The Subadditivity Of Risk Measure Of VaR

VaR is an important measure of risk which has been extensively recognized in the areaof finance, but it is not a Coherent Measure of Risk by reason of lack of subadditivity.Subadditivity is a basic property for a risk measure, because it ensures the diversificationprinciple of portfolio theory and it is a basic condition for the problem of portfolio decision-making. So people want to prove which asset classes are likely to satisfy the subadditivity.Student-t distribution is known to be a very important heavy tailed distributions in thearea of finance. Many empirical analysis results showed that it is a good approximation forthe distribution of returns of portfolio. Therefore, we choose it to explore VaR subadditivity.Our objective is to study it with positive integral number degrees of freedom of X and Y ,under the conditions that the loss probability larger than zero and smaller than one.Let X and Y are independent and conform to Student-t distribution with degrees offreedom n and m. we prove that:Conclusion 1 VaR_α(X +Y ) = VaR_α(X)+VaR_α(Y ) when n and m both equal to 1, andfor 0 <α< 1;Conclusion 2 VaR_α(X + Y ) = VaR_α(X) + VaR_α(Y ) = 0 whenα= 0.5;We change the other situations to the sup-additive of quantile, then we get conclusion3 by numerical analysis.Conclusion 3 VaR is sub-additive when 0 <α< 0.5, namely, VaR_α(X+Y ) < VaR_α(X)+VaR_α(Y ); But it is violated when 0.5 <α< 1, VaR_α(X+Y ) > VaR_α(X)+VaR_α(Y ).In order to prove conclusion 3, we change it to the one and only one root of function. withthe help of numerical analysis, we prove it and overcome the computed errors through directproof. Then, we make a supplement explain for the conclusion 3 by the way of substitutingapproximately. In this part, we get a good approximationψ(x,n) = forthe probability density function of Student-t distribution with degrees of freedom n, andthis approximation is better than the approximation of the probability density function ofNormal Distribution N(0, n/(n-2))[19] for the small degrees of freedom under the standard of square approximation errors[21]. Lastly, the simulation and empirical results confirm theconclusion 3 highly again.