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In this section we describe the concepts associated to the probability unbiased estimation of Value at Risk.

Let us suppose that X is an absolutely continuous random variable with distribution function Fθ, where θ is a parameter vector. The α quantile Qα of X is defined as

Qα = F_θ−1(α)

By definition, the quantile has the property that

Fθ(Qα) = α

This equation represents the intuitive concept of the quantile as a threshold that is exceeded with probability α. The quantile Qα of the distribution of returns of a given

financial asset or portfolio is known as the Value-at-Risk (VaR) at the level α or at the confidence level 1 − α.

We assume that the parameter vector θ can be estimated by any method like Maximum Likelihood, Generalized Method of Moments or others in such a way that the observed data are well described. We will assume that estimator to be at least consistent.

In a general estimation setup, a plug-in estimator for a function g(θ) is an estimator obtained by replacing the parameter θ in the function by an estimate, that is

d

g(θ) = g(bθ)

The quantile Qαcan be seen as a function of the parameter vector and the significance

level:

Qα= g(θ, α)

The plug-in VaR estimator is the only method to estimate VaR under a parametric approach:

[

V aRα= bQα= g(bθ, α)

We aim at estimating the risk of the future position where θ ∈ Θ are unknown. If θ were known, we could directly compute the corresponding VaR as a function of θ, g(θ), specifically with Fθ, and we would not need to consider the family of VaR, (g(θ))θ∈Θ.

Our aim is to estimate Qα in such a way that the estimator satisfies this probabilistic

’threshold property’ in the mean for a Fθ-distributed random variable X for all θ, i.e.

Eθ[Fθ( bQα)] = α

where Eθ denotes the expectation operator under probability measure Fθ.

This is a standard unbiasedness condition on the probability of exceeding the VaR estimate bQα. That probability is usually checked by backtesting. Unbiasedness would

imply that the VaR estimate bQα will be exceeded with an expected probability equal to

α.

Definition 1 An estimator dg(θ), obtained with sample observations (X1, ..., Xn) ∼ Fθ of

g(θ), is said to be probability unbiased with respect to a random variable Z with distribution function FZ, if

FZ(g(θ)) = Eθ[FZ( dg(θ))]

holds for all θ.

In the case of a quantile/VaR estimation where all Xi ∼i.i.d Fθ, i = 1, ..., n, g(θ) is

the α-quantile Qα, Z is the next sample observation Z = Xn+1, and FZ is the probability

distribution from which the sample has been obtained. Hence, a probability-unbiased VaR estimator with respect to Z = Xn+1 must satisfy:

Eθ[P (Xn+1< bQα)] = α (1.1)

Unfortunately, under nonlinear mappings of the parameter vector θ, as it is the case of the quantile, the plug-in procedure generally introduces a small sample bias: E(P (Xnew <

[

V aRα) 6= α. The reason is that it treats the estimated parameter vector as deterministic,

even though bθ is a random variable, a fact that must be incorporated into the estimation procedure in order to obtain probability-unbiasedness. As a consequence, the equation:

b

Qα = F−1 b θ (α)

where bθ is an estimator of the parameter θ, is only true asymptotically, i.e. as the number of observations goes to infinity, provided the plug-in estimator is consistent.

[

V aRα ≡ bQα n→∞

−→ V aR_α ≡ Q_α = F_θ−1(α)

almost surely for each θ ∈ Θ, so that it is asymptotically unbiased.

To obtain a probability-unbiased estimator for the quantile there are two approaches, 1. Estimating a probability αputo modify the quantile from the estimated distribution

for which the VaR is estimated. The VaR estimator will be

b

Qαpu = [V aRαpu = g(bθ, αpu) = F

−1 b θ (αpu)

where αpu is chosen so that equation (1.1) is fulfilled.

For example, if F is a Normal distribution, the VaR estimator can be written:

[

V aRα=µ +b bσzαpu

where µ and_b σ are the estimated mean and standard deviation, respectively, and_b zαpu is the inverse cumulative distribution function of the standard Normal(0,1) for

αpu.

2. Modifying the estimate of the parameter vector bθ of the distribution F to bθpu when

computing the plug-in estimator

b

Qα = F−1 b θpu(α)

If F is a Normal distribution: bθpu = (µbpu,bσpu), and the VaR estimator would then be written as follows:

[

V aRα =µbpu+σbpuzα

On the other hand, the plug-in estimator, which has been used in the calculation of quantile / VaR is:

[

V aRα=µ +b σzb α

In this chapter we follow the first of these two approaches to calculate the probability- unbiased VaR, and we use the second approach, whenever possible, to graph an approx- imation of the function F distorted by modifying the parameter bθ. Thus, we will be computing a probability-unbiased estimator of VaR, that is, an estimator:

b

Qα= F−1 b θ (αpu)