VaR Methods
description
Transcript of VaR Methods
![Page 1: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/1.jpg)
VaR Methods
IEF 217a: Lecture Section 6
Fall 2002
Jorion, Chapter 9 (skim)
![Page 2: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/2.jpg)
Value at Risk: Methods
• Methods– Historical– Delta Normal– Monte-carlo– Bootstrap
![Page 3: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/3.jpg)
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
![Page 4: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/4.jpg)
Delta Normal
• Estimate means and standard deviations• Use normal approximations• What if value is a function V(s)?• Need to estimate derivatives (see Jorion)• Computer handles this automatically in
monte-carlo• Also, derivatives are all local
approximations
![Page 5: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/5.jpg)
Monte-Carlo VaR
• Make assumptions about distributions• Simulate random variables• matlab: mcdow.m• Results similar to delta normal• Why bother with monte-carlo?
– Nonnormal distributions– More complicated portfolios and risk measures– Confidence intervals: mcdow2.m
![Page 6: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/6.jpg)
Value at Risk: Methods
• Methods– Historical– Delta Normal– Monte-carlo– Bootstrapping
![Page 7: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/7.jpg)
Bootstrapping
• Historical/Monte-carlo hybrid
• We’ve done this already– data = [5 3 -6 9 0 4 6 ];– sample(data,n);
• Example– bdow.m
![Page 8: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/8.jpg)
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 $
![Page 9: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/9.jpg)
Mark to Market Value(values in millions)
)1($15
)1(10
)$
($rr
BPBP
valBP
![Page 10: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/10.jpg)
Risk Factors
• Exchange rate ($/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
![Page 11: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/11.jpg)
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
![Page 12: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/12.jpg)
Mark to Market Value(current value in millions $)
300,331$
)055.0)360/91(1(
$15
)06.0)360/91(1(
10)5355.1(
BPval
![Page 13: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/13.jpg)
Mark to Market Value(1 day future value)
)055.0)360/90(1(
$15
)06.0)360/90(1(
10)5355.1)(1(
BPxval
X = % daily change in exchange rate
![Page 14: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/14.jpg)
X = ?
• Historical
• Delta Normal
• Monte-carlo
• Bootstrap
![Page 15: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/15.jpg)
Historical
• Data: bpday.dat
• Columns– 1: Matlab date– 2: $/BP– 3: British interest rate (%/year)– 4: U.S. Interest rate (%/year)
![Page 16: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/16.jpg)
BP Forward: Historical
• Same as for Dow, but trickier valuation
• Matlab: histbpvar1.m
![Page 17: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/17.jpg)
BP Forward: Monte-Carlo
• Matlab: mcbpvar1.m
![Page 18: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/18.jpg)
BP Forward: Bootstrap
• Matlab: bbpvar1.m
![Page 19: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/19.jpg)
Harder Problem
• 3 Risk factors– Exchange rate– British interest rate– U.S. interest rate
![Page 20: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/20.jpg)
3 Risk Factors1 day ahead value
)055.0)1)(360/90(1(
$15
)06.0)1)(360/90(1(
10)5355.1)(1(
z
y
BPxval
![Page 21: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/21.jpg)
Daily VaR AssessmentHistorical
• Historical VaR
• Get percentage changes for – $/BP: x– r(BP): y– r($): z
• Generate histograms
• matlab: histbpvar2.m
![Page 22: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/22.jpg)
Daily VaR AssessmentBootstrap
• Historical VaR
• Get percentage changes for – $/BP: x– r(BP): y– r($): z
• Bootstrap from these
• matlab: bbpvar2.m
![Page 23: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/23.jpg)
Bootstrap Question:
• Assume independence?– Bootstrap technique differs– matlab: bbpvar2.m
![Page 24: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/24.jpg)
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))
![Page 25: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/25.jpg)
New Challenges
• How do x, y, and z impact f()?
• How do x, y, and z move together?– Covariance?
![Page 26: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/26.jpg)
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
![Page 27: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/27.jpg)
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)
![Page 28: VaR Methods](https://reader036.fdocuments.in/reader036/viewer/2022062305/568150df550346895dbefe19/html5/thumbnails/28.jpg)
Value at Risk: Methods
• Methods– Historical– Delta Normal– Monte-carlo– Bootstrap