Materials for Lecture 17

1
Materials for Lecture 17

• Read Chapter 9
• Lecture 17 CV Stationarity.xls
• Lecture 17 Changing Risk Over Time.xls
• Lecture 17 VAR Analysis.xls
• Lecture 17 Simple VAR.xls

2
Value at Risk Analysis

• Value at Risk VAR
• Originally VAR was used to quantify market risk
  considered only 1 risk
• Now the focus is on analyzing multiple sources of
  risk including market risk
• By 2000 businesses were integrating their risk
  management systems across the whole enterprise
• Now market based VAR analyses extended to measure
  integrated mkt. and credit risk

3
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

4
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).

5
Value At Risk Model

• To estimate the VAR quantile for a risky business
  use these steps
• Develop a stochastic simulation model of the
  risky business decision
• Validate stochastic variables and validate the
  model
• Pick a c value, say, 5, so 1-c 95
• Simulate the model and analyze the KOV
• Calculate the quantile for the c value
• Calculate VAR Mean Quantile at 1-c
• Report the results

6
Value At Risk Model

• On selecting the c value generally talk about
  the 95 level
• This is to say we want to know the value of
  returns which we will exceed 95 of the time
• IF we are simulating 1000 iterations, the
  quantile will be the 50th value after we sort the
  stochastic results

7
VAR in Simetar

• Simulate the KOV and draw a PDF
• Change the Confidence level to 0.90
• Edit the title of the chart
• VAR value is the Lower Quantile

8
Valuation Models

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

9
VAR as Risk Capital

• VAR can be 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 available to meet shortfalls

10
VAR for Comparing Risky Alternatives

• Simulate multiple scenarios and calculate VAR for
  each alternative

11
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 Qunatile 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 needed to cover cash flow deficits

12
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
• Change the F(X) values for the low probability
• Change the minimum Xs

13
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
  (CVSim) to differ from historical CVHist
• CVHist sH / ?H Not Equal CVSim sH /
  ?S

14
Covariance Stationary

• CV stationarity can be a problem when simulating
  outside the sample period
• If Mean for X increases, CV declines, which
  implies less relative risk about the future as
  time progresses CVSim sH / ?S
• If Mean for X decreases, CV increases, which
  implies more relative risk as we get farther out
  with the forecast CVSim sH / ?S
• Chapter 9

15
CV Stationarity

• The Normal distribution is covariance stationary
  BUT it is not CV stationary if the mean changes
  from history
• 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

16
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 Jti value for each period (ti) to
  simulate as
• Jti ?ti / ?history
• The Jti value is then used to simulate the
  random variable in period ti as
• ?ti ?ti (Std Devhistory Jti SND)
• The resulting random values for all years ti
  have the same CV but different std dev than the
  historical data
• This is the result desired when doing multiple
  year simulations

17
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
• ?ti ?ti 1 Empirical(Sj , F(Sj),
  USD)

18
CV Stationarity and Empirical Distribution
19
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 volitility can change relative
  risk
• Create an Expansion factor or Eti value for each
  year to simulate
• Eti is a fractional adjustment to the relative
  risk for a random variable
• 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

20
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 Eti values for years 1-5 are, respectively,
  
• 0.0, 1.0, 1.15, 1.25, 1.25
• Apply the Eti expansion factors as follows
• Normal distribution
• ?ti ?ti (Std Devhistory Jti Eti
  SND)
• Empirical Distribution if Si are deviations from
  mean
• ?ti ?ti 1 Empirical(Sj , F(Sj),
  USD) EtI

21
Example of Expansion Factors