Historical Simulation with Component Weight

and Ghosted Scenario

Xinyi Liu

email: xinyilau@gmail.com

Historical simulation (HS) is a popular Value-at-Risk (VaR) approach that has the

advantage of being intuitive and easy to implement. However, its responses to most

recent news are too slow, its two “tails” (upper and lower) cannot learn from each

other, and it is not robust if there is insufficient data. In this article, we put forth

two strategies for improving HS in these weak areas at only minor additional com-

putational costs. The first strategy is a “ghosted” scenario, and the second is a

two-component (short-run and long-run) EWMA scheme. VaR is then calculated

according to the empirical distribution of the two-component weighted real and

ghosted scenarios.

1 INTRODUCTION

Since the 1996 Amendment of the Basel Accord (Basel Committee on Banking Supervision,

1996a, 1996b), value-at-risk (VaR) not only offers a measure of market risk, but it also forms

the basis for the determination of market risk capital. VaR is a quantile measurement of the

maximum amount that a portfolio is likely to lose over a given holding period under normal

market conditions. More specifically, based on information about market conditions up to a

given time t, the VaR for period t+∆t of one unit of investment is a negative α-quantile of the

conditional return distribution. That is:

V aRαt+∆t (x,Rt,θ,K) := − inf

z{z ∈ R : P [∆Pt,∆t (x,Rt,θ,K) ≤ z| Ft] ≥ α} ,

0 < α < 1 (1.1)

where Qα (•) denotes the quantile function, x are the exposures of the portfolio to various risk

factors, Ft represents the information available at date t, rt is the return of risk factors from t−1to t, Rt := {rt, rt−1, . . .}, ∆t is the holding period of the portfolio, α · 100% is the confidence

interval of the VaR, ∆Pt,∆t is the return on the portfolio from t to t + ∆t, θ is the parameters

used to construct the conditional return distribution and K is the number of historical return

observations used.

Researchers have developed a number of new approaches for calculating VaR, including

extreme value theory (EVT) (Longin 1999), filtered EVT (Ozun, Cifer and Yilmazer 2010),

mixture normal general autoregressive conditional heteroscedasticity (GARCH) (Hartz, Mittnik

and Paolella 2006), shortcut based GARCH-type processes (Krause and Paolella 2014), and

conditional autoregressive Value-at-Risk (CAViaR) (Engle and Manganelli 2004). But they are

often much more costly than a naı̈ve historical simulation. There are a few reasons.

First, most of the new approaches are either parametric or semi-parametric techniques that

require parameter estimation prior to forecasting VaR. However, the estimators used by many

of these new approaches, such as maximum likelihood estimators (MLE) and least absolute

deviations (LAD), are more computationally costly than the naı̈ve historical simulation.

1

Second, the cost may remains high even if the VaR quest is limited to a single risk factor,

and the reality in the financial services industry is that portfolios often include multiple risk

factors, even if an effort is made to consolidate risk into only a few key factors for each asset

class. Furthermore, with some models, including a number of risk factors renders multivariate

extensions computationally difficult. This problem is compounded by the fact that, computation-

wise, complex models that require large amounts of data are generally costly to build or to run.

Kuester, Mittnik and Paolella (2006) compare a number of VaR alternative prediction strategies

and conclude that a hybrid approach combining a heavy-tailed GARCH filter with an EVT-based

approach performs best overall. Their investigation, however, is limited to univariate time series,

and a high-dimensional EVT distribution model is not easy to work with (Andersen, Bollerslev,

Christoffersen and Diebold 2006).

Third, portfolios often include a number of non-linear products, such as options and other

derivatives, and complicated approaches are even harder to work with for such portfolios.

For practitioners, the two approaches that require the least computational costs and have

proven most popular are (1) simple historical simulation (HS) and (2) RiskMetrics’ IGARCH

EWMA. As noted by Andersen, Bollerslev, Christoffersen and Diebold (2006), industry prac-

tice largely follows “one of two extremely restrictive approaches: historical simulation or Risk-

Metrics.” According to a more recent international survey by Perignon and Smith (2010), HS

forecast model and its variant, the filtered HS, are the most currently used methods at commer-

cial banks. These assertions remain valid today, despite the fact that more advanced and suitable

models on the subject exist in the literature. There have been some disconnects between practi-

tioners’ and academics’ approaches to VaR modelling.

To bridge the academics and practitioners that use RiskMetrics’ IGARCH EWMA, Krause

and Paolella (KP) (2014) offer a new and promising approach. KP is a shortcut based GARCH-

type processes. It enables quick VaR and expected shortfall calculations, and it delivers highly

competitive VaR predictions, at the three common cutoff values and for all sample sizes.

To bridge the academics and practitioners that use historical simulation, Boudoukh, Richard-

son and Whitelaw (1998) (BRW) put forward a practical method named BRW-HS, a hybrid ap-

proach combining EWMA and HS. This method first allocates exponentially declining weights

to different scenarios and then constructs an empirical distribution. The authors assign a weight

[(1− λ) /

(1− λK

)],[(1− λ) /

(1− λK

)]λ, . . . ,

[(1− λ) /

(1− λK

)]λK−1, (1.2)

to each simulated scenario according to the most recentK returns: rt, rt−1, . . . , rt−K+1, where

rt is a vector denoting the historical returns from time t − 1 to t and K = 250 is the window

length. The VaR is then obtained according to the empirical distribution. Using the criteria set

by BRW (1998), the BRW-HS approach is easy to implement and outperforms both IGARCH

EWMA and HS (applied individually). However, BRW-HS approach performs poorly under

certain conditions. There are three reasons for this.

First, as Prisker (2006) argues, both the HS and BRW-HS approaches are under-responsive

to changes in conditional risk. This problem was illustrated by a short position of the S&P 500

index following the famous crash in Oct/1987. VaR forecasts derived by BRW-HS and HS fail

to raise an alarm, even when danger is imminent. This problem arises because both approaches

concentrate only on the historical observations on the lower tail of the portfolio return, when, in

fact, surprises are in the upper tail. This oversight has also been named as “the hidden danger

of historical simulation” by Prisker (2006).

Second, the approach is not robust enough when there are insufficient historical returns

available, such that only a short window length is attainable. Despite their simplicity, both

BRW-HS and HS fail to precisely forecast VaR at high percentiles (e.g., 99%). This effect is

2

empirically demonstrated in Section 5, where the window length K = 100 is used to test model

performance in the face of insufficient data.

Third, there is too great a trade-off between the number of relevant observations that can be

effectively employed and the speed of response to the most recent observations for BRW. Re-

sponse speed is governed by the value of the decay factor λ. If λ is far less than 1, then BRW-HS

responds aggressively to the most recent input but effectively employs less data. The opposite

also holds true. For example, if we set an aggressive decay factor {λ = 0.97,K = 250}, then

the weight assigned to the 125th observation is only 0.0007. This means that, for example, an

important event that took place six months ago would barely influence the current empirical

distribution. The limited effective size of data is even more problematic when an abundance of

historical data is employed with the intention of enhancing the reliability of the forecast. Empir-

ical evidence of how an aggressive decay factor affects results is provided in Section 5, where

the decay factor λ = 0.97 produces almost identical VaR forecasts when the window length

increases from K = 100 to K = 250. On the other hand, if we set a sluggish decay factor

{λ = 0.99,K = 250}, then the hidden danger is made more visible because the VaR forecast

fails to respond to the most recent observations in a timely manner. For more details on the

investigation of this scenario, please see Section 4.2.

With the aim of overcoming the above three deficiencies, two new strategies for the HS

are proposed in this article. The first involves a “ghosted” scenario, and the second is a two-

component EWMA that assigns weights to the simulated scenarios. It is implemented within

the context of portfolios with both long and short positions and submitted to powerful testing

criteria, including dynamic quantile tests proposed by Engle and Manganelli (2004) and Kuester,

Mittnik and Paolella (2006).

Our proposed method is useful for practitioners who prefer historical simulation. The pro-

posed method retains all the merits of historical simulation and imposes only limited additional

analytical or computational costs; however, it does greatly enhance the precision of the VaR

forecast.

This article is arranged as follows: The two strategies are presented in Section 2 and Sector

3 individually. The CEWG-HS approach, which integrates both, is introduced in Section 4. Sec-

tion 5 reviews VaR model selection criteria and presents empirical studies. Section 6 discusses

parameter optimization. Section 7 concludes.

2 HS WITH GHOSTED SCENARIO (GHS)

2.1 Symmetric empirical distribution

The naı̈ve HS is a nonparametric approach that assumes identical and independent distribution

(i.i.d.) and assigns a weight of 1/K to each scenario. HS is popular primarily because it is sim-

ple and it incorporates risk factor correlations without explicitly modelling them. However, the

i.i.d. assumption means that the empirical probability of one tail remains virtually unchanged

should an extreme event occur in the other. The i.i.d. assumption, therefore, is mostly respon-

sible for the “hidden danger” by Prisker (2006).

In this section, we propose a simple data augmentation technique that increases the num-

ber of scenarios simulated by HS and makes interaction between the two tails possible. The

technique is inspired by the work of Owen (2001), who applies a data augmentation approach

to mitigate the difficulties of empirical likelihood inferences. According to Owen, construct-

ing a family of symmetric distributions FS that puts a positive probability on every observation

serves as a natural approach to nonparametric inference under symmetry. Such a family can be

represented by: the center c of symmetry, and the weight wt attached to rt, which, by virtue of

3

symmetry, is also attached to r̃t = 2c− rt. This yields the equation∑K

t=1wt = 1/2. Then, the

probability that FS gives to rt is∑K

t=1 wt · (1rt=rt+ 1r̃t=rt

), such that rt is double counted,

as though it is both a data point and the reflection of a data point. This may provide a new and

non-degenerate method for resolving issues with empirical likelihood inferences.

Similarly, if the joint distribution of the returns of the risk factors is symmetric, we can

construct a family FS that assigns a positive probability to every historical observation of the

factor returns. Because the means are usually dwarfed by standard deviation when it comes

to high-frequency financial returns data, it is reasonable to let c = 0 and r̃t = −rt. Under

such an assumption, whatever happens to one tail affects the other in a similar way. We can

treat the reflection of a historical observation r̃t = −rt as if it were another true observation

and use it to simulate an imaginary scenario according to the reflection. Eventually, VaR is

obtained according to the 2K scenarios. The scenario that corresponds to the real historical

return is referred to as the “real” scenario, while the imaginary scenario that corresponds to the

reflective return is referred to as the “ghosted” scenario. The idea behind this approach is that the

extreme events in both tails tend to influence each other. We name this model “ghosted historical

simulation” (GHS). Note that GHS aims to construct a symmetric joint return distribution of the

risk factors but not of the portfolio return.

2.2 Reallocating weights to balance between real and ghosted sce-narios

Nevertheless, financial data often exhibits nonzero skewness and, therefore, it may not be ap-

propriate to allocate exactly the same weight 1/(2K) to both a real scenario and also its corre-

sponding ghost scenario. Although reflective data provides valuable side information, it is likely

to be less informative than real data. Therefore, we propose relaxing the symmetric assumption

by assigning a weight (1−G) /K to every real scenario and a weight G/K to every ghosted

scenario, where 0 ≤ G ≤ 0.5. The naı̈ve HS is a special case of GHS when G = 0, and the

symmetric GHS is also a special case when G = 0.5.

By allowing both tails to glean information from each other, it becomes possible for the VaR

of a short position in a financial asset to detect a surge of risk immediately when the market

crashes, and vice versa. This ability mitigates the hidden danger of under-response. Since the

sample size is literally doubled, GHS is rendered more robust than HS when there is insufficient

real data and when the confidence level of the VaR is particularly high.

3 HS WITH TWO-COMPONENT EWMA (CEW-HS)

3.1 The dilemma of using one decay factor

BRW-HS uses an EWMA scheme that looks similar to that of RiskMetrics. RiskMetrics uses

only one decay factor λ and a restrictive IGARCH that estimates long memory in volatility:

ht = (1− λ) rtr′t + λht−1 (3.1)

The term λ is the rate at which EWMA learns from new innovations, and the term (1− λ) stip-

ulates the rate at which the accumulated information ht−1 is released. An overly aggressive λwill utilize too few observations because the model begins to discount past information quickly.

On the other hand, if λ is too close to 1, it will learn slowly and perhaps be under-responsive. A

simple solution to this dilemma is to use a two-component (long-run and short-run) model.

4

3.2 Review of two-component (long- and short-run) models

An interesting model proposed by Engle and Lee (1999) helps solve the dilemma of using only

one decay factor. The authors decompose the volatility of a univariate time series into two

components,

(ht − qt) = γ(ε2t−1 − qt−1

)+ β (ht−1 − qt−1)

qt = ω + ρqt−1 + ϕ(ε2t−1 − ht−1

) (3.2)

The long-run (trend) volatility component qt is stochastic, and the difference between the con-

ditional variance and its trend (ht − qt) is called the short-run (transitory) volatility component.

Engle and Lee (1999) impose that 0 < γ + β < ρ < 1 so that the long-run component evolves

slowly over time and the short-run evolves quickly. Using daily data from some blue-chip stocks

listed on the S&P 500, they find that γ + β < 0.9 for most cases, meaning that the half-life of

a shock on the short-run volatility is less than log0.9 0.5 = 6.58 and dies out quickly. They

also report that 1 > ρ > 0.98 for most of the chosen time series, which implies a high level

of persistence in long-run volatility. Therefore, volatility is capable of responding quickly to

situations like the crash in Oct/1987, and also able to retain information on shocks further in the

past.

JP Morgan (1996) has also used different decay factors to calculate short- and long-run

variance-covariance matrices, but only individually. Given its emphasis on forecast, RiskMetrics

estimates the parameter λ in equation (3.1) by minimizing the mean square error (MSE) of the

our-of-sample volatility forecast. Then RiskMetrics suggests λ = 0.94 for daily and λ = 0.97for monthly forecasts, respectively. The different optimal λ values indicate the different patterns

of short- and long-run volatility evolution; the sluggish decay factor λ = 0.97 implies slow,

monthly (long-run) evolution while the aggressive soother λ = 0.94 implies active, daily (short-

run) evolution.

Inspired by Engle and Lee (1999), we believe it would be fruitful to use both the long-run

and short-run component weights on the empirical distribution to obtain the VaR forecast.

3.3 Two-component EWMA HS (CEW-HS)

In relation to the long-run and short-run component volatility model by Engle and Lee (1999),

we also propose that the importance of an observation (measured by its weight) be determined

by two components. In the case of a recent observation, the short-run component weight ensures

the quick response of the empirical distribution. For a long-dated but still relevant observation,

the long-run component weight generates a milder but still lasting influence.

Define C1 as the proportion of information that the empirical distribution draws from the

long-run component and C2 as that drawn from the short-run component, where 0 < C1 ≤ 1,

0 ≤ C2 < 1 and C1 + C2 = 1. Let λ1 and λ2 be the long-run decay factor and short-run decay

factor respectively and 0 ≤ λ2 < λ1 ≤ 1. To each of the scenarios simulated according to the

most recent historical returns, rt, rt−1, . . . , rt−K+1, assign a weight[(1− λ1)

/(1− λK1

)]C1 +

[(1− λ2)

/(1− λK2

)]C2

[(1− λ1)

/(1− λK1

)]λ1C1 +

[(1− λ2)

/(1− λK2

)]λ2C2, . . . , (3.3)

[(1− λ1)

/(1− λK1

)]λK−11 C1 +

[(1− λ2)

/(1− λK2

)]λK−12 C2,

respectively. The VaR is then obtained according to the empirical portfolio distribution. From

here forward, the above weighting scheme will be referred to as the two-component EWMA

5

Figure 1: Weights on the past observations with two-component EWMA

Figure 1 compares the single decay factors λ = 0.96 (dotted line) and λ = 0.995 (dashed line) against

CEW-HS with {C1 = 0.3, λ1 = 0.995, λ2 = 0.96} (solid line). Panel A and C show the weight allocated

to each individual observation at time t and Panel B and D shows the corresponding cumulative weight.

The window length is 250 for Panels A and B and is 100 for Panels C and Panel D.

historical simulation (CEW-HS). When λ1 = 1, all observations are equally weighted for the

long run component; when C1 = 1, it becomes the BRW approach; when C1 = 1 and λ1 = 1,

it becomes the naı̈ve HS approach.

The CEW-HS tries to obtain a good balance between timely response and long-run stabil-

ity. For example, consider the two-component weighting scheme plotted in Figure 1: {C1 =0.3, λ1 = 995, λ2 = 0.96,K = 250}. The single decay factor λ = 0.96 can only effectively

employ a limited number of observations as the weight converges to zero quickly, while the

single decay factor λ = 0.995is under-responsive to the most recent news. CEW-HS strikes

a good balance between these two extremes: the short-run decay factor λ2 = 0.96 enables an

aggressive response to the most recent observations while the long-run decay factor λ1 = 0.995ensures that all relevant data remains under consideration.

4 COMBINING TWO STRATEGIES

4.1 How GHS and CEW-HS work together

GHS and CEW-HS improve upon the original HS approach in different but highly compatible

ways. The combined approach, which we will refer to as CEWG-HS, is a hybrid of GHS and

CEW-HS approaches.

Denote rt as the realized factor returns from t − 1 to t and r̃t = −rt as the corresponding

ghosted returns. To each of the scenarios simulated according to the most recently realized K

6

returns, rt, rt−1, . . . , rt−K+1, assign a weight:

[(1− λ1)

/(1− λK1

)][1−G]C1 +

[(1− λ2)

/(1− λK2

)][1−G]C2,

[(1− λ1)

/(1− λK1

)]λ1 [1−G]C1 +

[(1− λ2)

/(1− λK2

)]λ2 [1−G]C2, . . . , (4.1)

[(1− λ1)

/(1− λK1

)]λK−11 [1−G]C1 +

[(1− λ2)

/(1− λK2

)]λK−12 [1−G]C2,

respectively. Similarly, to each of the scenarios simulated according to most recent ghosted Kreturns: r̃t, r̃t−1, . . . , r̃t−K+1, assign a weight:

[(1− λ1)

/(1− λK1

)]GC1 +

[(1− λ2)

/(1− λK2

)]GC2,

[(1− λ1)

/(1− λK1

)]λ1GC1 +

[(1− λ2)

/(1− λK2

)]λ2GC2, . . . , (4.2)

[(1− λ1)

/(1− λK1

)]λK−11 GC1 +

[(1− λ2)

/(1− λK2

)]λK−12 GC2,

respectively, and 0 ≤ G ≤ 0.5. The VaR was then obtained according to the empirical distribu-

tion.

The ghost strategy is particularly powerful when there is insufficient data and the VaR re-

quired has a high confidence level. In contrast, the two-component strategy is powerful when

there is abundance of data for the long-run component. Therefore, the two strategies strengthen

each other by functioning in complementary ways. The following section provides a general

summary of a portfolio VaR forecast for a holding period of one day using CEWG-HS.

4.2 How CEWG-HS mitigates the hidden danger

Within the context of the Oct/1987 market crash, let us compare the one-day 99% VaR forecast

of the portfolio with a short position in the S&P 500 using the HS, BRW-HS and CEWG-

HS approaches successively. The principal result of the analysis is that both HS and BRW

exhibit inertial responses at the time of the crash (Figure 2), and both VaR forecasts have two

consecutive “hits” right after the crash.

CEWG-HS, on the other hand, responds immediately when the market crashes. The unusu-

ally large and sudden capital gain warns that a large market correction is likely, and that the short

position will immediately become more vulnerable to a sizable loss. The information obtained

from the upper tail helps avoid VaR violation after the crash.

4.3 The implementation of CEWG-HS

Practitioners often work with a portfolio with a number of risk factors and perhaps with nonlin-

ear financial instruments. The following summarizes the general implementation of CEWG-HS

for a portfolio.

Step 1. Identify the key risk factors and map the portfolio positions accordingly.

Step 2. Obtain the historical return data of the risk factors for the past K trading days.

Step 3. Ghost the historical returns of those risk factors and obtain K ghosted return vectors.

Step 4. Assuming that a certain historical or ghosted return on the risk factors may be realized

the next day, price each asset, obtain a new portfolio value and calculate a portfolio return.

7

Figure 2: One-day 99% VaR forecasts for a portfolio with only a short position in the S&P 500

index from October 14th to 28th, 1987.

Figure 2 shows the one day 99% VaR forecasts for a short position in the S&P 500 index from October 14th

to 28th, 1987. The figure tracks the response of the VaR forecast during the period surrounding the market

crash on October 19th. The rolling window length for all three approaches is 100 days; the decay factor

is λ = 0.98 for BRW-HS; the parameters for CEWG-HS is {G = 0.4, C1 = 0.5, λ1 = 0.99, λ2 = 0.97}.

All approaches use the same kernel and linear interpolation rules as shown in section 4.4.

Step 5. If the portfolio return is simulated according to a historical (ghosted) return, label it as a

“real” (“ghosted”) scenario.

Step 6. Assign a weight to each scenario according to the CEWG-HS weight allocation scheme.

Step 7. Sort the portfolio returns so they are arranged in an ascending manner and find the VaR

by interpolation according to their weights and an applicable kernel rule.

In practice, the most computationally intensive step is 4, where the HS bottlenecks, if a

portfolio that contains non-linear financial instruments. Pricing those financial instruments is

the main cost contributor, and we should do as little as possible to further complicate it. The

new weighting scheme, which comes into effect in step 6, requires negligible additional com-

putational costs. Assigning the weight is also independent of steps 1-5, and the independence is

cost efficient. The ghosted scenarios increase the computational costs more than the weighting

scheme; however, the added costs is reasonable and would certainly be within the capability of

any financial institution running HS.

4.4 A simple example of implementation

Without losing generality and for the sake of simplicity, consider the example shown in Table 1,

where we examine the VaR of a given linear position for a risk factor at a given point in time and

then again two weeks later. We assume that the daily absolute returns during the two weeks stud-

ied are all less than 2%, and we set the parameters as {G = 0.3, C1 = 1, λ1 = 0.98,K = 100},

such that only one decay factor is used.

The left side of the table shows the sorted returns of real and ghosted scenarios on the initial

date. Since we assume that all absolute returns during the two weeks are less than 2%, the rank

of the sorted returns 10 days later remains the same. However, the original returns are further

in the past and therefore have less weight. Assuming an observation window of 100 days and

constant weight, the HS approach estimates that VaR is 2.06% with a confidence level of 95%for both cases.

A kernel rule can also be applied to spread out the weight of each observation, achieve

continuous distribution and find the quantile. According to Butler and Schachter (1998), there

are various kernel rules can also be applied to enhance HS performance. As the quantile of the

8

Table 1: An illustration of the weight allocation of the CEWG-HS approach

Order Return Periods Real or Real or Weight Cumul. Weight Cumul.

ago ghosted ghosted (G-BRW) weight (HS) weight

weight (G-BRW) (HS)

Initial date:

1 −3.30% 3 real 0.70 0.0155 0.0155 0.01 0.01

2 −2.90% 2 ghosted 0.30 0.0068 0.0223 0 0.01

3 −2.70% 15 ghosted 0.30 0.0052 0.0275 0 0.01

4 −2.50% 16 real 0.70 0.0119 0.0394 0.01 0.02

5 −2.40% 5 ghosted 0.30 0.0064 0.0458 0 0.02

6 −2.30% 30 real 0.70 0.0090 0.0548 0.01 0.03

7 −2.20% 10 real 0.70 0.0135 0.0682 0.01 0.04

8 −2.10% 60 real 0.70 0.0049 0.0731 0.01 0.05

9 −2.02% 32 real 0.70 0.0086 0.0818 0.01 0.06

10 days later:

1 −3.30% 13 real 0.70 0.0127 0.0127 0.01 0.01

2 −2.90% 12 ghosted 0.30 0.0055 0.0182 0 0.01

3 −2.70% 25 ghosted 0.30 0.0043 0.0225 0 0.01

4 −2.50% 26 real 0.70 0.0097 0.0322 0.01 0.02

5 −2.40% 15 ghosted 0.30 0.0052 0.0374 0 0.02

6 −2.30% 40 real 0.70 0.0073 0.0448 0.01 0.03

7 −2.20% 20 real 0.70 0.0110 0.0558 0.01 0.04

8 −2.10% 70 real 0.70 0.0040 0.0598 0.01 0.05

9 −2.02% 42 real 0.70 0.0070 0.0668 0.01 0.06

return distribution is a monotonous function of return distribution, the solution is always easy

to obtain. For simplicity of comparison, we use the same rule designed by BRW (1998), which

can be implemented with a simple spreadsheet.

An interpolation rule is required to obtain the quantile using the two data points of adjacent

quantiles. For simplicity, we use the linear interpolation method given by BRW (1998). For

example, under the CEWG-HS approach, the 5% quantile using the initial date lies somewhere

between −2.35% and −2.30%. Using the above allocation rule, −2.35% represents the 4.58%quantile and −2.25% represents the 5.48%quantile. We then assume the required VaR level is

a linearly interpolated return, where the distance between the two adjacent cumulative weights

determines the return. In this case, the one-day 95% VaR (5% quantile) is:

2.35%− (2.35%− 2.25%) · [(0.05− 0.0458)/(0.0548− 0.0458)] = 2.303%.

Similarly, the one-day 95% VaR 10 days later is:

2.25%− (2.25%− 2.15%) · [(0.05− 0.0448)/(0.0558− 0.0448)] = 2.203%.

Finally, the above two rules are insufficient for the one-day 99.5% VaR on the initial date.

Because the smallest observation,−3.30%, has a cumulative weight of only 1.5%/2 = 0.775%,

the 99.5% VaR must lie somewhere lower than the −3.30% level, a level at which no observa-

tions are available. In this situation, we assume that the distance between −3.30% and the upper

halfway is the same as the distance between −3.30% and the lower halfway, −2.90%, such that

the upper halfway to the left of −3.30% is:

−3.30%− (3.30%− 2.90%)/2 = −3.50%.

9

The 99.5% VaR is then:

3.50%− (3.50%− 3.10%) · [(0.005− 0)/(0.0155− 0)] = 3.371%.

5 VAR COMPARISON METHODS AND EMPIRICAL RESULTS

5.1 Unconditional and conditional coverage

It is hard to ascertain the accuracy of a VaR model based on real data since its “true” value

is still unknown, even if based on ex post information. There are several backtesting methods

available. For an excellent review of these methods please refer to Berkowitz, Christoffersen

and Pelletier (2011). We choose to use some popular methods here.

Define hit sequence as Ht = I (∆Pt < −V aRt). For unconditional coverage, Christof-

fersen (1998) suggests that if a VaR forecast is efficient, then Ht|ψt−1 should follow an i.i.d.Bernoulli distribution with the mean

E [Ht|ψt−1] = 1− α, ∀t. (5.1)

The hypothesis of test on unconditional coverage is then:

Hnull,unc : E [Ht] = 1− α versus Halter,unc : E [Ht] 6= 1− α. (5.2)

In order to test the hypothesis of independence, an alternative is defined in which the hit se-

quence follows a first order Markov sequence with a switching probability matrix:

Π =

[1− π01 π011− π11 π11

], (5.4)

where πij is the probability of an i on day t− 1 being followed by a j on day t. The hypothesis

of test on independence is then:

Hnull,ind : π01 = π11 versus Halter,ind : π01 6= π11. (5.5)

And the hypothesis of test on conditional coverage is:

Hnull,con : E [Ht] = 1− α and π01 = π11

versus Hnull,con : E [Ht] 6= 1− α or π01 6= π11(5.6)

The formula of the likelihood-ratio (LR) tests of the alternative hypothesis are provided in details

by Christoffersen (1998).

5.2 Dynamic quantile test

The independent test introduced by Christoffersen (1998) takes into account only the first-order

autocorrelation; as a result, its power is weak, especially when the confidence interval of a VaR

is high, for example, 99%. In this case, if a VaR model is correct, on average, there can be only

one incident of two consecutive hits for every 10,000 observations. However, in practice the

hit sequence is usually not long enough for an adequate assessment for Christoffersen’s (1998)

independence test.

The Dynamic Quantile (DQ) test proposed by Engle and Manganelli (2004) addresses this

complication. In addition to taking a greater autocorrelation of hits into account, Engle and

Manganelli (2004) remark that achieving the VaR confidence level is essential. Kuester, Mittnik

10

and Paolella (2006) draw upon this idea and suggest a simpler DQ test that can be achieved by

regressing Ht across a judicious choice of explanatory variables in ψt. According to Kuester,

Mittnik and Paolella (2006),

Ht = (1− α0) +

p∑

i=1

βiHt−i + βp+1V̂ aRt + µt, (5.7)

where, under the null hypothesis, α0 = α and βi = 0, i = 1, . . . , p + 1. Then, by converting

the formula to vector notation,

H− (1− α) ι = Xβ+ µ, ut =

{α− 1, with probability α,α, with probability 1− α,

(5.8)

where β0 = λ0 − λ and ι is a conformable vector of ones. The independence assumption leads

to the null hypothesis: H0 : β = 0. A suitable central limit theorem (CLT) is invoked that

yields:

β̂LS = (X′X)−1

X′(H− ιe)asy∼ N

(0, (X′X)

−1λ (1− λ)

), (5.9)

from which the following dynamic quantile (DQ) test statistic is established:

DQ =β̂

′

LSX′

Xβ̂′

LS

λ(1 − λ)

asy∼ χ2

p+2. (5.10)

In the empirical study that follows, we use two specific DQ tests in accordance with the

recommendations of Kuester, Mittnik and Paolella (2006). In the first test, DQhit, the regressor

matrix X contains a constant and four lagged hits: Ht−1, . . . , Ht−4. In contrast, the second test,

DQVaR, uses the contemporaneous VaR forecast.

5.3 Empirical study

Here, we examine the VaR forecasting performance of four portfolios, as listed in Table 2. To

maintain equity across the conditions, we use the same kernel and interpolation rules described

in the example in Section 4.4.

Table 2: Description of portfolios

Asset 1, S&P 500 index

Asset 2, Zero-coupon US Treasury 10-year Constant Maturity

Weight of Asset 1 (%) Weight of Asset 2 (%)

Portfolio 1 100 0

Portfolio 2 0 100

Portfolio 3 40 60

Portfolio 4 40 −60

The data for the portfolios comprises the daily closing prices, pt, of the S&P 500 index (SPX

henceforth) and zero-coupon 10-year US Treasury Bonds (T-Bond henceforth). We choose SPX

because the S&P 500 Total Return Index is not available prior to 1987. The price of SPX is

from finance.yahoo.com. The price of T-Bond is constructed according to the 5-year and 10-

year US treasury constant maturity Treasury yields from the FRED database of the Federal

11

Reserve Bank of St. Louis. T-Bond price is then estimated by linear interpolating the zero-

coupon yield curves. The data was sampled from Jan 2, 1962 to Aug 31, 2012 and yielded a

total of 12,585 observations of percentage log returns, rt := 100 ln (pt)− 100 ln (pt−1). Table

3 presents the summary statistics of {rt}. For the SPX, the sample skewness is −1.04 and

kurtosis is 29.8, indicating considerable violation of normality. The return distribution for the

T-bond shows violation of normality as well, though to a lesser degree than SPX. There seems

to be a structural break in movement between the two assets, as the correlation switches from

statistically positive to negative around the year 2000.

As shown in Table 3, the SPX has been more volatile than the T-Bond; therefore, for port-

folio 3, a greater percentage (60%) of assets is allocated to T-Bonds, such that the two assets

contribute similarly to overall portfolio volatility. Portfolio 4 resembles the risk of an equity

investor with bond-like liability. We assume that all the portfolios are rebalanced each day, such

that the percentage weight of each asset in Table 2 is constant at the end of each trading day.

In practice, we are also interested in the VaR forecast of the portfolios as that have the exact

opposite position of portfolio 1 to 4. Therefore, for portfolio 1 to 4, we report the VaR forecasts

for both tails represented by the following quantiles: {1%, 2.5%, 5%, 95%, 97.5%, 99%}.

Table 3: Summary statistics of portfolios

S&P 500 Index

Sample Size Mean Std. Dev. Skewness Kurtosis Min Max

12,585 0.0235 1.04 −1.04 29.8 −22.9 10.4

US Treasury 10-year Constant Maturity

Sample Size Mean Std. Dev. Skewness Kurtosis Min Max

12,585 0.0277 0.681 0.303 12.3 −6.45 7.5

Correlation from Jan/1962 to Dec/1999

0.263

Correlation from Jan/2000 to Aug/2012

−0.379

Table 4 reports the results of quantile estimates (VaR forecast) for a one-day horizon with

a rolling window length of 250. Unconditional coverage, first order independent test, uncondi-

tional coverage and the two versions of the DQ test are used to compare the performance of the

various models. Table 5 compares the performance of each approach under conditions of with a

rolling window length of 250 or 100 to forecast the VaR of all portfolios.

Compared to CEWG-HS, HS and BRW-HS approaches are under-responsive to changes in

conditional risk. For example, the average value of the one-day 99% VaR forecast using HS is

2.44, which is not lower than that of the CEWG-HS approach; however, the rate of violation of

HS is 1.41%, compared to 1.01% and 0.97% by using the CEWG-HS with two sets of param-

eters. This conclusion is confirmed by LRind, DQhit and DQVaR tests, both of which strongly

favor the CEWG-HS approach.

When the window length decreases from 250 to 100, CEWG-HS again performs better than

all other approaches listed in Table 5. Because only five months of data are used, the benefit of

using the long-run weight is insignificant. However, the ghosted scenario still enables CEWG-

HS to outperform the other approaches.

Compared to CEWG-HS, BRW-HS suffers greatly from the trade-off between the number

of effective observations and the response speed. For example, BRW λ = 0.97 is learning the

recent observations quickly, but this single aggressive decay factor fails to effectively employ

more relevant data. The same is true for IGARCH EWMA with λ = 0.94 and λ = 0.99.

12

Table 4: VaR prediction performance for Portfolio 1 with rolling sample size of 250

Model 100 (1− α) % Viol LRuc LRind LRcc DQhit DQV aR |V aR|

1 1.31 0.00 0.01 0.00 0.00 0.00 2.39

CEWG-HS2.5 2.59 0.51 0.00 0.00 0.00 0.00 1.87

{G = 0.3, C1 = 0.5,5 4.98 0.91 0.00 0.00 0.00 0.00 1.49

λ1 = 1, λ2 = 0.96}95 5.03 0.89 0.21 0.45 0.51 0.43 1.50

97.5 2.35 0.28 0.41 0.40 0.63 0.70 1.89

99 1.01 0.88 0.19 0.41 0.70 0.52 2.40

1 1.31 0.00 0.01 0.00 0.00 0.00 2.38

CEWG-HS2.5 2.63 0.34 0.00 0.00 0.00 0.00 1.87

{G = 0.4, C1 = 0.4,5 5.00 0.99 0.00 0.00 0.00 0.00 1.49

λ1 = 1, λ2 = 0.96}95 5.04 0.83 0.39 0.67 0.58 0.52 1.50

97.5 2.35 0.28 0.94 0.56 0.48 0.59 1.88

99 0.97 0.76 0.15 0.34 0.62 0.49 2.38

1 1.92 0.00 0.00 0.00 0.00 0.00 2.24

2.5 3.16 0.00 0.00 0.00 0.00 0.00 1.82

BRW 5 5.77 0.00 0.00 0.00 0.00 0.00 1.47

{λ = 0.97} 95 5.38 0.05 0.15 0.06 0.12 0.00 1.51

97.5 2.98 0.00 0.36 0.00 0.01 0.00 1.87

99 1.67 0.00 0.09 0.00 0.00 0.00 2.25

1 1.39 0.00 0.01 0.00 0.00 0.00 2.37

2.5 2.89 0.01 0.00 0.00 0.00 0.00 1.87

BRW 5 5.28 0.16 0.00 0.00 0.00 0.00 1.49

{λ = 0.99} 95 5.20 0.30 0.00 0.00 0.00 0.00 1.51

97.5 2.71 0.14 0.00 0.00 0.00 0.00 1.89

99 1.23 0.01 0.00 0.00 0.00 0.00 2.41

HS

1 1.45 0.00 0.00 0.00 0.00 0.00 2.42

2.5 3.08 0.00 0.00 0.00 0.00 0.00 1.87

5 5.51 0.01 0.00 0.00 0.00 0.00 1.50

95 5.50 0.01 0.00 0.00 0.00 0.00 1.50

97.5 2.98 0.00 0.00 0.00 0.00 0.00 1.89

99 1.41 0.00 0.00 0.00 0.00 0.00 2.44

1 1.81 0.00 0.00 0.00 0.00 0.00 2.10

2.5 3.33 0.00 0.01 0.00 0.00 0.00 1.77

IGARCH 5 5.41 0.04 0.00 0.00 0.00 0.00 1.49

EWMA(0.94) 95 5.48 0.02 1.00 0.05 0.03 0.00 1.49

97.5 3.01 0.00 0.97 0.00 0.00 0.00 1.77

99 1.39 0.00 0.70 0.00 0.00 0.00 2.10

1 1.61 0.00 0.00 0.00 0.00 0.00 2.20

2.5 2.95 0.00 0.00 0.00 0.00 0.00 1.85

IGARCH 5 4.78 0.27 0.00 0.00 0.00 0.00 1.56

EWMA(0.99) 95 4.55 0.02 0.01 0.00 0.00 0.00 1.56

97.5 2.54 0.79 0.00 0.00 0.00 0.00 1.85

99 1.41 0.00 0.00 0.00 0.00 0.00 2.20

Table 4 summarizes the VaR predictions for Portfolio 1. The results pertain to a 250-length rolling

window length. α is the VaR confidence level. CEWG-HS (0.96, 0.5 and 0.3) represents CEWG-HS with

{λ1 = 1, λ2 = 0.96, C1 = 0.5, G = 0.3}; IGARCH EWMA (0.94) represents IGARCH EWMA with

λ = 0.94. Entries in the last 6 to the last 2 columns are the significance level (p-Values) of the respective

tests. Bold type entries are not significant at the 1% level. For DQhit, Ht − (1− α) ι is regressed onto

a constant, while it is lagged 4 hit indicators for DQVaR; in addition, the contemporaneous VaR forecast

|V aR| denotes the average absolute value for the VaR forecasts.

13

Table 5: VaR prediction performance summary, number of tests that are insignificant

Port 1 Port 2 Port 3 Port 4 Port 1 Port 2 Port 3 Port 4 Average

window length = 250 window length = 100

CEWG-HS {G = 0.3,C1 = 0.5, λ1 = 1, λ2 = 0.96}

17 23 24 16 15 19 16 16 18.25

CEWG-HS {G = 0.4,C1 = 0.4, λ1 = 1, λ2 = 0.96}

17 19 24 15 15 20 16 16 17.75

BRW (0.97) 7 9 5 5 7 8 5 4 6.25

BRW (0.99) 5 8 9 4 3 3 5 3 5.00

HS 2 3 3 1 0 1 2 2 1.75

IGARCH EWMA (0.94) 7 6 8 17 7 5 8 16 9.25

IGARCH EWMA (0.99) 3 3 5 2 3 3 3 2 3.00

Table 5 summarizes the performance of various models for the four portfolios. The composite of the

four portfolios is listed in Table 2. It summarizes the number of tests that are not statistically significant

at the 1% level. For example, in Table 4, the CEWG-HS {G = 0.3, C1 = 0.5, λ1 = 1, λ2 = 0.96}for Portfolio 1 with a window length 250 has 17 bold entries, which means there are 17 tests that are not

statistically significant at the 1% level.

Overall, compared to the HS and IGARCH EWMA, the outperformance of CEWG-HS is

most significant at the 1% and 99% quantile levels, reasonably significant at 2.5% and 97.5%

levels, and almost ignorable at the 5% and 95% levels.

Examining the results in Table 5, overall, the performance of CEWG-HS prevails. In terms

of average number of tests that are insignificant, the two versions of CEWG-HS are 18.25 and

17.75 respectively, much higher than the two versions of BRW at 6.25 and 5.00 respectively, and

much higher than the two versions of IGARCH EWMA at 9.25 and 3.00 respectively. A closer

estimation of the results shows that the outperformances of the CEWG-HS are concentrated on

the 1% or 99% quantile levels, but not on the 10% or 90% levels.

6 PARAMETER OPTIMIZATION

The parameters used in our discussion so far are imposed. We will discuss the optimization of

the parameters briefly in this section.

The VaR estimation can be formulated as:

V aRαt,1 (x, rt,θ,K|ψt) = Qα (θ,∆Pt,1 (x, rt,K|ψt)) (6.1)

where∆Pt,1 (x, rt,K|ψt) is the portfolio return vector from time t to t+ 1,

x is the exposures of the portfolio to the risk factors,

rt is the historical returns of the risk factors,

K is the rolling window length,

ψt is the information up to t, and

θ represents the parameters of the proposed two approaches, θ = (G,C1, λ1, λ2),where G,C1, λ1 and λ2 are defined according to equation (4.1) and (4.2).

We use a method that is similar to the concept of quantile regression. Koenker (2005)

and Koenker and Xiao (2006) provide great references of quantile regression. Using the least

14

absolute deviations (LAD) estimators, the target function is

θ̂ = argminθ∈R

∑

∆Pt,1≥−V̂ aRα

t,1

(1− α)∣∣∣∆Pt,1 + V̂ aR

α

t,1

∣∣∣

+∑

∆Pt,1<−V̂ aRα

t,1

α∣∣∣∆Pt,1 + V̂ aR

α

t,1

∣∣∣

(6.2)

where

V̂ aRα

t,1 = V̂ aRα

t,1

(θ̂,x,Rt,K|ψt

), as defined by equation (1.1).

The objective function (6.2) is not smooth (i.e., not differentiable); it may have multiple

solutions, and the solution may not be stable. For an optimization with such objective functions,

researchers have been using genetic algorithms and other methods to derive a solution. However,

the purpose of our article is to propose simple solutions that are meant to be robust and intuitive.

As such, parameter optimization is discussed only very shortly here.

For simplicity and the sake of illustration, we impose values on some parameters:

Let λ1 = 0.98 and λ2 = 0.94, such that we only need to optimize two parameters, G and

C1. Further, we assume that G ∈ {0, 0.1, . . . , 0.5} and C1 ∈ {0, 0.2, . . . , 1}. We use portfolio

3 as an example, and let α = 99% and K = 100, such that the VaR quantile is high and the

observation window is short. We use the most simple exhaustive-search method. Table 6 lists the

absolute deviations for all the combinations of G and C1. The absolute deviation is minimized

when G = 0.5 and C1 = 0.4.

Table 6: Absolute deviations using different combination of G and C1

C1 \ G 0 0.1 0.2 0.3 0.4 0.5

0 262.415 243.161 225.830 222.579 221.944 221.025

0.2 256.318 240.282 224.671 221.070 219.724 219.247

0.4 251.674 238.861 225.859 220.305 218.832 218.142

0.6 248.558 238.871 227.837 220.773 218.657 218.144

0.8 247.051 239.698 230.486 223.280 219.005 218.415

1 246.530 240.419 233.730 226.669 222.172 219.251

Table 6 summarizes the absolute deviations by using different combinations of G and C1 for port-

folio 3 for 99% VaR. The rolling window is 100. Other parameters are imposed, where λ1 = 0.98and λ2 = 0.94.

7 CONCLUSIONS

Value-at-Risk (VaR) is one of the most widely accepted benchmarks for measuring future ex-

pected risk. One the on hand, recent literature on the subject has employed more advanced

econometric approaches to improve its accuracy. However, these tools are often less intuitive

for many practitioners and often more complex to implement. As a result, these approaches are

still largely limited to academic use, despite their outperformance compared to the approaches

commonly used in the investment world.

On the other hand, the naı̈ve historical simulation and the variance-covariance approaches

proposed by JP Morgan (1996), are still popular for practitioners, although the performances of

such simple approaches are often questioned.

15

In this article, our aim was to offer two new strategies for improving the popular historical

simulation which (1) incur minimal additional computational costs and (2) are practical and easy

to implement. The first strategy uses ghosted scenarios to augment data size and allows both tails

to learn from each other. The second incorporates two-component EWMA. One component is of

the two-component EWMA is the long-run weight, which employs older but more relevant data

effectively; the other component is the short-run weight, which responds quickly to the most

recent information. The CEWG-HS approach combines both strategies and greatly enhances

the performance of HS. Because the two strategies are independent and compatible, they can

work alone or simultaneously. Financial institutions should find it easy to upgrade their current

HS systems using one or both of these strategies.

Further research on VaR prediction may explore a number of directions. For example, a

simple and nature extension is the prediction of Expected Shortfall (ES). We are also interested

in employing an adaptive parameter as a transition variable in the spirit of Taylor (2004), such

that the quantile forecast adapts to the most recent observations more quickly when there is a

regime shift.

REFERENCE

Andersen, T., Bollerslev, T., Christoffersen, P. and Diebold, F. (2006). Practical volatil-

ity and correlation modeling for financial market risk management. Risks of Financial

Institutions, University of Chicago Press for NBER, 513-548.

Basel Committee on Banking Supervision (1996a). Amendment to the capital accord to

incorporate market risks. Bank for International Settlements.

Basel Committee on Banking Supervision (1996b). Supervisory framework for the use

of backtesting in conjunction with the internal models approach to market risk capital

requirements. Bank for International Settlements.

Basel Committee on Banking Supervision (2011). Revision to the Basel II market risk

framework. Bank for International Settlements.

Basel Committee on Banking Supervision (2013). Regulatory consistency assessment

programme (RCAP) – Analysis of risk weighted assets for market risk. Bank for Interna-

tional Settlements.

Berkowitz, J., Christoffersen, P., Pelletier, D. (2011). Evaluating value-at-risk models

with desk-level data. Management Science 57 (12), 2213–2227.

Bollerslev, T., and Mikkelsen, H. (1996). Modeling and pricing long memory in stock

market volatility.” Journal of Econometrics 73, 151-184.

Boudoukh, J., Richardson, M., Whitelaw, R. (1998). The best of both worlds. Risk 11,

64-67.

Butler, J., and Schachter, B. (1998). Estimating value at risk with a precision measure by

combining kernel estimation with historical simulation. Review of Derivatives Research

1, 371-390.

Christoffersen, P. (1998). Evaluating interval forecasts. International Economic Review

39, 271-292.

16

Christoffersen, P., and Goncalves, S. (2005). Estimation risk in financial risk manage-

ment. Journal of Risk 7, 1-28.

Engle, R., and Lee, G. (1999). A long-run and short-run component model of stock return

volatility. Cointegration, Causality and Forecasting, Oxford University Press 10, 475-

497.

Engle, R., and Manganelli, S. (2004). CAViaR: conditional autoregressive value at risk

by regression quantiles.” Journal of Business and Economic Statistics 22, 367-381.

Engle, R. and Ng, V. (1993). Measuring and testing of the impact of news on volatility,

Journal of Finance 48, 1749-1778.

Hartz, C., Mittnik, S. and Paolella, M. (2006). Accurate value-at-risk forecasting based on

the normal-GARCH model. Computational Statistics and Data Analysis 51, 2295-2312.

JP Morgan (1996). RiskMetrics-Technical Document, fourth ed. Morgan Guaranty Trust

Company, New York.

Jorion, P. (2006). Value at risk: the new benchmark for managing financial risk, third ed.

McGraw-Hill.

Koenker, R. (2005). Quantile regression, Econometric Society Monograph Series. Cam-

bridge University Press

Koenker, R., and Xiao, Z. (2006). Quantile autoregression. Journal of the American

Statistical Association 101, 980-990.

Krause, K., and Paolella, M. (2014). A fast, accurate method for value-at-risk and ex-

pected shortfall,” Econometrics, MDPI, Open Access Journal 2(2), 98-122.

Kuester, K., Mittnik, S., and Paolella, M. (2006). Value-at-risk prediction: a comparison

of alternative strategies. Journal of Financial Econometrics 4, 53–89.

Longin, F. (1999). From value at risk to stress testing: the extreme value approach. Work-

ing paper, Centre for Economic Policy Research, London, UK.

Mittnik, S., and Paolella, M. (2000). Conditional density and value-at-risk prediction of

Asian currency exchange rates. Journal of Forecasting 19, 313-333.

Owen, A. (2001). Empirical likelihood. Chapman & Hall/CRC.

Ozun, A., Cifter, A. and Yilmazer, S. (2010). Filtered extreme value theory for value-

at-risk estimation: Evidence from Turkey. The Journal of Risk Finance Incorporating

Balance Sheet 11 (2), 164-179

Perignon, C., and Smith, D. (2010). The level and quality of value-at-risk disclosure by

commercial banks. Journal of Banking and Finance 34 (2), 362–377.

Prisker, M. (2006). The hidden danger of historical simulation. Journal of Banking and

Finance 30, 561-582.

Taylor, J. (2004). Smooth transition exponential smoothing. Journal of Forecasting 23,

385-404.

17