Extra Risk

Study guide: Exam FAM

Extra Risk is mortality heavier than the standard basis, carried by a life who does not meet the insurer's standard underwriting criteria because of a medical impairment, a hazardous occupation or a dangerous pastime. Instead of building a separate table for such a substandard life, the insurer adjusts the standard table in one of the three ways set out in Dickson, Hardy & Waters (AMLCR, chapter 6): an age rating, a constant addition to the force of mortality, or a constant multiple of the rate of mortality.

$\text{Age rating: } {}_tp^{*}_x = {}_tp_{x+k}$

$\text{Constant addition: } \mu^{*}_{x+t} = \mu_{x+t} + \theta$

${}_tp^{*}_x = e^{-\theta t}\,{}_tp_x$

$\text{Constant multiple: } q^{*}_{x+j} = \lambda\,q_{x+j}$

Symbols: An asterisk marks a function of the substandard life. $k$ is the number of years of the age rating, $\theta > 0$ is the extra force of mortality, and $\lambda > 1$ is the multiple. Under a multiple, survival over several years is the product ${}_np^{*}_x = \prod_{j=0}^{n-1}\left(1 - \lambda\,q_{x+j}\right)$ .
Age rating: The life aged $x$ is priced as a standard life aged $x + k$ , so every age in the calculation shifts by $k$ . Its appeal is that premiums and policy values come straight from the existing tables.
Constant addition: The extra force is the same at every age, which suits a risk unrelated to age, such as a dangerous occupation. Because ${}_tp^{*}_x = e^{-\theta t}\,{}_tp_x$ , a substandard annuity value at rate $i$ equals the standard value at rate $j$ , where $1 + j = (1+i)e^{\theta}$ . Insurance values then follow from $A^{*}_x = 1 - d\,\ddot{a}^{*}_x$ , using the original rate of discount $d$ .
Constant multiple: The extra mortality is proportional to the standard rate, so it grows steeply with age, which suits a medical impairment. The multiple applies to $q$ rather than to $\mu$ , and it needs $\lambda\,q_{x+j} \le 1$ at every age used.

Five-Year Death Probability Three Ways A substandard life aged 50 is insured for five years. On the SULT, $q_{50}$ to $q_{54}$ are $0.001209$ , $0.001331$ , $0.001469$ , $0.001623$ and $0.001797$ , ${}_5p_{50} = 0.992593$ and ${}_5p_{55} = 0.987613$ . Find ${}_5q^{*}_{50}$ under (a) an age rating of 5 years, (b) a constant addition of $\theta = 0.001$ to the force of mortality, and (c) a multiple of $\lambda = 1.5$ on the rate of mortality.
Answer The standard probability is ${}_5q_{50} = 0.007407$ .
 $\begin{align*} \text{(a)}\quad {}_5q^{*}_{50} &= 1 - {}_5p_{55} \\ &= 0.012387 \\ \text{(b)}\quad {}_5q^{*}_{50} &= 1 - e^{-0.005}(0.992593) \\ &= 1 - 0.987642 \\ &= 0.012358 \\ \text{(c)}\quad {}_5q^{*}_{50} &= 1 - \prod_{j=0}^{4}\left(1 - 1.5\,q_{50+j}\right) \\ &= 1 - 0.988906 \\ &= 0.011094 \end{align*}$ 
All three raise the five-year death probability by about half or more. They part company at older ages: at 80, the one-year probability is $0.032658$ on the standard table, $0.057665$ with the age rating, $0.033625$ with the addition and $0.048987$ with the multiple. The addition adds about $0.001$ at every age, while the other two methods add more as the life ages.

Whole Life Premium under a Constant Addition A life aged 50 is subject to the SULT force of mortality plus $\theta = \ln(1.06/1.05) = 0.009479$ . Find the net annual premium for a fully discrete whole life insurance of $100,000 at $i = 5\%$ , given that on the SULT $\ddot{a}_{50} = 15.1511$ at $6\%$ . Compare it with the standard premium of $1,111.99 and with a 5-year age rating, for which $A_{55} = 0.23524$ and $\ddot{a}_{55} = 16.0599$ at $5\%$ .
Answer Since $1.05\,e^{\theta} = 1.06$ , the substandard annuity at $5\%$ is the standard annuity at $6\%$ .
 $\begin{align*} \ddot{a}^{*}_{50} &= 15.1511 \\ A^{*}_{50} &= 1 - \tfrac{0.05}{1.05}(15.1511) \\ &= 0.278519 \\ P^{*} &= \frac{100{,}000(0.278519)}{15.1511} \\ &= 1{,}838.28 \\ P_{\text{age}+5} &= \frac{100{,}000(0.23524)}{16.0599} \\ &= 1{,}464.77 \end{align*}$ 
The extra force of about $0.0095$ raises the premium by $65\%$ , much more than a 5-year age rating does. At age 50 the standard force is only about $0.0012$ , so the addition multiplies the mortality of the early policy years many times over.

How it appears on the exam

FAM gives a standard table, such as the SULT, and one of the three adjustments, then asks for a survival probability, a premium or a policy value for the substandard life, for example on a term or whole life insurance.

Interest adjustment: A question can choose $\theta$ so that $(1+i)e^{\theta}$ is a rate at which annuity values are given, and it then expects the candidate to value the substandard annuity at that rate.

Assumption changes: The same work answers FAM's objective on the effect of a change in the mortality assumption, by comparing the substandard premium or policy value with the standard one.

Exam traps

Adding to $q$ : A constant addition to the force gives $p^{*}_x = e^{-\theta}p_x$ , not $1 - (q_x + \theta)$ . With $q_x = 0.02$ and $\theta = 0.01$ , the correct value is $0.970249$ rather than $0.97$ , and the gap compounds over a policy term.

Multiplying $\mu$ : A multiple of the rate means $q^{*} = \lambda q$ . Using $\lambda\mu$ instead gives $1 - p^{\lambda}$ , which is smaller than $\lambda q$ .

Insurance at the adjusted rate: The rate $j$ applies to annuity values only. The standard $A_{50}$ at $6\%$ is $1 - \tfrac{0.06}{1.06}(15.1511) = 0.142391$ , about half the correct $A^{*}_{50} = 0.278519$ , because insurance values need the original $d$ .

Partial age rating: An age rating shifts every age in the calculation, including the annuity used for premiums and the ages in later policy values. Shifting only the insurance factor mixes two different lives.