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Backtesting

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Backtesting Assignment
Question1: Discuss the role of back testing of VaR models in portfolio management The growth of risk management as a sub-field in the theory of finance traces back to the increasing volatile markets of the 1970s. Risk management revolution crept up as fixed exchange rates were being demolished and new theory were advancing rapidly. As trading increased rapidly, unpredictable events such as financial disasters crept up to bring to light the need for improving risk management tools. Over the past few years, the Value-at-Risk (VaR) model has evolved into the most popular risk assessment tool in finance (Lucas, 2001). The VaR method captures market risks in an asset portfolio, which is the loss in portfolio value within a specific period using an specific confidence interval. Despite being widely used and accepted, it has attracted criticism over its incapability to produce reliable estimates of risks. Upon implementation, VaR systems involve various simplifications and assumptions as the tool forecasts future assets using historical market, which may not reflect the environmental scenario in future. This means the VaR is only useful when it predicts risks accurately (Lucas, 2001). To verify the consistency and reliability of VaR calculations, it is necessary to back test the model with appropriate statistical standards. Back testing entails comparing actual profits and losses to projected VaR estimates as an apt form of reality check (Lucas, 2001). Where the results reveal inaccurate estimates, the models are re-examined to identify the incorrect assumptions, parameters, or modeling. Various testing methods identified for back testing include Kupiec’s POF test, which examines the excessive losses of the VaR model and their frequency. The resulting failure rate has to correspond to a particular confidence level. For instance, computing VaR estimates using a 99% confidence level for a period of one year (with an equivalent of 250 trading days) may produce an average of 2.5 VaR violations during the period. The POF test calls for an observation of whether the resulting amount of exceptions is reasonable when compared to expected amounts. A regulatory back-testing framework has been set up to note the frequency of exceptions although it is not applicable in most internal validation processes because of other powerful approaches (Lucas, 2001). Apart from the acceptable exceptions, another important aspect of the VaR model is that observations have to be spread serially over time. A good model avoids exception clustering through quick reactions to changing instrument volatilities and correlations Today, back testing is an integral component of VaR reporting in risk management practices across financial environments. The absence of proper validation means that a supervisor cannot ascertain the capability of the VaR to yield accurate risk estimates (Lucas, 2001). VaR is an essential element of today’s current market environment where unpredictable market processes push investor interests in portfolio risk figures without acknowledging the accumulation of losses. Still, the VaR is poor at estimating losses and under these conditions because it is designed to measure expected losses under normal conditions (Lucas, 2001). The limitation is a drawback of VaR, though it is what makes back testing procedures interesting and challenging. Moreover, financial institutions that use the VaR model have back testing at hand, but still underreport true market risks. This occurs mainly because the applied monetary penalties on banks for these violations are often low (Lucas, 2001). To encourage these institutions to design adequate internal models, higher monetary penalties are needed on the excess violations. In sum, back testing plays significant roles in identifying whether internal models of financial institutions are valid or effective. The model assists supervisors and regulators in controlling moral hazard problems and encourages financial institutions to use appropriate models in computing accurate capital requirements.
Question 2: A backtesting exercise for 1997 A Value-at-Risk on a portolio of securies is calculated for 1997 using a confidence level of 99%. A normal (parametric) and a historical (simulation) method are applied to 253 trading days. If the prices of trading days exceed the calculated VaR model, it is appropriate and if it more than 1% of the prices of 2.5 traditing days, it is inappropriate. For the trading days, the actual profit and loss is claculated and backtesting of the stimates calculated using the two methods. Normality A variance-covariance approach is applied to claculating VaR and the maximum loss and profit for the period recorded. The maximum loss for the period was $24,523, maximum profit $22,277, and the average annual profit $ 731,275. At a confidence, level of 99%, the normality assumption genrates an estimate of $18,851 for ten days. The maximum loss for VaR was $ 21,434 while the minum $16,685. The results of the back testing highlight two case of where the realized loss exceeds the violations. The normality assumption restricts VaR violations to 253 days * the 1%, which equals 1.53 days for the entire years. A greater frequency that the biolation is reassessed. From the calculations, the frequency of violations is lesser than expected, and the accuracy of the model is validated.
|Table 1 |Mean |Min |Max |VaR Violation |
|VaR (Normal) |$18550.8 |$16685.3 |$21433.9 |2 |
|VaR (Historical) |$29104.4 |$20966.3 |$24,164.7 |0 |
|Realized P&L |$731.3 |$22276.6 |($24522.9) | |

From the table, the confidence level of 99% reveals a historical VaR average of $29,104 with the minimum loss at an average of $20966 and a maximum of $24,165. The violation for the historical method is zero, which means the model is robust. The differences between the two models lie in the higher VaR loss estimates that the historical simulation method generates. The minimum VaR loss using the historical method is $20,966 while that generated under normality is $176, 685. [pic]

[pic] The graph illustrates the higher VaR estimates for the historicla method ($3,387) and the estimates generated under normality ($1,217). Conclusively, the normality method produces lower VaR estimates as compared to the historical method. Both methods howevr prove to be valid and adequate in measuring market risks. Financial instititutions are more likely to choose the normality method because of the decreased volatiliy and risks. Question 3: A backtesting exercise for a portfolio of five stocks for 2006 The normality and historicla simulation methods are applied in estimating VaR on a portfolio of five securities with a value of $500,000. The stocks include CISCO, IBM, American Express, Time Warner, and Microsoft. The data is collected for a period of 251 trading days from and the VaR estimated using adjusted prices for the five securities. If the prices of trading days are exceed the calculated VaR model, it is appropriate and if it more than 1% of the prices of 2.5 traditing days, it is inappropriate. For the trading days, the actual profit and loss is claculated and backtesting of the stimates calculated using the two methods. Out of the 251 days in 2006, the maximum loss occuring was $32,680 while the maximum profit is $35848. The portfolio has gained an average of $5,329. Under normality, the average VaR is $28,792, the minimum VaR loss $26511, and the maximum loss $31,817. The 99% confidnence level limiting the violations equals 251 days * 1%, which equals 2.51 days pof the year. The data registers a frequency of VaR violations of two of the 2006, meaning that the VaR loss estimate is robust and does not need scaling.
|Table 2 |Mean |Min |Max |VaR Violation |
|VaR (Normal) |$28792.5 |$26510.9 |$31816.7 |2 |
|VaR (Historical) |$28825.9 |$21496.7 |$38371.6 |3 |
|Realized P&L |$5329 |$35848.1 |($32680) | |

The historical simulation of method generates an average VaR loss of $28,826, a minimum of $21,497, and a maximum of $38372. From the table, the violations exceed 2.51 violations at a confidence level of 99%, meaning that it has to be reassessed. However, scaling may be dismissed because the excess is only marginal. In the below graph, the difference between the values of the high volatility estimated at $2,793 and a standard deviation of $1,428 under the normality assumption are recorded. When compared to the estimates drawn from the previous study (question 2), the averages are almost similar. The maximum VaR loss remains large as compared to the historical method.

[pic] The near similar results of the back-testing model reveal that both methods can be used in measuring market risk and required capital. The normality approach generates an average VaR that is almost equivalent to the historical method. The graphical illustrations shows higher volatility when the historical method is used to generate data. The back testing of the historical method creates a frequency of VaR violations that exceed expected frequency, although it does not invalidate its use.

References
Lucas, A. (2001). Evaluating the Basle Guidelines for Backtesting Bank’s Internal Risk Management Models, Journal of Money, Credit and Banking, 33(3), 826-846.

Appendix 1
Table 1
VaR Multiplication Factors After Backtesting
|Zone |Number of exceptions |Scaling factor |
|Green Zone |0 - 4 |3 |
| | | |
|Yellow Zone |5 |3.4 |
| |6 |3.5 |
| |7 |3.65 |
| |8 |3.75 |
| |9 |3.85 |
| | | |
|Red Zone |>=10 |4 |

Appendix (Ox program code)
Question2 (Normality)
//program q2a.ox
//this program generates VaR calculations
//based on normality and also generates the
//actual loss over the 10-day period

#include

main()

{

ranseed(967537412);

decli,j,ww,meanp,meanpw,pdt,pdtdf,pdtdfu;

declpdtdfuv,pdtdfuvf,varn,realv;

pdt=new matrix[2275][19];

pdt=loadmat("pdatau.prn");

pdtdf=new matrix[2275][19];

pdtdf=diff0(pdt,1);

pdtdfu=new matrix[500][19];

ww=new matrix[19][1];

pdtdfuv=new matrix[19][19];

pdtdfuvf=new matrix[19][19];

meanp=new matrix[1][19];

varn=new matrix[1][1];

realv=new matrix[1][1];

ww[0][0]=10000/(pdt[1508][0]); ww[1][0]=10000/(pdt[1508][1]); ww[2][0]=10000/(pdt[1508][2]); ww[3][0]=10000/(pdt[1508][3]); ww[4][0]=10000/(pdt[1508][4]); ww[5][0]=10000/(pdt[1508][5]); ww[6][0]=10000/(pdt[1508][6]); ww[7][0]=10000/(pdt[1508][7]); ww[8][0]=10000/(pdt[1508][8]); ww[9][0]=10000/(pdt[1508][9]); ww[10][0]=10000/(pdt[1508][10]); ww[11][0]=10000/(pdt[1508][11]); ww[12][0]=10000/(pdt[1508][12]); ww[13][0]=10000/(pdt[1508][13]); ww[14][0]=10000/(pdt[1508][14]); ww[15][0]=10000/(pdt[1508][15]); ww[16][0]=10000/(pdt[1508][16]); ww[17][0]=10000/(pdt[1508][17]); ww[18][0]=10000/(pdt[1508][18]);
for(i=0;i

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