risk factors. The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio at the confidence level is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss, where is the inverse of the normal cumulative distribution function at probability level , are the individual positions returns variances and is the linear correlation measure between the two individual positions returns. Since variance is always positive, Thus the Gaussian VaR is subadditive for any value of and, in particular, it equals the sum of the individual risk exposures when which is the case of no diversification effects on portfolio risk. Thermodynamics Subadditivity occurs in the thermodynamic properties of non-ideal solutions and mixtures like the excess molar volume and heat of mixing or excess enthalpy. Combinatorics on words A factorial language is one where if a word is in , then all factors of that word are also in . In combinatorics on words, a common problem is to determine the number of length- words in a factorial language. Clearly , so is subadditive, and hence Fekete's lemma can be used to estimate the growth of . For every , sample two strings of length uniformly at random on the alphabet . The expected length of the longest common subsequence is a super-additive function of , and thus there exists a number , such that the expected length grows as . By checking the case with , we easily have . The exact value of even , however, is only known to be between 0.788 and 0.827. See also Notes References György Pólya and Gábor Szegő. "Problems and theorems in analysis, volume 1". Springer-Verlag, New York (1976). . Einar Hille. "Functional analysis and semi-groups". American Mathematical Society, New York (1948). N.H. Bingham, A.J. Ostaszewski. "Generic subadditive functions." Proceedings of American Mathematical Society, vol. 136, no. 12 (2008), pp. 4257–4266. External links Mathematical analysis Sequences and series Types of functions