Materials for Lecture 20 Read Chapter 9 Lecture 20 CV Stationarity.xlsx Lecture 20 Changing Risk Over Time.xlsx Lecture 20 VAR Analysis.xlsx Lecture 20 Simple VAR.xlsx

Value at Risk Analysis Value at Risk – VAR Originally VAR used to quantify market risk, but considered only 1 source of risk By year 2000 businesses were integrating their risk management systems across the whole enterprise –Focus on analyzing multiple sources of risk including market risk Now market based VAR analyses measure integrated market and credit risk

Value at Risk Model In an intuitive definition “VAR summarizes the worst loss over a target horizon with a given level of confidence” VAR defines the quantile of the projected distribution of gains and losses over the target horizon

Value At Risk Model If c is the selected confidence level, VAR corresponds to the 1-c lower tail of the probability distribution (the quantile).

Value At Risk (VAR) Model To estimate the VAR quantile for a risky business use these steps: 1.Develop a stochastic simulation model of the risky business decision 2.Validate stochastic variables and validate the model 3.Pick a ‘c’ value, say, 5%, so 1-c = 95% 4.Simulate the model and analyze the KOV 5.Calculate the quantile for the c value 6.Calculate VAR = Mean – Quantile at 1-c 7.Report the results

Value At Risk (VAR) Model On selecting the ‘c’ value – literature uses the 95% level This is to say we want to know the value of returns which the business will exceed 95% of the time If simulating 1,000 iterations, the quantile will be the 50 th value, so we can sort the stochastic results and read the 50 th value Or simply use the PDF in Simetar

VAR in Simetar Simulate the KOV and draw a PDF Change the Confidence level to 0.90 for “c” = 5% Edit the title of the chart VAR value is the Lower Quantile

Valuation Models A variation on VAR is the traditional valuation model Valuation models focus on the mean and the variation below the mean

VAR as Risk Capital VAR is the equity capital that should be set aside to cover most all potential losses with a probability of “c” Thus, the VAR is the amount of capital reserves that should be held to meet shortfalls

VAR for Comparing Risky Alternatives Simulate multiple scenarios and calculate VAR for each alternative

VAR for Calculation with Simetar Simulate multiple scenarios and use the SimData statistics and the QUANTILE function =QUANTILE(simulated values, 0.05) Can graph the PDFs and change confidence level

VAR Shortcomings VAR analyses generally used in business gives a false sense of security The literature assumes Normality for the random variables, why? –Normal is easy to simulate –Can easily calculate the Quantile if you know mean and std deviation Q = Mean – (2.035 * Std Dev) The chance of a Black Swan is ignored –This understates the Quantile and the equity capital reserve needed to cover cash flow deficits –Contributed to the Recession

Overcoming VAR Shortcomings Modify the probability distributions for the random variables that affect the business Incorporate low probability events that could cause major harm to the business. Use and EMP distribution and adjust the Probabilities and Sorted Deviates as a Fraction for Black Swan events –Change the F(X) values for the low probability –Change the minimum Xs

Covariance Stationary & Heteroskedasticy Part of validation is to test if the standard deviation for random variables match the historical std dev. –Referred to as “covariance stationary” Simulating outside the historical range causes a problem in that the mean will likely be different from history causing the coefficient of variation, CV Sim, to differ from historical CV Hist : CV Hist = σ H / Ῡ H Not Equal CV Sim = σ H / Ῡ S

Covariance Stationary CV stationarity likely a problem when simulating outside the sample period: –If Mean for X increases, CV declines, which implies less relative risk about the mean as time progresses CV Sim = σ H / Ῡ S –If Mean for X decreases, CV increases, which implies more relative risk about the mean as we get farther out with the forecast CV Sim = σ H / Ῡ S See Chapter 9

CV Stationarity The Normal distribution is covariance stationary BUT it is not CV stationary if the mean differs from historical mean For example: –Historical Mean of 2.74 and Historical Std Dev of 1.84 Assume the deterministic forecast for mean increases over time as: 2.73, 3.00, 3.25, 4.00, 4.50, and 5.00 CV decreases while the std dev is constant Simulation Results Mean Std. Dev CV Min Max

CV Stationarity for Normal Distribution An adjustment to the Std Dev can make the simulation results CV stationary if you are simulating a Normal dist. Calculate a J t+i value for each period (t+i) to simulate as: J t+i = Ῡ t+i / Ῡ history The J t+i value is then used to simulate the random variable in period t+i as: Ỹ t+i = Ῡ t+i + (Std Dev history * J t+i * SND) Ỹ t+i = NORM(Ῡ t+i, Std Dev * J t+i ) The resulting random values for all years t+i have the same CV but different Std Dev than the historical data –This is the result desired when doing multiple year simulations

CV Stationarity and Empirical Distribution Empirical distribution automatically adjusts so the simulated values are CV stationary if the distribution is expressed as deviations from the mean or trend Ỹ t+i = Ῡ t+i * [1 + Empirical(S j, F(S j ), USD)] Simulation Results Mean Std Dev CV Min0.00 Max

Empirical Distribution Validation Empirical distribution automatically adjusts so the simulated values are CV stationary –This is done by adjusting the standard deviation –This poses a problem for validation The correct method for validating Empirical distribution is: –Set up the theoretical mean and standard deviation –Mean = Historical mean * J –Std Dev = Historical mean * J * CV for simulated values / 100 Here is an example for J = 2.0

CV Stationarity and Empirical Distribution

Add Heteroskedasticy to Simulation Sometimes we want the CV to change over time –Change in policy could increase the relative risk –Change in management strategy could change relative risk –Change in technology can change relative risk –Change in market volatility can change relative risk Create an Expansion factor or E t+i value for each year to simulate –E t+i is a fractional adjustment to the relative risk –0.0 results in No risk at all for the random variable –1.0 results in same relative risk (CV) as the historical period –1.5 results in 50% larger CV than historical period –2.0 results in 100% larger CV than historical period Chapter 9

Add Heteroskedasticy to Simulation Simulate 5 years with no risk for the first year, historical risk in year 2, 15% greater risk in year 3, and 25% greater CV in years 4-5 –The E t+i values for years 1-5 are, respectively, 0.0, 1.0, 1.15, 1.25, 1.25 Apply the E t+i expansion factors as follows: –Normal distribution Ỹ t+i = Ῡ t+i + ( Std Dev history * J t+i * E t+i * SND ) Ỹ t+i =NORM (Ῡ t+i, Std Dev history * J t+i * E t+i ) –Empirical Distribution if S i are deviations from mean Ỹ t+i = Ῡ t+i * { 1 + [Empirical(S j, F(S j ), USD) * E t+I ]}

Example of Expansion Factors

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