VaR Methods IEF 217a: Lecture Section 6 Fall 2002 Jorion, Chapter 9 (skim)

Value at Risk: Methods Methods –Historical –Delta Normal –Monte-carlo –Bootstrap

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

Value at Risk: Methods Methods –Historical –Delta Normal –Monte-carlo –Bootstrapping

Bootstrapping Historical/Monte-carlo hybrid We’ve done this already –data = [ ]; –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 $

Mark to Market Value (values in millions)

Risk Factors Exchange rate ($/BP) r(BP): British interest rate r($): US interest rate Assume: –($/BP) = –r(BP) = 6% per year –r($) = 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 $)

Mark to Market Value (1 day future value) X = % daily change in exchange rate

X = ? Historical Delta Normal Monte-carlo Bootstrap

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: Bootstrap Matlab: bbpvar1.m

Harder Problem 3 Risk factors –Exchange rate –British interest rate –U.S. interest rate

3 Risk Factors 1 day ahead value

Daily VaR Assessment Historical Historical VaR Get percentage changes for –$/BP: x –r(BP): y –r($): z Generate histograms matlab: histbpvar2.m

Daily VaR Assessment Bootstrap Historical VaR Get percentage changes for –$/BP: x –r(BP): y –r($): z Bootstrap from these matlab: bbpvar2.m

Bootstrap Question: Assume independence? –Bootstrap technique differs –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))

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)

Value at Risk: Methods Methods –Historical –Delta Normal –Monte-carlo –Bootstrap