Ebook Historical Value at Risk models applied to Italian floating rate government bonds
Value at Risk (VaR) has become the standard measure of market risk in financial institutions such as banks. The main reason for its popularity is that it shows the total risk of a portfolio in a single number. Internally, banks use VaR as risk management tool by setting VaR limits for individual activities.
Externally, regulators impose capital requirements on banks based on their VaR values. Therefore, accurately estimating VaR is of great importance. VaR estimates the maximum expected loss over a certain period with a certain confidence level. That is, it estimates low quantiles of the profit and loss (P&L) distribution of an instrument or portfolio. In a trading environment, 1-day VaR figures at 99% confidence levels are widely used.
Different approaches can be used to estimate VaR. Manganelli and Engle (1998) give an overview of VaR models used in finance. In so-called parametric VaR models the underlying distribution of risk factors is specified. Risk factors can be interest rates, exchange rates, equity prices etcetera. In practice, it is often assumed that these risk factors are (multivariately) normally distributed. Unfortunately, this assumption is often not justified. Financial time series are usually fat-tailed, which means they have a number of extreme observations that is larger than is likely under the assumption of normality. Therefore, incorrectly assuming normality leads to underestimation of the number of extreme losses. The historical simulation approach does not have this issue since it relies on the non-parametric historical distribution of risk factors. Correlation between market variables, and non-normal behaviour such as fat tails are automatically incorporated in the estimate. We apply different versions of this Historical Value at Risk (HVaR).
Contents
1 Introduction
2 Problem formulation
3 Historical simulation Value at Risk
- 3.1 Simple HVaR
3.2 Weighted HVaR
3.2.1 Age weighted HVaR
3.2.2 Volatility weighted HVaR
3.2.3 Differences between age and volatility weighted HVaR
3.3 HVaR performance tests
3.3.1 Correct unconditional coverage
3.3.2 Correct conditional coverage
3.3.3 Mean absolute percentage error
3.4 VaR confidence bounds
4 Volatility updating schemes
- 4.1 Univariate models
4.1.1 GARCH
4.1.2 Kurtosis of a GARCH process
4.2 Multivariate models
4.2.1 Correlation models
4.2.2 Dynamic Conditional Correlation
5 Floating rate notes
- 5.1 Bond market definitions
5.2 Default-free floating rate notes
5.2.1 Pricing formula
5.2.2 Interest rate risk
5.3 Defaultable floating rate notes
5.3.1 Credit Spread
5.3.2 Interest rate and credit risk
5.4 Bootstrapping a yield curve
5.4.1 Interpolation and extrapolation
6 Empirical results
- 6.1 Data
6.1.1 Market data
6.1.2 CCT Credit Spread descriptives
6.2 Estimation results of volatility updating schemes
6.2.1 Univariate results
6.2.2 Multivariate results
6.3 Estimation results of HVaR approaches
6.3.1 Backtest results of zero coupon CCTs
6.3.2 Backtest results of a portfolio of CCTs
6.4 Bootstrapping HVaR confidence bounds
7 Summary
- A Overview CCT issues 2005-2009
B Motivation of sample size used for GARCH estimation
C Impact of using Euribor as risk-free curve
D Lists of symbols and abbreviations
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