Risk Management - Portfolio Investment - Assignment Example

PORTFOLIO INVESTMENT This project embarks on investing of 20 million Australian Dollars on a portfolio that is made up of for investments. This portfolio includes five stocks from four countries, Australia, France, U.S.A and Britain. The stocks are of the following companies, Aurizon FPO (AZJ.AX), Fair Media Limited (FXJ.AX), American International Group (AIG), Sanofi (SAN.PA) and the ARM Holdings PLC (ARM.L). We wish to invest this amount in the following way: AZJ.AX 3,000,000 FXJ.AX 2,000,000 AIG 8,000,000 ARM.L 5,000,000 SAN.PA 2,000,000 In order to model the assets for future investment we will simulate a similar investment program for the past years, 2012 – 2014. We will then proceed to calculate for the Value at Risk (VaR) and use that to carry out some tests. The data used to carry out this simulation was acquired from http://www.rba.gov.au/statistics/historical-data.html#exchange-rates and from au.finance.yahoo.com. then the data was cleaned so as to have holidays and none working days removed. QUESTION A1. Calculate the 95% daily portfolio VaR using the basic methodology of the historical simulation approach. This method takes historical returns and losses and calculates the cumulative distribution function (cdf). This simulation does not make any assumptions on the shape of the distribution. The historical method is easy to use. This is because if we are to find the 95% daily value at risk then we would first of all order our data, and then find the 95th percentile. For this simulation the 95% daily VaR is 306427.82. QUESTION A2 Calculate the 95% daily portfolio VaR using the exponential weighting methodology of the historical simulation approach The exponential weighted method is used to remove the assumption that all dates have the same importance. This is because recent days are more important those older days. In using this method, we first identify our parameter (lambda = 0.995). Then the weight for the data is calculated using whichever formula that is decided. This data should be ordered according to the returns/losses made. We will use the formula; This is used to calculate the weights for the various days in our portfolio. Then a 95% for our portfolio is found. Using this method the exponentially weighted var at 95% is 262823.2. 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 diversified VaR is the VaR that takes into account the portfolio. Here the fruits of diversification and spreading of risk is helpful. Undiversified VaR is the sum of all independently simulated vars for all the assets in the portfolio. With diversified and undiversified VaR we assume the data is normal. The co-variance Matrix   AZJ.AX FXJ.AX AIG ARM.L SAN.PA AZJ.AX 0.000156424 FXJ.AX 6.24723E-05 0.000655357 AIG 2.09022E-06 2.42374E-05 0.000277042 ARM.L 3.14229E-05 3.83249E-05 5.73909E-05 0.000385653 SAN.PA 4.68723E-06 3.68975E-06 7.12781E-05 7.97201E-05 0.000216366 The correlation Matrix is:   AZJ.AX FXJ.AX AIG ARM.L SAN.PA AZJ.AX 1 FXJ.AX 0.195117528 1 AIG 0.010040788 0.056881968 1 ARM.L 0.12793699 0.076233124 0.175578959 1 SAN.PA 0.025478243 0.009798604 0.291131531 0.27597863 1 The diversified VaR and undiversified VaR; Diversified VaR 345629.9379 Undiversified VaR 574853.799 The individual VaR for the various assets are; AZJ.AX 61716.43391 FXJ.AX 84216.28681 AIG 219023.0412 ARM.L 161508.5119 SAN.PA 48389.52515 II) Marginal VaR This is the amount of risk a new investment in the portfolio adds to the portfolio. a portfolio that has perfect correlation(unity) in its assets is hard to find. This is then where the concept of marginal VaR comes from. The marginal VaR for the data is: AZJ.AX 0.006085605 FXJ.AX 0.014802746 AIG 0.021139843 ARM.L 0.021274166 SAN.PA 0.011139027 III) The component VaR This is the change that would happen to a portfolio if a particular component was removed from the portfolio. The component VaR data is; AZJ.AX 18256.81554 FXJ.AX 29605.49244 AIG 169118.7472 ARM.L 106370.8296 SAN.PA 22278.05312 QUESTION B2 What is relationship between marginal VaR and incremental VaR? The marginal VaR is used by fund managers and portfolio managers to determine how best to minimize the overall portfolio Value at Risk. The marginal VaR is very useful in this, since a reduction or an increase in a components position is very vital. The marginal VaR tells us which component is to be increased and reduced. When this alteration, whether and increase or decrease happens, the portfolio var changes by some points (values). This change in the portfolio VaR due to changes in the assets position guided by the value of the marginal var, is called the incremental var. Question B3. On average, what is the relationship between component VaR and individual VaR for a particular position? The relationship between this two is clear. The component VaR is the change in portfolio VaR if the component in particular was to be removed from the portfolio, while the individual VaR is the VaR of that portfolio void of the benefits of diversification. Individual var is var of the component free from other components in the portfolio. 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 The marginal VaR is used by fund managers to get to know how altering the positions of the assets in the portfolio affect the overall value at risk. The way to interpret this is by taking the largest and smallest value of the marginal VaR for your portfolio components. Then subtract a value from the component with the largest marginal var. the value is say alpha, and then add the same alpha to the component with the smallest var. this alters the portfolios var. with a significant negative amount. This technique is used in risk minimization. For our portfolio, we altered it by a total of 500, 000 (AUD) and the portfolio var changed by 7594. 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 formula and its use is discussed further in the excel spread sheet. Here are the simulated results.   Historical Method Exponentially Weighted model building VAR value   no. of days. 722 722 722 exceptions 7 6 6 Lruc 0.006840452 0.220935331 0.220935331 The exceptions are discussed in the spreadsheet. 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 Method     Conditional     Days before     no exception exception unconditional Current day no exception 708 7 715   exception 7 0 7   total 715 7 722           The LRind value is 78.83752 and the LRcc value is 78.84436 Exponentially Weighted   Conditional     Days before     no exception exception unconditional Current day no exception 710 6 716   exception 6 0 6   total 716 6 722           The LRind value is 69.43319 and the LRcc value is 69.65412317 Model building     Conditional     Days before     no exception exception unconditional Current day no exception 710 6 716   exception 6 0 6   total 716 6 722           The LRind value is 69.43319 and the LRcc value is 69.65412317 The LRcc value is more than 5.33. This means that we reject the model for all the three cases. This goes further to show that there is bunching in the data. The LRuc is below 3.99. This then means that we accept the model for the unconditional basis. QUESTION D1 Back testing is usually conducted on a short horizon, such as daily returns. Explain why Back testing is the use of past data to determine its worthiness in prediction and simulation of the past. It is done for short horizons since it is a representation of derivatives of large data. This method is also done for short horizons since calculation for value at risk for larger horizons gives out large numbers. Short horizons are then manageable. 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. Exceptions in part c1 and c2 are binomially distributed with a mean of np and variance of npq. P is used to represent the error term, probability there is an error. In our simulation we used 0.01. N is the number of terms/days in the portfolio, in this case it is 722. The error is included since there is bound to be days that the value at risk for those particular days does not fall with the portfolio VaR or the confidence interval. For our simulation then we find that the upper value is not more than 8. This means that an exception of seven in our data using a two sided 90% confidence interval is acceptable. Using this exceptions and the model that was defined above, the value of LRuc is below 3.99. this means we accept the models. However the value of LRcc is above 5.3 and so we reject the model and say that there is bunching (clustering) in the data. Historical simulation is guided by a few assumptions. These assumptions include: The losses are independent and identically distributed (iid) The data is of equal importance. This is however not true in actual real life data. The method is not influenced by subjective information and cannot be used to predict large changes in the business environment. The data needs a lot of data for accuracy. If the data is not available then it is generated using known methods. The weighted method is useful in the removal of the second assumption and getting our simulation on the right path. The variance covariance method is also very useful. This method however makes an assumption on the shape of the distribution, normal. The various parts in the var-covar method have been discussed above. This method is however very hectic compared to the other methods. Read More