16.1 Forecast and VaR Models

Value at Risk (VaR) models are used in many financial applications. Their goal is to quantify the profit or loss of a portfolio which could occur in the near future. The uncertainty of the development of a portfolio is expressed in a ``forecast distribution'' $ P_{t+1}$ for period $ t+1$.

$\displaystyle P_{t+1} = {\cal L} (L_{t+1} \vert {\cal F}_t)$

is the conditional distribution of the random variable $ L_{t+1}$, which represents the possible profits and losses of a portfolio in the following periods up to date $ t+1$, and $ {\cal F}_t$ stands for the information in the available historical data up to date $ t$. An estimator for this distribution is given by the forecast model. Consequently the possible conditional distributions of $ L_{t+1}$ come from a parameter class $ {\cal P}_{t+1} = \{
P^{\theta(t)}_{t+1} \, \vert \, \theta(t) \in \Theta \}$. The finite-dimensional parameter $ \theta(t)$ is typically estimated from $ n=250$ historical return observations at time $ t$, that is approximately the trading days in a year. Letting $ \hat{\theta}(t)$ stand for this estimator then $ {\cal L} (L_{t+1}
\vert {\cal F}_t)$ can be approximated with $ P^{\hat{\theta}(t)}_{t+1}$.

An important example of $ {\cal P}_{t+1}$ is the Delta-Normal Model, RiskMetrics (1996). In this model we assume that the portfolio is made up of $ d$ linear (or linearized) instruments with market values $ X_{k,t}, k=1, ...,
d,$ and that the combined conditional distribution of the log returns of the underlying

$\displaystyle Y_{t+1} \in \mathbb{R}^d, Y_{k,t+1}= \ln X_{k,t+1} - \ln X_{k,t}, k=1, ..., d,$

given the information up to time $ t$ is a multivariate normal distribution, i.e.,

$\displaystyle {\cal L} (Y_{t+1} \vert {\cal F}_t) = {\text{\rm N}}_d(0,\Sigma_t)$ (16.2)

where $ \Sigma_t$ is the (conditional) covariance matrix of the random vector $ Y_{t+1}$. We consider first a single position $ (d=1)$, which is made up of $ \lambda_t$ shares of a single security with an actual market price $ X_t=x$. With $ w_t =
\lambda_t x$ we represent the exposure of this position at time $ t$, that is its value given $ X_t=x$. The conditional distribution of the changes to the security's value $ L_{t+1}=\lambda_t (X_{t+1}-X_t)$ is approximately:
$\displaystyle {\cal L} (L_{t+1} \vert {\cal F}_t)$ $\displaystyle =$ $\displaystyle {\cal L}(\lambda_t(X_{t+1}-x) \ \vert \ {\cal F}_t)$  
  $\displaystyle =$ $\displaystyle {\cal L} (w_t \frac{X_{t+1}-x}{x} \ \vert \ {\cal F}_t)$  
  $\displaystyle \approx$ $\displaystyle {\cal L} (w_t Y_{t+1} \ \vert \ {\cal F}_t) =$   N$\displaystyle (0, w_{t}^2 \sigma_{t}^2)$ (16.3)

with $ \sigma_{t}^2 = \mathop{\text{\rm Var}}( Y_{t+1} \ \vert \ {\cal F}_t) .$ Here we have used the Taylor approximation

$\displaystyle \ln X_{t+1} - \ln x = \frac{ X_{t+1}-x}{x} + {\scriptstyle \mathcal{O}}(X_{t+1} -x).$ (16.4)

The generalization to a portfolio that is made up of $ \lambda^1_t,
\cdots ,\lambda^d_t$ shares of $ d$ (linear) instruments is quite obvious. Let $ w_t$ be the $ d$-dimensional exposure vector at time $ t$

$\displaystyle w_t = (w_t^1, \cdots , w_t^d)^\top = (\lambda^1_t x^1 , \cdots ,\lambda^d_t x^d)^\top .$ (16.5)

$\displaystyle L_{t+1} = \sum_{k=1}^{d} \lambda^k_t (X_{k,t+1}-X_{k,t})$

is the change in the value of the portfolio. For a single position the conditional distribution of $ L_{t+1}$ given the information $ {\cal F}_t$ is approximately equal to the conditional distribution of

$\displaystyle w_t^\top Y_{t+1} = \sum_{k=1}^{d} w_t^k Y_{k,t+1}.$

In the framework of Delta-Normal models this distribution belongs to the family

$\displaystyle {\cal P}_{t+1} = \{$   N$\displaystyle (0, \sigma^2_t) : \sigma^2_t \in [ 0, \infty) \} ,$ (16.6)

with $ \sigma_t^2 = w_t^\top \Sigma_t w_t .$ The goal of the VaR analysis is to approximate the parameter $ \theta(t) = \sigma_t $ and thus to approximate the forecast distribution of $ P_{t+1}$.

Now consider the problem of estimating the forecast distribution from the view point of the following model's assumptions. The change in the value of the portfolio is assumed to be of the form

$\displaystyle L_{t+1}$ $\displaystyle =$ $\displaystyle \sigma_t \ Z_{t+1}$ (16.7)
$\displaystyle \sigma_t^2$ $\displaystyle =$ $\displaystyle w_t^\top \Sigma_t w_t,$ (16.8)

where $ Z_t$ is i.i.d. N(0,1) distributed random variable, $ w_t$ is the exposure vector at time $ t$ and $ \Sigma_t$ is the (conditional) covariance matrix of the vector $ Y_{t+1}$ of the log returns. We combine the last $ n$ realizations of $ Y_t= y_t, \ldots, Y_{t-n+1} = y_{t-n+1}$ from the log return vector with a $ (n \times d)$ matrix $ {\cal Y}_t =
( y_i^\top )_{ i=t-n+1, ... , t}$. From these observations we calculate two estimators from $ \Sigma_t$; first the naive RMA, i.e., rectangular moving average:

$\displaystyle \hat{\Sigma}_t = \frac{1}{n}{\cal Y}^\top _t {\cal Y}_t .$ (16.9)

Since the expected value of the vector of returns $ Y_t$ is zero according to the Delta-Normal model, this is exactly the empirical covariance matrix. The second so called EMA estimator, i.e., exponentially moving average, is based on an idea from Taylor (1986) and uses an exponential weighting scheme. Define for $ \gamma, \ 0 < \gamma < 1$

$\displaystyle \tilde{y}_{t-k} = \gamma^k y_{t-k} , k=0, ... , n-1, \quad \tilde{{\cal Y}}_t = ( \tilde{y}_i^\top )_{ i=t-n+1, ... , t}$

a log return vector is exponentially weighted over time and a $ (n \times d)$ matrix is constructed from this, then $ \Sigma_t$ is estimated with

$\displaystyle \hat{\Sigma}_t = (1-\gamma)^{-1} \tilde{\cal Y}^\top _t \tilde{\cal Y}_t .$ (16.10)

This normalization makes sense, since the sum $ \sum_{i=1}^n
\gamma^{i-1}=\frac{1-\gamma^n}{1-\gamma}$ for $ \gamma \to 1$ converges to $ n$, thus the RMA estimator is the boundary case of the EMA estimator. Both estimators can be substituted in (15.7) and (15.8), and we obtain with

$\displaystyle \hat{P}_{t+1} =$   N$\displaystyle (0, \hat{\sigma}^2_t) , \quad \hat{\sigma}^2_t = w_t^\top \hat{\Sigma}_t w_t$

an approximation of the forecast distribution, i.e., the conditional distribution of $ L_{t+1}$. It should be noted that the Bundesanstalt für Finanzdienstleistungsaufsicht ( www.bafin.de ) currently dictates the RMA technique.

The Value at Risk VaR is determined for a given level $ \alpha$ by

$\displaystyle VaR_t = F^{-1}_{t+1}(\alpha)\stackrel{\mathrm{def}}{=}inf \{ x ; F_{t+1}(x) \ge \alpha \}$ (16.11)

and estimated with

$\displaystyle \widehat{VaR}_t = \hat{F}^{-1}_{t+1}(\alpha)\stackrel{\mathrm{def}}{=}inf \{ x ; \hat{F}_{t+1}(x) \ge \alpha \}.$ (16.12)

Here $ F_{t+1}, \hat{F}_{t+1}$ represent the distribution function of $ P_{t+1}$, $ \hat{P}_{t+1}$. The quality of the forecast is of particular interest in judging the VaR technique. It can be empirically checked using the realized values $ (\hat{P}_t, L_t),
t=1, ..., N,$. In the event that the model assumptions, for example, (15.7) and (15.8), are correct for the form of the forecast's distribution, then the sample $ U_t = F_t(L_t),
t=1, ..., N,$ should have independent uniformly distributed random values over the interval $ [0,1]$ and $ \hat{U}_t = \hat{F}_t(L_t),
t=1, ..., N,$ approximately independent identically uniformly distributed random values. Then the ability of the forecasts distribution to fit the data is satisfied.