Quantitative Risk Management: Concepts, Techniques, and Tools - Coursework Example

RISK ANALYSIS by Question Computing VAR and ES A portfolio of a present value of 100 $; the P&L of the portfolio for the next week is illustrated by the subsequent discrete random variable. -5 -4 -3 -2 -1 0 1 2 3 4 5 0.03 0.05 0.08 0.1 0.12 0.25 0.15 0.1 0.06 0.04 0.02 Top line: possible realizations of the P&L Bottom line: corresponding The P&L Corresponding Returns Column1 -5 0.03 -4 0.05 0.666667 Mean 0.083333333 -3 0.08 0.6 Standard Error 0.17648838 -2 0.1 0.25 Median -0.066666667 -1 0.12 0.2 Mode #N/A 0 0.25 1.083333 Standard Deviation 0.55810526 1 0.15 -0.4 Sample Variance 0.311481481 2 0.1 -0.33333 Kurtosis -1.02517897 3 0.06 -0.4 Skewness 0.618098157 4 0.04 -0.33333 Range 1.583333333 5 0.02 -0.5 Minimum -0.5 Maximum 1.083333333 Normal Distribution VAR Sum 0.833333333 Mean 0.083333 Count 10 SD 0.558105 Confidence Level (95.0%) 0.399244451 VaR(%) ES Bottom 10% -0.63191 90% confidence -0.4 Bottom 5% -0.83467 95% confidence -0.38 Historical VaR n 10 Bottom 10% 1.0th Return Bottom 3rd return -0.4 Bottom 2nd return -0.4 bottom 1.0th Return -0.8 10%VaR Bottom 5% 0.5th Return Bottom 2nd return -0.4 Bottom 1st return -0.5 bottom 1.0th Return -0.82 5% VaR The 1-week VaR at 90% confidence level If the VaR on a portfolio is $ 100 million at a 1-week, 90% confidence level, only a 10% probability that the value of the portfolio will fall more than $ 100 million in that week. VaR at 90%=-0.8 The 1 week ES at 90% confidence level ES at 90% = -0.4 The 1 week VaR at 95% confidence level VaR at 95% = -0.82 The 1 week ES at 95% confidence level ES at 95% = -0.38 Bin Frequency Cumulative % -0.5 1 10.00% 0.027778 4 50.00% 0.555556 2 70.00% More 3 100.00% Monte Carlo simulation is applicable to compute VaR even if a portfolio is non-linear. The process is easy, if calculation intensive, and offers a vast deal of data since it provides the real distribution of outcomes that is computed. From this allocation, one can construe the predictable value of the collection, the extent in returns, assessments of the non-normality of the distribution, also VaR (Financial Engineering Associates, Inc. 2000). The application insight is imparted by observing the distribution of outcomes than is attained from reflection of a single integer such as VaR, which is determinant of the density of the extension of the distribution. The system is robust but computationally exhaustive. The choice of whether to utilize it rather than the variance-covariance technique involves deliberation of tradeoffs between preferred accuracy and obtainable computing time. First, it is necessary to note that if portfolio has just financial instruments whose costs differ linearly with reference to asset costs, then both techniques give the identical outcome. In this scenario, the variance-covariance technique is ideal because it compute faster (Financial Engineering Associates, Inc. 2000). Description of the Monte Carlo simulation procedure The steps are implemented by Outlook automatically; the learner or experts work is to interpret the outcome of the analysis. In case of a portfolio of financial distributions, whose values rely on a known approach on assets for which the learner knows the current and historical spots (vertices)? These instruments could be merchandise, foreign exchange rates, equities, and the like. 1. Express the entire instruments in the portfolio as a bearing of a set of encoded vertices (prices). Outlook provides a set of structures inbuilt to price several options and fiscal instruments. This condition is perhaps the most vital step in the Monte Carlo simulation procedure. If individuals identify the fundamental underlying financial points (vertices) on which the portfolio relies, they can simulate the activities of the portfolio for diverse market prices. 2. for existing prices of given risk factors; compute the price of the portfolio. The process is known as the mark-to-market worth of the portfolio 3. Create volatility, and correspondence matrices for the encoded points (correlation matrices and volatility can be generated with MakeVC or accessed from Risk metrics). From the historical awareness of asset price conduct, learners can create a distribution of precedent price alterations over a provided interval of instances (that data kept in the volatilities and correlations points) (Financial Engineering Associates, Inc. 2000). 4. The fourth step is to generate a market situation with simulated market costs considering the correlation and volatility model of those prices. In order to create each scenario, there is need to execute the subsequent steps: a) Create a series of random numerical (extract the numbers from standard normal supplies with standard deviation one and mean zero). The numeral of random figures for each case is determined by the numeral of determined vertices. b) Change the random figures into simulated returns by integrating covariance data incorporated in the volatility and correlation matrices. The different random numbers correlation accomplished by use of a Choleski breakdown of the correlation matrix. The Choleski breakdown is correspondent to captivating the square root of the correlation matrices (Financial Engineering Associates, Inc. 2000). c) Compute new prices for every of the determined vertices with presumptuous returns are lognormal. The outcome is a fresh set of costs for each of the determined risk factors. From Step 1, all the instruments defined in the collection as functions of those costs. Consequently, it is possible to compute the new imaginary value of the diverse instruments that create the portfolio for the fresh simulated prices (Financial Engineering Associates, Inc. 2000). d) Replicate steps a., b., and c., severally as the quantity of simulations needed to run. It is possible to generate a broad range of cases to explain possible prospect events. Because these cases are randomly created, there is inclusion of extremes, and cases that contradict accepted opinion. Consequently, results to not running the threat of instinctively being overly optimistic concerning possible prospect market situations. 5. From the horizon, recomputed “mark-to-horizon” charge of collection (mark-to-market price of the collection at horizon in the hypothetical potential), accessing the scenario’s profit or loss (P&L). The portfolio marked the portfolio to marketplace for the latest prices created. The fresh MTM value is contrasted with the initial MTM figure of the portfolio to acquire P&L for that give scenario (Financial Engineering Associates, Inc. 2000). The figure below is an example of how the outlook assists in generating desired values. Figure 1: Outlook QUESTION 2: IS VAR A COHERENT RISK MEASURE? 1. PROP: VaR gratifies the Positive homogeneity characteristics. PROOF: Let reflect on the function t: x → λ x having λ > 0, it is deducible that VaR has the translativity state. 2. The VaR satisfies the translation invariance characteristics. For every LεM and all lεR there is (L+l) = (L) + l. Axiom indicates that by subtracting or adding a deterministic magnitude l to a position resulting to the loss L company change its capital needs by exactly that sum (McNeil et al. 2005).The axiom is in reality possible for the risk capital elucidation of to bring logic. Consider a state having loss L and (L) > 0. Addition of the quantity of capital (L) to the state results to the attuned loss Ĺ =L - (L) with (Ĺ) = (L) - (L) =zero. Therefore, it concluded that the position Ĺ is tolerable without further change of capital (McNeil et al. 2005). 3. Solution: First, regard each position independently: in each individual state, there is just a 2.532% probability that the company pay out $ 10, 000. Put a different approach, the P&L supply is exactly P (P&L = -9000) = 2.532% and P (P&L ≤ 1000) = 1. 100% 5% 2.532%-9,000 1,000 Figure 2: P&L distribution for one binary option 100% 5% 0.0642%-18,000 -9,000 2,000 Figure 3: P&L distribution for a portfolio of two binary options It is not possible for the company to lose over $9000, so the condition is the 1% VaR, in that P (P&L ≤ -$9000) = 1%. Certainly, $9000 is as well the 2.5% VaR. However, that is the % VaR, in that the quantity X is that P (P&L ≤ -X) =5%? It is significant to acknowledge that the P&L is actually discrete for this binary selection; it is not only a discrete estimate to a continuous random figures. Either company makes a yield of $1000 or they lose $9000. The conditions are the just alternatives. It is logical to interpolate amid these results, as if the affected individuals could obtain a P&L among them. It is known the distribution utility is exactly as showcased in figure 2, therefore result readable and read off the 5% quantile, the outcome is +$1000. The summation of the given 5% VaRs is therefore the key lose is -$18,000 if both options are called. This result occurs with chance 0.025322 = 0.000642. If one alternative is called exactly $9000 is lose and the other is not. The chance for the condition is 2 x 0.02532 v (1-0.02532) = 0.049358. As a result, the probability that the company lose $9000 or extra is 4.9358% + 0.0642% = 5%. Therefore, as displayed in figure 2, the 5% VaR of the collection is $9000. This outcome is better than -$2000 for example the amount of the VaR on the two entity positions taken independently. Consequently, the VaR cannot be sub-additive (Alexander 2008). QUESTION 3: COMPUTING AND BACKTESTING VAR 1. A preliminary statistical analysis regarding the time series of the portfolio return. MSFT   AAPL   INTL   Mean 33.41196286 Mean 78.02927 Mean 24.76394 Standard Error 0.248265817 Standard Error 0.537381 Standard Error 0.170775 Median 31.485 Median 75.91 Median 23.695 Mode 28.64 Mode 113.54 Mode 24.06 Standard Deviation 6.817146075 Standard Deviation 14.75598 Standard Deviation 4.689328 Sample Variance 46.4734806 Sample Variance 217.7391 Sample Variance 21.9898 Kurtosis -0.753740629 Kurtosis -0.19786 Kurtosis 0.379569 Skewness 0.675018972 Skewness 0.614656 Skewness 1.12885 Range 24.53 Range 65.13 Range 19.48 Minimum 24.42 Minimum 53.4 Minimum 17.92 Maximum 48.95 Maximum 118.53 Maximum 37.4 Sum 25192.62 Sum 58834.07 Sum 18672.01 Count 754 Count 754 Count 754 2. Estimate the VaR at 90%, and 99% confidence levels The parametric Gaussian VaR: MSFT VaR at 90%=-0.10965 MSFT VaR at 99%= -0.12226 AAPL VaR at 90%=-0.1423692 AAPL VaR at 99%= -0.1593031 INTL VaR at 90%=-0.10694 INTL VaR at 99%= -0.14647 The parametric Student-t VaR: The empirical quantile 3. a Table showing for each model and each confidence level the number of VaR violations Normal Distribution VAR(MSFT)     Normal Distribution VAR(AAPL)     Normal Distribution VAR(INTL)     Mean -0.00075   Mean -0.0007667   Mean -0.00057   SD 0.013718   SD 0.0169721   SD 0.013434                       VaR (%)     VaR (%)     VaR (%)   Bottom 10% -0.01833   Bottom 10% -0.0225173   Bottom 10% -0.01778   Bottom 1% -0.03266   Bottom 1% -0.0402497   Bottom 1% -0.03182                     Historical VaR     Historical VaR     Historical VaR     n 753   n 753   n 753   Bottom 10% 75.3th Return   Bottom 10% 75.3th Return   Bottom 10% 75.3th Return   Bottom 3rd return -0.05351   Bottom 3rd return -0.0672214   Bottom 3rd return -0.04427   Bottom 2nd return -0.05632   Bottom 2nd return -0.0757538   Bottom 2nd return -0.06389   bottom 1.0th Return -0.10965 90%VaR bottom 1.0th Return -0.1423692 90%VaR bottom 1.0th Return -0.10694 90%VaR                   Bottom 5% 37.65th Return   Bottom 5% 37.65th Return   Bottom 5% 37.65th Return   Bottom 2nd return -0.05632   Bottom 2nd return -0.0757538   Bottom 2nd return -0.06389   Bottom 1st return -0.06781   Bottom 1st return -0.0848306   Bottom 1st return -0.08483   bottom 1.0th Return -0.12226 99%VaR bottom 1.0th Return -0.1593031 99%VaR bottom 1.0th Return -0.14647 99%VaR 4. Kupiec test (at significance level of 95%): a Table showing for each combination of model and confidence level the value of the likelihood ratio and your decision about the model accuracy 5. A conditional coverage test 6. historical volatility approximated using the last 60 days; option strike is set at $1750; Assumption: assume zero interest rate and zero dividend yield Option will expire in 30 days Simulate the portfolio value at the 5 days horizon Assumption: daily portfolio returns evolve according to the EWMA model with λ = 0:95. The 95%VaR and expected shortfall of the single components Findings The EWMA model shows the number of historical observations applicable by EWMA for a specified tolerance stage and a provided lambda. For instance, a decay issue of 0.85 at 1% lenience stage needs 28 daily proceeds, while a decay factor of 0.98 at 0.01% needs 456 every day returns. QUESTION 4: VAR OF A BOND The yield to maturity of the bond A coupon bond with 3 years to maturity semi-annual coupons Notional coupon at 4% Notional of 100$ Current price of the bond is 98 $. Assumptions: absolute daily transformation in the yield to maturity (ytm) is a Gaussian random unpredictable with daily volatility of 1% and means 0 (Here Δ = 1=250). An expression for the 10 days VaR of the bond Computing the 10-day, 99% VaR straight analysts normally calculate a 1-day 99% VaR and make assumptions Expression 10-day VaR = √10x1-day VaR 3(10 +250) –3(10) ~98(4%, (1%)2 ) QUESTION 5: SIMULATING RETURNS Assumption: that daily log-returns are iid The probability compactness function is provided by How to simulate 10 days returns Step 1: The core market factors influencing the portfolio worth would require identification. For instance, there could be 10 issues (exchange rate, prime rate, inflation, others.) that are recognized as the main drivers of a given portfolio. Step 2: For every main market issue, outline back its historical faction. Step 3: compute the alteration in portfolio value concerning on each set of historical figures of the key market factors. Step 4: An experiential allocation of the transformation in portfolio price generated and therefore VaR can be computed accordingly.  Simulate 1000 10-days returns according The simulated returns compute VaR and expected shortfall QUESTION 6: VAR IN THE BINOMIAL MODEL The stock price can shift down or up by a factor d = 0:9 and u = 1:1 If currently the stock is value at 100 in one period time it value will be 90 or 110 respectively with chance 0.55 and 0.45, (these are real-world chance not risk-unbiased ones). A portfolio of options: long put option with strike at 80; long call option with a strike of 120 The options expire in 20 periods No-arbitrage price of the two options The general strategy to alternative pricing is primary to suppose that prices do not offer arbitrage chances. Subsequently, the derivation of the alternative prices or pricing limits is acquired by providing the payoffs provided by the selection using the fundamental stock(asset ) and risk-free lending and borrowing. Suppose S = 120, X = 80, rf= 10% and T= 1 years. Subsequently S-PV(X)= 120 - 80/1.10 = 47.27. Given that the market value of the call provided as C = 20. (Take note that C> intrinsic worth = 20, but C< 47.27, which is the attuned intrinsic worth). Therefore , the outcome can . . . CF Buy the asset -20.00 Sell short the stock +120.00 Invest PV(X) at rf -72.72 Total +27.27 The 12 months probability distributions of the P&L of the portfolio and then the corresponding 90% VaR and Expected Shortfall Reference List Alexander, C 2008, Market risk analysis. Volume IV, Volume IV. Chichester, England, John Wiley. http://site.ebrary.com/id/10278223. Financial Engineering Associates, Inc. 2000, Introduction to Monte Carlo VaR with FEA VaRworks® Monte Carlo Simulation, Historical Simulation, Variance-Covariance VaR, VaRdelta® and Extreme Value Theory. Retrieved March 8, 2015, from http://www.fea.com/resources/tutorial_montecarlo_var.pdf McNeil JA, RüdigerFrey, R & Embrechts, P 2005, Quantitative Risk Management: Concepts, Techniques and Tools. Chapter 6. Retrieved March 8, 2015, from http://www.math.ethz.ch/~embrecht/RM/chap6.pdf Read More