## Daily risk methods

The risk forecast methods are historical simulation (HS), moving average normal (MA), exponentially weighted moving average normal (EWMA), normal GARCH, student-t GARCH and extreme value theory (EVT). The risk measures are Value-at-Risk (VaR) and expected shortfall (ES).

The oldest methods used to estimate them date back to the 1950s, at least, and the youngest method to the 1980s.

The information below comes from Financial Risk Forecasting, as well as and the code to estimate them.

### Notation

$$\DeclareMathOperator{\VaR}{VaR}$$ $$\DeclareMathOperator{\ES}{ES}$$ $$\DeclareMathOperator{\E}{E}$$ $$\DeclareMathOperator{\WE}{WE}$$

• $$p$$ Probability
• $$Y,y$$ return
• $$Q$$ Profit and loss
• $$\vartheta$$ Portfolio value
• $$Y_t$$ Returns
• $$y_t$$ Sample realization of $$Y_t$$
• $$\sigma$$ Unconditional volatility of returns
• $$\sigma_t$$ Conditional volatility of returns
• $$\lambda$$ Decay factor in EWMA
• $$Z_t$$ Residuals
• $$\alpha, \beta$$ Main model parameters
• $$L_1, L_2$$ Lags in volatility models
• $$\iota$$ Tail index
• $$\xi=1/\iota$$ Shape parameter
• $$q_n$$ Number of observations in the tail

### Non-parametric methods

#### HS

The VaR at probability $$p$$ is simply the negative $$(T \times p)^\text{th}$$ value in the sorted return vector, multiplied by the monetary value of the portfolio.

### Volatility based methods

#### MA

Possibly the simplest volatility forecast model is a moving average (MA) model.

$\hat{\sigma}^2_{t}=\frac{1}{\WE} \sum_{i=1}^{\WE}y_{t-i}^{2}$

#### GARCH

$\sigma^2_{t} =\omega +\sum_{i=1}^{L_1}\alpha _{i}Y_{t-i}^{2}+\sum_{j=1}^{L_2}\beta_j \sigma^2_{t-j}.$

The most common version only employs one lag in the GARCH(1,1) model.

$\sigma^2_{t} =\omega +\alpha Y_{t-1}^{2}+\beta \sigma^2_{t-1}.$

#### EWMA

$\hat{\sigma}_t^2 = (1-\lambda) y_{t-1}^2 + \lambda \hat{\sigma}_{t-1}^2.$

where $$0<\lambda < 1$$ is the decay factor and $$\hat{\sigma}_t^2$$ the conditional volatility forecast on day $$t$$.

#### tGARCH

$Z_{t}\sim t_{\left( \nu \right) }.$

### Tail methods

#### EVT

Hill

$\hat{\xi}=\frac{1}{\hat{\iota}}=\frac{1}{q_n}\sum_{i=1}^{q_n}\log \frac{x_{\left( i\right) }}{u},$ $\widehat{\VaR(p)}=u\left( \frac{q_n/T}{p}\right)^{\widehat{^{1/\iota }}}.$

### Others

APARCH, realized vol, implied vol