A Study on Risk Measurements Exceeding VaR: TCE, CVaR and ES

Because of VaR's limitations, three new risk measurements exceeding VaR were put forward, which are TCE, CVaR and ES. However, there is confusion about definitions of TCE, CVaR and ES and relationship between them. This article redefines TCE, CVaR and ES using terms system of return distribution, significance level and upper-quantile, and studies the relationship between them based on new definitions. The conclusion is that CVaR and ES are equivalent in any cases; TCE, CVaR and ES are equivalent when distributions of returns on risky securities are continuous.