You are doing a mortality study of insureds between ages 60 and 90 . Two specific lives contributed this data to the study:

Life	Age at Entry	Age at Exit	Cause of exit
1	60.0	74.5	Policy lapsed
2	60.0	74.5	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 log-likelihood function (using natural logs) associated with these two lives.

[math]L^{*}[/math] denotes the value of [math]L[/math] if the Gompertz parameters are [math]B=0.000004[/math] and [math]c=1.12[/math].

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

• -4,67
• -4.53
• -4.39
• -4.25
• -4.11

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

Answer: D

The contribution from Life 1 is [math]{ }_{14.5} p_{60}[/math]. With Gompertz and the selected parameters, the contribution is

[[math]]{ }_{14.5} p_{60}=\exp \left[-\frac{B}{\ln c} c^{60}\left(c^{14.5}-1\right)\right]=\exp \left[-\frac{0.000004}{\ln 1.12} 1.12^{60}\left(1.12^{14.5}-1\right)\right]=0.87619.[[/math]]

The contribution from Life 2 is [math]{ }_{14.5} p_{60} \times \mu_{74.5}[/math]. The contribution is

[[math]] { }_{14.5} p_{60} \times \mu_{74.5}=0.87619 \times 0.000004 \times 1.12^{74.5}=0.01627 [[/math]]

The contribution to the likelihood is [math]0.87619(0.01627)=0.01426[/math].

The contribution to the log-likelihood is [math]\ln (0.01426)=-4.25[/math].