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Help with contingencies question

A

AdamCll

Member
Sorry if this is in the wrong forum, I just finished an exam at University and even with the study material and lecture notes I can't seem to solve this question, if anyone would let me know how it's done, or what my next steps should be I'd appreciate it.

The question

A life office issues a 10-year endowment assurance policy to a male life aged 40 exact. The sum assured of £10,000 is payable immediately on death and premiums are payable annually in advance for the term of the policy or until his earlier death.

a) Calculate to 2 decimal places, the annual premium given:
Interest 3% per annum
The annuity (no idea how to insert it on here) part for the premium = 8.608 at 3% per annum interest
Expenses nil

b) Calculate to 2 decimal places, the prospective reserve of this policy on the premium basis after 3 years given (the annuity after 3 years, lasting 7 more years) = 6.329


My working out is as follows for part a
FiqjX8V.jpg


And then that's where I got stuck, I can't find a way to value the pure endowment part without using any sort of tables or conversions.
 
I think you can use the equation:

upload_2019-5-28_14-27-30.png

to get a value of the corresponding "annuity factor" payable continuously. This can then be used to derive the corresponding "assurance factor" function using:
upload_2019-5-28_14-30-7.png
 
Just a small correction to @KaustavSen's response here. The relationship between a-bar and a-due given above only works for whole life annuities. For example, we could use:

\[
\require{enclose}
{}\bar{a}_{40} \approx {}\ddot{a}_{40} - \frac{1}{2}
\]

However, if you have an annuity with a defined term (10 years in this question), then we need to use the following relationship:

\[
\require{enclose}
{}\bar{a}_{40:\enclose{actuarial}{10}} \approx {}\ddot{a}_{40:\enclose{actuarial}{10}} - \frac{1}{2} ( 1-v^{10} {_{10}p_{40}} )
\]

Or, in general:

\[
\require{enclose}
{}\bar{a}_{x:\enclose{actuarial}{n}} \approx {}\ddot{a}_{x:\enclose{actuarial}{n}} - \frac{1}{2} ( 1-v^{n} {_{n}p_{x}} )
\]
 
Just a small correction to @KaustavSen's response here. The relationship between a-bar and a-due given above only works for whole life annuities. For example, we could use:

\[
\require{enclose}
{}\bar{a}_{40} \approx {}\ddot{a}_{40} - \frac{1}{2}
\]

However, if you have an annuity with a defined term (10 years in this question), then we need to use the following relationship:

\[
\require{enclose}
{}\bar{a}_{40:\enclose{actuarial}{10}} \approx {}\ddot{a}_{40:\enclose{actuarial}{10}} - \frac{1}{2} ( 1-v^{10} {_{10}p_{40}} )
\]

Or, in general:

\[
\require{enclose}
{}\bar{a}_{x:\enclose{actuarial}{n}} \approx {}\ddot{a}_{x:\enclose{actuarial}{n}} - \frac{1}{2} ( 1-v^{n} {_{n}p_{x}} )
\]
Ah okay, but then there's still no way to value this without having any tables to find npx?
 
I'm pretty sure you'd need to be given some additional information to use for calculating the npx values here, unless I'm missing something!
 
Update: It was a typo in the paper, it was supposed to be payable end of year of death.
 
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