How to choose the return model for market risk? Getting towards a right magnitude of stressed VaR

Abstract

Value at Risk (VaR) and stressed value at Risk (SVaR) or expected shortfall are important risk measures widely used in the financial services industry for risk management and market risk capital computation. Fundamental to any (S)VaR model is the choice of the return type model for each risk factor. Because the resulting SVaR numbers are highly sensitive to the chosen return type model it is important to make a prudent choice on the return type modelling. We propose to estimate the return type model from historic data without making an a priori model assumption on the return model. We explain the fundamentals of return type modelling and how it impacts the magnitude of SVaR. We further show how to obtain a global return type model from a set of similar return type models by using geometric calculus. Numerical simulations and illustrations are provided. In this paper, we consider interest rate data, but the proposed methodology is general and can be applied to any other asset class such as inflation, credit spread, equity or fx.

Keywords:

• VaR
• SVaR
• Value at risk
• Stressed value at risk
• Fundamental review of the trading book
• FRTB
• Interest rates
• Local volatility
• Normal
• Lognormal
• Relative
• Absolute
• Displaced
• Risk factor returns
• Model risk
• Return modelling
• Geometric calculus
• Nature of interest rates

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

2 A shifted lognormal model with large shift could be used; this would be equivalent to an absolute return model.

4 10 day returns are required for FRTB regulation (BCBS 2016).

5 Or a Monte Carlo model which generates scenarios similar to historical returns.

7 Or a variant of a relative return model.

8 Returns can be simply historical or calibrated Monte Carlo.

10 For example σ can be equal to 1 (absolute returns), where 1 strictly speaking would not be considered as a one day absolute interest rate volatility.

12 It is assumed mathematically that $r(s_0,s)$ is weakly differentiable in s Lebesque almost everywhere.

14 In practice one often uses daily close of business market data.

16 A shift of 1% was used to be able to handle negative rates in EUR. In the following if we say relative we mean shifted relative in fact.