You are doing a mortality study of insureds between ages 70 and 90 . Two specific lives contributed this data to the study:

Life	Age at Entry	Age at Exit	Cause of exit
1	70.0	90.0	End of study
2	70.0	Between 89.0 and 90.0	Death

You assume mortality follows Gompertz law [math]\mu_{x}=B \times c^{x}[/math] and plan to use maximum likelihood estimation.

[math]L[/math] is the likelihood function associated with these two lives.

[math]L^{*}[/math] denotes the value of [math]L[/math] if the Gompertz parameters are [math]B=0.000003[/math] and [math]c=1.1[/math].

Calculate [math]L^{*}[/math].

• 0.0115
• 0.0131
• 0.0147
• 0.0163
• 0.0179

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Answer: A

The contribution from Life 1 is [math]{ }_{20} p_{70}[/math]. With Gompertz and the selected parameters, the contribution is [math]{ }_{20} p_{70}=\exp \left[-\frac{B}{\ln c} c^{70}\left(c^{20}-1\right)\right]=\exp \left[-\frac{0.000003}{\ln 1.1} 1.1^{70}\left(1.1^{20}-1\right)\right]=0.86730[/math].

The contribution from Life 2 is [math]{ }_{19} p_{70}-{ }_{20} p_{70}[/math]. The contribution is

[[math]] \begin{aligned} & { }_{19} p_{70}-{ }_{20} p_{70}=\exp \left[-\frac{B}{\ln c} c^{70}\left(c^{19}-1\right)\right]-0.86730=\exp \left[-\frac{0.000003}{\ln 1.1} 1.1^{70}\left(1.1^{19}-1\right)\right]-0.86730 \\ & =0.88058-0.86730=0.01328 \end{aligned} [[/math]]

The contribution to the likelihood is [math]0.86730(0.01328)=0.01152[/math].

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.