ADONGO'S MINIMUM UNCERTAINTY METHOD

ADONGO’S MINIMUM UNCERTAINTY METHOD

As an actuarial scientist, contingency is my concerned. I am much concern with contingency of deaths so as to construct a perfect standard mortality table for rate and premium making in life insurance.

Life insurance systems are established to reduce the adverse financial impact of random events of untimely death. Due to long-term nature of the insurance, the investment of earnings, up to the time of payment of any benefits or death claims, provides a significant element of uncertainty. In modelling the rating of insurances, I have had made an attempt to minimise these uncertainties by providing series of formulas which are very essential in life insurance.

Before I continue, I will like to consider the following questions:

1)Are the actual death of each age group equal to those expected?

2)It is possible to estimate the expected deaths in each age group without the initial exposures to risk in each age group?

The answer depends mainly on one factor which was discussed from my previous Diary(or Weblog) titled, “Adongo’s(or My) Growth Rate”.

Adongo’s (or My) Minimum Uncertainty equations are;

q_x=(1/x)ln(d_x/e^ln(dx)/2)           (1)

d_x=e^2xq)                                (2)

d_x=E_xq_x                                (3)

E_x=d_x/q_x                               (4)

∂_x=d_x/E_xq_x                            (5)

Z_x=(d_x-E_xq_x)/√(E_xq_x(1-q_x)    (6)

P_x=1-q_x                                    ( 7)

l_x=E_x(1-q_x)                             (8)

 


Where;

q_x=crude mortality rate

d_x=actual deaths at age x

E_x=initial exposed to risk at age x

E_xq_x=expected death at age x

∂_x=ratio of actual to expected death at age x

Z_x=standard deviation at age x

P_{x=crude rate of alive at age x}

l_{x=actual number of alive at age x}

NB: The formula  e^ln(dx)/2= d_x/e^ln(dx)/2 

The basic importances of equation (1), (2), (3), (4), (5), (6), (7)and (8) are to:

1)Construct mortality rate table.

2)Model the rate and premium of life insurance policy.

3)reduce the adverse financial impact of random events of untimely death.

4)determine the risk of death in each age group.

5)model lapse rates under life insurance policies.

6)model disability rates under life insurance policies.

7)model accidental death rates.

8)model rates of retirement under a large pension plan.

9)records of population and deaths by age obtained from census statistics.

10)model rates of marriage among bachelors

11)model fertility rates.

12)model gross reproduction rates.

13)model net reproduction rates.

14)determine reproduction-survival ratio.

CONSTRUCT MORTALITY TABLE

The following table shows the ages and actual deaths of age 35-39. Use Adongo’s(or My) Minimum Uncertainty Method to complet the table.

Attained Age(x)	Crude rate(qx)	Exposed to Risk(Ex)	Expected Deaths(Exqx)	Actual Deaths(dx)	Ratio of act. to Exp.(∂x)
35				26
36				32
37				31
38				43
39				84

Applying Adongo’s(or My) minimum uncertaintyMethod, the expected crude rate at age 35 is;

q_x=(1/35)ln(26/e^ln(26)/2)

 q_x=0.0465

Expected death at age 35 is;

d₃₅=e^2*35*0.0465

 d₃₅=25.92

Exposed to risk at age 35 is;

E₃₅=25.92/0.0465

 E₃₅=557.419

Ratio of actual to expected death at age 35 is;

∂_x=26/25.92

 ∂_x≈100%

We continue the same procedure to age 36, 37, 38, and 39.

PREMIUM MAKING IN LIFE INSURANCE

Adongo’s(or My) Minimum Uncertainty Method are used to determine the amount of money the insurer must have on hand for policyholder at the end of the year to pay death claim(Gross Single Premium[GSP]). The Gross Single Premium at age x is;

GSP_X=(C/x)ln(d_x/e^ln(dx)/2)

If we know deaths claim at age 35 to be $2000, then the Gross Single Premium at age 35 is;

GSP₃₅=(2000/35)ln(26/e^ln(26)/2)

 GSP₃₅=$93.00

Since the premium are paid at the beginning of the year( in advance), and claims amount need to be discounted for the one year to obtain the Net Single Premium at age x is;

NSP_x=[C/x(1+r_R)ⁿ]ln(dx/e^ln(dx)/2)

Where;

r_R=the expected rate of return

n=compounding period.

If we know the expected rate of return to be 10% then, Net Single Premium at age 35 is;

NSP₃₅=[2000/35(1+0.10)¹]ln(26/e^ln(26)/2)

 NSP₃₅=$84.546

NB: Adongo’s(or My) Minimum Uncertainty Method is not only restricted to Actuarial Science(or Insurance) but also applicable to Biostatistics and Demography.

REFERENCE

*Adongo Ayine William(Me), Diary(or Weblog) tiltled, “Adongo’s G