Calculationof 10-day portfolio VaR

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Valueat Risk is a measure of financial risk that is widely used. It givesa method of determining the quantity of risk of a portfolio andmanaging the risk (Marrison, 2002). In calculating the 10-day VaR,three ASX-listed companies were selected. The companies were AzureHealthcare Ltd (AZV.AX), Azonto Petroleum Ltd (APY.AX), and NationalAustralia Bank Limited (NAB.AX). The respective returns for therespective companies (in AUD) were obtained for the period between.The calculations were done using two different methods, thehistorical simulation and the Variance-Covariance parameter approach(VCV approach).

Justificationfor chosen calculation methods

Historicalsimulation

Thehistorical simulation was chosen because it offers an implementationof the full valuation in a straightforward manner. Market states aresimulated and produced through the addition of the period-to-periodchanges in market variables in a specific time series to the basecase. The historical simulation assumes primarily that there is afull representation of the set of possible future scenarios by theevents of a particular historical window. In this method, a set ofchanges in the risk factor is collected over a given historicalwindow. The scenarios obtained are then assumed to represent all thepossibilities that might happen in the immediate future (Alexander,2008).

Advantagesand disadvantages of historical simulation

Amajor advantage of historical simulation is that no assumptions aremade about the changes in risk factor being from a specificdistribution. As a result, this methodology is consistent withchanges in risk factor originating from any distribution. Thehistorical simulation also does not involve any estimation of thestatistical parameters like variances or covariance. It is alsocontinuously exempt from the unavoidable estimation errors. It iseasy to explain and defend even to an audience that is not technicalyet important, like the corporate board of directors. No assumptionsare needed on the distributions of the risk factors, there is easerevaluating the full portfolio based on the scenery data, and it isintuitively simplistic and obvious (Hull, 2008). The historical simulation also has certain disadvantages. Accomplishing the purest form of historical simulation is difficultsince the method poses the requirement of data about risk factorsspanning over a long historical period. This requirement is usuallynecessary so that what might happen in the future can be adequatelyrepresented. The method also does not involve distributionalassumptions. The scenarios used in calculating VaR are restricted tothe ones that took place in the historical sample (Bohdalova, 2007).The past is not always a good way of modeling for the future. Thereis also a huge probability of gaining results that are erroneous ifthe sample size is not sufficient (Boyarshinov, 2016).

Variance-covariancemethod

Thevariance-covariance (VCV) method makes use of a historical sample ofthe risk factors of a portfolio value. It functions almost like thehistorical simulation. However, unlike the historical simulation, thevariance-covariance method assumes that there are normallydistributed logarithmic yields for the risk factors. The estimate ofthe VaR is then equated to a sample quartile of the yield portfoliovolatility. The volatility of the portfolio yield is calculateddepending on the covariance between portfolio risk factor yields.

Advantagesand disadvantages of Variance-covariance method

Thereare many advantages of variance-covariance method. They are easy toimplement as compared to the historical simulation method and theMonte Carlo Simulation. The method requires less historical data incomparison to the historical simulation method. In most cases, it hasan acceptable precision and accuracy. The method alsohas its disadvantages. VCV method has a low quality of estimates forsecurities whose prices have a nonlinear dependence on the riskfactors. The assumption of the logarithmic distribution of the yieldsof the risk factors is not always correct. It also ignores the riskof extreme events that can cause significant losses in the value ofthe portfolio (Boyarshinov, 2016). The MonteCarlo Simulation method was not chosen because of its high dependenceon the resource which makes it very time-consuming. The sample sizesof observations that are used in deriving the distribution kind andparameters are also often insufficient. The insufficiency can lead toincorrect estimates of VaR.Calculationof VaR using Variance-covariance method Inthis method, the simple moving average (SMA) method was used. Thecalculations involved include the SMA daily volatility, SMA dailyVaR, and Portfolio holding SMA VaR. The portfolio comprises equalexposures of 1000 shares for all the stocks. The market price forAZV.AX, APY.AX and NAB.AX were AUD 0.068, 0.015 and 28.94respectively. The historical price data for all the three companieswere obtained for the period of 14^{th}July 2015 to 13^{th}May 2016. The daily time series are presented in the extract inFigure 1.this period is called the look-back period, which is theperiod over which the risk is evaluated.

Figure1: Time series data for AZV, APY, and NAB

Itis first important to determine the return series. To obtain this,the natural logarithms of the ratio of successive prices aredetermined. This is shown in Figure 2.

Theformula shown in the figure is LN(Cell B19/Cell B18). In cell B19,there is 0.15 while in cell B18 there is 0.15. Calculating theLN(0.15/0.15) gives 0 as shown in cell H19.

Figure2: Return Series

Thenext step is to calculate the daily volatility is calculated usingthe formula:

Rt= rate of return at the time, t.

E(R)= mean of return distribution

Thesquared differences of Rt are summed over E(R) across all data pointsthen the results are divided by the number of returns in the seriesminus one so that the variance can be obtained (Alexander, 2008). After this, the square rood to the result iscalculated, which is the standard deviation (SMA volatility) of thereturn series. The method used in this calculation, however, is theuse of the STDEV function in EXCEL, which is applied to the returnseries as illustrated in Figure 3. The daily volatility is determinedas 0.494429 for the portfolio.

Figure3: Calculating the daily volatility

Gettingthe SMA VaR is determining how much will be lost over a particularholding period with a particular probability. The daily volatility ismultiplied by the z-value of the inverse of the standard normalcumulative distribution function (CDF) which corresponds with aparticular confidence level, which is the NORMSINV (confidencelevel). For example, in our case, the confidence level is 99%, so itis expressed in excel as NORMSINV (99%). Figure 4 shows how the dailyVaR is calculated

Figure4: Daily Volatility

Figure4 show that the daily VaR is 1.150214. The VaR is the Daily VaRmultiplied by the square root of the holding period in days. Thesquare root function in Excel is SQRT. Figure 5 shows how thecalculation is done in excel.

Figure5: VaR

CalculatingVaR using the Historical Simulation method

Inthis approach, there is no assumption that is made about the returndistribution that is underlying. The first step is to obtain thereturn series and reorder them into ascending order from the smallestto the largest returns. This is done by use of the Sort function inexcel. After reordering the values, the number of returns in theseries is counted using the COUNTA formula as shown in Figure 6(Bohdalova, 2007).

Figure6: Determining the number of returns

Fromour series, there are 218 returns. The daily VaR is then calculatedas the return that is corresponding to the index number. It iscalculated by subtracting the confidence level then multiplying it bythe number of returns at the point where the result is rounded downto the nearest integer. The integer is the index number for aparticular return. This is shown in Figure 7.

Figure7: Historical Simulation

The10-day VaR is then calculated by multiplying the daily VaR by thesquare root of the holding period- 10 days. Figure 8 shows how thisis done.

Figure8: 10-day VaR.

Fromour calculations, the 10-day VaR for AZV.AX, APY.AX and NAB.AX are0.705641867, 0.909730591 and 0.150688793 respectively. The respectiveamounts of worst case loss are AUD 47.98364698, 13.64595886 and4360.933676.

Inorder to calculate the portfolio VaR through the historicalsimulation, the returns are combined to get the portfolio change asshown in Figure 9.

Figure9: Calculating portfolio change

Theobtained values are then sorted in ascending order from the least tothe largest value. The number of observation is equivalent to theindex, which is manually assigned. After this, the VaR is calculatedfor each portfolio change as shown in Figure 10.

Figure10: Calculating VaR historic

CellD10 contains the sum of all portfolio values it is shown in Figure11.

Figure11: Sum of portfolio values

Toget the VaR that corresponds to our rounded down Number ofObservation (Index), the IF function is used in column R as shown inFigure 12.

Figure12: Obtaining the VaR

Allpossible values of VaR obtained by the IF function are then summed upto get the overall VaR. This step is not very necessary since onlyone value is obtained by the IF function. However, it is required toset the result at an obvious position. It is shown in figure 13.

Figure13: VaR

Nowthe daily VaR has been obtained. It is multiplied by the square rootof the holding period (10 days) to get the portfolio 10-day holdingVaR. The last step is shown in Figure 14.

Figure14: 10-day Holding VaR.

Fromthe calculations, it can be seen that all the values obtained aredifferent. This shows how unreliable VaR is in determining the riskof business. If not keenly considered and used together with otherparameters, it may be misleading. As a result, investments made oncertain stock relying on the results of VaR calculations can end upbeing less profitable or even end up in losses. The VaR can be usedin calculating the capital requirements by banks in case they want toinvest in these stocks. The method used for these calculations is theBasel Accord, or the Basel II Framework.

BaselAccord

Themain goal of the Basel II Framework is promoting the propercapitalization of banks so that improvements in risk management canbe encouraged. This can, in turn, strengthen the stability of thefinancial system. The Basel committee imposed a requirement known asthe capital requirement for market risk in 1996 (Suarez, Dhaene,Henrard & Vanduffel, 2005). Capital requirements depend on therisk, which is a random variable faced by a bank (Dagher et al.,2016). The capital is related to the diverse categories of assetexposures. In this regard, a risk weight (RW) is used for theexposures. This is relative to the broad categories of relativeriskiness. The banks need to be more devoted to calculating the risksinvolved with any capital investment. In case this is ignored, theremay be risky investments that can end up being less profitable.

Calculationof capital requirement using Basel Accord

BaselII calculates the regulatory capital (CAP_{reg})as CAP_{reg}=8%×RW.The calculations for the three portfolio companies are presented inthe table below:

Exposure | RW | Required Capital |

1000 | 8.820523 | 705.6419 |

1000 | 11.37163 | 909.7306 |

1000 | 1.88361 | 150.6888 |

3000 | 22.07577 | 5298.184 |

TheRW is calculated by multiplying the 10-day VaR with a constant, 12.5.After obtaining the RW, it is multiplied by 8% and the resultingamount multiplied with the exposure to get the capital requirement.In the case of AZV, APY, and NAB, the total capital requirement is5298.18.

Differentmethods used in calculating the VaR immensely affect the value of thecapital requirement. This implies that VaR in itself cannot be a goodtool for calculating the capital requirement. In case there is a needto use the VaR, banks must develop a standard method of calculatinginternal VaR such that similar figures will be obtained. In case thisis overlooked, different banks will have different VaR values andtherefore different capital requirements values for a single stock orexposure.

Conclusions

Thecalculation of VaR is a complex process that requires a skilledapplication of financial and accounting knowledge. There are manydifferent methods of calculating VaR. Historical simulation andvariance-covariance methods were used in this study. This is becauseof their simplicity and ease of understanding. They are also not verytime-consuming. However, they also have their limitations that haveto be considered during the calculations. The VaR was found to be avery unreliable means of determining the health of a business or therisks that a company or stock may go through. The results of thecalculations were found to be varied and totally unrelated. Eventhough it has been applied to the calculation of capital requirementsby banks using the Basel Accord, the results are equally varied andunreliable. VaR should, therefore, be used together with othermechanisms for measuring the risks of business.

Bibliography

Alexander,C. (2008). *Valueat risk models: market risk analysis IV*,John Wiley & Sons.

Bohdalova,M. (2007). A comparison of Value-at-Risk Methods for measurement ofthe financial risk. *E-Leader,Prague*.

Boyarshinov,A.M. (2016). *Comparativeanalysis and estimation of mathematical methods of risk marketvaluation in application to Russian stock market*.

Dagher,Dell’Ariccia, Laeven, Ratnovski & Tong. (2016). *Benefitsand Costs of Bank Capital*.IMF Staff Discussion Note. SDN/16/14.

Hull,J. (2008). *Fundamentalsof futures and options markets,*6th end, Pearson International.

Marrison,C. (2002). *Thefundamentals of risk management*,McGraw Hill.

Suarez,F., Dhaene, J., Henrard, L. & Vanduffel, S. (2005). *BaselII: Capital requirements for equity investment portfolios*.Katholieke Universiteit Leuven: Department of Accountancy, Financeand Insurance (AFI).