CAS Exam 9Financial Risk & Rate of Return

The risk-free rate is 3%, the expected market risk premium is 6%, and a stock's beta is 1.2. Calculate the CAPM expected return.

An insurer writes premium equal to twice its surplus and holds reserves equal to three times surplus. Its combined ratio is 98% and it earns 4% on all assets (surplus plus reserves). Using Ferrari's framework and ignoring taxes, calculate the pre-tax return on surplus.

Losses are normally distributed with mean 0 and standard deviation 10. Calculate $TVaR_{99\%}$. (Use $z_{0.99}=2.326$, $\phi(2.326)=0.02665$.)

Assets of 130 have modified duration 4.0 and liabilities of 100 have modified duration 6.0. Calculate the modified duration of surplus.

Which property of a coherent risk measure does Value-at-Risk fail in general?

An insurer is deciding whether to allocate capital by (i) proportional VaR, (ii) marginal TVaR (Euler allocation), or (iii) Kreps' marginal standard deviation. For each, describe the allocation rule, one strength and one weakness, and state which is most consistent with a coherent risk measure.

A portfolio has an expected return of 11%, a risk-free rate of 3%, and a standard deviation of 16%. Calculate the portfolio's Sharpe ratio.

Aggregate losses $L$ are lognormal with $\ln L\sim N(10,\,0.5^2)$. Calculate $VaR_{95\%}(L)$. (Use $z_{0.95}=1.645$.)

An insurer's assets of 200 have modified duration 3.5 and its liabilities of 150 have modified duration 5.0. Calculate the modified duration of surplus.

An insurer writes premium equal to 1.5 times surplus and holds reserves equal to 2.5 times surplus. Its combined ratio is 100% (zero underwriting profit) and its investment yield on invested assets (surplus plus reserves) is 5%. Using Ferrari's total-return framework, calculate the pre-tax return on surplus.

An insurer is comparing VaR and TVaR at the 99% level as capital metrics for a catastrophe-exposed line. (a) Define each measure. (b) Explain, with an example structure of losses, why VaR can fail subadditivity while TVaR does not. (c) State one operational reason a regulator or rating agency might still prefer VaR-based metrics despite this theoretical weakness.