Value At Risk

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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

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
• Introduction
• Methods
• Easy example
• Harder example:
• Linsmeir and Pearson
• Monte-carlo methods and even harder examples
• Jorion
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:
• Regulatory
Why VaR?
• Risk summary number
• Relatively simple
• Relatively standardized
• Give high level management risk in 1 number
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
• Methods (Reading: Linsmeier and Pearson)
• Historical
• Delta Normal
• Monte-carlo
• Resampling
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
• Portfolio:
• \$100 in the Dow Industrials
• Perfect index tracking
• Problem
• What is the 5% and 1% VaR for 1 day in the future?
• 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 FilesKaplan: Appendix C
• All data in matrix format
• “Mostly” numerical
• Two formats
• Matlab format filename.mat
• ASCII formats
• Space separated
• Excel (csv, common separated)
• Data is in matrix dow
• Save data
• ASCII
• save -ascii filename dow
• Matlab
• save filename dow
Example: Load and plot dow data
• Matlab: pltdow.m
• Dates:
• Matlab datestr function
Back to our problem
• Find 1 day returns, and apply to our 100 portfolio
• Matlab: histdvar.m
Value at Risk: Methods
• Methods (Reading: Linsmeier and Pearson)
• Historical
• Delta Normal
• Monte-carlo
• Resampling
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
• Fatter tails
• Plot Comparison: twodowh.m
Longer Horizon: 10 Days
• Matlab: hist10d.m
Value at Risk: Methods
• Methods (Reading: Linsmeier and Pearson)
• Historical
• Delta Normal
• Monte-carlo
• Resampling
Monte-Carlo VaR
• 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: Methods
• Methods (Reading: Linsmeier and Pearson)
• Historical
• Delta Normal
• Monte-carlo
• Resampling
Resampling (bootstrapping)
• Historical/Monte-carlo hybrid
• Also known as bootstrapping
• data = [5 3 -6 9 0 4 6 ];
• sample(n,data);
• Example
• rsdow.m
• Introduction
• Methods
• Easy example
• Harder example:
• Linsmeir and Pearson
• Monte-carlo methods and even harder examples
• Jorion
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
• 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
• 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)

X = % daily change in exchange rate

X = ?
• Historical
• Normal
• Montecarlo
• Resampled
Historical
• Data: bpday.dat
• Columns
• 1: Matlab date
• 2: \$/BP
• 3: British interest rate (%/year)
• 4: U.S. Interest rate (%/year)
BP Forward: Historical
• Same as for Dow, but trickier valuation
• Matlab: histbpvar1.m
BP Forward: Monte-Carlo
• Matlab: mcbpvar1.m
BP Forward: Resampling
• Matlab: rsbpvar1.m
Harder Problem
• 3 Risk factors
• Exchange rate
• British interest rate
• U.S. interest rate
Daily VaR AssessmentHistorical
• Historical VaR
• Get percentage changes for
• \$/BP: x
• r(BP): y
• r(\$): z
• Generate histograms
• matlab: histbpvar2.m
Daily VaR AssessmentResample
• Historical VaR
• Get percentage changes for
• \$/BP: x
• r(BP): y
• r(\$): z
• Resample from these
• matlab: rsbpvar2.m
Resampling Question:
• Assume independence?
• Resampling technique differs
• matlab: rsbpvar2.m
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
• How do x, y, and z impact f()?
• How do x, y, and z move together?
• Covariance?
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
• 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