### Role of Generalised Linear Model in non-life pricing Phase3

Phase1: http://www.actuarysense.com/2018/10/role-of-generalised-linear-model-in-non.html
Phase2: http://www.actuarysense.com/2018/11/role-of-generalised-linear-model-in-non.html So we know that the purpose of GLM is to find the relationship between mean of the response variable and covariates.

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

1.CFM
·       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

2.UDD
·       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]

3.Balducci:
·       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.

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