Portfolio risk forecasting is subject to great parameter uncertainty, particularly for longer forward horizons. This simply reflects that large drawdowns are observed only rarely, making it hard to estimate their ‘structural’ properties. Bayesian forecasting addresses parameter uncertainty directly when estimating risk metrics, such as Value-at-Risk or Expected Shortfall, which depend on highly uncertain tail parameters. Also, the Bayesian risk forecasting method can use ‘importance sampling’ for generating simulations that oversample the high-loss scenarios, increasing computational efficiency. Academic work claims that Bayesian methods also produce more accurate risk forecasts for short- and long-term horizons.

Borowska, Agnieszka, Lennart Hoogerheide and Siem Jan Koopman (2019), “Bayesian Risk Forecasting for Long Horizons”, Tinbergen Institute Discussion Paper, TI 2019-018/III with one quote of Brewer, Brendon, “Introduction to Bayesian Statistics”

The below are excerpts from the papers. Headings and cursive text have been added.

### Conventional approaches to risk measures

“There are three main approaches to forecasting tail-related [risk] measures: **non-parametric historical simulations**, **parametric methods based on an econometric model** where the volatility dynamics are explicitly specified, [and] methods based on **extreme value theory**…Parametric models of volatility…allow prediction of risk based on the current volatility background.”

*For a summary of extreme value theory and related risk models **view post here**.*

“Generally __conventional parametric models are ill-suited for extreme events analysis because they focus on average scenarios in order to obtain a high goodness of fit__. This misperformance [becomes] more severe when the horizon of analysis increases….The global financial crisis stressed the necessity of precise predictions of long-term financial risk…Because most portfolios consist of assets that are held longer than just a few days…increased attention has been devoted to risk forecasting for one-month-ahead or even one-year-ahead horizons, and not only the standard, 1-day-ahead or 10-days-ahead measures…

### Bayesian estimation and ‘importance sampling’

“Risk forecasting, especially for long horizons, is subject to considerable parameter uncertainty. This is why the Bayesian approach seems to be particularly suited for long-run risk analysis.”

“__In Bayesian statistics, an unknown parameter looks mathematically like a ‘random variable’…the prior distribution and posterior distribution describe our uncertainty__ [including about mean or variance of other random variables].

In Bayesian statistics, the interpretation of what probability means is that it is a description of how certain you are that some statement, or proposition, is true…__Probabilities are in the mind, not in the world.__..When we get new information, we should update our probabilities. A Bayesian analysis starts by choosing some values for the prior probabilities. If something is more plausible before you get the data, it’s more plausible afterwards as well…Likelihood is the probability of the data that you actually got, assuming a particular hypothesis is true…To find the posterior probabilities [by taking] the product of the prior probabilities and the likelihoods and dividing it by its sum, producing numbers that do sum to one. Posterior probabilities are not the same as the prior probabilities, because we have more information now.” [Brewer]

“In order to estimate VaR and ‘Expected Shortfall’…__simulation-based methods need to be applied…__The drawback of the direct [conventional] approach is that it is subject to an inherent problem of rare events simulations, i.e. that __most of the generated scenarios are not the ones of the ultimate interest, the extremely negative ones__. This makes direct estimators very inefficient.

To overcome the inefficiency of the direct approach [one can apply] **importance sampling**, a well-known variance-reduction technique. Its main merit is the potential focus on the important subspace by adopting an appropriate sampling density,which in the context of VaR and Expected Shortfall should be tail-focused… the __key idea is to oversample the high-loss scenarios and to give them lower importance weights__.”

“[The figure below] illustrates the construction of the optimal importance density for the VaR estimation.”

“We suggest a novel sequential construction of the importance density, feasible thanks to employing [a] new algorithm. The __construction of importance densities allows for ‘guiding’ of the subsequent simulated returns overtime so that the cumulative return falls in the ‘high-loss’ region, making the analysis of long horizons feasible__…In our approach the properties of the subsequent conditional importance densities depend on the previous simulated returns in the sense that at each step we take into consideration the cumulative return up to that time point…to assess how much the situation still needs to deteriorate in order to qualify for being a ‘high-loss’ scenario.”

### The benefits of the method

“We present an __accurate and efficient method for Bayesian forecasting of two financial risk measures, Value-at-Risk and Expected Shortfall__, for a given volatility model[and] obtain precise forecasts not only for the 10-days-ahead horizon required by the Basel Committee but even for long horizons, like one-month or one-year-ahead…We analyse two benchmark models of volatility, the Generalized Autogressive Conditional heteroscedasticity model (GARCH) and Generalized Autoregressive Score model (GAS).”

“The __proposed method enables accurate forecasts even for long horizons, such as one-month or one-year-ahead__. We have carried out two empirical studies for daily S&P500 returns in different time periods, a tranquil period and a highly volatile crisis period. Both applications confirm that our method not only yields more accurate forecasts than the direct sampling approach, commonly used in practice but also achieves this in a time efficient way [requiring less computation time], resulting in a considerable gain in terms of time-precision trade-off.”