Value at Risk MGT 4850 Spring 2009 University of Lethbridge

Who can use VaR? Financial Institutions – not to expose themselves to expensive failure (Barings, Daiwa, Société Générale, Amaranth Advisors LLC)Société Générale Regulators – Basel Committee Nonfinancial corporations (cash flow at risk) Asset Managers - funds

Steps in Constructing VaR Current portfolio value Measure the variability per year Set time horizon Set the confidence interval Report the worst loss

Definition The worst expected loss under normal market conditions over a specific time interval at a given confidence level. –Confidence level –Time period Example – daily VaR equal to $1mil at 1% (i.e. only one chance in 100 that a daily loss bigger than 1 mil occurs under normal market conditions)

Portfolio example Value $100mil;mean return 20%; std 30%

Probability of 20 mil loss (9.12%)

PDF The probability density function of the normal distribution is a Gaussian functionprobability density functionGaussian function density function of the "standard" normal distribution:

Probability density vs. Cumulative

PDF and CDF

CALCULATING THE QUANTILES p

B6 B7

Continuous compounding Exponential function e x Natural logarithm - ln(x) Mean value of a portfolio in 1 year: –ln(int. value) + (mean ret.+σ 2 /2)T Now we need “loginv” function for the cutoff point

Inverse of the lognormal cumulative distribution function p.213

VaR for 3 asset problem p. 215

p.216