University of Eldoret (UoE) Group
Allan
Aswani
Emma
Okere
MODELLING OF POSTRETIREMENT MEDICAL
SCHEME PRODUCT USING
MULTIPLE STATE MARKOV
MODELS
(A Case Study of the Kenyan Health Insurance Industry)
INTRODUCTION
• Post-Retirement Medical Scheme (PRMS) is a contributory
insurance plan whereby members contribute premiums to a fund in
order to realize comprehensive benefits of a medical cover upon
retirement.
• In Kenya, civil servants enjoy a medical cover during employment.
This cover terminates upon retirement.
• This study used a multi-state Markov model within a continuoustime framework to design a Post-Retirement Medical
Scheme(PRMS) for civil servants and ensure that they enjoy the
benefits of a medical cover upon retirement without making any
further payments after they retire.
PROBLEM STATEMENT
• NHIF provides medical cover to
civil servants in their period of
service
• On retirement, the cover expires
hence the automatic deduction
from their payroll seizes.
• cohorts of ages 60 and above
tend to have high morbidity rate.
JUSTIFICATION
• Healthcare research has devoted
little attention to post-retirement
medical insurance
• Helps reduce the mortality rate of
retirees by providing accessible
medical services
• It is in line with the World Health
Organization requirement
OBJECTIVES
General
Objectives:
Specific
Objectives:
• To model a Post-Retirement Medical Scheme using
multiple state Markov models.
• To estimate transition probabilities and transition
intensities of retirees between health, inpatient, outpatient
and dead states.
• To estimate the rates of interest that will be used in pricing
of post-retirement medical scheme products.
• To price chargeable premiums for the Post-Retirement
Medical Scheme.
METHODOLOGY
• The preferred methodology is a multi state Markov model within
a continuous time framework(4 state markov model ).
• Transition intensities and probabilities are calculated using
chapman-Kolmogorov differential equations.
ij  pij (t ) |t 0  lim
d
dt
h 0
pij (h)   ij
h
pij( m,n )   pij( m,n ) pkj(i ,n )
• Spot rate modelling is used to approximate the interest rate to be
used in Premium calculation.
• The Premiums are then estimated using the equivalence principle.
µ13
µ31
  ( 12  13  14 )

 21

Q
31


0

12
13
14 

 (  21   23   24 )
 23
 24 
32
 ( 31  32  34 ) 34 
0
0
State 1:
Health State
µ12
State 2:
Outpatient State
µ21

0 
µ23
State 3: Inpatient
State
µ32
µ24
µ14
µ34
State 4:
Dead
p12 ( s, t  h)  p11 ( s, t ) p12 (t , t  h)  p12 ( s, t ) p22 (t , t  h)  p13 ( s, t ) p32 (t , t  h)  p14 ( s, t ) p42 (t , t  h)
INTEREST RATES
i
r e
1 e
• Approximated
interest rate
1
i  r 
2
• On bootstrapping, the
approximated interest
=7.0632066% p.a
convertible monthly
which is equivalent to
7.2964102%p.a
effective rate of
interest.
Money, Real and Approximated Rates
35
Money rates
Real Rates
30
Approximated
Rates
25
20
15
10
5
0
-5
-10
Jan
Mar
May
Jul
Sep
Nov
Jan
Mar
May
Jul
Sep
Nov
Jan
Mar
May
Jul
Sep
Nov
Jan
Mar
May
Jul
Sep
Nov
Jan
Mar
May
Jul
Sep
Nov
• Real interest rate
2011
2012
2013
2014
2015
NORMALITY TEST
0.02
0.01
0.00
Density
0.03
Kernel Density Ages
0
10
20
30
40
Ages
50
60
70
• Ho:No significant departure from
normality
• HA: There is significant departure
from normality
• Kolmogorov - Smirnov test at
95% confidence interval shows
that the p-value 0.01 is less than
0.05 thus we reject the null
hypothesis that the data conforms
to a normal distribution.
One-step transition probabilities for: males(t=1)
Health
p11
Age
p12
p14
p13
0.9915597
25
0.005629782
0.001969315
0.0008411894
0.006110433
0.0024664
0.002081
0.006893057
0.0026093
0.004492
0.001154376
0.0027071
0.006288
0.9893418
30
0.9860052
40
0.9794609
50
One-step transition probabilities for female (t=1)
Health
p11
Age
p12
p13
p14
0.9924859
25
0.005221074
0.001850414
0.000442597
0.005274923
0.0023191
0.000617
0.005314995
0.0024926
0.002009
0.001104483
0.0027845
0.003549
0.9917888
30
0.990183
40
0.9826212
50
PREMIUM CALCULATION
• Premiums are a series of payments made in advance.
• The amount and frequency of the payments depend on the terms of the policy.
Transition intensities obtained were graduated using Makeham’s Law;
u x  A  BC
x
t
B


p x  exp( At ) *  (
)C x (C t  1)
 ln C

Makeham’s parameters estimated from the data were;
MAKEHAMS PARAMETERS
A
0.006336636
B
-0.002049662
C
0.945334437
Transition probabilities are then used in calculation of premiums
using the stated formulae;
P s60  x|
60  x
px
 B  X
k  t  60
V
K
k 1
pijk
k
p60
Monthly chargeable premiums obtained for both males and Females are as
follows;
PREMIUMS
AGE
MALES
FEMALES
25
KES 103.90
100.45
30
KES 152.86
141.43
40
KES 356.27
348.57
50
KES 1,077.00
KES 15,496.01
1,069.30
15,488.31
60
CONCLUSION
• Transition probabilities for both males and females to inpatient and
outpatient states are higher at older ages.
• The mortality rate for males is higher than that of females
• The transition probabilities from healthy to outpatient and inpatient states
is higher for males than females whereas the recovery transition from
sickness to health states is higher for females than males.
RECOMENDATIONS
• A comprehensive study to be done on reserving should be done.
• Job groups can also be considered in determining the final premiums
so as to be progressive like the tax system.
THANK YOU