For a whole life insurance of 1000 with semi-annual premiums on (80), you are given:

(i) A gross premium of 60 is payable every 6 months starting at age 80

(ii) Commissions of [math]10 \%[/math] are paid each time a premium is paid

(iii) Death benefits are paid at the end of the quarter of death

(iv) [math]{ }_{t} V[/math] denotes the gross premium policy value at time [math]t, t \geq 0[/math]

(v) [math]\quad{ }_{10.75} V=753.72[/math]

(vi)

[math]t[/math]	[math]l_{90+t}[/math]
0	1000
0.25	898
0.50	800
0.75	706

(vii) [math]i^{(4)}=0.08[/math]

Calculate [math]{ }_{10.25} \mathrm{~V}[/math].

• 680
• 690
• 700
• 710
• 730

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

Answer: E

[math]i^{(4)}=0.08[/math] means an interest rate of [math]j=0.02[/math] per quarter. This problem can be done with two quarterly recursions or a single calculation.

Using two recursions:

[math]{ }_{10.75} V=\frac{\left[{ }_{10.5} V+60(1-0.1)\right](1.02)-\frac{800-706}{800}(1000)}{\frac{706}{800}}[/math]

[math]753.72=\frac{\left[{ }_{10.5} V+54\right](1.02)-117.50}{0.8825} \Rightarrow_{10.5} V=713.31[/math]

[math]{ }_{10.5} V=\frac{\left[{ }_{10.25} V\right](1.02)-\frac{898-800}{898}(1000)}{\frac{800}{898}} \Rightarrow 713.31=\frac{\left[{ }_{10.25} V\right](1.02)-109.13}{0.8909}[/math]

[math]{ }_{10.25} V=730.02[/math]

Using a single step, [math]{ }_{10.25} V[/math] is the [math]E P V[/math] of cash flows through time 10.75 plus [math]{ }_{0.5} E_{80.25}[/math] times the [math]E P V[/math] of cash flows thereafter (that is, [math]{ }_{10.75} V[/math] ).

[math]{ }_{10.25} V=(1000)\left[\frac{898-800}{898(1.02)}+\frac{800-706}{898(1.02)^{2}}\right]-(60)(1-0.1)\left[\frac{800}{898(1.02)}\right]+\left[\frac{706}{898(1.02)^{2}}\right](753.72)=730[/math]

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