# Practice Problem Set 10 – value at risk and tail value at risk

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: value-at-risk (VaR) and tail-value-at-risk (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 10-A Losses follow a paralogistic distribution with shape parameter $\alpha=2$ and scale parameter $\theta=1500$. Determine the VaR at the security level 99%.
 Practice Problem 10-B 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 10-C For a certain line of business for an insurer, the annual losses follow a lognormal distribution with parameters $\mu=5.5$ and $\sigma=1.2$. Evaluate the value-at-risk and the tail-value-at-risk at the 95% security level.
 Practice Problem 10-D Annual losses follow a normal distribution with mean 1000 and variance 250,000. Compute the tail-value-risk at the 95% security level.
 Practice Problem 10-E An insurance company models its liability insurance business using a Pareto distribution with shape parameter $\alpha=1.5$ and scale parameter $\theta=5000$. Evaluate the value-at-risk and the tail-value-at-risk at the 99.5% security level.
 Practice Problem 10-F Losses follow an inverse exponential distribution with parameter $\theta=2000$. Calculate the value-at-risk at the 99% security level.
 Practice Problem 10-G 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 value-at-risk and the tail-value-at-risk at the 95% security level.
 Practice Problem 10-H Losses follow a mixture of two Pareto distributions with equal weights where one Pareto distribution has shape parameter $\alpha=1.2$ and scale parameter $\theta=5000$ and the other has shape parameter $\alpha=2.4$ and scale parameter $\theta=5000$. Evaluate the value-at-risk and the tail-value-at-risk at the 99% security level.
 Practice Problem 10-I Losses follow a Weibull distribution with parameters $\tau=2$ and $\theta=1000$. Determine the value-at-risk at the security level 99.5%.
 Practice Problem 10-J Losses follow an inverse Pareto distribution with parameters $\tau=2.5$ and $\theta=5000$. Determine the value-at-risk at the security level 99%.

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10-A
• VaR = 4500
10-B
• VaR = 23025.85093
10-C
• VaR = 1761.639168
• TVaR = 2248.088854
10-D
• VaR = 1882.5
• TVaR = 2031.108111
10-E
• VaR = 165997.5947
• TVaR = 507992.784
10-F
• VaR = 198998.3249
10-G
• VaR = 47.80473823
• TVaR = 66.96553606
10-H
• VaR = 127375.8029
• TVaR = 257568.7795
10-I
• VaR = 2301.807413
10-J
• VaR = 1241241.206

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