Accelerated critical illness benefit and dependent probabilities
An insurer is pricing a two-year term assurance with an accelerated critical illness benefit for a life aged exactly 45. The sum assured is £150,000 and is paid at the end of the year of the first event, whether death or diagnosis of a specified critical illness. The independent annual rates are: in year one, q_death=0.0040 and q_CI=0.0060; in year two, q_death=0.0045 and q_CI=0.0070. The insurer assumes that, within each year, the independent decrements are uniformly distributed and uses an effective interest rate of 4% per year. A junior analyst has simply added the two independent rates to obtain a 1.0% first-year claim probability. The chief actuary asks for the dependent probabilities, the expected present value of the benefit, and a check on whether the junior analyst's approach is acceptable.
1. Calculate the dependent probability of a CI claim and of death in the first year, and the probability of surviving the first year, stating the assumption used.
Read the worked answer
Under the assumption of uniform independent decrements within each year, the dependent probability for cause j is approximately q_j(1-0.5 q_k), where q_k is the other independent rate. For CI in year 1: 0.0060 x (1-0.5 x 0.0040)=0.0060 x 0.9980=0.005988. For death in year 1: 0.0040 x (1-0.5 x 0.0060)=0.0040 x 0.9970=0.003988. The total first-year claim probability is 0.009976, and survival through year 1 is 1-0.009976=0.990024. The junior analyst's 0.0100 is the sum of the independent rates; it ignores the fact that a person cannot both die and be diagnosed in the same year, so it overstates the dependent probability. The overstatement is 0.000024, or about 0.24% of the true claim probability.
2. Calculate the expected present value of the accelerated benefit over the two years.
Read the worked answer
In year 2, conditional on survival, apply the same adjustment: CI=0.0070 x (1-0.5 x 0.0045)=0.0069843; death=0.0045 x (1-0.5 x 0.0070)=0.0044843. The total conditional year-2 claim probability is 0.0114685. The unconditional probability of a claim in year 2 is 0.990024 x 0.0114685=0.011354. The benefit is £150,000 at the end of the claim year, so EPV=150,000 x [0.009976v + 0.011354v^2], with v=1/1.04=0.961538 and v^2=0.924556. This gives 150,000 x [0.009592+0.010498]=150,000 x 0.020090=£3,013. The accelerated benefit is small in absolute terms because the two-year claim probabilities are small.
3. Explain why simply adding independent rates overstates the first-year claim probability and whether the overstatement is material here.
Read the worked answer
Adding independent rates double-counts the possibility of both decrements occurring in the same year, which is impossible under the single-decrement cause structure. The correct dependent probability is slightly lower because a person who dies early is removed from the CI risk for the rest of the year. The overstatement is immaterial here, about 0.0024 percentage points or 0.24% relative, because both rates are small. It would become material if the rates were large, for example in a high-mortality or high-morbidity older age group, or if the benefit were paid on both events. The actuary should still use dependent probabilities consistently in pricing and reserving, and should document the uniform-distribution assumption.