Q&A Bank 2.27

I am little bit confused with the premium equations so any help would be appreciated.
Expenses and premiums are quite clear so I will focus on the rest of the terms.

(i) Simple bonuses
The endowment term \[9,800A_{40:\require{enclose}{\enclose{actuarial}{20}}}\] should provide 9,800 in case of death or if the life survives at the end of the 20 years. So you need also to add 4,200 to get the correct maturity benefit.
However the increasing term shouldn't be for 19 years rather than 20? I.e. \[{(IA)}^1_{[40]:\require{enclose}{\enclose{actuarial}{19}}}\] ? Is this because the increasing term will pay nothing if the insured survives to year 20?

(ii) Compound bonuses
Probably similar to the above, why do we need to add the benefit amount on maturity?

Thanks!

 

Hi. With this type of question, I find it helpful to consider the death benefits and survival benefits separately.

In the simple bonus case, the death benefit is 10,000 on death in the first year, 10,200 on death in the second year, ..., up to a benefit of 13,800 on death in the 20th year. Since the increases are in steps of 200, I'd start with the IA term. This has to have a coeff of 200. So we start with 200(IA) increasing term assurance (with a term of 20 years). But to get the correct totals of 10000, 10200, etc, we need to add a level term assurance of 9800 to this.

Now consider the survival benefit. On survival to time 20, a total of 20 bonuses will have been added, bringing the benefit amount to 14000. The EPV of this is 14000 pure endowment.

Now add the death benefits and survival benefits together and evaluate in whichever way you find easiest.

Maybe you could try doing something similar with the compound bonus part and let us know if that works for you?

 

Thank you Julie!

I will try this way of thinking, it is very helpful.

 

Hi Julie,
Could I ask you about the solution to part (ii) of this question? Why is the maturity benefit in the question only 10,000 and not 10,000(1.04)^20. Is this an error in the solution or am I making some mistake??

 

\( \frac{10000×1.04^{20}}{1.04^{20}}=10000 \)
That was just adjusted, you can see D/D was taken at 0%

 

Hemant is correct. The benefit amount is 10,000 * 1.04^20. To calculate the EPV of this benefit, we multiply by a discount factor of v^20 and by the probability that the payment is made, which is 20p[40]. The factors of 1.04^20 and v^20 cancel out, so we're jsut left with 10,000 20p[40].