風險價值計算題(附答案)

考慮一個兩股票的組合，投資金額分別為60萬和40萬。問一、 下一個交易日，該組合在99%置信水平下的VaR是多少？二、 該組合的邊際VaR、成分VaR是多少？三、 如追加50萬元的投資，該投資組合中的那只股票？組合的風險如何變化?要求：100萬元投資股票深發(fā)展(000001),求99%置信水平下1天的VaR=?解：一、 歷史模擬法樣本數(shù)據(jù)選擇2004年至2005年每個交易日收盤價(共468個數(shù)據(jù))，利用EXCEL：獲取股票每日交易數(shù)據(jù)，首先計算其每日簡單收益率，公式為：簡單收益率=(Pt-Pt-1)/Pt-1,生成新序列，然后將序列中的數(shù)據(jù)按升序排列，找到對應的第468X1%=4.68個數(shù)據(jù)(謹慎起見，我們用第4個)，即-5.45%。于是可得，VaR=100X5?45%=5?45萬。如圖：MicrosoftExcel-015文件(E)編輯(E)視圖(V)插入(I)格式(0)工具(T)數(shù)據(jù)①)窗口(W)幫助(H)MxcrosoftOffice是非正版授權(quán)版本。點擊此處，立即行動。遠離潛在風險、享受正版卓越體驗。D3 〒任-（B3-B2）ZB2ABCDEFG1日期價格P簡單收益率YY由低至咼排序結(jié)杲22004-01-028.6432004-01-058.921 0.0324074071-0.09803921642004-01-069.330.045364126-0.08246073352004-01-079.390.006430868-0.06734006762004-01-089.410.002129925-0.05454545572004-01-099.03-0.040382572-0.05186020382004-01-129.190.017718715-0.04662379492004-01-139.18-0.001088139-0.043227666102004-01-149.03-0.016339869-0.042881647112004-01-159.10.007751938-0.041731066122004-01-169.280.01378022-0.041025641132004-01-299.530.026539655-0.040697674142004-01-309.28-0.026232949-0.040382572152004-02-029.670.042025862-0.040257649162004-02-039.830.016546019-0.038580247172004-02-0410.010.018311292-0.038461538182004-02-0510.060.004395005-0.038073908192004-02-0610.510.04473161-0.037735849202004-02-0910.590.007611798-0.036964981212004-02-1010.740.014164306-0.036020583222004-02-1110.62-0.011173184-0.035335689232004-02-1210.51-0.010357815-0.035262206

二、 蒙特卡羅模擬法(1)利用EVIEWS軟件中的單位根檢驗(ADF檢驗)來判斷股票價格序列的平穩(wěn)性，結(jié)果如下：NullHypothesis:SFZhasaunitrootExogenous:ConstantLagLength:0(AutomaticbasedonSIC,MAXLAG=0)t-StatisticProb.*AugmentedDickey-Fullerteststatistic-1.0382260.7407Testcriticalvalues:1%level-3.4441285%level-2.86750910%level-2.570012*MacKinnon(1996)one-sidedp-values.由于DF=-1.038226,大于顯著性水平是10%的臨界值-2.570012,因此可知該序列是非平穩(wěn)的。(2)利用EVIEWS軟件中的相關(guān)性檢驗來判斷序列的自相關(guān)性。選擇價格序列的一階差分(△P=Pt-Pt-i)和30天滯后期。結(jié)果如下：Date:10/20/09Time:17:03Sample:1/02/200412/30/2005Includedobservations:467AutocorrelationPartialCorrelationACPACQ-StatProb-|-|-|-|1-0.012-0.0120.06600.797.|.|.|.|2-0.020-0.0200.24620.884.|.|.|.|30.0060.0060.26370.967.|.|.|.|40.0440.0441.17280.883*|.|*|.|5-0.083-0.0824.44530.487*|.|*|.|6-0.070-0.0716.78800.341.|.|.|.|7-0.004-0.0096.79480.451.|*|.|*|80.0780.0759.67260.289.|.|.|.|90.0040.0149.67870.377.|.|.|.|10-0.023-0.0229.93030.447可知股票價格的一階差分序列AP滯后4期以內(nèi)都不具有相關(guān)性，即其分布具有獨立性(3)通過上述檢驗,我們可以得出結(jié)論,深發(fā)展股票價格服從隨機游走,即：Pt=Pt-1+￡十。下面，我們利用EXCEL軟件做蒙特卡羅模擬，模擬次數(shù)為10000次：首先產(chǎn)生10000個隨機整數(shù)，考慮到股市漲跌停板限制，以樣本期最后一天的股

價(6.14)為起點，即股價在下一天的波動范圍為(-0.614,0.614)。故隨機數(shù)的函數(shù)式為：RANDBETWEEN(-614,614)[用生成的隨機數(shù)各除以1000，就是我們需要的股價隨機變動數(shù)￡丿。然后計算模擬價格序列：模擬價格=卩0+隨機數(shù)*1000再將模擬后的價格按升序重新排列，找出對應99%的分位數(shù)，即10000Xl%=100個交易日對應的數(shù)值：5.539,于是有VaR=100X<5.539—6?14)*6?14=9.79萬?.F:E12初始價1&PDE_14龍初數(shù)(KAmBETWEEN(^614,614))-256模擬價裕-p姑隨初數(shù)門5.豳甸模擬價格的升序卅列5.526:B.M1736.3135.526ql>.1■!-跡ub-HHh.h恥L加難?JaJJ"±」■」丄春bJ14bSIb_byi-b.■S.&26b.14-0虬UUJb.b^r96.14846.2245.527356.14-54&5-5945-529416-14303S.53262［艮14-2225.'SISS.S35&3E.14-4S35.6575.B35胡6-14490瓦635.陽!56.Id2436.3835.535706.14-2795.8?15-536716.142926-4325-536{2b.丄刖->114b_也BII.6.146.20?S-537.匚b.936.0475一冏JLb.141(JU乩弱b.bsr96［亂143566.49?S.53i97臨上3666-S066.S34986?142636-403S.534I'.-b.地■冊&.&341■:1b.14-綁5.EG4￡&護1016.14232鼠貂25.5391026.143886.52S5.5391036.14-<025-7385-539104€.143286.46S5-539三」二zqI二g丄b直能7丄■?丿 ▼泗禺丄■z.*凹劉層:小忡. p ▼l11」址u=■[_E101 ■ &5.539三、 參數(shù)法(樣本同歷史模擬法)(一)靜態(tài)法：假設(shè)方差和均值都是恒定的簡單收益率的分布圖：R=(Pt-Pt-1)/Pt-1-0.10Series:SFZ3Sample1/02/200412/30/2005Observations467Mean-0.000490Median-0.001253Maximum0.100694-0.10Series:SFZ3Sample1/02/200412/30/2005Observations467Mean-0.000490Median-0.001253Maximum0.100694Minimum-0.098039Std.Dev.0.022079Skewness0.621496Kurtosis7.030289Jarque-Bera346.1299Probability0.000000-0.05-0.000.050.10對數(shù)收益率的分布圖：R=LN(P對數(shù)收益率的分布圖：R=LN(Pt)-LN(Pt-1)-0.10 -0.05 -0.00 0.05 0.10Series:SFZ2Sample1/02/200412/30/2005Observations467Mean-0.000731Median-0.001254Maximum0.095941Minimum-0.103184Std.Dev.0.021970Skewness0.436698Kurtosis6.794598Jarque-Bera295.0232Probability0.000000通過對簡單收益率和對數(shù)收益率的統(tǒng)計分析可知，與正態(tài)分布相比，二者均呈現(xiàn)出“尖峰厚尾”的特征。相對而言，對數(shù)收益率更接近于正態(tài)分布。因此，采用對數(shù)收益率的統(tǒng)計結(jié)果，標準差為0.02197。根據(jù)VaR的計算公式可得：VaR=2.33X0.02197X100=5.119萬(二)動態(tài)法：假設(shè)方差和均值隨時間而變化可以有多種不同的方法，下面簡單舉例：1、簡單移動平均法：取30天樣本，公式為：。2=(工R2)F30，通過EXCEL處理后結(jié)果為:。2=0.000211028,則有。=0.0145VaR=2.33X0.0145X100=3.379萬

ADc3EV-日期協(xié)格P簡單收益率簡單收益率平方方羞預測值22004-01-023.6432004-01-05a.32D.03240T407U.0010502452004-01-079.390.0064308684.13561E-051R2D04-02-051(1.Ofi0.OA499SOOS2.49501E-05202004-05-0910.鈾0.0076117985.79395E-05292004-02-2011.120.0500472140.002504724302004-02-2310.91-0.018884890.000356639J丄丄u.bl-u.uzrqyrriU.uuue?丄HQ522004-02-2510.3-0.029217720.0000536750.00061609出2D04-02-Z&10.340.0033834951.50815E-Q50.000581585542004-02-2710.360.0019342363.74127E-060.000511286驗2004-03-0110.370.0009652519.31709E-070.0005099392(J04-0^吆10.4L0.007714bGlb.0.O(JOt>11771北2004-03-0411.060.04241Z8180.00179S8470.000514723392004-03-0510.67-0.035262210.0012434230.000556131QU2004-0^-081(J.41-U. f旳0.00059^7690.U0U567U24602004-04-0510.59-0.013047530.0001702380.000360124-ZJU^-U4-dUu.cru^fwzod?^OfZfE-UDu.nuuzor(z丄豎上iro圧一va-￡6c一U?UU33UU333.OUi UDu.ulpulu丄乙2丄JLO■JJ-乙乙￡. 云u. J.龍uus-uf-ut>b.Tb-U.UlSbyKbjSU.UUUlKYbbj：u.uuut>a±yuKQUb2uut>-uy-uyb.%-u.UllUTbybU.UUUi22b7YU.J.U44632005-12-226.06000.000205973伽?nn^-i9-?sAHnmiFRii^Rftnnni3^4?Qnnnn?msi^4匹2005-12-266一230.0163132140.0002661210.0002152744G62005-12-276.2-0.004815412.31882E-050.000215944672005-12-286.17-0.004838712.34131E-050.0002140734682005-12-296.220.0081037286.56704E-050.0002158464692005-12-306.-0.012861740.0001654240.000211028]2、指數(shù)移動平均法：借鑒RISKMETRICS技術(shù)，令衰減因子入=0.94,在EVIEWS中做二次指數(shù)平滑,結(jié)果如下圖：Date:10/20/09Time:21:50Sample:1/05/200410/18/2005Ineludedobservations:467Method:DoubleExponentialOriginalSeries:SFZ4ForecastSeries:SFZ4SMParameters:Alpha0.9400SumofSquaredResiduals0.002756RootMeanSquaredError0.002429EndofPeriodLevels: Mean0.000165Trend9.24E-05方差的預測值。2=0.000165，則有。=0.0128VaR=2?33X0.0128X100=2.982萬3、GARCH通過觀察發(fā)現(xiàn)，該股票收益率的波動具有明顯的集聚現(xiàn)象，因而考慮其異方差性對殘差進行ARCH檢驗，結(jié)果表明存在著明顯的ARCH效應ARCHTest:F-statistic11.76612Probability0.000657Obs*R-squared11.52408Probability0.000687TestEquation:DependentVariable:RESIDEMethod:LeastSquaresDate:10/20/09Time:23:19Sample(adjusted):1/07/200410/18/2005Includedobservations:465afteradjustmentsVariableCoefficientStd.Error t-StatisticProb.C0.0004025.75E-05 6.9835550.0000RESIDA2(-1)0.1571270.045807 3.4301770.0007R-squared0.024783Meandependentvar0.000478AdjustedR-squared0.022677S.D.dependentvar0.001159S.E.ofregression0.001146Akaikeinfocriterion-10.70047Sumsquaredresid0.000608Schwarzcriterion-10.68266Loglikelihood2489.860F-statistic11.76612Durbin-Watsonstat2.022064Prob(F-statistic)0.000657利用EVIEWS建立GARCH(1,1)模型如下：R=-0.O515O1Rt-1+etot2=0?0000231+0?084672eb1+0.866212o/DependentVariable:SFZ2Method:ML-ARCH(Marquardt)-NormaldistributionDate:10/20/09Time:23:13Sample(adjusted):1/06/200410/18/2005Includedobservations:466afteradjustmentsConvergenceachievedafter14iterationsVariancebackcast:ONGARCH=C(2)+C(3)*RESID(-1#2+C(4)*GARCH(-1)CoefficientStd.Errorz-StatisticProb.SFZ2(-1) -0.051501 0.049748 -1.035249 0.3006

VarianeeEquationC2.31E-056.45E-063.5737520.0004RESID(-1^20.0846720.0162455.2121560.0000GARCH(-I)0.8662120.02061342.021720.0000R-squared-0.002784Meandependentvar-0.000801AdjustedR-squared-0.009295S.D.dependentvar0.021941S.E.ofregression0.022043Akaikeinfocriterion-4.895787Sumsquaredresid0.224484Schwarzcriterion-4.860214Loglikelihood1144.718Durbin-Watsonstat1.924864進而可根據(jù)上述方程來預測下一期的收益Rt+i和方差ot+i2，在EVIEWS中的處理如下圖：^EViewa-EEquation：UNTITLEDWorkfile：SFZ\Untitled]DFileEdi1ObjectITieirProcQuickOnionsVindowHelp矗w|