Part-1-discrete-331-2011-F

# Part-1-discrete-331-2011-F - Review Notes – ACTSC 331...

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Unformatted text preview: Review Notes – ACTSC 331, FALL 2011 Part 1 – Policy Values/Benefit Reserves (Discrete Basis) 1. (Prospective/future) loss function or future loss random variable : A life in- surance policy is issued to a life ( x ) or [ x ] (if the life is selected at age x ). Given that the life is still alive after t years or T x > t or K x ≥ t (if t is an integer), the (prospective/future) loss function or the future loss random variable of the policy at time t , denoted by t L , is defined as t L = Present value ( PV ), at time t, of the future benefits after time t or from age x + t + PV , at time t, of the future expenses (if any) after time t- PV , at time t, of the future premiums after time t. 2. Policy value or benefit reserve (BR) of a policy at time t , denoted by t V , is defined as the conditional expectation of the future loss random variable, given that ( x ) is still alive, namely t V = E ( t L | T x > t ) , t ≥ or t V = E ( t L | K x ≥ t ) , t = 0 , 1 , 2 ,.... 3. Prospective principle : Policy value/benefit reserve at time t ≥ = Expected present value ( EPV ) at time t of the future loss after time t or from age x + t, given that ( x ) is still alive at time t = EPV at time t of the future benefits after time t or from age x + t + EPV at time t of the future expenses (if any) after time t- EPV at time t of the future premiums after time t, given that ( x ) is still alive at time t. A policy value/benefit reserve formula derived from the prospective principle is called a prospective formula for the policy value or benefit reserve. 1 4. Aggregate Mortality Law: T x- t | T x > t = d T x + t , T [ x ]- t | T [ x ] > t = d T [ x ]+ t , t ≥ , and K x- k | K x ≥ k = d K x + k , K [ x ]- k | K [ x ] ≥ k = d K [ x ]+ k , k = 0 , 1 , 2 ,..., where = d means that equality in distribution . 5. Note that the future loss function t L and the policy value/benefit reserve t V depend on premiums, benefits, and expenses (if any) of a policy. When a policy has ex- penses, if expenses are included in the calculations, the premiums, t L and t V are called gross premiums, gross future loss function , and gross premium pol- icy value/benefit reserve , respectively. If not, they are called (net) premiums, (net) future loss function , and (net) premium policy value/benefit reserve ,...
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Part-1-discrete-331-2011-F - Review Notes – ACTSC 331...

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