Chapter 5 Flashcards
(34 cards)
What are the three methods of graduating crude mortality
The parametric formula for the whole age range
By reference to a standard table
Or spline functions/ piecewise formulae
How many parameters need to be estimated to fit a GM(m,n) curve
M+n
In practice what sort of parametric formula does Mew x usually take
Mew x typically follows an exponential curve quite closely for middle to late ages so modelling human mortality we always try include a gompertz term
What are Gompertz and Makeham in GM(m,n) form
GM(0,2) - Gompertz
GM(1,2) - Makeham
How did the 00 series tables come about
Data was collected by most life companies during 1999-2002 and were analysed by the CMI - produced tables for males and females and for different classes of insurance and pensions business. Graduating formula was of the form GM(r,s)
What way can you find the parameter estimated for a GM(r,s) type of formula for mew
Find likelihood ignoring constants of deaths observed given the central exposed to risk and maximise it to find the MLEs of the r+s parameters
What are three methods to fit a parametric formula
MLE
Minimise the chi-squared statistic
Minimise the value of the weighted least squares
What are three other considerations to take into account when fitting a model
Using additional information form other investigations
Financial risks
Changes in Mortality with time
Explain what we should look out for when considering : Using additional information form other investigations when we are fitting a model
Graduated estimates should be compared with other experiences we know of.
Could check: mortality of males is higher than females, mortality of life insurance policy holder is lower than the population as a whole, the mortality of a person recently taking out a policy is lower than a person who took out policy ages ago. etc
Explain what we should look out for when considering : financial risks when we are fitting a model
Be aware depending on if its life isnurer or pensions provider where losses will be made.
Life insurance - dont underestimate mortality
Pensions - Dont overestimate mortality!
Explain what we should look out for when considering : changes in mortality over time when we are fitting a model
Some companies use graduated mortality tables to estimate future mortality - but mortality investigations must be based on past mortality
Trend is important and past mortality is likely to be on the conservative side for insurance but not adequate for pensions/annuities
For pensions - will have m=to make a projection of future improvement in mrotality
What are the steps in graduation by mathematical formula
- select graduation formula - never choose more than 5 parameters to estimate
- determine best-fitting parameters
- calculated the graduated rates at each age once the formula is fitted
- test for goodness of fit and sense checks. - if tests fail retrun to step 1
Why dont we check for smoothness of graduation when using parametric formula
Formula chosen is usually already smooth
What are the pro of graduation by mathematical formula
Using a small number of parameters the graduation will be acceptably smooth
When comparing several experiences its useful to fit the same parametric formula to all - difference in parameter show differences in experience
Well suited method to production of standard tables
What are the cons of graduation by parametric formula?
Can be difficult to find a curve that fits experience at all ages well
Care is required when extrapolating - may struggle to fit curve at ages with sparse data so these results may need adjustment
If you need more information to fill in for sparse data - try to reference a standard tables if there is one suitable
What is graduation by reference to a standard table base don?
The assumption that if the lives involved in the graduation is sufficiently similar to the lives whose experience is the basis of a standard table then the true mortality of our class of lives should be similar to the table.
When do we use this method of graduation with reference to a standard table
When there is not a large amount of data
When there is a standard table we think is appropriate to the population - must have a similar shape and a simple relationship evident
What are the steps to graduating with reference to a standard table?
- select the standard table
- Find simple relationship to the standard table (from crude rates)
- Determine best fitting parameter values
- Calculate the graduated rates given assumed relationship
- Test result graduation for goodness of fit and sense checks
To find the simple relationship between tables and crude rates what should you do?
Aim is to find a function with standard rates as a parameter to get the graduated rates - unless youre dealing wiht a short age range plot the crude data on a logarithmic scale against the standard rates. Usually you’ll fidn a lienar relationship between crude rates (now graduated) and standard table rates
What are the two ways you can find the best fitting parameters when graduating with reference to a standard table
MLE by maximising the likelihoods or could find least squares estimates
What are the pros of graduation with reference to a standard table
Can be used to fit small data sets where suitable table exists
If a simple function is chosen and the table rates are smooth a smooth graduation will result
Information obtained from the standard table cna be useful in deciding the shape of graduation at extreme ages
What are the cons of graduation with reference to a standard table?
Method is not suitable for preparation of standard tables
Its not easy to choose a standard table always - features of experience we observe must be very similar to the tables experience
Define a spline function
Polynomials of a specified degree defined on a piecewise basis across an age range. Piece join at knots where certain continuity conditions are required for smoothness
What type of spline is usually used for mortality rates and what does more knots mean?
Commonly cubic splines are used (polynomial degree 3 ). The more knots the more closely the graduated rates adhere to the crude rates but the less smooth the graduation will be.
