SOA FAMFundamentals of Actuarial Mathematics

Mortality follows a constant force $\mu=0.02$. Calculate ${}_{10}p_x$.

With constant force of mortality $\mu=0.02$ and force of interest $\delta=0.05$, calculate the continuous whole-life net premium rate $\bar P(\bar A_x)$.

Aggregate claims $S$ are compound Poisson with $\lambda=3$. Severity has mean 2,000 and second moment 6,000,000. Calculate the standard deviation of $S$.

Losses follow a Pareto distribution with $\alpha=3$ and $\theta=2{,}000$. Calculate $E[X\wedge 1{,}000]$.

For a whole-life policy with death benefit 1,000 payable at the end of the year of death, the policy value at time 10 is 100, the annual net premium is 20, $i=5\%$ and $q_{x+10}=0.01$. Calculate the policy value at time 11.

A policy has an ordinary deductible of 500, a policy limit (maximum covered loss) of 5,000 and 80% coinsurance. Losses are exponential with mean 2,000. Derive the expected payment per loss and per payment.

Five i.i.d. exponential claim amounts sum to 800. Calculate the maximum likelihood estimate of the mean $\theta$.

Under limited-fluctuation (full) credibility for claim counts, the standard requires $P(|\hat\lambda-\lambda|\le k\lambda)\ge p$. With $z_{(1+p)/2}=1.96$ and $k=0.05$, calculate the expected number of claims needed for full credibility.

An expected loss ratio of 0.65 was used to set current rates; the actual loss ratio for the period is 0.75. Using the loss-ratio method, calculate the indicated rate change.

A stock trades at 50 and pays no dividends. A 1-year European call with strike 48 costs 6.50. The annual effective risk-free rate is such that $e^{-rT}$ discounting applies with $r=5\%$. Using put–call parity $C-P=S-Ke^{-rT}$, calculate the price of the corresponding put.

Distinguish whole life, term, and endowment insurance in terms of the benefit trigger and the present-value random variable, and explain why $A_x < \bar A_{x:\overline{n}|}$ can occur for an $n$-year endowment relative to whole life at the same age.

Under a constant force of mortality $\mu=0.03$ and force of interest $\delta=0.05$, calculate $\operatorname{Var}(Z)$ for the continuous whole-life insurance present-value random variable $Z=v^{T}$.

A pension obligation consists of a single lump-sum payment of 500,000 due in 12 years, valued at 5% annual effective. Calculate its modified duration.