CPT AND ACTUARIAL SCIENCE

THE CENTRAL PREDICTION THEORY(CPT) AND ACTUARIAL 

 The actuary who concerns with the contengencies of death, retirements, sickness, withdrawals, marriage, etc. may want to know the mean (or exact) probailities or mean (or exact) rates as a representative of individuals occurrence of such events in order to predict the exact future occurence so as to calculate exact premiums and exact annuities for insurance and other financial operations without account of random errors.

Taking into consideration, the mortality rates over certain range of ages can be fitted as Central Binary logistic prediction model to a given set of data so as to determine future exact estimates of the actual deaths d_xm, future exact crude rates q_xm, provided the exact exposed to risk E_xm, for each year of age is known.

Algebraically, the central binomial logistic prediction model is given as:

q*_xm=1/[1+e^-(α+βxm)]

α=Σd/2n

β=Σd/2Σ(x)

d*_xm=E_xmq*_xm

d and (x) represent death and age respectively.

Examle, mortality rates over 30-34 were estimated fitting the central binary logistic prediction model to the data below.

Ages(x)	Deaths (d)
30	335
31	391
32	428
33	436
34	458

If the mean expose to risk is 140000 estimate:

1)      The parametres α and β.

2)      The exact crude rate of an insured of exact age 42

3)      The exact estimate of actual deaths of an insured of exact age 42

SOLUTION

1)α=204.8

   

   β=6.4

2) q*₄₂=1

even though central binary logistic prediction is applied here, the Central Poisson Prediction is appropriate.


REFERENCE
Galton, Francis.(1886). 'Regression Towards Mediocrity in Hereditary Stature'. Volume 15.