In probability theory and mathematical finance, the Vasicek distribution is a continuous probability distribution on the unit interval that describes the fraction of a large portfolio of loans lost to default. It is also called the large homogeneous portfolio (LHP) distribution or the asymptotic single risk factor (ASRF) loss distribution. The distribution has two parameters: the probability of default p {\displaystyle p} of an individual borrower, and the correlation ρ {\displaystyle \rho } between the asset values of any two borrowers. The distribution was obtained by the Czech-American mathematician Oldrich Vasicek in two technical notes written for KMV Corporation in 1987 and 1991, and set out in full in a 2002 article in Risk. It arises as a limit: individual defaults are generated by a structural credit risk model in the manner of Robert C. Merton, borrowers are linked by a single normally distributed common factor representing the state of the economy, and the number of loans is allowed to grow without bound. Because defaults are dependent, the central limit theorem does not apply and the loss fraction is not asymptotically normal; instead the law of large numbers applies conditionally on the common factor, and the limiting distribution inherits the shape of the factor's effect on the default rate. The distribution is strongly right-skewed and leptokurtic, with a mean equal to p {\displaystyle p} but with high percentiles far beyond what a normal distribution of the same variance would give. This property is the reason it is used to set capital: it is the distribution underlying the internal ratings-based (IRB) risk-weight formulas of the Basel II and Basel III accords, and it also underpins the large-portfolio approximation used to price tranches of collateralized debt obligations.
Definition A random variable L {\displaystyle L} taking values in ( 0 , 1 ) {\displaystyle (0,1)} has the Vasicek distribution with parameters p {\displaystyle p} and ρ {\displaystyle \rho } , both in ( 0 , 1 ) {\displaystyle (0,1)} , if its cumulative distribution function is
F ( x ; p , ρ ) = Φ ( 1 − ρ Φ − 1 ( x ) − Φ − 1 ( p ) ρ ) , 0 < x < 1 , {\displaystyle F(x;p,\rho )=\Phi \left({\frac {{\sqrt {1-\rho }}\,\Phi ^{-1}(x)-\Phi ^{-1}(p)}{\sqrt {\rho }}}\right),\qquad 0<x<1,}
where Φ {\displaystyle \Phi } is the cumulative distribution function of the standard normal distribution and Φ − 1 {\displaystyle \Phi ^{-1}} its inverse, the probit function. Vasicek's own papers write N {\displaystyle N} for this function. Differentiating gives the probability density function
f ( x ; p , ρ ) = 1 − ρ ρ exp ( 1 2 [ Φ − 1 ( x ) ] 2 − 1 2 ρ [ 1 − ρ Φ − 1 ( x ) − Φ − 1 ( p ) ] 2 ) . {\displaystyle f(x;p,\rho )={\sqrt {\frac {1-\rho }{\rho }}}\,\exp \left({\frac {1}{2}}\left[\Phi ^{-1}(x)\right]^{2}-{\frac {1}{2\rho }}\left[{\sqrt {1-\rho }}\,\Phi ^{-1}(x)-\Phi ^{-1}(p)\right]^{2}\right).}
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