In actuarial applications, an important focus is on developing loss distributions for insurance products. It is also critical to employ risk measures to evaluate the exposure to risk. This post provides practice problems on two risk measures that are useful from an actuarial perspective. They are: valueatrisk (VaR) and tailvalueatrisk (TVaR).
Practice problems in this post are to reinforce the concepts of VaR and TVaR discussed in this blog post in a companion blog.
Most of the practice problems refer to parametric distributions highlighted in a catalog for continuous parametric models.
Practice Problem 10A 
Losses follow a paralogistic distribution with shape parameter and scale parameter .
Determine the VaR at the security level 99%. 
Practice Problem 10B 
Annual aggregate losses for an insurer follow an exponential distribution with mean 5,000. Evaluate VaR and TVaR for the aggregate losses at the 99% security level. 
Practice Problem 10C 
For a certain line of business for an insurer, the annual losses follow a lognormal distribution with parameters and . Evaluate the valueatrisk and the tailvalueatrisk at the 95% security level. 
Practice Problem 10D 
Annual losses follow a normal distribution with mean 1000 and variance 250,000. Compute the tailvaluerisk at the 95% security level. 
Practice Problem 10E 
An insurance company models its liability insurance business using a Pareto distribution with shape parameter and scale parameter . Evaluate the valueatrisk and the tailvalueatrisk at the 99.5% security level. 
Practice Problem 10F 
Losses follow an inverse exponential distribution with parameter . Calculate the valueatrisk at the 99% security level. 
Practice Problem 10G 
Losses follow a mixture of two exponential distributions with equal weights where one exponential distribution has mean 10 and the other has mean 20. Evaluate the valueatrisk and the tailvalueatrisk at the 95% security level. 
Practice Problem 10H 
Losses follow a mixture of two Pareto distributions with equal weights where one Pareto distribution has shape parameter and scale parameter and the other has shape parameter and scale parameter . Evaluate the valueatrisk and the tailvalueatrisk at the 99% security level. 
Practice Problem 10I 
Losses follow a Weibull distribution with parameters and . Determine the valueatrisk at the security level 99.5%. 
Practice Problem 10J 
Losses follow an inverse Pareto distribution with parameters and . Determine the valueatrisk at the security level 99%. 
Problem  Answer 

10A 

10B 

10C 

10D 

10E 

10F 

10G 

10H 

10I 

10J 

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Daniel Ma Mathematics
Actuarial exam
2018 – Dan Ma