SOA ALTAMAdvanced Long-Term Actuarial Mathematics

In a two-decrement model the total force of decrement is constant at 0.03. Calculate the probability that a life remains in the active state for 5 years.

A plan pays 2% of final salary per year of service. A member aged 45 has 20 years of service and current salary 80,000. Salaries grow 3% per year and retirement is at 65. Calculate the projected-unit-credit accrued annual benefit.

A profit test gives the profit vector $Pr_1=-50,\;Pr_2=40,\;Pr_3=40$ with ${}_1p_x=0.98$ and ${}_2p_x=0.96$. Calculate the profit signature component $\Pi_3$.

A permanent disability model has states 0 (healthy), 1 (disabled) and 2 (dead) with constant forces $\mu^{01}=0.02$, $\mu^{02}=0.01$, $\mu^{12}=0.05$. Derive ${}_tp_x^{00}$ and ${}_tp_x^{01}$, and evaluate both at $t=10$.

In a double-decrement model with constant forces $\mu^{(w)}=0.05$ (withdrawal) and $\mu^{(d)}=0.01$ (death), calculate the probability a life remains active for 3 years.

A 4-year profit test has profit vector $Pr_1,\dots,Pr_4=-80,30,30,30$ with survival probabilities ${}_0p_x=1,{}_1p_x=0.95,{}_2p_x=0.90,{}_3p_x=0.85$. Using a 10% hurdle rate, calculate the NPV of the profit signature.

A member age 40 earns 50,000 currently, retiring at 65 (25 years). Salary grows 3% annually; the benefit is 2% of final salary per year of service, valued at retirement with an annuity-due factor of 12, discounted at 6% and multiplied by a 0.90 probability of surviving in service to retirement. Calculate the projected-unit-credit normal cost for the one year of accrual being valued.

Under a Lee–Carter model, $\ln m_{x,t}=\alpha_x+\beta_x\kappa_t$ with $\beta_x=0.5$ and $\kappa_t$ falling by 1.0 in total over a projection horizon. Calculate the resulting multiplicative change in $m_x$.

For the double-decrement model of altx-1 (constant forces $\mu^{(w)}=0.05,\mu^{(d)}=0.01$), derive an expression for ${}_tq_x^{(d)}$, the probability of decrement by death within $t$ years, and evaluate it at $t=3$.

Explain why an insurer writing single-premium immediate annuities is exposed to longevity risk in the opposite direction from a term-life insurer, and describe one financial instrument used to hedge it.