARCH models to adapt the risk measures to the current
estimate of volatility. Also, risk measures can be easily computed for a portfolio of stocks if
we assume that their returns have a joint parametric distribution such as a multivariate t-
distribution. Nonparametric estimation using sample quantiles works best when the sample
size and α are reasonably large. With smaller sample sizes or smaller values of α, it is
preferable to use parametric estimation.
VaR for normally distributed returns:
(3)
4. Semi parametric estimation of VaR
There is an interesting compromise between using a totally nonparametric estimator of VaR
and a parametric estimator. The nonparametric estimator is feasible for large α, but not for
small α. For example, if the sample had 1000 returns, then reasonably accurate estimation
of the 0.05-quantile is feasible, but not estimation of the 0.0005-quantile. Parametric
estimation can estimate VaR for any value of α but is sensitive to misspecification of the tail
when α is small. Therefore, a methodology intermediary between totally nonparametric
and parametric estimation is attractive. The approach used in this section assumes that
the return density has a polynomial left tail, or equivalently that the loss density has a
polynomial right tail.
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Because the return distribution is assumed to have a Pareto left tail, for y >0,
Pr (R ≤ - y) = L(y)y- α, (4)
Where L(y) is slowly varying at infinity and α is the tail index. Therefore, if y1 > 0 and y2 >
0, as y →∞ then
(5)
Now suppose that y1 = VaR(α1) and y2 = VaR(α0), where 0 < α1 < α0. Then we have:
(6)
or
(7)
so, now dropping the subscript “1" of α1 and writing the approximate equality as exact, we
have
(8)
Equation (8) becomes an estimate of VaR(α) when VaR(α0) is replaced by a nonparametric
estimate and the tail index a is replaced by one of the estimates discussed soon in the
following section. Notice another advantage of (8), that it provides an estimate of VaR(α) not
just for a single value of α but for all values. This is useful if one wants to compute and
compare VaR(α) for a variety of values of α. The value of α0 must be large enough that
VaR(α0) can be accurately estimated, but α can be any value less than α0.
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A model combining parametric and nonparametric components is called semiparametric, so
estimator (8) is semiparametric because the tail index is specified by a parameter, but
otherwise the distribution is unspecified.
Regression Estimator of the Tail Index
It follows from (4) that
(9)
where
If R(1),…,R(n) are the order statistics of the returns, then the number of observed returns
less than or equal to R(k) is k, so we estimate log{P(R≤R(k)} to be log(k/n). Then, from (9),
we have
(10)
or, rearranging (10)
(11)
The approximation (11) is expected to be accurate only if -R(k) is large, which means k is
small, perhaps only 5%, 10%, or 20% of the sample size n.
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If we plot the points for m equal to a small percentage of n, say
10%, then we should see these points fall on roughly a straight line. Moreover, if we fit the
straight-line model (11) to these points by least squares, then the estimated slope, call it
, estimates 1/α. Therefore, we will call -1/ the regression estimator of the tail index.
Example – VaR estimation for Colombian Stock Exchange ret urns
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This code computes a non-parametric VaR for a given alpha quantile %% Load the data IGBC=xlsread('IGBC'); StockPrices=IGBC(:,2); Date=IGBC(:,1)+datenum('30-Dec-1899'); % Colombian Stock Index Chart plot(Date,StockPrices),title('IGBC') datetick('x',12) grid on
%% Returns
Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan150
2000
4000
6000
8000
10000
12000
14000
16000
18000IGBC
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n=length(StockPrices); logIGBC=log(StockPrices); returns=logIGBC(2:n)-logIGBC(1:n-1); plot(Date(2:n),returns),title('Daily IGBC Returns') datetick('x',12) grid on
%% Kth order statistic of the sample return, R(k) cutoff=0.1;% alpha quantile
% Cuttoff is the percentage of data sample included in the tail of the
% distribution, meaning a 90% interval confidence. K=round((n-1)* cutoff); % K observations rounded to the nearest integer order_returns=sort(returns); plot(order_returns) grid on worst_return=min(order_returns); maximum_return=max(order_returns); quant_nonparametric=order_returns(K);
Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15Daily IGBC Returns
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% Output answers highlighted in blue (obtained directly from %
the command window by running the parameters)
K =
273
worst_return =
-0.1105
maximum_return =
0.1469
quant_nonparametric =
-0.0132
%% Non parametric Value at Risk (VaR) % figures in COP millions Exposure=100000 VaR_nonparametric=-Exposure*quant_nonparametric
0 500 1000 1500 2000 2500 3000-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
# of observations
Sort returns
%
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Exposure =
100000
VaR_nonparametric =
1.315 (mil trescientos quince millones de pesos por cada cien mil millones de exposición al IGBC
pueden perderse en el mejor del 10% de los peores eventos del índice accionario de Colombia)
%% Parametric VaR. Estimation under the assumption that returns are