You are given the following information about a special fully discrete 2-payment, 2-year term insurance on (80):

(i) Mortality follows the Standard Ultimate Life Table

(ii) [math]\quad i=0.03[/math]

(iii) The death benefit is 1000 plus a return of all premiums paid without interest

(iv) Level premiums are calculated using the equivalence principle

Calculate the net premium for this special insurance.

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

Answer: D

The equation of value is given by

Actuarial Present Value of Premiums = Actuarial Present Value of Death Benefits.

The death benefit in the first year is [math]1000+P[/math]. The death benefit in the second year is [math]1000+2 P[/math].

The formula is [math]P \ddot{a}_{80: 2}=1000 A_{80: 2}^{1}+P(I A)_{80: 2}^{1}[/math].

Solving for [math]\mathrm{P}[/math] we obtain [math]P=\frac{1000 A_{80: 21}^{1}}{\ddot{a}_{80: 21}-(I A)_{80: 21}^{1}}[/math].

[math]\ddot{a}_{80: 21}=1+p_{80} v=1+\frac{0.967342}{1.03}=1.93917[/math]

[math]1000 A_{80: 21}^{1}=1000\left(v q_{80}+v^{2} p_{80} q_{81}\right)=1000\left(\frac{0.032658}{1.03}+\frac{(0.967342)(0.036607)}{1.03^{2}}\right)=65.08552[/math]

[math](I A)_{80: 21}^{1}=v q_{80}+2 v^{2} p_{80} q_{81}=\frac{0.032658}{1.03}+(2) \frac{(0.967342)(0.036607)}{1.03^{2}}=0.09846[/math]

[math]P=\frac{65.08552}{1.93917-0.09846}=35.36 \rightarrow D[/math]

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