Role of Generalised Linear Model in non-life pricing Phase3

Before reading this article, make sure that you read phase1 and phase2. Here are the link:
Phase2: So we know that the purpose of GLM is to find the relationship between mean of the response variable and covariates.

In this Article we are going to talk about Linear Predictors.
Linear Predictor: Let’s denote it with, “η” (eta). So, linear predictor is actually a function of covariates. For example, in the normal linear model where function is Y = B0 + B1x. So linear predictor will be η = B0 + B1x. Always note that linear predictor has to be linear in its parameter. In this case parameters are B0 and B1. But still the question is how I came up with B0 + B1x as a function? First of all, note that broadly there are two types of Covariates. 1. Variables: It takes the numerical value. For example: age of policyholder, years of ex…

CFM vs UDD vs Balducci

Life Tables:
It is a computational tool based on a specific survival model. Our task is to generate a survival model and our output will be a life table. Life table is based on Unitary method.
lx = Expected no. of lives at age x
dx= Expected no. of deaths between exact age x and exact age x+1
NOTE: We call it expected and not actual because most of the values will be in decimal too, so how can it be actual.

But there is a limitation of Life Tables: It is defined only for integer ages. If we have to calculate probability of death/survival at any non-integer age we will use 3 assumptions:
1.CFM= Constant Force of Mortality
2.UDD= Uniform Distribution of death
3.Balducci assumption

·       It assumes that the force of mortality is constant, i.e. ux+t = ux
·       If the force of mortality of a newborn is constant, it means that the expected future lifetime of this life is 1/μ no matter what age he is in.
·       Survival Function in this case will decrease exponentially.
·       If 0<s<t<1 then t-spx+s = tpx/spx = (px)t-s

·       It assumes that force of mortality increases between integer ages and here tqx = t.qx  where 0<t<1
·       It assumes that deaths are uniform over a period of time. For ex: There are 24 deaths in a year then it means there are 2 deaths in a month or 1 death in every 15 days.
·       It assumes that PDF of Tx to be constant if it follows uniform distribution.
·       Survival Function in this case will decrease linearly.
·       If  0<s<t<1 then t-sqx+s  = [qx(t-s)]/[1-s.qx]

·       Here force of mortality decreases between integer ages and assumption is 1-tqx+t = (1-t)qx
·       It means that mortality is heaver in the first half of the year than it is in the second half.
·       The force of mortality decreases in late teenage years/early twenties, so the Balducci assumption may be more appropriate there. Look at the values for mu in the AM92 tables.
·       If 0<s<t<1 then t-sqx+s = [qx(t-s)]/[1-(1-t)qx]

Now let’s see the human mortality graph:

·       You can see here the hump between ages 15 -20 years of age. This effect is called the "accident hump", which is described in CT5. Basically, mortality comes to a small peak around 17/18/19, and then decreases. The peak is usually associated with deaths from car accidents of inexperienced drivers, and drug/alcohol-related deaths.

Follow us on LinkedIn : Actuary SenseFollow me on LinkedIn: Kamal Sardana

But there is something more related to mortality:

Gompertz and Makeham Laws of Mortality:

Gompertz law is an exponential function and it is often a reasonable assumption for middle ages and older ages. Formula is ux = Bcx. Now we say that it is a reasonable assumption between 35-65 years of age because if we take the log of the equation i.e logux= logB + xlogc , then we can see that increase in rate of mortality with age is constant which is often a reasonable assumption.

Makeham law formula is ux= A+Bcx. It incorporates a constant term, which is sometimes interpreted as an allowance for accidental detahs, not depending on age.

Gompertz and makeham family curves equation is (polynomial{1})+exp(polynomial{2})

Initial and central rates of mortality

qx = is called initial rates of mortality, because it is the probability that a life alive at exact age x dies before the exact age x+1

mx = is the probability of dying between exact ages x and x+1 per person year lived between exact ages x and x+1

Follow us on LinkedIn : Actuary Sense
Follow me on LinkedIn: Kamal Sardana


  1. Nice post,thanks for giving this post this is very useful to every one and like this types also good explanation.thank you
    Trading erp software in chennai


Post a Comment

Popular posts from this blog

Role of Generalised Linear Model in Non Life Pricing - Phase1

How to Calculate Minimum Required Contribution that has to be paid by Employer to fund the Plan