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Value At Risk. IEF 217a: Lecture Section 5 Fall 2002 Jorion Chapter 5. Outline. Computing VaR Interpreting VaR Time Scaling Regulation and VaR Jorion 3, 5.2.5-5.2.6 Estimation errors. VaR Roadmap. Introduction Methods Reading: Linsmeier and Pearson Easy example Harder example:

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value at risk

Value At Risk

IEF 217a: Lecture Section 5

Fall 2002

Jorion Chapter 5

outline
Outline
  • Computing VaR
  • Interpreting VaR
  • Time Scaling
  • Regulation and VaR
    • Jorion 3, 5.2.5-5.2.6
  • Estimation errors
var roadmap
VaR Roadmap
  • Introduction
  • Methods
    • Reading: Linsmeier and Pearson
  • Easy example
  • Harder example:
    • Linsmeir and Pearson
  • Monte-carlo methods and even harder examples
    • Jorion
value at risk var history
Value at Risk (VaR)History
  • Financial firms in the late 80’s used it for their trading portfolios
  • J. P. Morgan RiskMetrics, 1994
  • Currently becoming:
    • Wide spread risk summary
    • Regulatory
why var
Why VaR?
  • Risk summary number
    • Relatively simple
    • Relatively standardized
  • Give high level management risk in 1 number
what is var
What is VaR?
  • Would like to know maximum amount you stand to lose in portfolio
  • However, the max might too large
  • 5% VaR is the amount that you would lose such that 5% of outcomes will lose more
value at risk methods
Value at Risk: Methods
  • Methods (Reading: Linsmeier and Pearson)
    • Historical
    • Delta Normal
    • Monte-carlo
    • Resampling
historical
Historical
  • 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
easy example
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?
data dow industrials
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 (1000’s of shares)
matlab and data files kaplan appendix c
Matlab and Data FilesKaplan: Appendix C
  • All data in matrix format
  • “Mostly” numerical
  • Two formats
    • Matlab format filename.mat
    • ASCII formats
      • Space separated
      • Excel (csv, common separated)
loading and saving
Loading and Saving
  • Load data
    • “load dow.dat”
    • Data is in matrix dow
  • Save data
    • ASCII
      • save -ascii filename dow
    • Matlab
      • save filename dow
example load and plot dow data
Example: Load and plot dow data
  • Matlab: pltdow.m
  • Dates:
    • Matlab datestr function
back to our problem
Back to our problem
  • Find 1 day returns, and apply to our 100 portfolio
  • Matlab: histdvar.m
value at risk methods1
Value at Risk: Methods
  • Methods (Reading: Linsmeier and Pearson)
    • Historical
    • Delta Normal
    • Monte-carlo
    • Resampling
delta normal
Delta Normal
  • Make key assumptions to get analytics
    • Normality
    • Linearization
  • Dow example:
    • Assume returns normal mean = m, std = s
      • 5% return = -1.64*s + m
      • 1% return = -2.32*s + m
    • Use these returns to find VaR
    • matlab: dnormdvar.m
compare with historical
Compare With Historical
  • Fatter tails
  • Plot Comparison: twodowh.m
longer horizon 10 days
Longer Horizon: 10 Days
  • Matlab: hist10d.m
value at risk methods2
Value at Risk: Methods
  • Methods (Reading: Linsmeier and Pearson)
    • Historical
    • Delta Normal
    • Monte-carlo
    • Resampling
monte carlo var
Monte-Carlo VaR
  • Make assumptions about distributions
  • Simulate random variables
  • matlab: mcdow.m
  • Results similar to delta normal
  • Why?
    • More complicated portfolios and risk measures
    • Confidence intervals: mcdow2.m
value at risk methods3
Value at Risk: Methods
  • Methods (Reading: Linsmeier and Pearson)
    • Historical
    • Delta Normal
    • Monte-carlo
    • Resampling
resampling bootstrapping
Resampling (bootstrapping)
  • Historical/Monte-carlo hybrid
    • Also known as bootstrapping
  • We’ve done this already
    • data = [5 3 -6 9 0 4 6 ];
    • sample(n,data);
  • Example
    • rsdow.m
var roadmap1
VaR Roadmap
  • Introduction
  • Methods
    • Reading: Linsmeier and Pearson
  • Easy example
  • Harder example:
    • Linsmeir and Pearson
  • Monte-carlo methods and even harder examples
    • Jorion
harder example
Harder Example
  • Foreign currency forward contract
  • 91 day forward
  • 91 days in the future
    • Firm receives 10 million BP (British Pounds)
    • Delivers 15 million US $
risk factors
Risk Factors
  • Exchange rate ($/BP)
  • r(BP): British interest rate
  • r($): US interest rate
  • Assume:
    • ($/BP) = 1.5355
    • r(BP) = 6% per year
    • r($) = 5.5% per year
    • Effective interest rate = (days to maturity/360)r
find the 5 1 day var
Find the 5%, 1 Day VaR
  • Very easy solution
    • Assume the interest rates are constant
  • Analyze VaR from changes in the exchange rate price on the portfolio
mark to market value 1 day future value
Mark to Market Value(1 day future value)

X = % daily change in exchange rate

slide31
X = ?
  • Historical
  • Normal
  • Montecarlo
  • Resampled
historical1
Historical
  • Data: bpday.dat
  • Columns
    • 1: Matlab date
    • 2: $/BP
    • 3: British interest rate (%/year)
    • 4: U.S. Interest rate (%/year)
bp forward historical
BP Forward: Historical
  • Same as for Dow, but trickier valuation
  • Matlab: histbpvar1.m
bp forward monte carlo
BP Forward: Monte-Carlo
  • Matlab: mcbpvar1.m
bp forward resampling
BP Forward: Resampling
  • Matlab: rsbpvar1.m
harder problem
Harder Problem
  • 3 Risk factors
    • Exchange rate
    • British interest rate
    • U.S. interest rate
daily var assessment historical
Daily VaR AssessmentHistorical
  • Historical VaR
  • Get percentage changes for
    • $/BP: x
    • r(BP): y
    • r($): z
  • Generate histograms
  • matlab: histbpvar2.m
daily var assessment resample
Daily VaR AssessmentResample
  • Historical VaR
  • Get percentage changes for
    • $/BP: x
    • r(BP): y
    • r($): z
  • Resample from these
  • matlab: rsbpvar2.m
resampling question
Resampling Question:
  • Assume independence?
    • Resampling technique differs
    • matlab: rsbpvar2.m
risk factors and multivariate problems
Risk Factors and Multivariate Problems
  • Value = f(x, y, z)
  • Assume random process for x, y, and z
  • Value(t+1) = f(x(t+1), y(t+1), z(t+1))
new challenges
New Challenges
  • How do x, y, and z impact f()?
  • How do x, y, and z move together?
    • Covariance?
delta normal issues
Delta Normal Issues
  • Life is more difficult for the pure table based delta normal method
  • It is now involves
    • Assume normal changes in x, y, z
    • Find linear approximations to f()
      • This involves partial derivatives which are often labeled with the Greek letter “delta”
      • This is where “delta normal” comes from
  • We will not cover this
monte carlo method
Monte-carlo Method
  • Don’t need approximations for f()
  • Still need to know properties of x, y, z
    • Assume joint normal
    • Need covariance matrix
      • ie var(x), var(y), var(z) and
      • cov(x,y), cov(x,z), cov(y,z)
  • Next section, and Jorion