Paper 1 March 2015

[nyaman]

Hi Tutors,

In the question am failing to understand how they come to the conclusion under sense checks of lower annuity rates since greater chance of dying during the policy term and not paying all premiums. Can someone please explain?

My check on this had been only on the premiums being lower for the younger ages than the older ones due to a higher probability of dying.

 

[Lucy England]

The annuity rates are effectively calculated by taking:
\[
\sum (\text{payment} \times \text{probability of payment} \times \text{discounting}) \]
Annuity payments depend on the life being alive. Since the probabilities of death are higher at older ages, probabilities of payment reduce as lives get older. This brings down the value of the annuity factors at older ages.

The logic for your check sounds fine to me and I expect it would have gained a mark. The solutions go into a bit more detail by commenting on the effect of mortality on the assurance and annuity factors, which both feed into the premium calculation:
\[
\require{enclose}
{} P = S \times (1 + \text{loading}) \times \frac{A^{1}_{x:\enclose{actuarial}{n}}} {\ddot{a}_{x:\enclose{actuarial}{n}}}\]
With the formula above coming from the additional guidance document (rearranged slightly) / row 24 of the 'Projection' tab in the sample solution.

For an older life we have higher probabilities of death, which means the annuity factors have lower values and the assurance factors have higher values (payment depends on death for term assurances so higher prob of death => higher value). So that fraction at the end of the premium equation will go up in value as lives get older.