Help with contingencies question

[AdamCll]

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

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.

 

[KaustavSen]

I think you can use the equation:



to get a value of the corresponding "annuity factor" payable continuously. This can then be used to derive the corresponding "assurance factor" function using:

 

[AdamCll]

I think you can use the equation:

View attachment 1229

to get a value of the corresponding "annuity factor" payable continuously. This can then be used to derive the corresponding "assurance factor" function using:
View attachment 1230

Ahh I see, that makes sense, thank you.

 

[Lucy England]

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:

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

 

[AdamCll]

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?

 

[Lucy England]

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!

 

[AdamCll]

Update: It was a typo in the paper, it was supposed to be payable end of year of death.