var methods ief 217a: lecture section 6 fall 2002 jorion, chapter 9 (skim)
Post on 22-Dec-2015
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TRANSCRIPT
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
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
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
Bootstrapping
• Historical/Monte-carlo hybrid
• We’ve done this already– data = [5 3 -6 9 0 4 6 ];– sample(data,n);
• Example– bdow.m
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(current value in millions $)
300,331$
)055.0)360/91(1(
$15
)06.0)360/91(1(
10)5355.1(
BPval
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
Historical
• Data: bpday.dat
• Columns– 1: Matlab date– 2: $/BP– 3: British interest rate (%/year)– 4: U.S. Interest rate (%/year)
Daily VaR AssessmentHistorical
• Historical VaR
• Get percentage changes for – $/BP: x– r(BP): y– r($): z
• Generate histograms
• matlab: histbpvar2.m
Daily VaR AssessmentBootstrap
• Historical VaR
• Get percentage changes for – $/BP: x– r(BP): y– r($): z
• Bootstrap from these
• matlab: bbpvar2.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))
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)