MAFS5220 Final Reserach Project

8/13/2019 MAFS5220 Final Reserach Project

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Department of Mathematics

Hong Kong University of Science and Technology

MAFS 5220

Quantitative and Statistical Risk analysis

A Sequential Algorithm on Robust Value at Risk Approach with

application to indexes and migration matrix risk control

Huang Yuelun

[email protected]

UG hall6 room 704

I would like to thank Prof. Ling Shiqing, Prof Jing Bingyi for their kindly

and precious advice on this time series and statistical technique and Prof.

Chan for his kindly guidance on the working optimization problem.
mailto:[email protected]:[email protected]:[email protected]


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A Sequential Algorithm on Robust Value at Risk Approach With application to

indexes and migration matrix risk control

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Content

1. Abstract ................................................................................................................ 3

2. background and data selecction certerion .......................................................... 4

3. data indetifcation and resources ......................................................................... 4

4. Value at Risk approach revisited ............................................................................ 6

5. priliminary data analysis and problem description ............................................... 9

6. brief introduction on time series forecasting technique ..................................... 11

7. The pesudo Robust Var algorithm ...................................................................... 15

8. Conclusion on HSI index study ............................................................................. 21

9. further application ............................................................................................... 23

10. Appendix & reference .......................................................................................... 26


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1. AbstractValue at Risk approach has been one of the most widely known risk control

techniques for its simplicity, easily and widely applied nature, However, the

dark side of this approach is very often criticized for itsloose control in the tail

events and non-sub-additivity. Although other risk measures like CVaR and

EVaR are developed to successfully conquer such inconsistency. But the

problem is such measures form a strict sub-additive upper bound, and it is

fundamentally too conservative. It trades the profitability with risk in the strict

sense. In essence, the relationship of risk and profit are not clearly solved.

Both methods have advantages in certain situations but fail in other cases.

Therefore they are not robust.

This research project embraces an original idea, by utilizing the advanced timeseries technique to redevelop the VaR approach into Robust VaR (Also called

for General VaR ) approach. With the advanced economy downturn detecting

mechanism and volatility prediction, the methods high accuracy better

balances the risk and business level than the non-robust methods. Moreover,

it gives more robust defense on the downturn risk signaling and control. The

method of the paper is carried in a pseudo algorithm format, it is neat and

easy to implement for simplicity. Throughout the paper HSI index study will be

demonstrated. Moreover the application and the prospect are also oriented.

Key words: Markov switching, volatility, downturn risk, Value at risk, Robust,

empirical Bayesian, asymmetric optimization

2.Background and data selection criterionData Selection criterion:

To carry out the strategy and validate the empirical result of improved

Robustized Value at risk approach, one needs appropriate data selection to

back up the idea and demonstrate the result. To find the data, we need the

data satisfy the following properties:

1. High liquidity with large trading volume.2. There is no closed pricing formula and the return should subject to

stochastic changes and not easy to monitor.


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3. The data should be taken in long time horizon and compute withquarterly horizon to stabilize the information shocks.

The reason behind first requirement is that price of certain underlying are

subject to effects of a lot of factors in reality. The high liquidity and large

trading volume means other risks, for instance, liquidity risk and relevant are

less significant so we can better monitor the risk with basic prediction of the

underlying asset volatility movements. The second requirement is that the risk

analytics are applied in real in the sense that people cannot form consistent

predictions for the future and the pricing are at stochastic dynamics changes.

The last one is a way of eliminating the white noises, since the daily return

might subject to strong speculation purposes, while the quarterly returns are

more stable and highly associate with the real behavior and realization of themarket expectation.

In below, it can be demonstrated that the HSI index is a good candidate data

set and we will use it to carry out the method for demonstration purpose.

3.Data identification and collectionHeng Seng Index:

A brief introduction:

The Hang Seng Index ("HSI") is one of the earliest stock market indexes in

Hong Kong. Publicly launched on 24 November 1969, the HSI has become the

most widely quoted indicator of the performance of the Hong Kong stock

market. To better reflect the price movements of the major sectors of the

market, HSI constituent stocks are grouped into Finance, Utilities, Properties,

and Commerce and Industry Sub-indexes.i

High liquidity and trading volume:


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Figure 1 HSI trading volume

It can be seen that the trading volume are at least millions level per workday

and the volume subjects to a steady growth in the past 10 years. With such

high turnover rate, we conclude the liquidity requirement is satisfied.

Unpredictability and Risk control

To quote my second case study report, I examine through the efficient market

hypothesis and found out that in reality it does worked for most of the cases,

the arbitrage opportunities are eliminated to almost zero. Due to the highly

diversified asset holding and sectors, the predictability of single component is

extremely difficult and unstable. The time series lags (one lag unit is defined

as a quarter) are very weak. This is verified by figure2. Since no strong lag

implies exists no serially predictability. We can insure that Heng Seng index

satisfies our second requirement. Moreover, we can also reassert that the

market is efficient and any transactions associate with it subject to stochastic

risk.


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Figure 2 ACF and PACF of the HSI quarterly return

Data Sources and basic descriptions

Heng Seng index has rather solid historical data and it is easy to found the

stable result. The data below is cached from Yahoo Financeii, with the horizon

from 1986/12/31 to 2013/5/1 in a monthly format and then use natural

logarithmic percentage to denote quarterly rate.

4.Value at Risk approach revisited:From my personal first report, I have been discussing the problem of Value


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at risk approach and the standardized result attached as follow:


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The pros and cons are demonstrated here, the VAR lacks the information of

the tail distribution beyond the%, the VAR approach are Robustly practicaland gives a very good result when the economics performs normally, where

outliers are eliminated. The CVAR is very sensitive to outliers. It works better

during the recession because the extreme values are more frequent and this

method tends to overestimate the risk in the normal period. With such

insights, it is natural to question whether returns come from the same

stationary distribution and extreme value occurs equally by chance is a

reasonable assumption. Based on my work, I suggest use empirical Bayesian,


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trying to utilize the best features of both by conditioning the type of the tails,

i.e. the mode of economy, where whether the extreme values are possible.

5.Preliminary data analysis and problem description

Compute the VaR and CVaR of the HSI index by the normal method.


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Figure 3 empirical quantile plot of return Figure 4 kernel density distribution

From the empirical result figure 3, we can compute

VaR at 5%: -21%. CVaR at 5%: ==10610.5=-39.057%

Practice remark:

Those results are too inconsistent, below graph(figure 4) shows the VaR, CVaR

and 95% volatility level. The upper line is CVaR threshold, the lower arrow is

the VaR threshold, while the shock are calculated at 95% of confidence level.

The VaR underestimate the risk extensively, but the CVaR overestimate risk at

all time.

Figure 5 95% volatility level, VaR and CVaR level


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The Common VaR and CVaR exists three big problems:

The inconsistency of VaR and CVar:

VaR and CVaR are different measures, CVaR is a strict upper bound of VaR, and

therefore the value of both, especially associate with large volatility, will have large

difference.

whether the weights should be given in the form of exponential decay or

polynomial decay is unknown:

One way to solve it is people can give different weights to historical volatility, the

method shows a concern of conditioning on different situations, but it didnt go deep

enough, there is no theoretical foundation given what and why the weight should begiven in this way other than manipulation. Further than that, once the decay is

determined, it becomes deterministic and no more flexibility is given to predict about

future dynamics.

Lastly, the VaR and CVaR is not local optimal:

Clearly there are three major shocks in 86-88, 97-99,08-10, however our prediction is

static and hence it is not really optimal at the local level, and for 00-06, the HIS are

less volatile, but we predict the VaR and CVaR at the same level, it seems to be more

overall efficient but less local level efficient.

A Robust result for such VaR and CVaR approach can very easily create an

over-specification problem. And it works very badly out of sample because of low

flexibility since the method is created to fit into the overall sample and less

idiosyncratic feature is considered.

To conquer such problem, we will use an entirely different approach based on the

very James D Hamiltons masterpiece iii on Switched Markov Process, using

binjamini Hochbergs false discovery rate on detecting mechanism and Engers Arch

and Garch model for volatility forecastingiv

.

6. Brief introduction of on time series forecasting techniques

A brief introduction on Arch and Garch model:


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In econometrics, AutoRegressive Conditional Heteroskedasticity (ARCH) models are

used to characterize and model observed time series. They are used whenever there

is reason to believe that, at any point in a series, the terms will have a characteristic

size, or variance. In particular ARCH models assume the variance of the current error

term or innovation to be a function of the actual sizes of the previous time periods'

error terms: often the variance is related to the squares of the previous innovations.

The Garch model denotes Robustized Arch model with the setup list below arch

model .In our case , we will use Garch to predict the volatility movement as a

building block for risk control.

A brief introduction on Switched Markov Process:

Markov switching model specifics the mechanism of a random process the can be equal to either 0(bad economy) or 1(good economy), this denotes the

different modes of the economy (in our case). Each case will be a simple AR(1)

process.


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Then if we can use the data to determine the stationary Markov transition probability

kernel, which at every node, we will look into future and compute whether next

node will be good economy or bad economy.

00 1001 11=

[ ]

Now with the basic setup we can carry out more Robust setup with different mode.


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For completeness, the derivation of the mechanism is listed, the expected duration

of good state or bad state are for very useful result to predict short term economics

movement, the filtering probability and the optimal stationary measure is derive in

the following way to detect the markov trend of economics .


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A brief introduction on Benjamini Hochberg procedure and FDRv:

The BH procedure is applied for large scale inference, the definition of

False Discover Rate greatly enhance the power of the test. Below

attaches the construction of such a test. In our case, we will test the


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threshold probability level for recession and use such inference as an

alternative detecting mechanism to optimization method.


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7.The Pseudo Robust VaR Algorithm

Step 1: check for GARCH effect.

{ , , , ARCH effect is method is used to check whether the volatility are constant

during the process, this is very important for Value at Risk approach since the

standard-deviation accounts for the loss we are interested in, therefore we

must forecast the volatility movement. For the HSI index future, the ARCH

effect is very clearly exists (see figure 5), the constant volatility prediction is no

good. Since the p-value is around 0.3 we dont reject the arch effect exists

(more in appendix). The volatility effect existed and non-constant is also

verified by (figure 6). And the predictability is not very good and it subjects to

strong volatility movement.

Figure 6 Arch statistics


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Figure 7 fitted and risdual graph assume no arch effect

Step 2: Fit an Optimal grach model to predict the volatility.

Based on AIC criterion, the model fitted below Cgarch(1,1) (figure 8) ,The model

forecast statistics (figure 9) conditional standard deviation (figure 10)

Figure 8 Cgarch model

Model prediction:


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Figure 9 model statistics and forecasting

Figure 10 conditional standard variance prediction

Using the model prediction we can demonstrate the path dependent conditional

variance with 1 quarter ahead forecasting above. Denote the predicted conditional

variance Step 3: Work out Markov Switching model

Here is the estimated HSI index Markov transition probability kernel (figure 11) and

the expected duration (figure 10) which accounts for the mean expected time for

each mode. The regime specification on (figure 12), state 1 is good and state 2 is bad.

To demonstrate the power of such method, the HSI historical index price are plot and

shaded for the crisis on (figure 13) the Markov switching auto regression are

specified on (figure 14) with the filtered probability


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Figure 11 expected mode duration

Figure 12 markov transition probability kernel

Figure 13 markov regime specification


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Figure 14 historical HSI price shaded for crisis

Figure 15 Filtered probability shaded for stated 2

Step 4: Robust VaR threshold model and BH procedure

Basic notation and description:

S(t)=1 denotes good state, S(t)=2 denotes bad state

x denotes the one-step forecast probability ending in state 2

y denotes the realized filtered probability for state 2

{ , . , VaR =1.645*

Where

is the result derived as constant from step 1 or Garch

prediction step 2

For demonstration purpose the is set at 5%


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The threshold model:

Function specification:

In below we define the objective function,: , ,

, { , ., Where 0.4 is a cut-off probability value in mode 2.

Threshold= ,}From the historical data, the threshold is 0.24 meaning that:

predited eoo tte { St + , St + 2St .2St + 2, St + 2St .2

The BH procedure:

In this case, the FDR is set to be 25%. This means the wrong detection

(good state is predicted to be bad) has a probability of 25%, we dont


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want to make too much wrong detection because this will eliminate

the business opportunity, but too few detection will cause a higher

significant level of downturn prediction, which will not be accurate.

This is trade-off for the user, but it does allow great flexibility. In this

setting the C is 22%, compare to the asymmetric model.

Robust VaR application:

Robust VaR prediction definition condition on good state:

ot r { r., St + } r + , St t }

Where is the result derived as constant from step 1 or Garch predictionstep 2If the economy is at state 2, the bad state:

The kernel is very hard to monitor and for simplicity we use expectation.

( + 2| 2) .3897( + | 2) .3

Robust VaR prediction condition on bad state:

ot r + r8. Conclusion on HSI index study

Out of sample validation:

The data is utilized till the 2013/4/1, with the fitted model above. The one step

bad economy forecast probability is see (figure

16):( + 2| ) .9, which is below both thresholdssolved for both detection bound, therefore we conclude there is no sign

indicates downturn trend. The HSI out of sample performance is listed below

(figure 17). One can fit an up-trend line for validation; therefore we conclude

the result is valid. But if on the other hand, the company is more focused on

high-frequency trading, more delicate risk control technique for short time

horizon could be build based on the framework.


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Figure 16 one step 2013 second quarter bad economy probability forecasting

Figure 17out of sample validationFinal result Demonstration and discussion:

(In graph the General VaR denotes Robust VaR)

In figure 18, the HSI return, CVaR,VaR and the Robust VaR is plot for

comparison, it is seen that the Robust Var provides good performance. Firstly,

it provides accuracy, at 95% level; the crossing is only twice, for large-scale

speculation and extreme downturn. Secondly, it gives business dynamics by

lower risk bound. Thirdly, unless extreme downturn occurs, the model handles


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normal business cycle downturn well. e.g.1998, 2000-2003, 2011. Last but not

least, more flexibility is allowed for extreme downturn without hurt the

normal period. In Figure(19 and 20), CVaR captures the extreme value. But

reconditioning on CVaR level, we can give more tolerance on extreme crisis.

The NNSD is predicted by Garch, the difference between it with Robust VaR is

determined by downturn detecting mechanism and the risk aversion level or

trade off prediction accuracy level against the bad periods. By reconditioning

the parameter the method gives a better downturn management

performance.

Figure 18 Method comparison


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Figure 19 Garch prediction and MS model switching model CVar=39

Figure 20Garch prediction and MS model switching model CVar=45

The HSI study result is very good; this method not only can be converted

into sequential algorithm but also introduce dynamics adjustment into

Risk management,

9. Further Application and prospect:


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The key result embedded in this study is the economy downturn detecting

mechanism, this method is fundamentally important in stress testing and

scenario analysis, we can build different scenarios directly into the model and

system without needs to re- parameterize the model and concern about the

inconsistency of the estimation.

On migration matrix:

It is well-known that many securities are subject to economy downturn risk;

corporate bond defaults, CMO defaults and various are all subject to the

business cycle and the effect is significant. For migration matrix, we can apply

the same argument: consider normal predicted migration matrix A

and distressed predicted migration matrixD .Follow similar argument. Firstly, detect the downturn risk and then determine

a downturn transition probability threshold from historical data.

The new migration matrix can be carried out as followings:

{ A, t + dthreholdA + D, otherwieWhere is a weight coefficient depending on the risk aversion level or useBH procedure set

equals to the trade off FDR to validate the argument for

more flexibility.

Prospect:

For the moment, all the work are carried out for single setting, if the portfolio

are highly diversified, in a high dimension space, one can either run a Markov

switching Vector auto regression model, and the state space can be more than

2 to catch up with the situation closely. Or one can still predict only the

economy downturn risk, use the component analysis to work out the portfolio

risk conditioning on the downturn. Survival analysis can also be used for the

expected duration to measure the switching pressure. If on the other hand,

the company is more focused on high-frequency trading, more delicate risk

control technique for short time horizon could be build based on the

framework use the Garch model to decompose the white noise process.

10. Appendix & Reference:


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HSI data analytics:

Figure 21 arch lag test

Figure 22 Cgarch parameters estimation table


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Figure 23 Markov switching AR proces

i http://www.hsi.com.hk/HSI-Net/HSI-Net

ii http://finance.yahoo.com/q/hp?s=%5EHSI+Historical+Prices

iii Hamilton J D. A new approach to the economic analysis of nonstationary time

series and the business cycle[J]. Econometrica: Journal of the Econometric Society,

1989: 357-384.iv Bollerslev T, Engle R F, Nelson D B. ARCH models[J]. 1994.

Engle R F, Kroner K F. Multivariate simultaneous Robustized ARCH[J]. Econometric

theory, 1995, 11(01): 122-150.v Benjamini Y, Hochberg Y. Controlling the false discovery rate: a practical and powerful

approach to multiple testing[J]. Journal of the Royal Statistical Society. Series B(Methodological), 1995: 289-300.
http://www.hsi.com.hk/HSI-Net/HSI-Nethttp://www.hsi.com.hk/HSI-Net/HSI-Nethttp://finance.yahoo.com/q/hp?s=%5EHSI+Historical+Priceshttp://finance.yahoo.com/q/hp?s=%5EHSI+Historical+Priceshttp://finance.yahoo.com/q/hp?s=%5EHSI+Historical+Priceshttp://www.hsi.com.hk/HSI-Net/HSI-Net

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