Pricing Whole-life Insurance

Mar 2, 2018

Link to complete code on Github

Introduction

This program was modified from one of the insurance products that I devised. It launched the market on November 30th 2017, and had a pretty good sales record. The insurance contract has a premium paying term of 20 years and provides benefits as follows,

1. Death/ Total disability benefit:
- during the premium paying term (<=the 20th policy year):
max(policy value, 1.06 times the total premium paid, 3 times of amount insured)
- after the premium paying term (>=the 21st policy year):
max(policy value, 1.06 times the total premium paid, 1 time of amount insured)

2. Life annuity:
- payable annually
- during the premium paying term ( <=the 20th policy year):
5% of the annual premium
- after the premium paying term ( >=the 21st policy year):
50% of the annual premium

3. Endowment:
When the insured reaches age 105, the endowment is paid as 1.06 times the total premium paid. And the policy ends.

The mortality rate was based on the Fifth Life Experience Table of Taiwan Life Insurance Industry and provided as a CSV file in the Github project link page.


An Insurance Policy Catered to Human Life Cycle

The main target client group of this contract are people aged between 20 to 55. People in such age range are often those who bring home the bacon; they go out to work, on business travels frequently, which exposed them to higher risks on streets or abroad. Once accidents strike, he/she may lose the ability to work, affecting the financial situation of the entire family. Therefore during the premium paying term, which is 20 years, the policy provides high leverage of death coverage compared to the amount of premium paid. After 20 years, it would be the time of retirement, and the policy provides 50% of the annual premium until the insured reaches age 105. The annuity helps policy beneficiaries prevent money shortage in face of prolonged life expectancy.

financial-plan(photo from omtatsat.tk)


Pricing Algorithm

In addition to reading the mortality rate from CSV at the beginning, the pricing program can be mainly divided into two parts: the benefit and the present value discounting process.

1. Benefits
The benefits part is where I set the amount of coverage should death or total disability occur, and the amount of annuity to give out to the beneficiary upon the existence of the insured. The benefits part is written as follows,

# benefit of death/ total disability coverage and annuity payment 
  
  termpB[pt-1]= 1.06*np.ceil(GP*unit)/unit*ppp 
  
  for i in range(0,pt):
       if i>0:
            pricingVs[i]=pricingV[i]+termpB[i-1]+survB[i-1] #pricing, policy value with annuity amount included
            pricingV[i]=max((deathPVlist[i]+termPlist[i]+survPlist[i])-aDuelist[i]*p2,0)
            if i<=ppp-1:deathBV[i]=(pricingV[i]+pricingVs[i+1]+p2)/2
            else: deathBV[i]=(pricingV[i]+pricingVs[i+1])/2
            
       else: #i=0
            pricingV[i]=max((deathPVlist[i]+termPlist[i]+survPlist[i])-aDuelist[i]*NP,0)
            deathBV[i]=(pricingV[i]+pricingVs[i+1]+p1)/2
            
       if i<=ppp-1:  ###death benefit with premium included to take max.
            deathB[i]=max(deathBV[i],1.06*np.ceil(GP*unit)/unit*(i+1),3)
            survB[i]=np.ceil(GP*unit*0.05)/unit
            
       else:
            deathB[i]=max(deathBV[i],1.06*np.ceil(GP*unit)/unit*ppp,1)
            survB[i]=np.ceil(GP*unit*0.5)/unit
            
  deathBV[pt-1]=(pricingV[pt-1]+termpB[pt-1]+survB[pt-1])/2
2. Discounting the Insurance Benefits
Using backward recursion, the program can generate the expected present value of the benefit outgo, which is the expected present value of the net premium income.

# discounting the expected value of death/ total disability coverage and annuity payment 
  for i in range(pt-1,-1,-1):
       survP= (survB[i]*(popSurv[i]-popDeath[i])*v+survPlist[i+1]*popSurv[i+1]*v)/popSurv[i]
       termP=(termpB[i]*(popSurv[i]-popDeath[i])*v+termPlist[i+1]*popSurv[i+1]*v)/popSurv[i] 
       if i>=ppp: aDue=0
       else: aDue=(popSurv[i]+aDue*popSurv[i+1]*v)/popSurv[i]
       aDuelist[i]=aDue
       termPlist[i]=termP
       survPlist[i]=survP     
       deathPV= (deathB[i]*popDeath[i]*v**0.5+deathPVlist[i+1]*popSurv[i+1]*v)/popSurv[i]  
       ##pricing death value of the policy   
       deathPVlist[i]=deathPV


Mathematical Statement

Writing the expected present values of the benefits provided in this policy in mathematical equations would be as follows,
n: premium paying term
i: assumed interest rate
x: age insured
1. Death/ Total Disability Benefit


screen shot 2018-12-07 at 11 31 20 pm

2. Annuity


screen shot 2018-12-07 at 11 31 43 pm

3. Endowment


screen shot 2018-12-07 at 11 31 50 pm

4. Insurance Premium


screen shot 2018-12-07 at 11 31 55 pm

Pricing Program Applications

The program starts by asking you to enter information about the age, loading, and the sex you want to test. The loading is the markup of net premium, which is one source of profits that insurance companies earn, and most of the time the source of agent commissions. After entering the information you want to know, the program would demonstrate the asymptotic process and generate the result if the premium converges under the conditions you entered.

For example, to price the insured who is

1. male, aged 35, and the loading set to be 10%

screen shot 2018-12-08 at 12 08 48 am

2. female, aged 62, and the loading set to be 9.3%

screen shot 2018-12-16 at 8 24 43 pm



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