Rates vary by

- Age
- Gender
- Geographical Area
- Social Class
- Time

Important causal Factors (may be correlated)

- Occupation
- Nutirition
- Housing
- Climate
- Educational Attainment
- Lifestyle
- Genetics

How do you think L.A. companies can rate their polices for these sort of factors

This is the tendency for high risk individuuals to be more likely to buy insurance

What do you think insurance companies can do about this:

Risk classification and detailed underwriting

Exclusions for certain risks

Raising defences for certain claims

Charging additional premiums for higher risk activities

Moral Hazard refers to the situation where a person takes more risk because he knows he has insurance

Is this a big issue for life insurance

Probably not - as most people don't want to die just to make a claim

How should companies respond to this risk

Similar to adverse selection

The purpose of single figure indices is to summarise a death rate for a whole population for the purpose of comparing it with another population

Typically weighted averages would be used as they are better served for comparisons between different populations

What are the limitiations of such indices

They are 'summary' statisitics and therefore involve a loss of information

If they are weighted this can introduce biases and distortions

This is just the 'average' death rate

$$CDR=\frac{\displaystyle\sum_xE^c_xm_x}{\displaystyle\sum_xE^c_x}$$

That is simply the total observed deaths divided by the total exposed to risk

You can also think of this as:

$$CDR=\frac{\displaystyle\sum_x d_x}{\displaystyle\sum_xE^c_x}$$

$$DSDR=\frac{\displaystyle\sum_x {^sE^c_xm_x}}{\displaystyle\sum_x {^sE^c_x}}$$

If we write this as $DSDR=\frac{\displaystyle\sum_x {^sE^c_xm_x}}{\displaystyle\sum_x {^sE^c_x}} = \displaystyle\sum_x \beta_xm_x$ where $\beta_x=\frac{^sE^c_x}{\sum_x {{ }^{s}E^c_x}}$ then we can directly compare two populations as follows:

$DSDR^I - DSDR^{II} = \sum \beta_x (m^I_x - m^{II}_x)$

Trying to do this for a crude mortality rate produces a distortion effect

The* comparative mortality factor* allows us to create a dimensionless index. It is defined as follows:

$$CMF = \frac{DSDR}{CDR^s}=\frac{\sum { }^sE_x^c m_x}{\sum { }^sE_x^c { }^sm_x}$$

which can also be written:

$$CMF = \frac{DSDR}{CDR^s}=\frac{\sum { }^sE_x^c { }^sm_x \left( \frac{m_x}{^sm_x}\right)}{\sum { }^sE_x^c { }^sm_x}$$

The* standardized mortality ratio* is the ratio of the 'total observed deaths' to the 'total expected deaths'

$$SMR = \frac{\sum E^c_x m_x}{\sum E^c_x { }^sm_x}$$

It is similar to the CMF but note the different weights used

Enables easy comparison to be made between many different experiences, whereas a comparison of age specific mortality rates would be difficult to assimilate, and often subject to large sampling error.

Some indices (such as the SMR) allow mortality of different experiences to be compared against a common standard.

Some indices have practical advantages: e.g. the CDR can be calculated entirely without age specific data; the SMR can be calculated without the need for regional age specific mortality rates; i.e. the individual $m_x$.

Some indices (such as the CDR and SMR) may be sensitive to differences in age structure between populations as well as to differences in mortality, making the interpretation uncertain.

Single figure indices cannot show how the differences in mortality between regions may vary by age, hence important features of the comparisons may be overlooked.

Single figure indices are often biased towards a certain group.