CHAPTER 2:LITERATURE REVIEW
2.5 Reduced-Form Method
exceed the VaR and such events tend to be clustered. Figure 1 reuses a picture from their article that shows the VaR exceedences from the six banks reported in standard deviations of the portfolio returns. It shows that the exceedences are large and appear to be clustered in time across banks. The majority of violations appear to take place during the August 1998 Russia default and ensuing Long-Term Capital Management (LTCM) debacle. This suggests that the banks’ models, besides a tendency toward conservatism, have difficulty forecasting changes in the volatility of P&L.
Moreover, the empirical performance of current models reflects difficulties in structural modelling when the portfolios are large and complex. Large trading portfolios have exposures to several thousand market risk factors, with individual positions numbering in tens of thousands. It is almost impossible to output daily VaRs that measures the joint distribution of all material risks conditional on current information. To estimate the portfolio’s risk structure, the banks make many approximations and parameters are often estimated only roughly. While this may appear to give representation to a wide range of potential risks, the various compromises tend to reduce any forecasting advantage.
The limitations of structural modelling extend to capturing time-varying volatility. None of the structural-based models makes any systematic attempt to capture time variation in the variances and covariances of market risks. As for evaluating exposure to liquidity or other market crises, banks are mostly limited to performing stress exercises on their portfolios.
2.5 REDUCED-FORM METHOD
Berkowitz and O’Brien (2001) claimed the clustering of violations suggests that the volatility of P&L may be time varying to a degree not captured by the bank’s internal models. To adjust this and predict the volatility, they formulate an alternative VaR model determined from an ARMA(1,1) plus GARCH(1,1) model of portfolio returns. They reduced the risk factors to a univariate time series, and their reduced-form model offers a more tractable approach to estimating P&L mean and volatility dynamics. While the reduced-form approach does not account for changes in portfolio composition, they claimed that limitation can be relaxed by estimating
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GARCH effects for historically simulated portfolio returns to current positions, rather than historically observed returns.
This was note the debut of reduced-form VaR forecasting approach in the academic society. Zangari(1997) suggested it in the RiskMetrics Monitor. And, Lopez and Walter (1999) reported a favorable results applying GARCH to portfolio returns as against applying GARCH at the risk factor level. Engle and Manganelli (1999) suggested reduced-form forecasting alternatives to GARCH.
The advantage of fitting the time series model to reported P&L is that any systematic errors in the reported numbers are incorporated into the model. This would provide the reduced-form model an advantage over the banks’ internal models if the latter were not calibrated to reflect reported P&L. This following article adopts this reduced form model with the historical data of one life insurance company in Taiwan to testify its forecasting ability.
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l C h engchi U ni ve rs it y Chapter 3: Data Description
3.1 DAILY TRADING PROFIT AND LOSS
This paper collects daily profit and loss (P&L) associated with trading activities from one insurance company from Jan 1, 2011 to Feb 27, 2014 in Taiwan.
The trading revenue is based on position values recorded at the close of day and, unless reported otherwise, represents the insurance company’s consolidated trading activities. These activities include trading in interest rate, foreign exchange, and equity assets, liabilities, and derivatives contracts. Trading revenue includes gains and losses from daily marking to market of positions. However, for the financial products like bond, which does not mark to market daily, we use the theoretical P&L calibrated by the company.
Summary statistics for daily P&L from Jan 1, 2011 through Feb 27, 2014 are reported in Table 2. During this period, the company had negative average profits. In column 5, the Kurtosis estimate is large relative to Normal distribution, i.e. 3. Both the skewness estimated in Table 2 and the histograms in Figure 2 suggest that the portfolio returns tend to be left-skewed. The histogram of P&L in Figure 2 also exhibits extreme outliers in left tail. We take a close look at the outlier, it happened in June 30, 2011. And we both find out the P&L in 2011 are really volatile compared to 2012 and 2013. The possible reason is that the investments of the company are affected by the European debt crisis in 2009 and the following effects in this period.
Table 2. Daily P&L Summary Statistics
Obs Mean STD 99th percentile Kurtosis Skew
777 -0.07 1.00 -2.34 28.22 -1.76
Notes: Daily profit and loss data are divided by its sample standard deviation to protect company’s confidentiality.
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Figure 2. Daily P&L Distributions
Notes: Histogram of daily profit and loss data reported by insurance company from 1/Jan/2011 to 27/Feb/2014. Data are de-meaned and divided by its standard deviations.
3.2 DAILY VAR
The daily VaR estimates are generated by insurance company for the purpose of forecast evaluation or back-testing and are required by regulation to be calculated with the same risk model used for in internal measurement of trading risk. Generally, the VaRs are for one-day ahead horizon and a 99% confidence level for losses.
However, this paper also test a 95% confidence level VaR. Since, there are only few violations of 99% confidence interval VaR. With statistical concerns, we choose 95%
confidence level VaR to have more observation units. In our case, the insurance company’s internal model with a VaR confidence level of 99% only has four exceptions during the sample period. With 95% confidence interval VaR, the internal model has 19 exceptions during this period.
At 95th and 99th percentile, P&L would be expected to exceed VaR 38 and 7 times in 777 trading days. However, the numbers of violation are only 19 and 4 times in this period. With this sense, the internal VaR forecasts happen to be conservative.
We can drill down this phenomenon further by looking at the mean violation at Table 3. Column 4 shows that the mean violations of 95% and 99% VaR are more than one and two standard deviations beyond the VaR. To get a sense of the size of these violations, we take Normal distribution as a benchmark. Under a Normal distribution the probability of a loss just one standard deviation beyond a 99% VaR is 0.04%.
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And the probability of a loss two standard deviations beyond 99% VaR is virtually 0.
With that in mind, while violations of VaR are infrequent, the magnitudes of violations can be surprisingly large. In Figure 3, we present the time series of insurance company’s P&L and corresponding one-day ahead 95th and 99th percentile VaR forecast (expressed in terms of the standard deviation of the insurance company’s P&L). This plot tends to confirm the conservativeness of the VaR forecasts where violations of VaR are relatively few but large.
Table 3. Daily VaR Summary Statistics Confidence
Interval Mean VaR Number of
Violation Mean Violation
95% -2.07 19 -1.03
99% -3.10 4 -2.78
Notes: Daily VaR data are divided by its sample standard deviation to protest the confidentiality. Mean violation refers to the loss in excess of the VaR.
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Figure 3. Internal Daily 95% & 99% VaR Models and Actual P&L Notes: The upper model is used to forecast the one-day ahead 95% percentile of P&L. The lower model is used to forecast the one-day ahead 99% percentile of P&L. Daily P&L are plotted by dotted lines, and VaR are plotted by lines. Data are expressed in standard deviations.
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Figure 4. Violations of Internal 95% & 99% VaR Models
Note: The upper plot shows the daily P&L for those days on which P&L drops below the forecast 95th percentile given the internal models, and the lower one shows the 99th percentile.
Data are expressed in standard deviations.
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N a tio na
l C h engchi U ni ve rs it y Chapter 4: Research Method
4.1 VALUE-AT-RISK (VAR)
Jorion (2001) mentioned two ways of computing VaR. They are nonparametric and parametric VaR. In this paper, we use the parametric method to compute VaR.
We pick a normal distribution to fit the data. First, we need to translate the general distribution f(w) into standard normal distribution , where has mean zero and standard deviation of unity. We associate W* with cutoff return R* such that W*=(1+R*). Generally, R* is negative and can be written as -|R*|. Further, we can associate R* with standard normal deviate by setting
| | (1-1)
It is equivalent to set
∫ ∫ | | ∫ (1-2)
Thus the problem of finding VaR is equivalent to finding the deviate such that the area to the left of it is equal to 1-c. For a defined probability p, the deviate can be found from table of cumulative standard normal distribution function, that is,
∫ (1-3)
We then retrace our steps, back from we just found to cutoff return R* and VaR. From equation (1-1), the cutoff return is
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(1-4)
From more generality, assume now that the parameters and are expressed on the annual basis. The time interval considered is , in years. We find the VaR relative to the mean as
√ (1-5)
In other words, the VaR figure is simply a multiple of the standard deviation of the distribution times an adjustment factor that is related directly to the confidence level and horizon. When VaR is defined as an absolute dollar loss, we have
√ (1-6)
Set the return , the 99% VaR forecast is then given by ̂ ̂ , and the 95% VaR forecast is given by ̂ ̂ .
4.2 TIME SERIES MODEL
Figure 4 shows the violations of 95% VaR tend to be clustered5. That suggests the volatility of P&L may be time varying to a degree not captured by the internal models. To capture and predict the volatility, we formulate an alternative VaR model determined from time series models of portfolio return. Time series models allow us to have and ; hence, we can use delta-normal method to compute daily VaR for the trading positions.
5 With total 19 violations, there are 8 violations happened in July 2011. And among them, there are 4 violations followed by previous-day violation.
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4.2.1 The ARMA Process
We need sample mean to compute VaR. In this section, we will introduce ARMA process to generate the mean we need. Enders (2010) demonstrates the ARMA process with white-noise at the beginning part. A sequence { } is a white-noise process if each value in the sequence has a mean of zero, has a constant variance, and is uncorrelated with all other realizations. Formally, if the notation E(x) denotes the theoretical mean value of x, the sequence is a white-noise process if, for each time period t,
[or var( )=var( )=…= ] ( )
=0 for all j and s [or ( ) ]
Have white-noise in mind, for each period t, is constructed by taking the values and multiplying each by the associated value of . A sequence formed in this manner is called a mobbing average of order q and is denoted by MA(q).
∑ (2-1-1)
It is possible to combine a moving-average process with a linear difference equation to obtain an autoregressive moving-average model. Consider the pth order difference equation
∑ (2-1-2)
Now let { } be the MA(q) process given by (2-1-1), so that we can write
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∑ ∑ (2-1-3)
We follow the convention of normalizing units so that is always equal to unity. If the characteristic roots of (2-1-3) are all in the unit circle, { } is called an autoregressive moving-average (ARMA) model for . If we take ARMA(1,1) as an example, we can write the equation as following and take as .
4.2.2 The ARCH/ GARCH Process
With mean at hand, we still need standard deviation to compute daily VaR. In this section, we will introduce two time series process which can generate time-varying volatility for the P&L. They are ARCH and GARCH processes.
ARCH
Engle (1982) let { ̂ } denote the estimated residuals from the model
so that the conditional variance of is
| [ ]
To this point, we have set equal to the constant . Suppose that the conditional variance is not constant. One simple strategy is to forecast the conditional variance as an AR(q) process using squares of the estimated residuals
̂ ̂ ̂ ̂ (2-2-1)
where is a white-noise process.
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If the values of all equal zero, the estimated variance is simply the constant . Otherwise, the conditional variance of evolves according to the autoregressive process given by (2-2-1). As such, you can use (2-2-1) to forecast the conditional variance at t+1 as
̂ ̂ ̂ ̂
For this reason, an equation like (2-2-1) is called an autoregressive conditional heteroskedastic (ARCH) model. So, ARCH(1) can be expressed by
|
among them and .
GARCH
Bollerslev (1986) extended Engle’s original work by developing a technique that allows the conditional variance to be an ARMA process. Let the error process be such that
√
where , and
∑ ∑ (2-2-2)
Since { } is a white-noise process, the conditional and unconditional means of are equal to zero. Taking the expected value of , it is easy to verify that
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[ ]
This important point is that the conditional variance of is given by
. Thus, the conditional variance of is the ARMA process given by the expression in (2-2-2). This general ARCH(p,q) model- called GARCH(p,q)- allows for both autoregressive and moving-average components in the heteroskedastic variance. Hence, GARCH(1,1) can be expressed by
|
among them and .
Now, we understand the ARMA and ARCH/GARCH processes, and we can combine two processes to generate the mean and standard deviation for computing VaR. Take ARMA(1,1)- GARCH(1,1) as an example, it can be represented by the following equations
, ,
The stands for mean, , and stands for standard deviation. With these two series, we can compute 95% VaR forecast at time t by ̂ ̂ and 99%
VaR forecast by ̂ ̂ .
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4.3 MODEL SELECTION
After fitting P&L with time series models, we have to pick the best among them. And, we use AIC and SBC as the standard to verify the fitting performance of models.
The AIC and the BIC
In Enders (20100, for a given sample size T, selecting the values of p and q so as to minimize AIC (Akaike Information Criterion) equivalent to selecting p and q so as to minimize the sum:
AIC = T ln(SSR)+2(1+p+q)
Minimizing the value of the AIC implies that each estimated parameter entails a benefit and a cost. Clearly, a benefit of adding another parameter is that the value of SSR is reduced. The cost is that degrees of freedom are reduced and there is added parameter uncertainty. Thus adding additional parameters will decrease ln(SSR) but will increase (1+p+q). The AIC allows you to add parameters until the marginal cost (i.e., the marginal cost is 2 for each parameter estimated) equals the marginal benefit.
The BIC (Schwartz Baysian Information Criterion) incorporates the larger penalty (1+p+q) lnT. To use the BIC, select the values of p and q so as to minimize
BIC = T ln(SSR) + (1+p+q) ln(T)
For any reasonable sample size, ln(T) > 2 so that the marginal cost of adding parameters using the BIC exceeds that of the AIC. Hence, the BIC will select a more parsimonious model than the AIC. As indicated in the text, the BIC has superior large simple properties. It is possible to prove that the BIC is asymptotically consistent while the AIC is biased toward selecting an overparameterized model.
However, Monte Carlo studies have shown that in small samples, the AIC can work better than the BIC.
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In order to compare the performance of models, we have to do the backtesting to verify the accuracy of VaR models. Backtesting is a formal statistical framework that consists of verifying that actual losses are in line with projected losses. This involves systematically comparing the history of VaR forecasts with their associated portfolio return.
4.4.1 Kupiec
Kupiec (1995) develops approximate 95 percent confidence regions for verification test, which are reported in Table 4. These regions are defined by tail points of the log-likelihood ratio:
Table 4. Model Backtesting, 95% Non-rejection Test Confidence Regions Nonrejection Region for Number of Failures N Probability Note: N is the number of failures that could be observed in a sample size T without rejecting the null hypothesis that p is the correct probability at the 95 percent level of test confidence.
Source: Adapted from Kupiec (1995)
[ ] [ ]
which is asymptotically, (i.e., wheh T is large) distributed Chi-square with one degree of freedom under the null hypothesis that p is the true probability. Thus we would reject the null hypothesis if LR > 3.841.
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4.4.2 Christoffersen
Christoffersen (1998) extends the LRuc statistic to specify that the deviation must be serially independent. The test is set up as following:
Deviation indicator= 0 if VaR is not exceeded;
Deviation indicator= 1 otherwise.
Then we define Tij as the number of days in which state j occurred in one day while it was at I the previous day andπi as the probability of observing an exception conditional on state i the previous day. Table 5 shows how to construct a table of conditional exceptions.
If today’s occurrence of an exception is independent of what happened the previous day, the entries in the second and third columns should be identical. The relevant test statistic is
[ ] [ ]
Here, the first term represents the maximized likelihood under the hypothesis that exceptions are independent across days, or
The second term is the maximized likelihood for the observed data.
Table 5. Building an Exception Table: Expected Number of Exceptions Conditional
Day Before
No Exception Exception Current day
No exception
Exception )
Total
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The combined test statistic for conditional coverage then is
LRcc = LRuc + LRind
Each component is independently distributed as x2(1) asymptotically. The sum is distributed as x2(2). Thus we should reject at the 95 precents test confidence level if LR>5.991. We would reject independence alone if LRind>3.84.
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l C h engchi U ni ve rs it y Chapter 5: Results
The time series models are estimated each day with data available up to that point. To obtain stable estimates for the model, forecasts for 2011 (days 1 through 243) are in-sample. Rolling out-of-sample forecasts starts after 2012. Out-of-sample estimates are updated daily.
Here, we adopt two kinds of reduced-form models; they are fitted model and Berkowitz & O’Brien model (BO Model). The fitted model uses the first 243-day data as the in-sample to fit a time series model; and the BO model follows Berkowitz
& O’Brien (2002) using a ARMA(1,1)-GARCH(1,1) as the time series model.
Given parameters estimates, we forecast the next day’s 95% and 99% VaR.
The results of the forecast, both within and out-of-sample, are shown in Figure 5 by the grey line, along with P&L by the dotted line and internal model by solid line. As we can see one-day ahead reduced-form forecasts appear to track the lower tails of P&L really well compared to the internal structural model. It tracked the huge P&L drop in 2011, which did not caught by the internal model. This shows that time series model does better at adjusting in volatility through time.
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95% VaR of Fitted Model
95% VaR of BO Model
Figure 5. P&L, 95% Internal VaR, Fitted VaR, and BO VaR Note: Data are expressed in standard deviations.
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99% VaR of Fitted Model
99% VaR of BO Model
Figure 6. P&L, 99% Internal VaR, Fitted VaR, and BO VaR
Figure 6. P&L, 99% Internal VaR, Fitted VaR, and BO VaR