Introduction to Value at Risk Models von Edu Pristine

Dozent des Vortrages Introduction to Value at Risk Models

Edu Pristine

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... for Market Risk Capital - Demonstrate Analytical VaR Model - Explain Monte Carlo Simulation VaR model - Demonstrate Historical Simulation VaR model - Describe Risk Factor Mapping - Demonstrate ...

... Risk management perspective, VaR is losing its charm; calculation of VaR using normal distribution remains one of the major topics in PRM exam. This may be attributed to the fact that estimation of VaR using normal distribution is easily examinable. Estimation of worst-case loss using ...

... loss that might be incurred on a position 99 out of 100 days. VaR doesn’t indicate what would be the quantum of loss when VaR is exceeded. For instance, if 99% 1- day VaR is Rs. 10; it means that for 99 out of 100 days, maximum loss would be Rs. 10. However, it doesn’t tell what would be the ...

... the same as used for day-to-day internal risk management. VaR confidence level was 99%. The time horizon was two weeks or 10 business days. Market risk capital calculation as per Basel guidelines -Banks capital requirement set at 3 times ...

... Local Revaluation Method Full Revaluation Method Monte ...

... by delta, gamma, duration etc.) Monte Carlo simulation: Simulate the underlying variables and value the instruments at different possible values of the underlying variables, to arrive at ...

... occur at: 95% VaR = Mean - 1.65 Standard deviation (because there is 5% probability that a normally distributed variable would decline below 1.65 Std dev from mean. Check in Excel using Normsinv(05)) -97.5% VaR = Mean - 1.96 Standard deviation (because there is 2.5% probability that a normally ...

... of the normal curve constitutes 5% of the total area under the curve. There is 5% probability that the losses will lie in the ...

... Market Value is USD 10 mn. What is VaR (%) at 99% ...

... below: 10 day VaR = 1-day VaR sqrt(10) 1-day VaR = 10-day VaR / sqrt(10) 1-day VaR = Yearly VaR / sqrt(250) assuming 250 trading days in a year Weekly ...

... value for 10 day VaR in the earlier case ...

... is known as volatility clustering (high returns followed by another higher return in either direction and vice-versa). We can test for volatility clustering by examining autocorrelation of squared returns (i.e. square the returns data and calculate its correlation with itself with a lag. For instance, correlation of squared returns at time ‘t’ with time ‘t-1’. ...

... Jumps (sharp rise or decline). Fat tails (high chance of extreme values) Non-linear or complicated pay-offs from the instrument (exotic options) MCS allows VaR estimation for almost all types of instruments and pay-offs. All risk-factors/variables (interest rate, forex rate) impacting price of each instrument are identified. Random processes underlying these risk-factors are identified (for instance, ...

... a straddle, maximum loss happens when underlying price stay close to strike price). Any portfolio that consists of many such instruments can also be handled. However, MCS is not suitable ...

... simply assumes that past data is a good indicator of likely future scenarios. For instance, to compute 1-day 99% VaR based on 1001 days data: We have to estimate worst-case loss over 1-day period using past 1001 trading days data (four years). Changes in market ...

... weightage is assigned to recent events). This method is similar to using EWMA for estimating volatilities. Vol-adjusted Historical VaR: Actual return data is updated based on recent volatility data. For instance, if recent daily volatility is 1% as compared to 2% volatility one-month ago, month-old data overstates what is likely to happen in future. Therefore, historical returns are revised downwards. If recent daily volatility is higher than historical volatility, historical returns data ...

... yrs, 1 yr, 1.5 yrs, 2 yrs…10 yrs yields) or all variables might not have substantial historical data available to compute VaR. In such cases, all exposures are mapped to limited number of risk-factors. For instance, a portfolio might consist of 0-Coupon Bonds of maturity 1.5 yrs and 7 yrs. The ...

... in form of a foreign currency term loan. The company prepares P&L in INR and the current forex rate is INR 40 = 1 USD. Volatility in USD/INR forex rate is 10% p.a. What is daily VaR at 99% confidence interval. Exposure in base currency = INR 40 million -1-day volatility = 10%/sqrt(250) = 0.632% -1-day 99% VaR in base currency because of ...

... cumbersome -To calculate portfolio variance, one needs n*(n-1)/2 covariance, if there are ‘n’ stocks in the portfolio. Instead, all equity positions can be mapped to market index. Weighted average Beta of ...

... Portfolio size = Rs. 15 million -1-day 97.5% Portfolio VaR = 1.96 * 2.5% * 15 * 1.07 = Rs. 0.78 million -Yearly 97.5% Portfolio VaR = 1-day Portfolio VaR * sqrt(250) = Rs. 12.43 million. This method assumes that at portfolio level, all firm-specific (idiosyncratic ...

... 1 bps. If 5-Year Spot rate is 7%. Calculate PV01 for ZCB with maturity payment of USD 1 mi. Present Value at 7% = 1,000,000 * e(-.07*5) Present Value at 7.01% = 1,000,000 * e(-.0701*5) Therefore, PV01 = 1,000,000 *(e(-.0701*5) - e(.07*5)) = - USD 345. It means ...

... = 6.3825%. Let us map USD 1m such that C2 is the cash-flow at 2nd Yr and C3 is the cash flow at 3rd Yr. We want to ensure that sensitivity of USD 1m at 2.75 Years to 2-Year Spot and 3-Year Spot remain same after it is mapped to 2-Year and 3-Year. Let us take C2 first. PV01 of ...

... C2 = 325274. Similarly, PV01 of C3 (cash flow at 3rd Year) to 3-Year Spot = PV01 of 1,000,000 (cash-flow at 2.75 Yr) to 3-Year Spot -i.e. C3*(e(-.0651*3)-e(-.065*3)) = 1000000(e(063825*2.75)-e(06375*2.75)) -C3 = 701202. Note that C2+C3 ...

... 26%1% 36.50%1.25% Assume correlation between 2-Year and 3-Year rate is 0.85 Solution MaturitySpot 26.000%6.0100% 2.756.375% 36.500%6.5100% Sensitivity of 2.75Year Spot to 2Year Spot Sensitivity of 2.75Year Spot to 3Year Spot MaturitySpot MaturitySpot 26.0100%26.0000% 2.756.3775%2.756.3825% 36.5000%36.5100% i.e. if 2Year Spot ...

... a 2-Year bond paying 10% interest semi-annually can be considered as a portfolio of 0.5-Yr ZCB of USD 5, 1-Yr ZCB of USD 5, 1.5-Yr ZCB of 5 and 2-Yr ZCB of USD 105. Once we have mapped cash flows to standard maturities, calculation of VaR requires application ...

... Libor at time ‘0’ was 4%: In this case, we know that after 4 more months, FRN would be selling at par (because interest would be reset). Also, we know the interest rate that would be earned for the current period of 6 months (4% + 1.5%). Therefore, FRN can be ...

... It is a first order approximation and is accurate only for small changes in value of the underlying. Change in value of Option = Delta * change in Underlying * Value of Underlying -Illustration: A portfolio consist of options on ABC ...

... 2727 Delta-Gamma approximation: It is a 2nd-order approximation (i.e. gamma is 2nd derivative of derivative’s price w.r.t. underlying) & holds for even large change in value of underlying. Change in value of option = (Delta * Value of Underlying * Change in value of underlying) + 0.5*(Gamma* Value ...

... 1, Strike Price = 1, Annual volatility of the underlying = 25%, Delta of option = 0.591 and Gamma = 2.198 Solution: 10-day volatility in underlying = 25%*sqrt(10/250) = 5%. Since it is a call option, losses will be recorded ...

... underestimating worst-case losses. Stress testing -Testing a portfolio under extreme scenarios that: Might have actually happened in past (say, what would be the decline in value of an equity portfolio if market index declines by 6.8 standard deviation, as it did in Jan 1988 with S&P) ...

... normally distributed. Central Limit Theorem states that averages of large number of independent realizations of a random event follow normal distribution. This follows even when the random event is not normally distributed. The assumption of normality holds when events (like default event, or ...