Investing on a Series of Stocks - Assignment Example

Information on the portfolio We are to invest a total of AUD 20,000,000 on a series of stocks in our portfolio. There are five stocks in our portfolio from different countries. We will simulate this investment over the last three years, from 1 January 2012 to 31 December 2014. The exchange rates are gotten from http://www.rba.gov.au/statistics/historical-data.html#exchange-rates while the stock prices for the five stocks are gotten from au.finance.yahoo.com. data cleaning was accurately done so as to come up wth the appropriate parameters to carry out our tests AZJ.AX 3,000,000 FXJ.AX 2,000,000 AIG 8,000,000 ESSR.L 5,000,000 SAN.PA 2,000,000 QUESTION A1. Calculate the 95% daily portfolio VaR using the basic methodology of the historical simulation approach. The historical approach, we can use the ranked losses then calculate a percentile for the loss that we intend to have. This approach then becomes very easy to use since we need the 95th percentile for our losses. Using this approach you will find that our 95% daily portfolio VAR is 337397.7. QUESTION A2 Calculate the 95% daily portfolio VaR using the exponential weighting methodology of the historical simulation approach. Using the weighted approach, we use this to weigh all our data then use that weight to calculate our var. this then is useful since it is not only a percentile but also a way in which we can actually see the value. The weighted historical approach is then very easy to use and simulate. We use the formula, In our data we are using lambda as 0.995 to calculate our weight. Using this approach we have calculated our loss to be approximately, 303,053. QUESTION B1 Calculate the 95% daily VaR for the portfolio using the model-building approach. Generate a VaR report for the portfolio based on the model-building approach, which should contain the following elements: I) An analysis of the diversification effect (undiversified VaR and diversified VaR). The model building approach is also called the variance co variance method. The co variances are;   COVARIANCE     AZJ.AX FXJ.AJ AIG ESSR.L SAN.PA AZJ.AX 0.000155311 FXJ.AJ 6.17547E-05 0.000651748 AIG 1.10297E-06 2.47181E-05 0.000275802 ESSR.L 1.25972E-05 -3.72472E-06 6.37902E-05 0.000897072 SAN.PA 5.37635E-06 8.08459E-06 6.84366E-05 5.07951E-05 0.000209927 And the correlation coefficients are   CORRELATION     AZJ.AX FXJ.AJ AIG ESSR.L SAN.PA AZJ.AX 1 FXJ.AJ 0.194101654 1 AIG 0.00532923 0.058301028 1 ESSL.R 0.033748971 -0.004871254 0.12824539 1 SAN.PA 0.029775072 0.021856709 0.284417156 0.117050737 1 From this, the variance covariance matrix can easily we generated. The table below shows individual VaR together with diversified and undiversified var. DIVERSIFIED VAR 386625.6656 AZJ.AX VAR 61496.41735 FXJ.AJ VAR 83984.07782 AIG VAR 218532.403 ESSR.L VAR 246326.4279 SAN.PA VAR 47664.05652 UNDIVERSIFIED VAR 658003.3825 The undiversified VaR shows the VaR that would have affected the portfolio was it not diversified. The diversified VaR is also the portfolio var. This approach assumes that our data is normally distributed. II) Marginal VaR This is especially useful in getting to know with component has the highest marginal var, so that we may know which component we would reduce and increase to reduce the risk at same portfolio value. AZJ.AX 0.004702586 FXJ.AJ 0.011784704 AIG 0.018999017 ESSR.L 0.035882223 SAN.PA 0.008772624 III) Component VaR stock CVaR percentage AZJ.AX 14107.75714 3.65% FXJ.AJ 23569.40715 6.10% AIG 151992.1369 39.31% ESSR.L 179411.1169 46.40% SAN.PA 17545.2476 4.54% QUESTION B2 What is relationship between marginal VaR and incremental VaR? Let’s take this simulation as a case study, then the marginal var. is largest for ESSR.L with a VaR of 0.036. The smallest marginal var. is for AZJ.AX. If we were to reduce the overall VaR, and not affect the overall portfolio and the amount of the portfolio, then we would reduce the amount of the one with the largest VaR, and place it in the asset with the smallest var. This would then maintain the amount of the portfolio at 2 million but reducing the overall value at risk. This reduction is then known as the incremental var. In this case, we alter the portfolio with 300,000 and the incremental var. becomes -9.3538913. QUESTION B3 On average, what is the relationship between component VaR and individual VaR for a particular position? The two are related. This is because they both use marginal VaR for their calculations. However the two are different. The component VaR is an approximation measuring the change in the portfolio VaR if the entity (stock) is removed from the portfolio. The individual VaR however does not consider the portfolio. The individual VaR is VaR if the said entity (stock) was to be on its own. QUESTION B4 Marginal VaR in Part B1 are useful tools for risk management. Discuss how to change the portfolio positions to minimizing the portfolio VaR while keeping the portfolio fully invested. Generate a new VaR report based on the risk-minimizing positions. As discussed earlier marginal VaR can be used to reduce risk of a portfolio, this is by use of the least and largest marginal var. The smallest and largest marginal VaR are picked, then you reduce the amount for the largest portfolio by lets say a, then increase the smallest by the same amount. This way the portfolio is fully invested but the portfolio VaR has been reduced. In this portfolio, we altered it this way by a sum of 300,000. This in turn reduced the portfolio VaR by an amount of more than 9300. QUESTION C1 Implement back testing procedures to evaluate the portfolio VaR determined in Part A1, Part A2 and Part B1, respectively with unconditional coverage In this we will use the formula, This then gives the results as follows:   historical simulation ewma model building VAR value no. of days. 726 726 726 exceptions 5 4 4 Lruc 0.797679724 1.76608 1.766080385 This is done with a probability of 1%. More in the excel spread sheet. QUESTION C2 Implement back testing procedures to evaluate the portfolio VaR determined in Part A1, Part A2 and Part B1, respectively with conditional coverage. Historical Approach   Conditional     Days before   no exception exception unconditional Current day no exception 716 5 721 exception 5 0 5 total 721 5 726 EWMA   Conditional     Days before   no exception exception unconditional Current day no exception 718 4 722 exception 4 0 4 total 722 4 726 Model building   Conditional     Days before   no exception exception unconditional Current day no exception 718 4 722 exception 4 0 4 total 722 4 726 The LRind values are; Historical approach: 59.74661 The EWMA: 49.58797 The model building: 49.58797 More of this in the excel LRcc = LRind + LRuc So the values for the various approaches are: Historical Model; 60.54429 The EWMA: 51.35405 The Model building approach: 51.35405. The LRcc for all the approaches is more than 5.99 so we reject the models. This translates to there is presence of buching. QUESTION D1 Back testing is usually conducted on a short horizon, such as daily returns. Explain why. Back testing is the process of testing a trading strategy on prior time periods so as to use that as a basis for the future. It is done for short horizons because It is a process that uses derivative of securities held by a large sample. To achieve this the horizon then much be as small as possible. From the stochastic volatility of stocks and exchange rates. This then enables the capture of the rates as they change. The VaR numbers are too high for swap portfolios. This is however not applicable on options and futures since their var values are very small for short horizons. QUESTION D2 Contrast and compare your findings in Part C1 and Part C2 and further comment on the performance of the market risk measurement approaches used in Part A1, Part A2 and Part B1. As stated in the above section, back testing is used so as to know if we can use the past to project the future. The method is used to verify is the actual losses are in line with projected losses. This method compares the overall observations with the ones that are within the confidence interval and the ones that are exceptions to these. The exceptions are the values that are outside the VaR and are not with the confidence interval. Using the Kupiec (1995) test, the approach then tells us not to reject the model because the values from our model are below 3.84. However Using Christofferson (1998) test, LRcc is more than 5.88 for all approaches. This then means that we reject the model. This is because there is presence of bunching.\ For the historical approach, the approach recognizes the data. The data is ordered from the largest to smallest or the other way around. It is then assumed that history will repeat itself. Using 95% the daily VaR then becomes 337397.7. The weighted approach to historical method above is an advanced method. It gives more importance to recent days than order days. A weight is used to give this importance, in this case we use, 0.995. From this the 95 daily portfolios is 303,053. In B1 the approach is very simple. This is because we use simple statistical methods. We simulate the variance, covariance, standard deviations, correlations and the variance covariance matrix. This intern helps to calculate the variance of the portfolio and its standard deviation. The marginal var. is a tool used to give the var. if the assets were operating void of the portfolio. This is very important especially in risk minimization as done above. The component var. is used to measure how much would our portfolio var. be affected if individual entities (stocks) in the portfolio were to be removed. The incremental var. is used to show how much the risk would reduce or increase if investments in the various assets were to be increased and reduced. It is also a factor in risk minimization. Read More