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Value at Risk and Monte Carlo Simulation

Value at Risk and Monte Carlo Simulation. Agenda for Today. Home work Financial engineering Introduction to VaR Demo of Monte Carlo simulation and VaR in SAS. What is VaR?.

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Value at Risk and Monte Carlo Simulation

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  1. Value at Risk and Monte Carlo Simulation

  2. Agenda for Today • Home work • Financial engineering • Introduction to VaR • Demo of Monte Carlo simulation and VaR in SAS VaR and Monte Carlo Simulation

  3. What is VaR? • VaR stands for Value at Risk, which is an attempt to provide a single number to summarize the total risk in a portfolio of financial assets for senior management. • VaR is widely used by corporate treasurers and fund managers in financial institutions. Central bank regulators also use VaR to determine the capital a bank is required to keep to reflect the market risk it is bearing. VaR and Monte Carlo Simulation

  4. Definitionof VaR • VaR is the amount of losses that will not be exceeded in a specified period of time, with a pre-defined amount of certainty. • VaR is a function of 2 parameters: the time horizon, n; and confidence interval, x. • It is a loss level over n days that we are x percent certain will not be exceeded. • VaR assumes the loss distribution is normally distributed. VaR and Monte Carlo Simulation

  5. VaR’s Role in Banking Regulation • The Basel Committee on Banking Supervision is a committee of the world’s banking regulators that meet regularly in Switzerland. • The Committee’s 1996 Amendment to the BIS Accord calculates capital for banks’ trading book using the VaR with n=10 and x=99. VaR and Monte Carlo Simulation

  6. VaR’s Role in Banking Regulation • This means it focuses on the revaluation loss over a 10-day period that is expected to be exceeded only 1% of the time. • Banks are required to hold k times VaR capital with minimum k=3.0 VaR and Monte Carlo Simulation

  7. Time Horizon n • The time horizon N is measured in days. In practice n is usually set to 1 because there is not enough data to estimate directly the market behavior over periods of longer than 1 day. • Therefore, N-day VaR=1-day VaR * √n assuming the changes in portfolio value on successive days are independent identical normally distributed. VaR and Monte Carlo Simulation

  8. Calculating VaR • There are 2 ways to calculate VaR: • Historical Simulation • Modeling-building Approach VaR and Monte Carlo Simulation

  9. Historical Simulation • Historical simulation uses past data as a guide to predict what will happen in the future. • Suppose we want to calculate VaR for a portfolio using a one-day time horizon, with 99% confidence interval. • The first step is to identify the market variables (e.g. exchange rates, equity prices, interest rates, etc.) that affect the portfolio. • We can then collect data on these market variables over the most recent say 500 days. VaR and Monte Carlo Simulation

  10. Historical Simulation (cont.) • This provide us with 500 alternative scenarios for what can happen between today and tomorrow. • Scenario 1 is where the percentage changes in the values of all variables are the same as they were on the first day for which we have collected data. • Scenario 2 is where they are the same as on the second day we have data , and so on. VaR and Monte Carlo Simulation

  11. Historical Simulation (cont.) VaR and Monte Carlo Simulation

  12. Historical Simulation (cont.) • Assume variable 1’s value today is vm=$25.85 and the portfolio value is $23.5 million. The value of the market variable 1 tomorrow is vm(vi/vi-1) 25.85*(20.78/20.33)=26.42 • If the value of the portfolio is known today, we can calculate the change in the value of between today and tomorrow for different scenarios. VaR and Monte Carlo Simulation

  13. Historical Simulation (cont.) • For each scenario we calculate the dollar change in the value of the portfolio between today and tomorrow. • This defines a probability distribution. If we chose the 1 percentile worst losses, then we can be 99% certain we will not take a loss greater than the VaR. • The fifth-worst daily change is the first percentile of the distribution, which is the first percentile point. VaR and Monte Carlo Simulation

  14. Model-building Approach • Consider a portfolio consisting of $10 million of Microsoft shares and $5 million of AT&T shares. Suppose the returns on the 2 shares have bivariate normal distribution with a correlation of 0.3 • Suppose the daily volatility of MS stock is 2% per day and AT&T is 1% per day. So the standard deviation of MS and ATT is $200,000 and $ 50,000, respectively. VaR and Monte Carlo Simulation

  15. Model-building Approach (cont.) • The standard deviation of the portfolio is σp = √ σx2 +σy2 +2 ρijσxσy = √200,0002+50,0002+2*0.3*200,000*50,000 =220,227 • The one-day 99% VaR is: 220,227 * 2.33=513,129 • The 10-day 99% VaR is: 513,129 * √10=1,622,657 VaR and Monte Carlo Simulation

  16. The standard deviation of each stock by itself (assuming there is only 1 stock in the portfolio) is: MS:  10MM * .02 =200,000 at 99% level,                   200,000*2.33=464,000 ATT: 5MM * .01=50,000, at 99% level,                   50,000*2.33=116,500 • So holding 1 portfolio with 2 correlated assets is less risky than holding 2 separate portfolios of the same assets since 464,000+116,500=580,500 which is greater than 513,129. VaR and Monte Carlo Simulation

  17. Model-building Approach (cont.) • Suppose a portfolio worth P consisting of n assets with an amount wi being invested in asset i (1 ≤ i≤ n). • If the return on asset i in one day is denoted as ∆xi. The dollar change in the value of the investment in asset i in one day is wi∆xi and n ∆P= ∑ wi∆xi i=1 assuming∆xi and ∆P are normally distributed and the value of the portfolio ∆P is linearly dependent on percentage changes in the market variables. VaR and Monte Carlo Simulation

  18. Model-building Approach (cont.) • To calculate the standard deviation of ∆P, we define σi as the standard deviation and daily volatility of ith asset and ρij as the coefficient of correlation between return on asset i and asset j. • The variance of ∆P is σp2 = ∑ ∑ ρij wiwjσiσj or = ∑wi2σi2 + 2 ∑ ∑ ρij wiwjσiσj VaR and Monte Carlo Simulation

  19. Model-building Approach (cont.) • With options in the portfolio, consider first a portfolio consisting of options on a single stock whose current price is S. Suppose that the delta of the position is δ, the rate of change of the value of the portfolio with S so that δ =∆P/∆S thus ∆P = δ ∆S where ∆S is the dollar change in the stock price in one day and ∆P is the dollar change in the portfolio in one day. VaR and Monte Carlo Simulation

  20. Model-building Approach (cont.) • Now define ∆x as the percentage change in the stock price in one day so that ∆x = ∆S/S thus ∆P=Sδ ∆x • With a position in several underlying market variables that includes options, we can approximate linear relation ship between ∆P and∆x as n ∆Pi= ∑ Siδi∆xi i=1 where i is the value of ith market variable. Let wi= Siδi then we have n ∆Pi= ∑ wi ∆xi i=1 VaR and Monte Carlo Simulation

  21. Model-building Approach (cont.) • Consider an example: a portfolio consists of options on MS and ATT. The option on MS have delta of 1,000 and the options on ATT have a delta of 20,000. MS share is $120, and ATT share price is $30. Therefore, ∆P =120*1000* ∆x1 +30*20,000* ∆x2 where ∆x1 and ∆x2 are return from MS and ATT in one day. Assume the portfolio is equivalent of investment in $120,000 in MS and $60,000 in ATT. VaR and Monte Carlo Simulation

  22. Model-building Approach (cont.) • Now assume the daily volatility of MS is 2% and ATT is 1%, and the correlation between the daily changes is 0.3, the standard deviation of ∆P is √(120*0.02)2+(60*0.01)2+2*0.3*120*0.02*60*0.01 =7.099 since N(-1.65)=0.05, the 5 day 95% VaR is 1.65*√5*7,099=$26,193 VaR and Monte Carlo Simulation

  23. Quadratic Model • The linear model is an approximation. It does not take account of the gamma of the portfolio. • Gamma id defined as the rate of change of the delta with respect to the market variable. • When gamma is positive, the probability distribution of ∆P tends to be positively skewed, when Gamma is negative, it tends to be negatively skewed. VaR and Monte Carlo Simulation

  24. A long call is positively skewed and has a positive gamma VaR and Monte Carlo Simulation

  25. A short call is negatively skewed and has a negatively gamma VaR and Monte Carlo Simulation

  26. Quadratic Model • A positive gamma portfolio tends to have a less heavy left tail than the normal distribution. If we assume the distibution is normal we will tend to overestimate VaR. • A negative gamma portfolio tends to have a heavier left tail than the normal distribution. If we assume it to be normal, we will tend to underestimate VaR. • Combing both delta and gamma to relate ∆Pto ∆xi. VaR and Monte Carlo Simulation

  27. Quadratic Model • Consider a portfolio dependent on a single asset whose price is S. Suppose that the delta of a protfolio is δ and its gamma is γ . ∆P = δ ∆S + (½)γ(∆S)2 Where ∆x = ∆S/S Therefore, ∆P =S δ ∆x + (½)S2γ (∆x)2 • The change in the portfolio value is then calculated form the quadratic equation. VaR and Monte Carlo Simulation

  28. Estimating Daily volatilities • Let σn = the volatility per day of a market variable on day n, as estimated at the end of day n-1. Suppose that the value of the market variable at the end of day i is Si. • The variable ui is defined as the percentage change in the market variable between the end of day i-1 and the end of day I so that ui = (Si – Si-1) / Si-1 Using the most recent m observation, we can find an estimate of σ2n m ^ σ2n =(1 / m-1) ∑ (un-i – u ) 2 i=1 VaR and Monte Carlo Simulation

  29. Estimating Daily volatilities • Let E(u)=0 and m-1=m to simplify calculation, m σ2n =(1/m) ∑ un-i2 i=1 • Given the objective is to monitor the current level of volatility, it is appropriate to give more weight to recent data m σ2n = ∑ xiun-i2 i=1 where xi is the amount of weight given to the observation i days ago. xi>0 and xi <xj , when i > j and m ∑ xi =1 i=1 VaR and Monte Carlo Simulation

  30. The Exponentially Weighted Moving Average (EWMV) Model • Let xi+1 = λxi where λ is a constant between 0 and 1. It follows σ2n = λ σ2nn-1 + (1- λ) un-i2 m σ2n = λm σ2n-m + (1- λ) ∑ λi-1 un-i2 i=1 For large m, λm σ2n-m is sufficiently small to be ignored. So the equation is same as before: m σ2n = ∑ xiun-i2 i=1 with xi = (1- λ) λi-1 The weights for ui decline at rateλ as we move back through time. Each weight is λ times the previous weight. VaR and Monte Carlo Simulation

  31. The Exponentially Weighted Moving Average (EWMV) Model • At any given time, we need to remember only the current estimate of the variance rate and most recent observation on the value of the market variable. • When we get a new observation on the value of the market variable, we calculate a new u2 and use the equation to update our estimate of the variance rate. • The value of λ governs how responsive the estimate of the daily volatility is to the most recent observation on the daily changes. A low value of λ leads to a great deal of weight being given to the un-i2 when σ2n is calculated. VaR and Monte Carlo Simulation

  32. Estimating Correlations • The correlation between 2 variables X and Y can be defined as Cov(X,Y)/ σxσy • Consider 2 market variables U, V. Define ui and vi as the percentage changes in U and V between the end of day i-1 and the end of day i. ui= (Ui - Ui-1) / Ui-1 vi= (Vi - Vi-1) / Vi-1 where Ui and Vi are the value of U and V at the end of day i. We also define: σu,n : Daily volatility of variable U, estimated for day n σv,n : Daily volatility of variable V, estimated for day n covn : Estimate of covariance between daily changes in U and V, calculated on day n. The correlation between U and V on day n is: covn /σu,n σv,n VaR and Monte Carlo Simulation

  33. Estimating Correlations • Using an equal-weighting scheme and assuming that the means of ui and vi = 0, we can estimate the variances of U and V as before: m σ2u,n = (1/m) ∑ un-i2 i=1 m σ2v,n = (1/m) ∑ vn-i2 i=1 m Covn =(1/m) ∑ un-i vn-i i=1 • Using EWMA model as before: Covn = λ Covn-1 + (1- λ) un-ivn-i VaR and Monte Carlo Simulation

  34. Monte Carlo Simulations in SAS

  35. SAS Demo 1 Estimation of Default Risk • Evaluate the default risk of holding companies for whom most, or all, of their income derives from the upstreaming of cash from subsidiaries in the form of dividends. • The likelihood that the company will default is the joint likelihood that a subset of subsidiaries will stop paying dividends, such that the combined value of dividends from still-paying subsidiaries falls below a threshold at which the holding company is able to service its debts. • This situation can be decomposed into a set of conditional default probabilities associated with the complete set of outcomes for dividend payment/non-payment at the subsidiary level, multiplied by the likelihood of each specific outcome. VaR and Monte Carlo Simulation

  36. SAS Demo 1 Estimation of Default Risk • Mathematically, P(Hd) = ∑ P(Hdj) P(Oj) = ∑ P(Hdi|Oid) + ∑ ∑ P(Hdi|{Oi1d,Oi2d) P({Oi1d, Oi2d})+…+ P(Hdi|{O1d,O2d , O7d) P({O1d,O2d ,…, O7d} • There are 127 terms corresponding to the set of all possible outcomes of at least one subsidiary defaulting. • The probability of each subsidiary defaulting can be found by the bond rating. VaR and Monte Carlo Simulation

  37. SAS Demo 1 Estimation of Default Risk • The next required input is a correlation matrix, which describes the potential for correlated outcomes, e.g. more than one subsidiary stopping its dividend payments in the same year. • Generate random standard normal variables that conforms to the predefined correlation structure. • By simulating a very large number of outcomes that are consistent with our inputs, we can simply use the frequency that the company default threshold is exceeded as an estimate of the default likelihood of the company. VaR and Monte Carlo Simulation

  38. SAS Demo 2 VaR Estimation for Credit Risk • VaR calculations for credit risk are frequently directed to longer, multi-period time horizons. • Credit movements are generally defined with respect to a manageable number of subsets of the portfolio as opposed to each individual exposure. • The most common subsetting is by credit quality. • Losses are commonly driven by default rate and recovery rate. VaR and Monte Carlo Simulation

  39. SAS Demo 2 VaR Estimation for Credit Risk • The simulation needs to incorporate both the stochastic nature of default rates year-to-year and the stochastic rating migrations that affect the evolution of the distribution of ratings within the portfolio. VaR and Monte Carlo Simulation

  40. SAS Demo 2 VaR Estimation for Credit Risk • The rating migration depends on the randomly drawn default rate and a preset threshold. If the default rate is greater than the threshold, the rating migration will switch to the one for the high default period. VaR and Monte Carlo Simulation

  41. SAS Demo 3 VaR Estimation for Market Risk • For a portfolio of short time horizon, we need to calculate the distribution of future returns for each asset or asset class in the portfolio and the correlation of price movements across assets or asset classes. • To do this, factor approach, in which individual asset movements are modeled as a combination of response movements to a set of correlated factors and to a a set of independent idiosyncratic movements, is used. VaR and Monte Carlo Simulation

  42. SAS Demo 3 VaR Estimation for Market Risk • The factors are the systematic component of asset price movement, usually from financial and macroeconomic data such as the S&P 500 Index, 90-day T-bill yield, or exchange rate. • The variance and covariance matrix of the factors are calculated and the simulations of the underlying security will be based on the matrix. VaR and Monte Carlo Simulation

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