Introduction to Value At Risk VaR

1
Introduction to Value At Risk VaR

• FIN285 Lecture 5
• Fall 2003
• Readings Dowd Chapter 2

2
Value-at-Risk (VaR)

• Probabilistic worst case
• Almost perfect storm
• 1/100 year flood level

3
VaR Advantages

• Risk -gt Single number
• Firm wide summary
• Handles futures, options, and other complications
• Relatively model free
• Easy to explain
• Deviations from normal distributions

4
Value at Risk (VaR)History

• Financial firms in the late 80s used it for
  their trading portfolios
• J. P. Morgan RiskMetrics, 1994
• Currently becoming
• Wide spread risk summary
• Regulatory

5
Value at Risk Methods

• Methods
• Delta Normal
• Historical
• Monte-carlo
• Bootstrap

6
Outline

• Definitions
• Parameters
• Regulation
• Limitations
• Expected tail loss (ETL)

7
Defining VaR

• Mark to market (value portfolio)
• 100
• Identify and measure risk
• Normal mean 0, std. 10 over 1 month
• Set time horizon of interest
• 1 month
• Set confidence level 95

8
Portfolio value today 100, Normal returns (mean
0, std 10 per month), time horizon 1 month,
95 VaR 16.5
0.05 Percentile 83.5
9
VaR Definitions in Words

• Measure initial portfolio value (100)
• For 95 confidence level, find 5th percentile
  level of portfolio values (83.5)
• The amount of this loss (16.5) is the VaR
• What does this say?
• With probability 0.95 your losses will be less
  than 16.5

10
Increasing the Confidence Level

• Increase level to 99
• Portfolio value 76.5
• VaR 100-76.5 23.5
• With probability 0.99, your losses will be less
  than 23.5
• Increasing confidence level, increases VaR

11
Choosing VaR Parameters

• Holding period
• Risk environment (depends)
• Portfolio constancy/liquidity (short)
• Data quantity (short)
• Confidence level
• Estimate of extreme tails (high)
• Min return
• Data quantity (low)

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Outline

• Definitions
• Parameters
• Regulation
• Limitations
• Expected tail loss (ETL)

13
Regulation

• International bank capital requirements
• Basle accord (1996 amendment)
• Internal models
• Capital requirement k(Average VaR over the
  last 60 days)k is between 3 and 4
• VaR parameters
• 99 confidence
• 10 day holding period

14
More Thoughts on Regulation and VaR (see box 2.2)

• Somewhat consistent approach (but arbitrary)
• Variability across firms (trusting banks to get
  risk management right)
• Might banks be able to figure out how to get
  around this regulation?
• Does this make capital markets more or less
  stable?

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Outline

• Definitions
• Parameters
• Regulation
• Limitations
• Expected tail loss (ETL)

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VaR Limitations

• Uniformative about extreme tails
• Bad portfolio decisions
• Might add high expected return, but high loss
  with low probability securities
• Might discourage diversification
• Basically (VaR/Expect return) calculations still
  not well understood
• VaR is not Sub-additive

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Not Subadditive

• Sub-additive risk
• VaR doesn't meet this requirement
• Mean/variance does

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Why is This a Problem?

• Simple adding of risks underestimates risk
• Over aggressive behavior
• Financial firm might do better (in terms of
  capital requirement regulation) if it split
  itself up
• Also, individual investors might split up their
  trading accounts to get better margin requirement
  deals

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Outline

• Definitions
• Parameters
• Regulation
• Limitations
• Expected tail loss (ETL)

20
Expected Tail Loss

• Expected loss, given that loss is greater than VaR

21
Portfolio value today 100, Normal returns (mean
0, std 10 per month), time horizon 1 month,
95 VaR 16.5, Expected Tail 79.2, ETL 20.8
0.05 Percentile 83.5
22
Matlab Code for Expected Tail Loss
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ETL versus VaR

• Advantages
• Better information on possible tail losses
• Some better properties (sub-additive)
• Disadvantages
• Sensitive to outliers
• Difficult to estimate (for high confidence
  numbers)
• More difficult to explain

24
Normal Distributions

• Many VaR calculations can be done using tables
• Find percentile value for confidence level for
  normal, mean 0, std 1 using standard tables
• For 0.05 level, this is 1.64
• Critical return (R) std(percentile value)
  0.1(-1.64) -0.164
• W W(1R) 100(1-0.164) 83.6
• VaR Loss W W 100-83.6 16.4

25
Normal DistributionsNonzero Mean (Absolute VaR)

• Assume std 0.10, mean 0.05
• Critical return (R)
• mean std(percentile value)
• 0.050.1(-1.64) -0.114
• W W(1R) 100(1-0.114) 88.6
• VaR Loss W W 100-88.6 11.4
• This is known as Absolute VaR
• Absolute dollar loss

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Normal DistributionsNonzero Mean (Relative VaR)

• Assume std 0.10, mean 0.05
• Critical return (R)
• mean std(percentile value)
• 0.050.1(-1.64) -0.114
• W W(1R) 100(1-0.114) 88.6
• Relative VaR is measured relative to expected
  wealth in the future
• VaR Loss E(W) W 100(1.05)-88.6 16.4
• This is known as Relative VaR

27
Absolute versus Relative VaR

• Absolute
• Measure total loss possible against todays
  wealth
• Relative
• Measure loss against expected increases in
  todays wealth.
• If portfolio is expected to grow by 10 percent,
  measure loss relative to this growth
• If means are positive, then relative VaR will be
  larger (more conservative)
• If means are near zero (short horizons) then they
  are the same

28
Normal Distributions in Practice

• Assume returns are normal
• Estimate mean and std using data
• Then get VaR using tables or monte-carlo

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Historical VaR

• Use past data to build histograms
• Method
• Gather historical prices/returns
• Use this data to predict possible moves in the
  portfolio over desired horizon of interest

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Easy Example

• Portfolio
• 100 in the Dow Industrials
• Perfect index tracking
• Problem
• What is the 5 and 1 VaR for 1 day in the
  future?

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DataDow Industrials

• dow.dat (data section on the web site)
• File
• Column 1 Matlab date (days past 0/0/0)
• Column 2 Dow Level
• Column 3 NYSE Trading Volume (1000s of shares)

32
Matlab and Data Files

• All data in matrix format
• Mostly numerical
• Two formats
• Matlab format filename.mat
• ASCII formats
• Space separated
• Excel (csv, common separated)

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Loading and Saving

• Load data
• load dow.dat
• Data is in matrix dow
• Save data
• ASCII
• save -ascii filename dow
• Matlab
• save filename dow

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Example Load and plot dow data

• Matlab pltdow.m
• Dates
• Matlab datestr function

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Back to our problem

• Find 1 day returns, and apply to our 100
  portfolio
• Matlab dnormdvar.m
• Methods used
• Delta normal (tables)
• Historical
• Note difference

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Outline

• Computing VaR
• Interpreting VaR
• Time Scaling
• Regulation and VaR
• Jorion 3, 5.2.5-5.2.6
• Estimation errors

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Interpreting VAR

• Benchmark measure
• Compare risks across markets in company
• Flag risks appearing over time
• Potential loss measure
• Worst loss
• Equity capital

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Outline

• Computing VaR
• Interpreting VaR
• Time Scaling
• Regulation and VaR
• Jorion 3, 5.2.5-5.2.6
• Estimation errors

39
Time Scaling

• VaR calculations can be made beyond 1 period in
  the future
• Time scaling
• Analytic
• Monte-carlo

40
Scale Factors and Analytics (Jorion)

• Reminder
• Let r(t) be a random return (independent over
  time)

41
Scale Factors and Analytics
42
Scaling in Words

• Mean scales with T
• Std. scales with sqrt(T)
• Reminder needs independence

43
Three Methods

• Approximate scaling
• Exact (log normal) scaling
• Bootstrap/monte-carlo

44
Approximate

• Assume that long horizon returns are the sum of
  the short horizon returns

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Computing VaR

• Mark to market (value portfolio)
• 100
• Identify and measure risk factor variability
• Normal mean 0, std. 0.1 over 1 month
• Set time horizon
• 6 months (before 1 month)
• Std sqrt(6)0.10.245
• Set confidence level
• 5

46
6 Month VaR

• Many VaR calculations can be done using tables
• Find percentile value for confidence level for
  normal, mean 0, std 1 using standard tables
• For 0.05 level, this is 1.64
• Critical return (R) std(percentile value)
  sqrt(6)0.1(-1.64) -0.40
• W W(1R) 100(1-0.40) 60
• VaR Loss W W 100-60 40

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Exact Methods

• Assume that prices are a geometric random walk
  with normal increments

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Value of Portfolio at T
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Critical Return

• Let R be the alpha critical value for the T
  period log return
• Now define the future wealth level at the alpha
  level by

50
Computing VaR

• Mark to market (value portfolio)
• 100
• Identify and measure risk factor variability.
  Assume log returns are distributed
• Normal mean 0, std. 0.1 over 1 month
• Set time horizon
• 6 months (before 1 month)
• Std sqrt(6)0.10.245
• Set confidence level
• 5

51
6 Month VaR Exact(approximate numbers)

• Find percentile value for confidence level for
  normal, mean 0, std 1 using standard tables
• For 0.05 level, this is 1.64
• Critical return (R) std(percentile value)
  sqrt(6)0.1(-1.64) -0.40
• W W(1R) 100exp(-0.40) 67 (60)
• VaR Loss W W 100-67 33 (40)

52
Bootstrap Methods

• If the 1 period return distribution is unknown,
  and you dont want to hope the central limit
  theorem is working at T periods, then a bootstrap
  might be a good way to go
• Resample 1 period returns, T at a time, and build
  a histogram for the T period returns
• Use this to find the alpha critical value for
  wealth

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Examples From Data

• Matlab
• hist10d.m
• hist10dln.m

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Outline

• Computing VaR
• Interpreting VaR
• Time Scaling
• Regulation and VaR
• Jorion 3, 5.2.5-5.2.6
• Estimation errors

55
Regulation and Basel Capital Accord

• 1988
• Minimum capital requirements
• Agreed minimum for signing central banks
• Why?
• Avoid global systemic risk

56
The Early Basel Formulas

• Capital back must be at least 8 of risk
  weighted assets
• Risk weighting increases arbitrarily across asset
  classes

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Criticism

• Ignores risk mitigation (hedging) methods
• Ignores diversification effects
• Ignores term structure effects
• Too few risk classes
• Ignores market risk

58
Standardized Model (1993)

• More classes
• New formulaic risk measures
• Problems
• Still arbitrary formulas and classes
• Misses diversification effects
• Ignores internal risk management methods

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Internal Models Approach1995

• Radical Change
• Core component (VaR)
• 10 trading day VaR
• 99 percent confidence
• Max ( last 60 days VaR, todays VaR)
• Use at least 1 year of historical data
• Scale factor (3 or more)
• Plus factor if banks numbers look unreliable

60
Scale Adjustment

• Find 99 quantile return for 10 day period
• R
• Adjust this by a factor of 3
• 3R
• Why 3?
• Trying to eliminate failures

61
An Example Using the Delta-Normal Approximation

• Estimate distribution of 1 day returns
• Normal, mean 0, std 0.01
• Find the 10 day std.
• sqrt(10)0.01 0.032
• Mean 010 0
• Get the 99 return level from tables
• 2.330.032 0.075

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An Example Using the Delta-Normal Approximation

• Get the 99 return level from tables
• 2.330.032 -0.075
• Critical R (k)0.075 (3)-0.075 -0.225
• 22.5 loss Basel requires cushion for
• 100 portfolio -gt Capital required 22.5
• All is standard VaR except for k

63
Outline

• Computing VaR
• Interpreting VaR
• Time Scaling
• Regulation and VaR
• Jorion 3, 5.2.5-5.2.6
• Estimation errors

64
Estimation Errors

• Value at Risk is only an estimate
• What are its confidence bands?
• Methods
• Analytics (Jorion 5)
• Monte-carlo

65
Precision of Mean and Std Estimators (Jorion page
123)
66
Quantile Std. Errors
67
Normal Quantile Estimates
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Precision

• Note mean more precise than std
• Can use as input into VaR estimates to get
  confidence bounds
• We wont do this.
• Monte-carlo methods
• mcdow2.m

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Outline

• Computing VaR
• Interpreting VaR
• Time Scaling
• Regulation and VaR
• Jorion 3, 5.2.5-5.2.6
• Estimation errors