Cardinales invariantes
5.2. El n´ umero de Nagami
Recall that by calculating next-day VaR we are interested in making the following statement:
We are C% certain that we will not lose more than $V in the next X day.
By now we have 3 methods (5 approaches total) of calculating this at-risk amount, but which method should be our ultimate choice? In the following sections we will review the results of these methods using our equal allocation portfolio using back testing. In simple terms, we calculate a series of next-day VaR and compare these calculated numbers with what the actual loss was in history. With this comparison, we are able to back test whether we indeed have only 5% of the times that the predicted VaR’s (losses) are of greater magnitude than the actual loss. We use this failure rate to assess which model is the “best”.
In Table 5.3 we have a summary of all of our models’ back test results un-der a 95% confidence level over different data window lengths. We use our R-application (details in Chapter 6) to calculate these series of VaR’s during year 2001-2007 using the above criteria, and compared them with what the actual
gains/losses were in history. With such comparisons, we come up with the back test summary table. The percentage amounts shown on Table 5.3 represent the percentage of calculated VaR that underestimated the actual loss (actual loss greater than what as “predicted”).
Skimming briefly, all the failure rates are within reasonable range which as-sures us that the models are reasonably constructed. Next, we proceed to the determination of which model yields the most ideal results. First of all, we need to take a step back and recall the purpose of back testing. By back testing, we are trying to assess which model we can trust when coming up with a next-day value-at-risk amount of our particular portfolio. We want something that will give us an amount that is 95% of the time correct. In other words, we want the model to give us a failure rate no more than 5%, but at the same time not ex-tremely below the threshold either, since that will mean that the model is overly conservative and that resources might be wasted for unnecessarily high capital requirement (recall current regulatory framework discussed in Chapter 3). Hav-ing this in mind, we can reasonably say that the historical bootstrappHav-ing method is overly conservative. In Figure 5.11, we see an interesting pattern when com-pared to the graphs of all other methods. As discussed in Chapter 3, the random movements we observe here is related to the breakdown of time variation, which means total return independence are assumed. In investment language, we can see it as a breakdown of “time memory” of the trend. On the other hand, going back to Table 5.3, the historical simulation method yields failure rates that are all above the 5% threshold, so we can conclude that this method is too lax for our purpose. Similarly, the Monte Carlo method also gives us coverage rate that are unsatisfactory (all above 5%). We are then left with the parametric mod-els. Recall that we want failure rates that are close to but not over 5%, with this criteria, the “winner” in this model competition is parametric with equally
Window Parametric (Eq) Historical (Eq) Historical (BS) Monte Carlo Lambda Parametric (Exp.)
12-mo 0.0482 0.0561 0.0331 0.0568 0.97 0.0510
6-mo 0.0489 0.0568 0.0309 0.0554 0.94 0.0482
3-mo 0.0510 0.0633 0.0331 0.0575 0.90 0.0510
1-mo 0.0518 0.0891 0.0618 0.0640 0.80 0.0611
Table 5.3: Back Testing Results for VaR Models - Failure Rates
Parametric VaR − Equally Weighted Moving Average
Time
Returns
2001 2002 2003 2004 2005 2006
−0.10−0.050.000.050.100.15
Actual Gain/Loss
Parametric VaR 12−mo Window Parametric VaR 6−mo Window Parametric VaR 3−mo Window Parametric VaR 1−mo Window
Figure 5.8: Back Testing: Parametric VaR Equally Weighted weighted moving average using a 6-month data window.
Of course, this is a very rudimentary exercise of demonstrating how each method is used and validated. More data and portfolio varieties will be needed if we were doing this exercise for a real financial institution with complex investment portfolios. And with more complicated portfolios that consist of complex financial instruments, we would definitely be able to see the advantage of using the Monte Carlo method over the linear parametric method.
Parametric VaR − Exponentially Weighted Moving Average
Time
Returns
2001 2002 2003 2004 2005 2006
−0.10−0.050.000.050.100.15
Actual Gain/Loss Parametric VaR lambda 0.97 Parametric VaR lambda 0.94 Parametric VaR lambda 0.90 Parametric VaR lambda 0.80
Figure 5.9: Back Testing: Parametric VaR Exponentially Weighted
Historical VaR − Equally Weighted Moving Average
Time
Returns
2001 2002 2003 2004 2005 2006
−0.10−0.050.000.050.100.15
Actual Gain/Loss
Historical VaR 12−mo Window Historical VaR 6−mo Window Historical VaR 3−mo Window Historical VaR 1−mo Window
Figure 5.10: Back Testing: Historical VaR Equally Weighted
Historical VaR − Bootstrapped From All Available History
Time
Returns
2001 2002 2003 2004 2005 2006
−0.10−0.050.000.050.100.15
Actual Gain/Loss
Bootstrapped VaR 2−Year Window Bootstrapped VaR 1−Year Window Bootstrapped VaR 1/2−Year Window Bootstrapped VaR 1/2−Year Window
Figure 5.11: Back Testing: Historical VaR Boostrapping
Monte−Carlo Simulated VaR − 1000 Iterations Each
Time
Returns
2001 2002 2003 2004 2005 2006
−0.10−0.050.000.050.100.15
Actual Gain/Loss
Monte−Carlo VaR 12−mo Window Monte−Carlo VaR 6−mo Window Monte−Carlo VaR 3−mo Window Monte−Carlo VaR 1−mo Window
Figure 5.12: Back Testing: Monte Carlo Simulated VaR
5.4 Some Critiques
The criticisms made by Nassim Taleb in his international best sellers as mentioned in Section 3.4 also apply here. In our exercise, we found that using the parametric method with equally weighted moving average is the most reasonable way to calculate VaR’s for our stock portfolio. With that, we conclude that we are 95%
confident that the next-day loss (the trading day following 12/31/2007) will be no worse than -1.54%. However, what happens to the remaining 5%? For this remaining 5%, we can experience losses of -2%, or -20%, or even -200% and below! In risk management, we are interested in knowing and preparing for the worst possible scenario, and it is these tail events that we should pay attention to. In other words, value-at-risk method gives us only the beginning of the story.
The detail and possible ending of this story will need to be further explored.
Nevertheless, value-at-risk is a value-adding tool in terms of making investors or portfolio managers aware of possible danger, and it is certainly playing an important role in risk management for financial institutions today.