The higher the confidence level, the greater the Value-at-Risk for the portfolio. Realized day-to-day return on the bank's portfolio is compared with VaR for the bank's portfolio. In particular, these so-called variance methods all relate the Value-at-Risk of a portfolio directly to the variance or standard deviation of the portfolio return.
Static Models
Another interesting consequence of the additivity feature is the fact that the marginal distribution of the multivariate normal distribution is also normally distributed. As a result, the risk of a large multivariate linear portfolio can easily be expressed in the risks of individual components. Applying a normal distribution to our AEX portfolio using an estimation sample consisting of the last 1000 days and an estimation sample consisting of prior data, we obtain the following parameter estimates.
This means that the mass in the tails of the distribution is underestimated, which is consistent with the claim that stock returns are heavy-tailed. In the static normal model, we were able to express T-periods-ahead Value-at-Risk in terms of one-period-ahead Value-at-Risk. One can say that a large part of the data is generated by a distribution with a relatively low variance, and that the rest follows a distribution with a relatively high variance.
This means that 92 percent of the time we find ourselves in the low variance distribution, and 8 percent of the time the high variance distribution prevails. Regarding the evaluation of the model, far out in the tail the performance is quite good.
The GARCH Model
If the market was volatile in the current period, the next period's variance will be high, which is amplified or offset according to the size of the return deviation in that period. However, in light of the diminishing power of the back-test in the tail, one can state the conjecture that far out in the tail the normal GARCH model still underpredicts the probability mass. Therefore, in practice it can be fixed quite arbitrarily to the sample variance of the entire series, for example.
The RiskMetricsTM approach essentially involves tracking the return data and using it, along with the decay factor, to update the conditional volatility estimates. Note that imposing the constraint that α and β must together form one implies a persistence in the conditional variance, i.e. a shock that takes the conditional variance to a higher level does not die out over time, but 'lasts eternal'. To overcome the problems in the tails, one can try different assumptions regarding the distribution of the innovations.
For any of the left-tail probability levels, we are unable to reject the model based on a Kupiec posterior test. We will assume that the variance of the first normal distribution follows a GARCH process, while the second is equal to the first plus the variance of the jump.
Historical simulation
Plain
To this end, we set the evaluation period equal to the last thousand trading days and predict the VaR for different window sizes and left tail probabilities. In the tables below, we report the average of the VaR estimates over time for these window sizes and left tail probabilities, as well as the accompanying error rates. The error rates are not always close to their corresponding left tail probabilities and usually exceed them.
Apparently, the Value-at-Risk estimates for the last thousand days in the sample are too low for most window sizes and left-tail probabilities. Overall, in this sample the HS approach appears to accurately predict VaR for the more conservative left tail probabilities, but only for a window size of 250 days and for the maximum window size. Note that in the case of our 0.1 percent left tail probability, HS makes no sense for windows less than 1000 days because for these windows it is not possible to determine the 0.1 sample percentile.
In general, it is not possible to make VaR predictions using HS for left-tail probability levels smaller than the reciprocal of the window size. The VaR forecasts in Figure 3 are based on a left tail probability level of 1 percent and window sizes of 250 and 1000 days.).
Tail index estimators
There is no reason to assume that the model over- or undervalues the Value-at-Risk. Note that the corresponding Value-at-Risk estimates are significantly higher than those we saw in the previous chapter. Ergo, if we are willing to believe that we are dealing with a heavy-tailed distribution with finite variance, the square root of the time rule overestimates the T-periods-ahead Value-at-Risk.
For example, if we consider the AEX Value-at-Risk estimate at a left tail probability of 1 percent, the 10-day Value-at-Risk will be approximately 2.81√. Another approach to tail index estimation was proposed by Van den Goorbergh (1999). The tail index is estimated at 2.42, which implies that the mean and variance of stock returns are finite, but all higher moments are limitless.
Furthermore, the square root of the time rule for converting the Value-at-Risk one period forward into a Value-at-Risk with a term of T periods overvalues the Value-at-Risk with a term of T periods. The performance of the least squares tail approach is not very convincing for the higher left tail probability levels; the Kupiec backtest rejects the model at the important 1 percent level.
Review of the AEX Results
However, due to the possibility of applying the time root rule instead of the square root of time, the resulting capital requirement may well be lower. For the historical simulation method, the results are reported for the maximum window size to make a more or less fair comparison with the other VaR techniques that all use an estimation sample of that size. Once again, the superiority of the GARCH(1,1) model with Student-t innovations is demonstrated: at none of the left tail probabilities examined, this model is rejected.
Moreover, it outperforms the other models at almost all probability levels, as its error rates are closest to ideal error rates in most cases. For example, if we had focused on the bond market instead of the stock market, we might have come to different conclusions. Interestingly, the actual error rates are almost always higher than expected for all methods and all left-tail probabilities.
For none of the methods is the actual failure rate within the 95% confidence band for all left-tailed probabilities, although that for GARCH is very close. As for GARCH models, the normal distribution performs well for confidence levels above 1%.
4 An Application to the Dow Jones
Introduction
Estimation results
Recall that in the Green Zone the VaR scale factor is three, in the Yellow Zone it is gradually increased and in the Red Zone it reaches its maximum, four. In the Red Zone, a bank will likely be required to completely revise its internal model. The Red Zone is not entered once and the number of VaR violations in the Yellow Zone is limited, especially for the Student distribution.
Apparently, this model best describes the downside risk of investing in the Dow Jones Industrial Average, as was the case for the AEX portfolio, due to its ability to handle volatility clustering using a GARCH(1,1) variance specification, and the underlying tongue-tailed Student-t distribution that takes care of the residual leptokurtosis caused by extreme returns. As for the tail index estimation techniques that appeared to be tailored to estimate large quantiles, the results are very disappointing. The results for the important 1 percent level clearly show that the 1 percent extremes are still correlated over time.
Given the poor performance of the tail index estimators, it is all the more striking that these methods result in the lowest capital requirement (if a maximum stress factor of 4 is maintained). Based on these results for linear equity portfolios, more differentiation in the penalty rate seems desirable.
5 Conclusions
Consequently, the tail index estimation methods we have seen in this chapter should be able to predict value at risk even though there is statistical evidence that the distribution of stock returns changes over time (volatility clustering). Even at the most conservative left tail probabilities, such as 0.1 and 0.01 percent, where you would expect tail index techniques that focus on predicting extreme events to excel, the results are not very convincing. GARCH Student is clearly optimal at the 1% level, but GARCH-normal provides the lowest imputation on average.
By far the most important feature of stock returns for value-at-risk modeling is volatility clustering. Even at lower left-tail probabilities (up to 0.01%), GARCH modeling effectively reduces average failure rates and failure rate volatility over time, while at the same time average VaR is lower. Tail index techniques are not successful, due to the fact that they do not deal with the phenomenon of volatility accumulation.
At the 1% level the assumption of a constant VaR over one year resulted in up to 35 VaR violations in one year (1974). Even at the 0.1% level, the average number of VaR violations over the past 39 years was significantly higher than expected.
The generous availability of historical daily return data on this index allowed us to mimic the behavior of banks more realistically, namely by re-estimating and re-estimating the Value-at-Risk models every year. For left-tail probabilities of 1% or lower, the assumed conditional distribution for the stock returns must be fat-tailed.
