Journal of Risk Model Validation

Shrunk volatility value-at-risk: an application on US balanced portfolios

Stefano Colucci

  • We  generalize, using a t-student distribution,  the simple prediction strategy to model VaR with performance comparable to that of sophisticated simulation models.
  • We observe that the generalization of the model joint with the choice of the shrinkage estimator between realized and implied volatilities works well in the VaR estimation of the whole tail.
  • The performance of the models is treated using both statistical accuracy and efficiency evaluation tests according to the Basel II.
  • We examine the VaR forecasts performance on 39 US Balanced portfolios over an out-of-sample period that covers different financial crises.
  • The Shrunk Volatility VaR Model is a flexible model to estimate VaR that requires only two parameter namely the shrinkage parameter (α) and the t-student degrees of freedom (ν).

We test the naive model to forecast ex ante value-at-risk (VaR) using a shrinkage estimator between realized volatility estimated on past return time series and implied volatility quoted on the market. Implied volatility is often indicated as the operator’s expectation about future risk, while the historical volatility straightforwardly represents the realized risk prior to the estimation point, which by definition is backward looking. Therefore, the VaR prediction strategy uses information both on the expected future risk and on the past estimated risk. We examine Cesarone and Colucci’s 2016 model – Shrunk Volatility VaR (ShVolVaR) – and generalize it, assuming that returns are conditionally Student t distributed (the authors assume that returns are normally distributed, but normality is a particular case of Student t distribution with degrees of freedom that tend to infinity). The ShVolVaR results are compared with those of six benchmark industry VaR models. The performance of all VaR models is validated using both statistical accuracy and efficiency evaluation tests on thirty-nine equally spaced, balanced portfolios composed by US equity and bonds over an out-of-sample period that covers different crises. We evaluate model performances on four VaR confidence levels (95%, 99%, 99.5% and 99.9%). We also validate the models under loss function backtests; our results confirm the efficacy of implied volatility indexes as inputs for a VaR model, combined with realized volatilities. Further, we confirm Cesarone and Colucci’s 2016 conclusion in almost all balanced portfolios. In our empirical analysis, we find that a VaR model that correctly estimates VaR for all ε and all the balanced portfolios considered does not exist. However, for each confidence level, the ShVolVaR model, with appropriate values for the parameters α  and v, can achieve the same results as the common VaR models that are widely used in the finance industry.

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