Models with Expenses

\[[{_tV^g}+G_t(1-\%G_t)-e_t](1+i)=(b_{t+1}+E_{t+1})q_{x+t}+{_{t+1}V^g}\ p_{x+t}\]

Thiele’s Differential Equation

\[\frac{d}{dt}{_tV^g}=G_t(1-\%G_t)-e_t+(\delta_t+\mu_{x+t}){_tV^g}-(b_t+E_t)\mu_{x+t}\]

Expense Reserves

\[_tV^e+\text{APV at }t\text{ of }\textbf{future}\text{ expenses}=\text{APV at }t\text{ of }\textbf{future}\text{ expense premiums}\]

The values might seem counter-intuitive due to deferred acquistion costs (DAC).

FPT Reserve

• 1st year premium: \(P_1=vb_1q_x\)
• Subsequent years premium is determined to cover for the remaining term.