# Prediction of healthcare costs on consumer direct health plan in the
Brazilian context

Peixoto  Instituto de Matemática e Estatística, Universidade de São Paulo,
Brazil<EMAIL_ADDRESS>Claudia M  Instituto de Matemática e Estatística,
Universidade de São Paulo, Brazil<EMAIL_ADDRESS>Marcondes  Instituto
de Matemática e Estatística, Universidade de São Paulo, Brazil
<EMAIL_ADDRESS>Diego  Instituto de Matemática e Estatística,
Universidade de São Paulo, Brazil<EMAIL_ADDRESS>Melo  Instituto de
Matemática e Estatística, Universidade de São Paulo, Brazil<EMAIL_ADDRESS>Mariana P  Instituto de Matemática e Estatística, Universidade de São Paulo,
Brazil<EMAIL_ADDRESS>Maia  Departamento de Contabilidade e Atuária,
Universidade de São Paulo, Brazil<EMAIL_ADDRESS>Ana C  Departamento de
Contabilidade e Atuária, Universidade de São Paulo, Brazil<EMAIL_ADDRESS>Correia  Instituto de Matemática e Estatística, Universidade de São Paulo,
Brazil<EMAIL_ADDRESS>Luis A  Instituto de Matemática e
Estatística, Universidade de São Paulo, Brazil<EMAIL_ADDRESS>
###### Abstract

The rise in healthcare costs has led to the adoption of cost-sharing devices
in health plans. This article explores this discussion by simulating Health
Savings Accounts (HSAs) to cover medical and hospital expenses, supported by
catastrophic insurance. Simulating 10 million lives, we evaluate the
utilization of catastrophic insurance and the balances of HSAs at the end of
working life. To estimate annual expenditures, a Markov Chains approach -
distinct from the usual ones - was used based on recent past expenditures, age
range, and gender. The results suggest that HSAs do not create inequalities,
offering a viable method to sustain private healthcare financing for the
elderly.

## Keywords

health savings accounts, consumer directed health plans, prediction of
healthcare expenses, Markov Chain

##  Introduction

Consumer directed health plans (CDHP) are products intended to fund healthcare
expenses, which have been consolidated in the United States since the end of
the nineties as a possible approach to contain the rise in healthcare costs.
In CDHP, the insured has a personal account to pay for his or her medical
procedures, what leads to the increase in awareness about healthcare costs
([1] and [2]).

According to [2], there are three features often associated to CDHP:
relatively high deductibles, a personal account to accumulate resources, and
availability of information about healthcare costs. On the one hand, these
features diverge from models based purely on mutualism, as they seek to
accomplish a higher perception of well-being and economy efficiency, while
mitigating the disruptive impacts due to moral hazard. On the other hand, the
attachment of individual expenses to a personal account raises questions about
justice, as individuals could persistently experience health shocks during
their work life, making it unlikely for them to save enough to cover their
expenses after retirement. This would cause, from a population point of view,
a great variation on the balance of these accounts, causing a part of the
population to have a great amount of resources at retirement (savings), while
other part would not have accumulated anything (the product would have
features of a self-insurance)[3].

In recent years, a great rise in healthcare costs has been observed around the
world. In the Brazilian supplementary health system, for example, the mean
annual healthcare expense, including dental benefits, increased 73% between
2013 and 2018 according to the National Supplementary Health Agency (ANS)[4],
what has led to a public debate about the sustainability of the sector[5].
Following the example of other countries, it has been proposed an alternative
to the system in an attempt to establish norms and legal certainty for
products with co-payment and deductible devices, as it is a reasonable
assumption that Brazil faces similar challenges as other countries in what
concerns the essential cause of healthcare costs increase, that is, spending
more than needed to treat a specific health problem. This over-consumption and
oversupply pattern is understood as the result of two factors. On the one
hand, consumers may partake in more risky activities when protected by a
health plan, increasing the probability of needing more care (ex-ante moral
hazard) or simply consuming more care than necessary (ex-post moral hazard).
On the other hand, providers may supply more care than needed, inducing the
demand ([6] and [7]).

In order to mitigate the impact of these factors, in 2018 the Brazilian
regulatory agency (ANS) proposed a resolution to regulate co-payment and
deductibles in health plans (RN nº 433, June 27 2018), which had a negative
public repercussion, that led ANS to suspend the norms, leaving the market
uncovered in what concerns the understanding of these devices. The negative
repercussion was caused by a misunderstanding about the real use of these
devices in health plans, what evidences the need to further the discussion
about them, including the possibility of considering them as part of a new
market segment.

This paper aims to subsidize the discussion with the analysis of empirical
data by simulating health savings accounts (HSA). A product derived from CDHP,
the HSA are individual accounts set to exclusively cover expenses with
healthcare goods and services, which is composed by a resources accumulation
phase, typically subsidized by the employer with participation of the
employee, followed by a decumulation phase, generally after retirement. The
objective of this form of funding is the implementation of a risk pooling over
the individual cycle of life, mobilizing resources from various sources to
fund the decumulation phase.

In the literature, several studies ([8], [9] and [10]) emphasize that the
theoretical perspective of HSA addresses incentives leading to moral hazard, a
significant inefficiency in the healthcare goods and services market. This
moral hazard has been identified as one of the primary causes of cost
inefficiencies in the sector [11]. Furthermore, in mobilizing resources to a
savings account for the stages of the cycle of life when the increase in
healthcare expenses extrapolate that in income, the HSA are an answer to
policies which aim to maintain private funding of healthcare for the elderly
population.

Since the end of the nineties, countries as the United States, China,
Singapore and South Africa have taken steps to incorporate these alternatives
in their health systems. In the US experience, the HSA products combine a high
deductible health plan with tax breaks to form a savings account. These
products have a mixed framework, which seeks to reduce the individual
exposition to the risk of extreme events, by combining two devices: a
catastrophic health insurance associated with a personal savings account.

In this paper, we simulate a HSA product combined with a catastrophic health
insurance during a labor period of forty years (from 25 to 65 years-old) in
the Brazilian context. The main objective of the simulation is to study, from
a population point of view, the variation of HSA balances at the end of the
accumulation phase. A great discrepancy in these balances may be interpreted
as the result of persistent health shocks, understood here as the cause of the
financial need for healthcare goods and services, and measured through the
costs covered by a health plan under the individual perspective. The
persistence of shocks is a fundamental aspect in the debate about CDHP since
one could argue that individuals with poor health would not be able to
accumulate savings over time, so these accounts would reflect inequalities
between individuals with distinct overall health conditions.

In order to simulate the annual healthcare expenses of an individual, we
propose an approach based on Markov Chains [12], which differs from the usual
methods of predicting healthcare expenses based on regression models [13]. In
this approach, we define levels of annual expense, which are ranges of
expense, and apply a prediction technique which estimates the probability of
an individual to have annual expenses in each level, based on his or her
levels in the previous two years, age range and sex. This model does not try
to predict the exact value of the annual expenses by assuming some kind of
functional relation between the expenses and the independent variables, but
rather predicts the probability of each level of expense for each combination
of categories of the independent variables sex, age range and previous expense
levels.

Freed from assumptions about regression error distributions and the restricted
form of the functional relation between dependent and independent variables,
which are often not satisfied in real datasets, a Markov Chain approach makes
a mild assumption about the dependence between the expenses in a year with
that in previous years, and tries to achieve a more realistic goal, which is
to predict an expense level rather than the exact value of the expense. This
more realistic approach may lead to improved prediction models, as it can be
applied in the context of any country and naturally considers their specific
behaviors.

In Section 2 we present the dataset used in the simulation, and discuss the
considered CDHP product, the modelling of healthcare expenses and the proposed
Markov Chain approach. In Section 3 we present the results of the simulation
studies, and in Section 4 we comment on how they may be interpreted to
subsidize the debate about CDHP products in Brazil.

##  Materials and Methods

###  Dataset

The dataset contains claims of a Brazilian self-management health plan. There
are six types of health plans operators in Brazil, which are characterized by
the juridical nature of their operations. The self-management health plans are
similar to the self-insured group health plans in the US, which offer plans
for employees and dependents, assuming the financial risk, which can be
partially funded by the employees. This portfolio was followed longitudinally
for five years (2005 to 2009) and all claims (expenses) are corrected by the
inflation index IPCA (Extended National Consumer Price Index) to the December
2009 value.

As the objective of this paper is to study the persistence of health shocks
during the work life, we will consider only working age individuals, that is,
those with 25 years completed at January 2005 and with at most 65 years
completed at December 2009, which amounts to approximately 39,000 lives.
During the five years of study, the portfolio size changed because of the
entrance of new individuals, death and other motives, so from all these lives,
around 11,000 were not followed during all the period, so there will be
considered in the analysis only the 27,780 individuals between 25 and 65
years-old which were in the portfolio for the whole period.

Table 1 presents some descriptive statistics of the annual expenses of the
individuals within the considered age range which stayed in the portfolio
during the five-year period. The percentage of individuals with zero annual
expenses is between 5 and 6% in all years. As Table 1 portrays the behavior
over time of a closed cohort, it is expected that a shift in the mean expense
occurs due to the aging of the group. However, even though the variation in
the mean expense from 2005 to 2009 was around 36% and the expenses are
corrected by a general inflation index, there is no guarantee that the real
variation due to aging can still be observed in the costs of the healthcare
goods and services of this portfolio. Therefore, in reality, both the aging
effect and the real variation in the costs are reflected in these values.

As expected ([14], [15] and [16]), large expenses are concentrated in a small
parcel of individuals: in 2008, lesser than 1% of the individuals were
responsible for expenses ranging from R$ 31,158 to R$ 1,044,525 where we
observe that the maximum expense is 385 times the mean one. Also, in 2005,
around 5% of the individuals were responsible for annual expenses ranging from
R$ 7,083 to R$ 426,772, and the maximum expense was 206 times the mean one.

In order to compare men and women according to their annual expenses, we
present in Tables 2.1 and 2.2 in Appendix A some descriptive statistics for
the annual expenses for each sex. From the total of individuals, there are
13,539 (49%) women and 14,241 (51%) men. Over time, the percentage of women
without expenses ranged between 5.1 and 5.5%, while the percentage of men
ranged from 5.8 to 6.8%. When considering only the individuals with positive
expense, we see that the mean annual expenses of women is greater than that of
men, as it varied between R$ 2,341 and R$ 3,062 for the female sex, and
between R$ 1,795 and R$ 2,576 for the male sex. In the same manner, the median
annual expenses of women ranged from R$ 998 to R$ 1,104 and that of men from
R$ 580 to R$ 721.

Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  27,780 |  27,780 |  27,780 |  27,780 |  27,780
PctNoExpense |  5.97 |  5.52 |  5.77 |  5.55 |  5.76
p25 |  319 |  303 |  332 |  368 |  359
p50 |  764 |  736 |  791 |  904 |  872
p75 |  1,746 |  1,666 |  1,789 |  2,090 |  2,008
p90 |  3,993 |  3,735 |  4,130 |  4,861 |  4,761
p95 |  7,083 |  6,531 |  7,477 |  8,726 |  9,114
p96 |  8,497 |  7,960 |  8,864 |  10,247 |  10,780
p97 |  10,450 |  9,853 |  10,935 |  12,910 |  14,190
p98 |  13,538 |  13,134 |  15,380 |  18,237 |  20,160
p99 |  22,910 |  21,683 |  25,851 |  31,158 |  33,730
p995 |  33,905 |  35,695 |  43,747 |  54,407 |  55,601
p999 |  92,962 |  105,585 |  92,549 |  139,811 |  181,947
max |  426,772 |  528,293 |  355,220 |  1,044,525 |  965,287
mean |  2,064 |  2,041 |  2,220 |  2,710 |  2,813
sd |  6,833 |  8,127 |  7,536 |  11,612 |  12,673

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 1. Descriptive statistics of the annual expenses of individuals between
25 and 65 years-old which stayed on the portfolio during the whole five-year
period. The percentiles, mean and standard deviation are calculated
considering only the individuals with positive expenses.

In Tables 3.1 to 3.18 in Appendix A we present the descriptive statistics of
the annual expenses by sex and age range, where it can be seen in all years
that the mean annual expenses increase with age, highlighting the effect of
age range on healthcare expenses. When we assess the factors which influence
individual healthcare expenses, age is always presumed to have a positive
effect since, as age increases, so does the probability of occurrence of
chronicle diseases and loss of functional capabilities. Therefore, it is
expected that high expenses are related to advanced age ([17] and [18]).

In Figure 7 in Appendix A we present the mean annual expenses for each
combination of sex and age range, with an error bar representing one standard
error. On top of each error bar, we present the size of each group. When we
compare men and women, we see a distinct pattern, as women tend to spend more
in the early age ranges, a tendency which inverts itself in later ranges. This
feature is found in other references in the literature [19]. Over time, we
observe a greater increment in the expenses of the later age ranges.

While the mean annual expenses of men increase constantly with age, the mean
annual expenses of women is stable until the age of 50, when we see an
increase in the mean of the age ranges 51-55 and 56-60, followed by a small
reduction in age range 61-65. Thus, we see that women spend more than men in
practically all age ranges, although this difference decreases with age. In
2006 and 2008, the mean annual expenses of men in the age range 61-65
surpassed that of women.

###  Persistence of costs

Persistence of costs is defined as the continuity (permanence) of elevated
health costs of an individual over time. In order to analyze the persistence
of costs, we divide the individual annual expenses of each year
$i\in\\{2005,2006,2007,2008,2009\\}$ into four levels, namely:

  * •

$F_{1,i}$: annual expenses lesser or equal to R$ 300 at year i;

  * •

$F_{2,i}$: annual expenses greater than R$ 300 and lesser or equal to R$ 1,000
at year i;

  * •

$F_{3,i}$: annual expenses greater than R$ 1,000 and lesser or equal to R$
5,000 at year i; and

  * •

$F_{4,i}$: annual expenses greater than R$ 5,000 at year i.

Dividing the expenses this way we have, for each individual, a sequence of
five levels describing his or her expenses throughout the years. For example,
an individual with costs $\\{0;200;500;1,500;200\\}$ has sequence
$F_{1,2005},F_{1,2006},F_{2,2007},F_{3,2008},F_{1,2009}$ as expense levels.

Figure 1: Transition matrix of expenses for each two years combination.

In Figure 1 we present the sample proportion of individuals which transitioned
between each pair $(F_{k,i},F_{l,j}),j>1$ of states, in which rows represent
the origin state $F_{k,i}$ and columns the destination state $F_{l,j}$. The
color of the matrices entries is related to the estimated transition
probability: red means great probability (around 0.5), gray means medium
probability (around 0.2) and white means low probability (around zero). In
this figure, there are ten 4 by 4 matrices, one for each pair of distinct
years, which present the transition probabilities from the ranges of the early
year to that of the later. For example, the matrix in the lower-left corner
presents the transition probabilities between the expense’s levels of 2005 and
2006, and the matrix in the upper between the expense’s levels of 2008 and
2009. The leading diagonal of each matrix is the proportion of individuals
which stayed at the same expense level at both years.

In Figure 1 we see, for example, that within individuals with expenses greater
than R$ 5,000 in 2008, 46% had expenses between R$ 1,000 and R$ 5,000, and 27%
maintained the expenses at level greater than R$ 5,000, in 2009; within
individuals with annual expenses lesser or equal to R$ 300 in 2007, 52% had
expenses lesser than R$ 300, and only 3% had expenses greater than R$ 5,000,
in 2008. We also note that high values of probabilities are concentrated
around the leading diagonals, except at the greatest expense level. However,
these high values decrease with the distance between the years, highlighting
the fact, also observed in Eichner, Mcclellan & Wise [20], that persistence of
costs decays over time. It is important to note that the matrices were
constructed considering only the individuals between 25 and 65 years-old that
were alive in 2009. If the individuals which died in the considered period
were also considered, then there would be greater probabilities in the
diagonal point of the greatest expense level, as it is known that individuals
tend to have consistently high expenses in the period leading up to their
death.

In order to evaluate the effect of age on the persistence of costs, we present
in Fig. 8 in Appendix A the transition matrices calculated considering only
the younger (21 to 40 years-old) and older (41 to 65 years-old) individuals,
respectively. When comparing the diagonal of the matrices of both figures, we
see that, for younger individuals, the probability of those in expense level 3
remaining in the same level or moving up to expense level 4 is approximately
56%. However, for the population aged 40 and above, this proportion is around
66%, i.e., 10 percentage points higher, in average.

###  Health Savings Accounts (HSA)

The health savings accounts (HSA) are devices in which a savings account
attached to each individual receives annual contributions with the exclusive
purpose of covering healthcare expenses. The annual expenses covered by HSA
are limited by a predefined value while its balance is positive. In case
expenses exceed the limit or the account balance, then a catastrophic
insurance is activated to cover them. Although there is a limitation on the
expenses covered by funds from the savings account, in this device there is no
limitation on individual annual healthcare expenses, as the insurance covers
any expenses beyond the threshold.

The CDHP product considered in this paper is a HSA in which each individual
has an account, started at 25 years-old, from which are deducted his or her
healthcare expenses. The account dynamic is as follows:

  * •

At the beginning of each year, the employer deposits R$ 2,500111[Include the
rationale of choosing this amount as reference for HSA deposits – w/ Ana] on
the individual account.

  * •

In case the annual healthcare expenses of an individual do not exceed the
account balance or R$ 5,000, they are fully paid by funds from the account. If
the annual expenses surpass R$ 5,000 or the balance of the account, then R$
5,000, or the balance, is withdrawn to partially cover them, and the remaining
value is covered by a catastrophic insurance. Therefore, all healthcare
expenses up to R$ 5,000 or the account balance, if lesser, are covered by the
individual, and the remaining value is covered by the insurance.

In what follows, we present an approach based on Markov Chain to simulate the
product described above, which seeks to predict the expense level of each
individual at each year of his or her work life in order to assess the balance
of the accounts at age 65, and the frequency and severity of catastrophic
insurance use, from a population point of view. In the next section, we
discuss the approaches to healthcare expenses prediction at the individual
level used in the literature, presenting their qualities and shortcomings.

###  Prediction of future healthcare expenses at individual level

Modelling techniques to assess the risk associated with events which incur in
healthcare expenses are relatively recent [13], for their development is
dependent on the volume and quality of available information, which have only
improved in the last couple of decades. Also, the sector has been demanding
new frameworks for covering and management of health risks, in contrast to its
historical foundation, based purely on refunding, increasing the need for
improved quantitative models.

In order to model healthcare expenses patterns in the US, Frees, Gao &
Rosenberg [21] considered more than thirty factors related to an individual’s
demography, socioeconomic level, health condition, employment and availability
of health insurance. There were considered two dependent variables, namely,
frequency and severity of expenses, and the significant factors found for each
of them differed. Also, Duncan, Loginov & Ludkovski [17] explored the effects
on healthcare expenses of approximately one hundred independent variables
related to demography, age and comorbidities history, among others.

Pope et al [22] and Duncan, Loginov & Ludkovski [17] pointed out the
limitations of models which consider only demographic factors and health
condition history, affirming the importance of taking into account past
expenses while developing prediction models. Indeed, according to Duncan,
Loginov & Ludkovski [17], one of the best sources of information for
predicting healthcare expenses are past expenses.

About issues encountered when modelling individual healthcare expenses,
Duncan, Loginov & Ludkovski [17] evidenced that: (a) annual expenses follow
approximately a log-normal distribution and are characterized by high
variance; (b) the error distribution is clearly heteroscedastic: the
conditional variance for individuals with high expenses is greater than that
of individuals with low expenses; (c) there exists interaction between the
independent variables, and the relationship between expenses and health
condition is expected to be non-linear, what implies non-linear relationships
between dependent and independent variables; (d) many of the independent
variables are highly correlated and some of them have very rare categories.
Due to the great variability of annual expenses in a population and the great
number of independent variables, the quality of fitted models is quite low.

Other problems encountered when modelling annual healthcare expenses are the
excess of zeros and the presence of extreme values. The first problem is due
to individuals which do not use their health insurance in the period of a
year. The second is caused by the fact that a very small portion of a
population has annual expenses hundreds of times the mean one. In order to
attack these problems, Marcondes, Peixoto & Maia [23] proposed a hurdle model
for heavy-tailed data, which is specifically useful for annual healthcare
expenses data. A hurdle model takes special care of the zero values, while a
heavy-tail model takes into account values far from the mean. The mixture of
models treating these two features is a better fit for annual healthcare
expenses than the usual regression models, as evidenced by the application in
Marcondes, Peixoto & Maia [23], which considered a dataset of annual
healthcare expenses.

Based on the literature, we see that there is a need to develop better
techniques to model healthcare expenses in a dynamic scenario characterized by
the rapid increase in the amount of available information and data about
individuals. With this purpose, we consider a Markov Chain approach to model
healthcare expenses in order to simulate an HSA product.

###  Methodology for prediction of future healthcare expenses at individual
level

In this section, we present the methodology proposed in this paper to predict
future healthcare expenses at the individual level, which is based on Markov
Chains of order 2 that are defined in Appendix B. The prediction technique
will be employed to estimate annual individual healthcare expenses in order to
simulate the expenses which will be covered by a HSA during the labor life of
simulated individuals.

####  Simulation of individual annual healthcare expenses over time

Our simulation study is based on the assumption that the pattern of annual
healthcare expenses of an individual is similar to that of those of the same
sex, age range and expenses history. The study consists in the simulation of
10,000 lives, starting at 25 years-old, which are followed annually until 65
years-old, what corresponds to the forty-one years of a work life. The annual
expenses of these individuals in their first year (25 years-old) are sampled
from the 2009 annual expenses empirical distribution of age range 25-30. There
are 1,683 individuals in the age range 25-30 in 2009, so in order to obtain
10,000 lives from them, we will sample 10,000 lives with repetition from the
1,683 individuals, setting the age of all new sampled individuals to exactly
25 years-old.

From the sex, age range, expenses history and expenses in the first year of
the 10,000 lives, we simulate their personal HSA over time, a process
fulfilled by iterating two consecutive procedures. In the first step, we
predict to which expense level each individual will go, based on probabilities
depending on his or her sex, age range at the present year and expense level
on the current and previous year. On the second step, we sample the exact
value to be deducted from each individual account to cover the expenses from
the empirical distribution of the annual expenses of the individuals of the
same sex, age range and predicted expense level. This empirical distribution
is the one in Fig. 9 which is related to the individual’s age range and sex.
The distribution is built by combining individual’s medical expenses from
years $2005$ to $2009$, stratified by age and sex, obtaining the possible
points that can be sampled within the expense level sampled in the first step,
i.e., between the respective dotted lines in Fig. 9. The simulation was
replicated $1,000$ times. In Appendix B, we further explain the simulation
dynamics.

##  Results

In order to assess the effectiveness of the HSA, we present in this section
the results of the simulation study regarding the balance of the individual
accounts over time, and the frequency and severity of the catastrophic
insurance use.

###  Balance of the individual health savings accounts

In Table 5 we present descriptive statistics of the individual HSA of the
simulated 10,000 lives at every five years between 25 and 65 years-old.

| 25-30 | 31-35 | 36-40 | 41-45 | 46-50 | 51-55 | 56-60 | 61-65
---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
n0 |  2,693 |  46.38 |  496 |  21.93 |  227 |  14.92 |  144 |  11.69 |  121 |  10.89 |  123 |  10.53 |  126 |  10.61 |  133 | 11.61
p5 |  1,058 |  14.09 |  2,907 |  71.85 |  5,138 |  125.04 |  7,434 |  174.39 |  9,238 |  214.90 |  10,232 |  248.86 |  10,734 |  278.85 |  10,706 | 294.34
p10 |  1,593 |  16.48 |  4,711 |  72.43 |  7,961 |  122.21 |  11,123 |  166.51 |  13,734 |  202.02 |  15,413 |  237.51 |  16,479 |  263.42 |  16,807 | 280.47
p15 |  1,960 |  12.16 |  6,165 |  77.60 |  10,056 |  118.58 |  13,810 |  161.58 |  16,966 |  190.51 |  19,139 |  221.65 |  20,622 |  251.52 |  21,249 | 271.23
p25 |  2,385 |  6.84 |  8,467 |  75.27 |  13,298 |  111.78 |  17,930 |  141.20 |  21,876 |  173.91 |  24,800 |  204.01 |  26,927 |  233.74 |  28,062 | 254.14
p40 |  3,993 |  22.48 |  11,214 |  70.06 |  17,117 |  101.10 |  22,734 |  127.61 |  27,598 |  157.55 |  31,423 |  188.78 |  34,341 |  217.03 |  36,126 | 238.85
p50 |  4,723 |  14.96 |  12,846 |  65.88 |  19,359 |  95.41 |  25,535 |  123.00 |  30,934 |  150.65 |  35,309 |  179.50 |  38,721 |  206.08 |  40,932 | 231.85
p75 |  7,843 |  39.48 |  16,608 |  53.84 |  24,777 |  82.13 |  32,278 |  109.82 |  39,016 |  135.52 |  44,831 |  167.13 |  49,603 |  193.85 |  53,051 | 227.11
p85 |  9,404 |  23.11 |  18,587 |  50.79 |  27,201 |  73.21 |  35,361 |  102.83 |  42,753 |  133.39 |  49,286 |  164.03 |  54,800 |  189.26 |  58,949 | 230.20
p95 |  11,792 |  22.48 |  21,663 |  56.93 |  30,869 |  76.88 |  39,728 |  101.04 |  48,068 |  132.23 |  55,699 |  168.59 |  62,406 |  209.51 |  67,718 | 253.78
p98 |  13,193 |  29.56 |  23,301 |  46.22 |  32,987 |  77.81 |  42,280 |  103.93 |  51,064 |  140.66 |  59,313 |  181.59 |  66,747 |  238.78 |  72,770 | 291.57
max |  14,621 |  50.65 |  26,407 |  114.55 |  37,976 |  194.88 |  49,347 |  318.00 |  60,427 |  526.57 |  71,231 |  762.78 |  81,510 |  1,064.25 |  89,269 | 1,342.32
mean |  5,446 |  20.08 |  12,571 |  50.98 |  18,858 |  74.16 |  24,831 |  95.64 |  30,115 |  115.29 |  34,478 |  134.69 |  37,971 |  151.71 |  40,345 | 168.52
sd |  3,410 |  8.89 |  5,613 |  24.45 |  7,780 |  39.52 |  9,799 |  53.11 |  11,769 |  65.71 |  13,756 |  79.44 |  15,599 |  91.95 |  17,186 | 102.92

$n_{0}$: total of individuals with zero balance in HSA; p(k): k-th sample
percentile; SD: standard deviation.

Table 5. Descriptive statistics of the simulated individual mean HSA balance
at every five years between 25 and 65 years-old. We present the mean () and
standard deviation () over the 1,000 simulations.

According to the simulation, at 30 years-old 26% of the lives had a zero
balance in their HSA and, at 40 years-old, this percentage was lesser than 3%,
in average. As the annual health expenses are smaller at young age, the mean
increase in the individual HSA balance is greater in early ages, and in later
age ranges this balance is more spread around the mean. At 40 years-old, only
5% of the lives had a balance greater than R$ 30,000 in average and, at 45
years-old, between 50% and 75% of the lives had this kind of balance. At 55
years-old only 15% of the lives had a balance lesser than R$ 20,000, while 15%
had a balance greater than R$ 50,000 in average and, at 60 years-old, only 25%
of the lives had a balance lesser than R$ 30,000 in average and 50% had a
balance greater than R$39,000 in average.

At 65 years-old, the end of the work life, only 1% of the lives had zero
balance and 5% a balance lesser than R$11,000 in average, while half the lives
had a balance greater than R$41,000 in average, with 25% with a balance
greater than R$53,000 in average. The mean balance at 65 years-old is around
R$40,000 in average, with a maximum balance around R$89,000 in average. In
Fig. 2 we present the empirical distribution of the individual HSA balance in
the 10,000,000 simulated individuals at 65 years-old. This distribution is
approximately symmetrical (Skewness coefficient = -0.1202) and 01 outlier was
detected (R$93,142.83). According to the simulation, the HSA is a good
alternative to fund healthcare costs at old age, as a great parcel of the
simulated population had a considerable balance at retirement. Note that the
simulation is based on the expenses of a portfolio in which there is no
individual HSA, and that we considered a savings account without interest.
Therefore, the balance of the individual HSA could in reality be greater than
that simulated.

Figure 2: Empirical distribution of the simulated HSA balances at 65 years-old
of the 10,000,000 simulated individuals.

###  Frequency and severity of the catastrophic insurance use

In this section we analyze the use of the catastrophic insurance, more
specifically its frequency and severity. In Table 6 we present the percentage
(frequency), and the total value covered (severity), of the lives which used
the catastrophic insurance each number of times.

| n | (%) of Usage | Tot. CI Usage |  | Mean Usage per live
---|---|---|---|---|---
No. of Times |  |  |  |  |  |  |  (%) CI Usage |  Cum.(%) of Usage |  Cum.(%) CI Usage |  |
0 |  559 |  23.19 |  5.59 |  0.23 |  0 |  0 |  0.00 |  5.58 |  0.00 |  0 |  0
1 |  1152 |  31.78 |  11.52 |  0.32 |  14,706,212 |  1,106,298 |  3.01 |  17.10 |  2.99 |  12,760 |  882
2 |  1474 |  36.82 |  14.74 |  0.37 |  36,720,303 |  1,876,230 |  7.50 |  31.82 |  10.46 |  24,919 |  1,079
3 |  1510 |  36.43 |  15.10 |  0.36 |  55,345,252 |  2,439,868 |  11.31 |  46.91 |  21.72 |  36,648 |  1,323
4 |  1361 |  33.12 |  13.61 |  0.33 |  65,156,889 |  2,701,662 |  13.32 |  60.50 |  34.97 |  47,885 |  1,603
5 |  1121 |  32.67 |  11.21 |  0.33 |  65,797,506 |  2,987,801 |  13.45 |  71.71 |  48.35 |  58,667 |  1,975
6 |  867 |  28.40 |  8.67 |  0.28 |  60,042,438 |  2,939,841 |  12.27 |  80.36 |  60.56 |  69,293 |  2,533
7 |  636 |  24.31 |  6.36 |  0.24 |  50,631,601 |  2,721,570 |  10.35 |  86.72 |  70.86 |  79,608 |  3,089
8 |  446 |  20.18 |  4.46 |  0.20 |  39,977,371 |  2,576,693 |  8.17 |  91.17 |  78.99 |  89,615 |  4,112
9 |  302 |  16.89 |  3.02 |  0.17 |  29,969,236 |  2,245,427 |  6.12 |  94.19 |  85.09 |  99,320 |  5,125
10 |  200 |  14.02 |  2.00 |  0.14 |  21,602,668 |  2,021,014 |  4.41 |  96.19 |  89.48 |  108,085 |  6,792
11 |  129 |  11.41 |  1.29 |  0.11 |  15,237,936 |  1,766,926 |  3.11 |  97.48 |  92.58 |  117,705 |  8,513
12 |  84 |  9.04 |  0.84 |  0.09 |  10,595,814 |  1,449,420 |  2.16 |  98.32 |  94.73 |  125,683 |  10,782
13 |  55 |  7.57 |  0.55 |  0.08 |  7,431,749 |  1,289,946 |  1.52 |  98.87 |  96.24 |  134,133 |  13,990
14 |  37 |  6.02 |  0.37 |  0.06 |  5,205,901 |  1,061,262 |  1.06 |  99.24 |  97.30 |  141,704 |  17,071
15 |  24 |  4.78 |  0.24 |  0.05 |  3,651,611 |  915,835 |  0.75 |  99.48 |  98.05 |  149,867 |  21,897
16 |  16 |  3.97 |  0.16 |  0.04 |  2,549,838 |  749,741 |  0.52 |  99.64 |  98.56 |  159,669 |  29,516
17 |  10 |  3.32 |  0.10 |  0.03 |  1,747,222 |  693,394 |  0.36 |  99.75 |  98.92 |  166,207 |  37,701
18 |  7 |  2.61 |  0.07 |  0.03 |  1,168,382 |  553,658 |  0.24 |  99.82 |  99.16 |  177,096 |  49,625
19 |  4 |  1.88 |  0.04 |  0.02 |  744,571 |  416,481 |  0.15 |  99.86 |  99.31 |  181,694 |  60,022
20 |  3 |  1.43 |  0.03 |  0.01 |  504,427 |  338,090 |  0.10 |  99.88 |  99.41 |  192,019 |  91,845
21 |  2 |  0.95 |  0.02 |  0.01 |  368,242 |  239,624 |  0.08 |  99.90 |  99.49 |  202,041 |  96,327
22 |  1 |  0.67 |  0.01 |  0.01 |  308,767 |  210,895 |  0.06 |  99.91 |  99.55 |  219,688 |  128,687
23 |  1 |  0.59 |  0.01 |  0.01 |  267,208 |  177,913 |  0.05 |  99.93 |  99.60 |  212,600 |  122,867
24 |  1 |  0.32 |  0.01 |  0.00 |  260,201 |  173,980 |  0.05 |  99.94 |  99.66 |  231,627 |  137,556
25 |  1 |  0.25 |  0.01 |  0.00 |  257,426 |  140,819 |  0.05 |  99.95 |  99.71 |  241,421 |  119,099
26 |  1 |  0.12 |  0.01 |  0.00 |  303,843 |  168,739 |  0.06 |  99.96 |  99.77 |  301,320 |  169,538
27 |  1 |  0.21 |  0.01 |  0.00 |  349,976 |  283,372 |  0.07 |  99.97 |  99.84 |  339,873 |  283,236
28 |  1 |  0.00 |  0.01 |  0.00 |  307,167 |  167,335 |  0.06 |  99.98 |  99.90 |  307,167 |  167,335
29 |  1 |  0.00 |  0.01 |  0.00 |  237,283 |  109,358 |  0.05 |  99.99 |  99.95 |  237,283 |  109,358
30 |  1 |  0.00 |  0.01 |  0.00 |  239,257 |  71,911 |  0.05 |  100.00 |  100.00 |  239,257 |  71,911

Table 6. Mean Frequency and severity of the catastrophic insurance use by the
simulated lives during the 40 years period. We present the mean () and
standard deviation () over the 1,000 simulations.

In average, only 5.6% of the simulated lives did not use the insurance at the
40 years period, approximately 50% used at most three times, and 80% used at
most six times. For 0.5% of the lives, it was necessary to use the insurance
13 times or more, and for 13 lives the insurance was triggered between 20 and
30 times among the 40 possibles. Furthermore, we note that around 50% of the
expenses covered by the catastrophic insurance were with lives that used it at
most five times, which corresponds to 72% of the lives, while 10% of the
expenses covered were with lives that used it 10 times or more, which
corresponds to 3.5% of lives in average. Also, around 0.5% of the expenses
covered by the insurance were with only 11 lives in average , which used it at
least 21 times.

From a population point of view, the simulation of 1,000 repetitions of the
portfolio during 40-years each, revealed a mean healthcare expense of
R$1,082,350,725 with standard deviation of R$7,688,268, from which
R$489,367,594 (45.2%) with standard deviation of R$6,862,835 was covered by
the catastrophic insurance (in average), and the remaining R$592,983,131
(54.8%) with standard deviation of R$1,784,781 was covered by the individual
HSA (in average).

Figure 3: Distribution of percentage of coverage for healthcare expenses from
HSA and catastrophic insurance in the 10,000,000 simulated individuals.

In figure 3 we present the distributions of percentage of coverage for
healthcare expenses, either by HSA or catastrophic insurance.

###  Savings account balance, coverage by HSA and coverage by catastrophic
insurance

The proposed CDHP product has three main features: the HSA balance at 65
years-old, the expenses covered by the HSA and the expenses covered by
catastrophic insurance. The first two features are more susceptible to the
effects of a HSA on an individual behavior, as the decision to consume
healthcare goods and services may be inhibited by this CDHP product. Table 7
displays descriptive statistics of these three features.

| HSA Balance | HSA Coverage | Cat, Insurance Coverage
---|---|---|---
|  |  |  |  |  |
pctnul |  0.73 |  - |  0.00 |  - |  5.59 |  -
p5 |  10,536 |  329 |  30,669 |  290 |  2,022 |  99
p10 |  16,737 |  300 |  35,956 |  263 |  4,281 |  145
p15 |  21,267 |  288 |  39,834 |  259 |  6,638 |  177
p25 |  28,263 |  271 |  45,996 |  249 |  11,970 |  249
p40 |  36,568 |  256 |  53,738 |  241 |  21,650 |  363
p50 |  41,535 |  248 |  58,645 |  248 |  29,666 |  459
p75 |  54,106 |  249 |  72,079 |  274 |  63,926 |  948
p85 |  60,242 |  257 |  79,243 |  298 |  94,228 |  1,586
p95 |  69,381 |  290 |  90,573 |  357 |  178,278 |  4,494
p98 |  74,616 |  342 |  96,831 |  359 |  264,692 |  6,711
max |  89,251 |  1,355 |  100,000 |  - |  1,176,242 |  119,167
mean |  41,000 |  176 |  59,298 |  178 |  51,833 |  713
sd |  17,739 |  108 |  18,015 |  111 |  69,719 |  2,297

p(k): k-th sample percentile; SD: standard deviation.

Table 7. Descriptive statistics of the HSA balance at 65 years-old, the
expenses covered by the HSA and the expenses covered by catastrophic insurance
over the work life. The percentiles, mean and standard deviation are
calculated considering only the values greater than zero (n=10,000). We also
present the mean () and standard deviation () over the 1,000 simulations.

If a life did not have any expenses in the 40 years period, its account
balance would be R$ 100,000, although the simulated mean and median balance at
65 years-old were between R$41,000 and R$41,535, in average with s.d. of
R$248, and the maximum was R$ 89,251 in average, with s.d. of R$ 1,355.
Therefore, the mean account balance at retirement is in average 42% the value
which has been deposited. The mean expense covered during the work life by the
insurance was around R$51,800, in average with s.d. of R$713, and the maximum
was around R$1,176,000, in average with s.d. of R$ 119,167, which is 22 times
the mean value, evidencing the presence of outliers. For 25% of the lives, the
insurance covered lesser than R$ 12,000, and for half the lives it covered at
most R$29,700 in average with s.d. of R$459. On the other hand, the 5% of
lives with the greatest expenses covered by the insurance had more than R$
175,000 covered in average. When considering the percentage of expenses
covered by the insurance, half the lives had at most 40% of them covered by
the insurance in average, while 25% of the lives had almost half of their
expenses covered by it in average, and 5% of the lives had more than 75% of
their expenses covered by the insurance in average.

Figure 4: Severity of catastrophic insurance use in the 10,000,000 simulated
individuals. Figure 5: Logarithm transform of severity of catastrophic
insurance usage in the 10,000,000 simulated individuals.

Figures 4 and 5 display the empirical distribution of the severity of
catastrophic insurance use, and of its logarithm transformation, where it can
be noted the presence of outliers, evidencing the coverage of great healthcare
expenses by the insurance. Coverages under R$1.00 were ignored in the
logarithmic transformation.

Figure 6: Dispersion plot between the total expenses at the 40 year period of
each life in the 10,000,000 simulated individuals and the percentage of these
expenses which were covered by catastrophic insurance. The color refers to the
number of times in which the insurance was used over the work life.

In Figure 6 we present the dispersion plot between the total expenses at the
40 year period of each life and the percentage of these expenses which were
covered by catastrophic insurance. The color of the points refers to the
number of times in which the insurance was used over the work life.

##  Discussion and Conclusions

An increase in healthcare costs has been observed recently in several
countries, especially in Brazil, where it has led to a discussion about co-
payment and deductible devices for health plans, which culminated in a
resolution by the regulatory agency (ANS) to incorporate these devices to the
Brazilian supplementary health system. A negative public repercussion based on
misunderstandings followed the resolution, forcing ANS to revoke it, leaving
the market uncovered regarding the understanding of these devices. In this
context, this paper aimed to subsidize the discussion about co-payment and
deductibles by simulating a consumer directed health plan product which has
never been commercialized in Brazil, the health savings accounts, which
combines a high deductible health plan with tax breaks to form a savings
account aiming to reduce the individual exposition to the risk of extremal
events by combining a catastrophic insurance with an individual savings
account.

This mixed product benefits both the employer and employee. From the employee
point of view, the accredited network of healthcare providers is superior than
the standard, and the individual is not limited to the accredited network.
Moreover, the balance may be invested to increase income, but due to the
purpose of this work, this issue of investment has not been considered. From
the employer point of view, it is expected lower individual healthcare
expenses, since the employees, when paying for their own healthcare goods and
services, tend to be more selective in choosing their providers and more
thrifty when using the available services.

In order to carry out our simulation, we had to first establish a prediction
technique for annual healthcare expenses. As evidenced in the literature, this
is not an easy endeavor due to peculiarities of healthcare expenses data:
great number of correlated independent variables, enormous variability, non-
linearity, excess of zeros and presence of extreme values [17]. In order to
circumvent these difficulties, we proposed an approach to healthcare expenses
prediction based on Markov Chains, which seeks to predict the annual expenses
of an individual based on his or her sex, age range and expenses in the last
two years. In this approach, first the expense level is estimated and then,
the exact expense is chosen from an empirical distribution.

Although classical in Statistics and Probability Theory, and applied in health
economics to predict health conditions and costs related to it ([24], [25],
[26], [27], [28], [29] and [30]), it seems that a Markov Chain approach may
also be useful to predict annual healthcare expenses, as it is known that
expenses history is an important factor in predicting healthcare expenses
[17]. A Markov Chain approach based on previous expenses, sex, and age range
may be more viable for pricing health plans than a method based on health
condition due to ethical issues.

Based on a dataset containing annual expenses of a portfolio of a Brazilian
self-management health plan, we simulated the proposed HSA product for 10,000
lives from 25 to 65 years-old in order to study the balance of the individual
savings accounts over time and at retirement, and the frequency and severity
of catastrophic insurance use. In this simulation, we took into account the
persistence of costs phenomenon, widely known to be part of individual
healthcare expenses. The results evidenced a low prevalence of persistence of
costs at the individual level, as at 65 years-old the balance of the
individual savings accounts is symmetrically spread around the average of R$
44,000, supporting that the adoption of this model is not a mechanism of
inequalities generation. During the 40 years period in the 10,000,000
simulated individuals, the total healthcare expenses of the simulated
portfolio was R$1,082,350,725 in average with s.d. of R$7,688,268, of which
R$489,367,594 in average (45.2%) with s.d. R$6,862,835 was covered by
catastrophic insurance. From the individual perspective, half the lives had
40% of their expenses covered by catastrophic insurance and only 5% of them
had more than 75% of their expenses covered by catastrophic insurance. At
retirement, the mean balance of the individual accounts was 45.2% of the value
deposited over the work life, so the mean expense covered by the HSA during
the work life was 54.8% that deposited, in average.

Therefore, we may conclude that a mixed CDHP product combining a HSA and a
catastrophic insurance may be viable in Brazil from a population point of view
as it does not create inequalities caused by persistence of costs. In order to
implement such a product, one needs to adapt the value deposited by the
employer every year and the limit covered by the HSA to suit the health system
costs.

We leave some topics for future research. An interesting topic would be to
compare a Markov Chain approach to healthcare expenses prediction with the
usual methods based on regression. Also, the development of a method which
incorporates the best features of both approaches seems promising. About the
proposed CDHP, there are some hyperparameters which can be optimized in order
to obtain a more realistic simulation: the break points of the expense levels,
the value deposited by the employer every year and the limit of expenses
covered by the HSA. In order to optimize these quantities, it would be needed
data with better quality and greater quantity than that of this paper.
Finally, to further the discussion in Brazil about co-payment and deductible
devices, it would be interesting to study other kinds of CDHP products in the
Brazilian context.

## Appendix A - Descriptive Statistics and Transition Matrix

### Descriptive Statistics of Annual Expenses, by sex

In this section, it is presented descriptive statistics by sex of the annual
expenses individuals by sex, between 25 and 65 years-old which stayed on the
portfolio during the whole five years period. The percentiles, mean and
standard deviation are calculated considering only the individuals with
positive expenses.

Table 2.1: Descriptive Statistics Annual Expenses (Female)  Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  13,539 |  13,539 |  13,539 |  13,539 |  13,539
PctNoExpense |  5.09 |  5.26 |  5.16 |  5.15 |  5.55
p25 |  457 |  422 |  449 |  487 |  490
p50 |  998 |  940 |  987 |  1,130 |  1,104
p75 |  2,193 |  2,029 |  2,152 |  2,539 |  2,464
p90 |  4,808 |  4,462 |  4,848 |  5,603 |  5,453
p95 |  8,402 |  7,501 |  8,247 |  9,338 |  9,662
p96 |  9,781 |  8,818 |  9,393 |  10,659 |  11,122
p97 |  11,293 |  10,376 |  11,416 |  12,757 |  13,915
p98 |  14,425 |  13,252 |  15,886 |  17,114 |  19,095
p99 |  24,076 |  21,371 |  24,681 |  27,696 |  30,718
p995 |  33,027 |  34,656 |  44,429 |  51,048 |  53,649
p999 |  68,154 |  125,250 |  98,976 |  137,450 |  203,682
max |  282,993 |  528,293 |  355,220 |  1,044,525 |  550,309
mean |  2,341 |  2,327 |  2,494 |  2,956 |  3,062
sd |  6,068 |  8,897 |  7,828 |  13,247 |  11,958

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 2.2: Descriptive Statistics Annual Expenses (Male)  Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  14,241 |  14,241 |  14,241 |  14,241 |  14,241
PctNoExpense |  6.81 |  5.77 |  6.35 |  5.93 |  5.97
p25 |  237 |  231 |  252 |  289 |  271
p50 |  580 |  572 |  625 |  721 |  681
p75 |  1,351 |  1,336 |  1,432 |  1,677 |  1,594
p90 |  2,983 |  2,914 |  3,208 |  3,927 |  3,868
p95 |  5,588 |  5,491 |  6,259 |  7,680 |  8,212
p96 |  6,847 |  6,838 |  7,833 |  9,690 |  10,494
p97 |  8,799 |  8,865 |  10,461 |  12,985 |  14,589
p98 |  12,720 |  12,836 |  14,960 |  19,653 |  21,593
p99 |  22,163 |  21,840 |  26,596 |  34,427 |  36,397
p995 |  35,525 |  36,021 |  42,975 |  59,041 |  57,155
p999 |  100,317 |  100,298 |  90,679 |  139,391 |  146,580
max |  426,772 |  371,231 |  266,119 |  286,716 |  965,287
mean |  1,795 |  1,768 |  1,956 |  2,473 |  2,576
sd |  7,490 |  7,306 |  7,235 |  9,786 |  13,316

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

### Descriptive Statistics of Annual Expenses, by sex and age category

In this section, it is presented descriptive statistics by age range of the
annual expenses of individuals between 25 and 65 years-old which stayed on the
portfolio during the whole five years period. The percentiles, mean and
standard deviation are calculated considering only the individuals with
positive expenses.

Table 3.1: Descriptive Statistics Annual Expenses (Female) 21a24 Description |  2005 |  2006 |  2007 |  2008
---|---|---|---|---
n |  555 |  372 |  210 |  75
PctNoExpense |  1.98 |  3.76 |  5.24 |  9.33
p25 |  339 |  349 |  390 |  291
p50 |  833 |  730 |  888 |  699
p75 |  1,974 |  1,740 |  2,278 |  1,334
p90 |  4,661 |  3,678 |  5,621 |  3,088
p95 |  6,495 |  5,171 |  9,039 |  6,091
p96 |  7,803 |  6,268 |  11,516 |  6,626
p97 |  8,963 |  7,937 |  12,664 |  6,749
p98 |  10,295 |  10,176 |  19,717 |  8,433
p99 |  14,738 |  11,447 |  29,037 |  10,056
p995 |  19,713 |  15,185 |  30,401 |  10,823
p999 |  74,328 |  32,586 |  63,712 |  11,437
max |  130,804 |  41,884 |  72,040 |  11,591
mean |  2,030 |  1,617 |  2,609 |  1,410
sd |  6,198 |  3,051 |  6,446 |  2,122

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.2: Descriptive Statistics Annual Expenses (Male) 21a24 Description |  2005 |  2006 |  2007 |  2008
---|---|---|---|---
n |  375 |  250 |  149 |  65
PctNoExpense |  10.93 |  12.00 |  14.77 |  15.38
p25 |  182 |  127 |  139 |  123
p50 |  415 |  322 |  366 |  296
p75 |  997 |  803 |  809 |  848
p90 |  2,338 |  1,710 |  2,080 |  1,682
p95 |  6,259 |  3,174 |  3,354 |  1,891
p96 |  8,641 |  3,473 |  4,105 |  1,938
p97 |  10,881 |  3,739 |  4,756 |  3,092
p98 |  13,469 |  4,524 |  8,094 |  4,712
p99 |  16,587 |  7,974 |  10,062 |  10,423
p995 |  25,255 |  20,678 |  10,517 |  13,634
p999 |  94,741 |  23,561 |  10,945 |  16,203
max |  128,157 |  24,017 |  11,052 |  16,846
mean |  1,688 |  889 |  923 |  899
sd |  7,596 |  2,362 |  1,762 |  2,330

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.3: Descriptive Statistics Annual Expenses (Female) 25a30 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  1,541 |  1,371 |  1,267 |  1,160 |  1,002
PctNoExpense |  4.67 |  4.60 |  3.95 |  4.31 |  5.29
p25 |  393 |  328 |  330 |  372 |  358
p50 |  907 |  737 |  769 |  880 |  880
p75 |  2,427 |  1,751 |  1,797 |  2,011 |  1,873
p90 |  5,535 |  4,493 |  4,694 |  5,093 |  4,647
p95 |  8,885 |  8,096 |  7,014 |  8,777 |  9,509
p96 |  9,876 |  8,901 |  7,996 |  10,444 |  10,602
p97 |  10,801 |  10,246 |  8,702 |  11,730 |  12,047
p98 |  12,642 |  11,821 |  10,653 |  13,740 |  14,197
p99 |  17,993 |  17,090 |  14,164 |  18,273 |  19,864
p995 |  24,900 |  21,846 |  27,135 |  29,606 |  27,090
p999 |  52,313 |  38,304 |  83,367 |  74,171 |  42,511
max |  85,433 |  52,910 |  118,517 |  80,355 |  57,494
mean |  2,279 |  1,887 |  2,002 |  2,223 |  2,114
sd |  4,416 |  3,656 |  5,623 |  5,058 |  4,204

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.4: Descriptive Statistics Annual Expenses (Male) 25a30 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  1,243 |  1,072 |  915 |  793 |  681
PctNoExpense |  7.48 |  7.74 |  8.09 |  10.47 |  11.01
p25 |  179 |  160 |  180 |  195 |  177
p50 |  406 |  379 |  433 |  450 |  416
p75 |  990 |  858 |  906 |  1,079 |  1,014
p90 |  2,192 |  1,927 |  1,802 |  2,349 |  2,500
p95 |  3,635 |  4,106 |  3,435 |  4,680 |  4,462
p96 |  4,197 |  5,306 |  3,999 |  6,085 |  5,243
p97 |  6,112 |  7,026 |  5,596 |  6,553 |  6,861
p98 |  9,181 |  9,000 |  7,460 |  11,110 |  10,335
p99 |  16,453 |  18,774 |  13,060 |  20,153 |  19,082
p995 |  26,250 |  21,917 |  19,561 |  35,258 |  54,969
p999 |  63,068 |  67,481 |  66,575 |  95,351 |  201,729
max |  229,290 |  128,291 |  199,011 |  147,129 |  323,604
mean |  1,388 |  1,290 |  1,255 |  1,578 |  2,038
sd |  7,579 |  5,432 |  7,288 |  7,035 |  14,706

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.5: Descriptive Statistics Annual Expenses (Female) 31a35 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  1,872 |  1,806 |  1,707 |  1,566 |  1,454
PctNoExpense |  5.02 |  4.32 |  4.98 |  4.92 |  5.85
p25 |  368 |  335 |  393 |  373 |  341
p50 |  845 |  817 |  863 |  942 |  804
p75 |  1,948 |  1,884 |  1,909 |  2,245 |  1,877
p90 |  4,659 |  4,735 |  4,757 |  5,259 |  4,887
p95 |  8,907 |  7,946 |  8,319 |  9,141 |  8,771
p96 |  10,201 |  9,012 |  9,227 |  10,004 |  9,550
p97 |  11,406 |  9,878 |  10,565 |  10,924 |  10,444
p98 |  13,272 |  10,757 |  13,401 |  13,968 |  14,317
p99 |  17,819 |  15,736 |  20,232 |  18,703 |  19,230
p995 |  28,856 |  24,549 |  31,337 |  27,130 |  33,526
p999 |  59,213 |  64,758 |  70,975 |  71,472 |  52,075
max |  87,084 |  101,389 |  98,995 |  141,717 |  69,533
mean |  2,146 |  1,964 |  2,174 |  2,342 |  2,101
sd |  4,815 |  4,583 |  5,234 |  5,756 |  4,532

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.6: Descriptive Statistics Annual Expenses (Male) 31a35 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  1,908 |  1,801 |  1,664 |  1,462 |  1,248
PctNoExpense |  7.39 |  6.89 |  8.17 |  7.25 |  7.05
p25 |  195 |  181 |  188 |  206 |  176
p50 |  460 |  436 |  475 |  527 |  483
p75 |  1,026 |  976 |  1,106 |  1,205 |  1,123
p90 |  2,010 |  2,044 |  2,342 |  2,862 |  2,561
p95 |  3,546 |  3,299 |  4,449 |  5,792 |  4,816
p96 |  4,413 |  3,778 |  5,336 |  8,042 |  6,489
p97 |  5,803 |  4,659 |  6,675 |  10,330 |  8,563
p98 |  8,308 |  7,118 |  9,835 |  12,979 |  12,292
p99 |  13,512 |  15,364 |  19,401 |  22,030 |  18,377
p995 |  22,473 |  21,891 |  25,576 |  42,934 |  21,802
p999 |  67,187 |  107,083 |  63,819 |  92,329 |  42,142
max |  130,614 |  172,207 |  90,771 |  113,230 |  61,267
mean |  1,243 |  1,295 |  1,357 |  1,792 |  1,396
sd |  4,771 |  6,417 |  4,446 |  6,534 |  3,639

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.7: Descriptive Statistics Annual Expenses (Female) 36a40 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  2,242 |  2,204 |  2,126 |  2,056 |  1,959
PctNoExpense |  5.62 |  6.72 |  6.40 |  5.54 |  5.77
p25 |  363 |  344 |  346 |  379 |  390
p50 |  793 |  762 |  791 |  916 |  937
p75 |  1,871 |  1,628 |  1,802 |  2,141 |  2,053
p90 |  4,520 |  3,644 |  4,375 |  5,012 |  4,378
p95 |  8,043 |  6,276 |  7,735 |  8,502 |  8,314
p96 |  9,237 |  7,244 |  8,981 |  9,400 |  9,402
p97 |  10,519 |  8,840 |  10,607 |  10,728 |  10,757
p98 |  12,118 |  10,912 |  13,138 |  13,291 |  13,554
p99 |  18,267 |  15,634 |  20,821 |  21,374 |  19,857
p995 |  25,417 |  31,130 |  56,363 |  40,889 |  27,142
p999 |  75,004 |  102,143 |  85,783 |  176,175 |  113,373
max |  157,006 |  528,293 |  92,919 |  318,932 |  221,151
mean |  2,058 |  2,154 |  2,153 |  2,745 |  2,336
sd |  5,657 |  13,605 |  6,006 |  12,851 |  8,471

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.8: Descriptive Statistics Annual Expenses (Male) 36a40 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  2,286 |  2,195 |  2,139 |  2,088 |  2,052
PctNoExpense |  7.09 |  6.33 |  6.69 |  6.90 |  7.26
p25 |  209 |  216 |  214 |  229 |  204
p50 |  513 |  496 |  500 |  553 |  512
p75 |  1,211 |  1,111 |  1,098 |  1,299 |  1,170
p90 |  2,682 |  2,424 |  2,386 |  2,679 |  2,744
p95 |  5,035 |  4,506 |  4,694 |  4,616 |  5,537
p96 |  6,375 |  5,409 |  5,287 |  5,164 |  6,540
p97 |  7,534 |  6,514 |  6,862 |  6,745 |  8,344
p98 |  10,579 |  9,115 |  9,601 |  9,955 |  11,419
p99 |  19,902 |  14,195 |  17,052 |  16,145 |  20,056
p995 |  24,752 |  21,963 |  34,069 |  22,558 |  31,768
p999 |  99,053 |  42,568 |  83,016 |  71,215 |  115,938
max |  230,437 |  162,392 |  211,526 |  286,716 |  142,242
mean |  1,564 |  1,327 |  1,535 |  1,558 |  1,635
sd |  6,815 |  4,665 |  6,983 |  7,595 |  6,326

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.9: Descriptive Statistics Annual Expenses (Female) 41a45 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  2,557 |  2,494 |  2,454 |  2,382 |  2,339
PctNoExpense |  6.14 |  6.01 |  5.30 |  5.58 |  6.67
p25 |  455 |  413 |  434 |  443 |  447
p50 |  949 |  859 |  912 |  985 |  977
p75 |  2,031 |  1,804 |  1,887 |  2,125 |  2,021
p90 |  4,369 |  3,735 |  4,018 |  4,842 |  4,648
p95 |  7,502 |  6,142 |  6,514 |  8,198 |  8,096
p96 |  8,718 |  7,183 |  7,572 |  9,834 |  9,821
p97 |  10,362 |  9,017 |  9,128 |  11,537 |  10,956
p98 |  13,856 |  11,100 |  12,013 |  14,644 |  13,852
p99 |  25,158 |  18,653 |  19,821 |  23,148 |  21,052
p995 |  34,825 |  30,076 |  28,599 |  35,298 |  50,353
p999 |  48,097 |  81,079 |  115,517 |  118,538 |  177,314
max |  66,973 |  162,119 |  196,400 |  201,497 |  254,654
mean |  2,140 |  1,993 |  2,188 |  2,503 |  2,563
sd |  4,509 |  6,006 |  7,444 |  8,070 |  9,987

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.10: Descriptive Statistics Annual Expenses (Male) 41a45 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  2,489 |  2,446 |  2,376 |  2,381 |  2,317
PctNoExpense |  7.35 |  5.27 |  6.02 |  5.50 |  4.92
p25 |  233 |  225 |  230 |  270 |  246
p50 |  559 |  534 |  566 |  673 |  573
p75 |  1,312 |  1,272 |  1,280 |  1,498 |  1,306
p90 |  2,765 |  2,692 |  2,832 |  3,318 |  2,987
p95 |  4,860 |  5,530 |  5,428 |  6,306 |  6,220
p96 |  5,590 |  6,698 |  6,697 |  7,610 |  7,884
p97 |  7,162 |  8,705 |  8,475 |  9,842 |  9,736
p98 |  10,509 |  11,371 |  12,263 |  14,657 |  13,208
p99 |  17,289 |  16,961 |  19,340 |  25,622 |  24,125
p995 |  32,462 |  25,310 |  39,384 |  40,011 |  32,908
p999 |  91,013 |  54,602 |  71,849 |  94,264 |  75,851
max |  117,357 |  267,607 |  244,532 |  140,615 |  114,218
mean |  1,584 |  1,600 |  1,718 |  2,005 |  1,735
sd |  5,447 |  6,817 |  7,229 |  7,051 |  5,285

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.11: Descriptive Statistics Annual Expenses (Female) 46a50 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  2,337 |  2,440 |  2,464 |  2,520 |  2,531
PctNoExpense |  4.45 |  5.00 |  5.24 |  5.40 |  5.33
p25 |  561 |  507 |  545 |  543 |  532
p50 |  1,131 |  1,058 |  1,121 |  1,234 |  1,133
p75 |  2,225 |  2,186 |  2,259 |  2,588 |  2,473
p90 |  4,625 |  4,572 |  4,471 |  5,128 |  5,336
p95 |  7,772 |  7,324 |  8,005 |  8,436 |  9,916
p96 |  9,564 |  8,366 |  9,267 |  9,812 |  12,467
p97 |  11,216 |  10,515 |  11,043 |  11,603 |  14,748
p98 |  14,077 |  13,077 |  14,062 |  15,898 |  17,793
p99 |  24,609 |  24,373 |  22,171 |  25,993 |  25,841
p995 |  27,863 |  37,853 |  35,379 |  55,079 |  45,159
p999 |  50,564 |  70,935 |  70,728 |  119,113 |  129,367
max |  89,188 |  177,710 |  207,664 |  203,918 |  348,062
mean |  2,296 |  2,322 |  2,434 |  2,795 |  2,946
sd |  4,482 |  6,132 |  6,664 |  8,123 |  10,707

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.12: Descriptive Statistics Annual Expenses (Male) 46a50 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  2,905 |  2,855 |  2,767 |  2,625 |  2,515
PctNoExpense |  5.61 |  4.59 |  5.35 |  5.22 |  5.61
p25 |  263 |  257 |  281 |  299 |  281
p50 |  646 |  617 |  674 |  740 |  680
p75 |  1,439 |  1,440 |  1,477 |  1,661 |  1,525
p90 |  3,139 |  3,108 |  3,250 |  3,873 |  3,545
p95 |  5,968 |  5,723 |  6,672 |  7,680 |  8,102
p96 |  7,164 |  7,470 |  8,149 |  9,585 |  10,445
p97 |  9,182 |  9,022 |  11,134 |  15,316 |  15,606
p98 |  13,455 |  12,634 |  15,129 |  23,472 |  22,146
p99 |  20,150 |  23,003 |  27,865 |  37,801 |  45,070
p995 |  30,414 |  35,122 |  48,033 |  62,346 |  70,808
p999 |  98,447 |  86,831 |  89,023 |  113,012 |  160,275
max |  426,772 |  137,333 |  124,276 |  275,813 |  397,015
mean |  1,921 |  1,808 |  2,005 |  2,585 |  2,747
sd |  9,858 |  5,929 |  6,495 |  10,411 |  13,107

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.13: Descriptive Statistics Annual Expenses (Female) 51a55 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  1,491 |  1,646 |  1,847 |  2,031 |  2,204
PctNoExpense |  4.63 |  3.65 |  4.44 |  4.38 |  4.95
p25 |  703 |  656 |  584 |  656 |  620
p50 |  1,394 |  1,265 |  1,207 |  1,388 |  1,337
p75 |  2,625 |  2,517 |  2,547 |  2,880 |  2,900
p90 |  5,317 |  5,024 |  5,686 |  6,299 |  6,104
p95 |  9,427 |  8,570 |  9,812 |  10,487 |  10,003
p96 |  11,851 |  10,229 |  11,274 |  12,000 |  12,111
p97 |  15,409 |  12,954 |  14,602 |  13,995 |  17,558
p98 |  20,293 |  19,269 |  19,709 |  21,897 |  27,194
p99 |  34,516 |  34,645 |  35,600 |  43,418 |  43,935
p995 |  56,503 |  78,735 |  55,339 |  53,830 |  75,109
p999 |  73,357 |  197,347 |  137,637 |  202,005 |  204,799
max |  282,993 |  297,188 |  355,220 |  1,044,525 |  279,515
mean |  3,130 |  3,355 |  3,188 |  3,859 |  3,680
sd |  9,842 |  13,890 |  12,160 |  25,844 |  12,885

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.14: Descriptive Statistics Annual Expenses (Male) 51a55 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  1,972 |  2,263 |  2,552 |  2,755 |  2,928
PctNoExpense |  5.27 |  4.33 |  5.21 |  4.39 |  4.64
p25 |  314 |  312 |  330 |  391 |  345
p50 |  750 |  754 |  789 |  884 |  836
p75 |  1,642 |  1,658 |  1,785 |  1,963 |  1,874
p90 |  3,515 |  3,551 |  3,787 |  4,571 |  4,220
p95 |  6,266 |  6,527 |  7,079 |  8,413 |  9,370
p96 |  7,223 |  8,288 |  9,252 |  10,396 |  12,245
p97 |  10,201 |  10,694 |  13,255 |  13,862 |  15,997
p98 |  13,822 |  15,581 |  20,222 |  23,473 |  24,546
p99 |  31,181 |  25,131 |  32,155 |  39,463 |  36,614
p995 |  58,903 |  40,441 |  44,668 |  67,096 |  51,064
p999 |  109,224 |  100,505 |  90,346 |  157,042 |  171,368
max |  112,453 |  371,231 |  129,112 |  267,595 |  965,287
mean |  2,177 |  2,221 |  2,273 |  2,893 |  3,027
sd |  7,391 |  10,533 |  6,786 |  11,132 |  20,648

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.15: Descriptive Statistics Annual Expenses (Female) 56a60 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  831 |  980 |  1,098 |  1,211 |  1,309
PctNoExpense |  6.26 |  6.33 |  5.19 |  4.54 |  3.97
p25 |  627 |  662 |  685 |  780 |  732
p50 |  1,366 |  1,346 |  1,432 |  1,615 |  1,648
p75 |  2,563 |  2,705 |  2,863 |  3,477 |  3,520
p90 |  5,487 |  6,278 |  5,818 |  6,573 |  7,741
p95 |  9,145 |  11,310 |  9,959 |  9,957 |  15,414
p96 |  11,318 |  13,667 |  11,264 |  11,932 |  19,931
p97 |  14,696 |  17,068 |  17,858 |  16,031 |  24,509
p98 |  19,522 |  20,310 |  23,083 |  19,755 |  29,453
p99 |  28,883 |  27,819 |  29,923 |  37,098 |  51,170
p995 |  56,251 |  40,600 |  40,949 |  43,065 |  94,078
p999 |  137,346 |  140,851 |  113,612 |  120,279 |  274,734
max |  209,437 |  222,438 |  148,681 |  340,890 |  550,309
mean |  3,157 |  3,258 |  3,175 |  3,659 |  4,931
sd |  10,157 |  9,886 |  8,071 |  12,243 |  21,379

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.16: Descriptive Statistics Annual Expenses (Male) 56a60 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  931 |  1,089 |  1,255 |  1,469 |  1,689
PctNoExpense |  7.73 |  5.88 |  5.58 |  4.49 |  5.09
p25 |  377 |  353 |  421 |  457 |  442
p50 |  965 |  857 |  1,019 |  1,072 |  1,000
p75 |  2,293 |  1,995 |  2,353 |  2,415 |  2,381
p90 |  6,331 |  4,743 |  6,129 |  6,873 |  6,665
p95 |  13,567 |  10,440 |  12,530 |  12,824 |  15,095
p96 |  17,177 |  15,045 |  16,844 |  14,812 |  21,437
p97 |  20,182 |  19,150 |  20,385 |  19,038 |  26,588
p98 |  34,044 |  26,450 |  28,116 |  24,707 |  38,919
p99 |  42,818 |  40,854 |  43,186 |  39,265 |  57,110
p995 |  75,095 |  68,072 |  54,982 |  61,353 |  70,828
p999 |  98,802 |  108,815 |  106,887 |  132,189 |  165,958
max |  105,272 |  168,989 |  144,749 |  153,098 |  201,089
mean |  3,397 |  2,973 |  3,259 |  3,311 |  3,791
sd |  9,557 |  9,741 |  8,983 |  9,333 |  11,983

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.17: Descriptive Statistics Annual Expenses (Female) 61a65 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  113 |  226 |  366 |  538 |  741
PctNoExpense |  3.54 |  6.64 |  5.19 |  6.69 |  6.48
p25 |  675 |  732 |  643 |  824 |  784
p50 |  1,250 |  1,415 |  1,509 |  1,803 |  1,732
p75 |  2,954 |  2,864 |  3,455 |  4,261 |  3,444
p90 |  6,080 |  5,454 |  8,467 |  9,869 |  7,437
p95 |  10,928 |  8,319 |  17,779 |  20,360 |  15,702
p96 |  11,122 |  11,551 |  20,114 |  25,948 |  19,334
p97 |  11,801 |  14,764 |  26,854 |  28,537 |  26,256
p98 |  12,380 |  16,414 |  33,172 |  37,319 |  30,266
p99 |  17,442 |  20,332 |  56,645 |  63,507 |  42,492
p995 |  19,103 |  44,430 |  60,399 |  71,529 |  126,540
p999 |  20,255 |  56,734 |  150,422 |  132,337 |  240,232
max |  20,543 |  59,669 |  195,519 |  135,830 |  274,612
mean |  2,603 |  2,898 |  4,505 |  5,116 |  4,905
sd |  3,544 |  5,850 |  13,086 |  12,251 |  18,070

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Table 3.18: Descriptive Statistics Annual Expenses (Male) 61a65 Description |  2005 |  2006 |  2007 |  2008 |  2009
---|---|---|---|---|---
n |  132 |  270 |  424 |  603 |  811
PctNoExpense |  8.33 |  8.89 |  8.25 |  7.63 |  7.52
p25 |  361 |  313 |  411 |  539 |  480
p50 |  744 |  940 |  1,026 |  1,283 |  1,133
p75 |  2,198 |  2,243 |  2,449 |  3,129 |  2,733
p90 |  4,427 |  5,697 |  5,275 |  8,083 |  7,854
p95 |  7,887 |  14,443 |  11,139 |  25,074 |  19,228
p96 |  8,416 |  16,886 |  13,019 |  31,824 |  23,151
p97 |  8,923 |  18,602 |  15,341 |  41,806 |  28,318
p98 |  10,875 |  38,905 |  24,083 |  49,420 |  35,593
p99 |  20,273 |  49,467 |  29,582 |  107,065 |  90,216
p995 |  26,916 |  53,310 |  59,040 |  152,434 |  112,752
p999 |  32,392 |  106,560 |  197,939 |  229,425 |  229,212
max |  33,761 |  123,830 |  266,119 |  237,957 |  320,963
mean |  2,021 |  3,500 |  3,398 |  5,923 |  4,872
sd |  4,053 |  10,600 |  14,966 |  20,614 |  18,301

n: total of individuals in portfolio;p(k): k-th sample percentile;SD: standard
deviation

Figure 7: Profile plot of the mean annual expenses by year, for each sex and
age range considering only individuals between 25 and 65 years-old which
stayed on the portfolio during the whole five years period. Figure 8:
Transition matrices of expenses for each two years combination, i.e. from
$2005$ to $2009$, considering individuals with ages between (a) 21 to 40
years-old, and (b) 41 to 65 years-old.

## Appendix B - Markov Chains and HSA simulation

The Markov Chain approach proposed in this paper aims to predict the expense
level of an individual in a given year by estimating the probability of his or
her annual expenses being at every level given the expense level of previous
years. Formally, considering the definition of $F_{j,i}$ presented in the
section 2.5.1, the probability of an individual with annual expenses in
$F_{k,i}$ at year $i$ having annual expenses in $F_{l,j}$ at year $j$ is
estimated as:

$\hat{P}(F_{k,i},F_{l,j})=\dfrac{\text{Number of individuals in $F_{k,i}$
which are also in $F_{l,j}$}}{\text{Number of individuals in
$F_{k,i}$}},\text{ with }j>i$ (1)

In Markov Chain Theory, $\hat{P}(F_{k,i},F_{l,j})$ is an estimator of the
transition probabilities between states $F_{k,i}$ and $F_{l,j}$. Indeed, the
ratio above is the proportion of individuals which were in $F_{k,i}$ at year
$i$ and transitioned to $F_{l,j}$ at year $j$. Therefore, if we were to
estimate the probability of an individual in $F_{k,i}$ to be in $F_{l,j}$ at
year $j$ we could use the proportion $\hat{P}(F_{k,i},F_{l,j})$, which is a
consistent estimator for such probability [31].

The approach above may also be applied to estimate the transition
probabilities for individuals of a given sex and/or age range. This
probability would allow to study the persistence of costs in distinct groups
of individuals, as it is believed to behaviour differently for each age range
and sex. The estimation of these probabilities would be done in the same
manner as above, but the number of individuals considered in the ratio of
$\hat{P}(F_{k,i},F_{l,j})$ would be that within the group of interest.
Proceeding this way, we have a prediction for the probability of an
individual, with given sex, age range and expense level in a previous year, to
be in each expense level in the current year.

A transition matrix characterizes a Markov Chain, that is a sequence of random
variables in which the distribution of the current random variable depends
only on the value of the previous $k$ and is given by the probabilities of the
transition matrix, in which $k$ is called the order of the Markov Chain. If
$k=1$ then the transition matrix is calculated as the ratio given in (1) when
$j=i+1$. If $k=2$, then the probability of being in a given state after
visiting some in the past depends only on the last two states visited. In this
case, the Markov Chain is generated by the transition probabilities estimated
as

$\hat{P}(F_{k,i-2},F_{m,i-1};F_{l,i})=\dfrac{\text{Number of individuals in
$F_{k,i-2}$ which are also in $F_{m,i-1}$ and $F_{l,i}$}}{\text{Number of
individuals in $F_{k,i-2}$ which are also in $F_{m,i-1}$}}$ (2)

which refers to the transition to $F_{l,i}$ after being in
$F_{k,i-2},F_{m,i-1}$ at the previous two years. These probabilities may also
be estimated for a given group of individuals, considering the numbers in the
ratio to be that within the group.

With a Markov Chain of order 2 we may estimate the probability of an
individual with given sex and age range to be in each expense level as a
function of the levels he or she was in the last two years. This is the method
we use to predict the expense level of each life in the simulated portfolio.
Observe that it diverges from the usual methods based on regression as we do
not try to predict the exact value of the annual expense, but rather the
expense level, so we do not have to assume a distribution for the expenses,
nor a functional relation between the current year expenses and the
independent variables sex, age range and previous two years expenses.

Transition Matrices: In order to perform the first step of the simulation, we
need to estimate a transition matrix between the expense levels. For this
purpose, we suppose that the expense level follows a homogeneous Markov Chain
of order 2 [12]. This means the transition from one state to another does not
depend on the year, but only on the states. Formally, this means that

$P(F_{k,i-2},F_{m,i-1};F_{l,i})=P(F_{k,j-2},F_{m,j-1};F_{l,j})$ (3)

for any $i,j$, in which $P$ is the population transition probability, in
contrast to the estimated transition probability $\hat{P}$. With this
assumption, we may estimate the transition probabilities by that of the time
period 2007 to 2009. Therefore, we estimate the transition from expense levels
$F_{k},F_{m}$ to $F_{l}$ as

$\hat{P}(F_{k},F_{m};F_{l})=\hat{P}(F_{k,2007},F_{m,2008};F_{l,2009})$ (4)

in which we consider only the individuals within the respective sex and age
range group in the ratios that define these transitions. This assumption is
supported by the matrices in Figure 1, where we see that those which compare
years with the same distance are similar (these are the matrices in a same
diagonal of Figure 1).

Under this approach, we have 16 transition matrices, one for each combination
of sex and age range (26-30, 31-35, 36-40, 41-45, 46-50, 51-55, 56-60, 61-65).
Each matrix has 16 rows, one for each pair of expense levels in the current
and last year, and 4 columns, one for each possible expense level in the next
year. The entries of these matrices are transition probabilities from the
state given by the pair to each one of the expense levels. To estimate the
transitions, we consider that the age range of each individual in the dataset
is the one which he or she was part of for the most number of months in the
triennial 2007-2009.

First step: The expense level of an individual next year is predicted based on
his or her sex, age range and expense level last year and in the current year.
This prediction is sampled from the estimated conditional distribution of the
expense levels of given sex, age range and two last expense levels, which is
the row associated to the last two expense levels of the transition matrix of
the given sex and age range. We start this process at the initial values for
24 and 25 years-old to estimate the level at 26 years-old. We then iterate
this process to estimate the level for the following ages: use the estimated
values for 25 and 26 to predict 27 years-old; that of 26 and 27 to predict 28
years-old, and so on until the age of 65. Therefore, for each one of the
10,000 lives, we have a sequence of 41 levels corresponding to its predicted
expense level for each work life year. Observe that, from age 27 on, the
expenses history is predicted in the previous two years.

Second step: After we simulate the expense level of a life in a given year, we
need to simulate the value to be withdrawn from its account to cover a
healthcare expense in such level. This is done by sampling a point from the
empirical distribution of the aggregated annual expenses of all years which
contain only the individuals with the same sex and age range of the life.
Within this empirical distribution, we sample one of the points inside the
expense level predicted on step one, i.e., inside the respective dotted lines
in Figure 9, which shows the aggregated empirical distributions of the
logarithm of the annual healthcare expenses between 2005 and 2009 for each sex
and age range.

Figure 9: Empirical distributions of the logarithm of the annual healthcare
expenses between 2005 and 2009. The distributions aggregate points from the
five years. The dotted lines are respectively $\log(300)$, $\log(1000)$ and
$\log(5000)$, which are the break points of the expense levels $F_{i,j}$.

In Table 4 we see the number of expenses in the dataset at each expense level
by sex and age range. These percentages aggregate all expenses of the period,
so they refer to the percentage of expenses, rather than the percentage of
individuals, as each individual has five expenses, one for each year. In the
dataset, 25% of women’s expenses and 45% of men’s in the age range 25-30 are
in the first level (up to R$ 300), percentages which decrease with age,
attaining the values of 15% and 23% for women’s and men’s, respectively, at
the age range 61-65. As the age increases, the same reduction is observed in
the percentage of expenses in the second expense level (R$ 300 - R$ 1,000) for
both sexes. On the other hand, the percentage of expenses in the third expense
level (R$ 1,000 - R$ 5,000) increases with age and the difference between
these percentages in age ranges 25-30 and 61-65 is 14 and 17 percentage points
for women’s and men’s, respectively. The same is observed for the fourth
expense level (greater than R$ 5,000): 10% of women’s expenses and 4% of men’s
in the age range 25-30 are greater than R$ 5,000 in opposition to 16% of
women’s and 12% of men’s in the age range 61-65. We see in the last three age
ranges (after 51 years-old) that around 58% to 62% of women’s expenses, and
40% to 48% of men’s, are greater than R$ 1,000.

|  |  Female |  |  |  | Male |  |
---|---|---|---|---|---|---|---|---
Age Range |  [0-300] |  (300-1000] |  (1000-5000] |  >5000 |  [0-300] | (300-1000] | (1000-5000] | >5000
25a30 |  1,597 |  2,071 |  2,076 |  597 |  2,138 | 1,531 | 861 | 174
|  (25.2%) |  (32.7%) |  (32.7%) |  (9.4%) |  (45.5%) | (32.5%) | (18.3%) | (3.7%)
31a35 |  2,091 |  2,741 |  2,790 |  783 |  3,374 | 2,690 | 1,703 | 316
|  (24.9%) |  (32.6%) |  (33.2%) |  (9.3%) |  (41.7%) | (33.3%) | (21.1%) | (3.9%)
36a40 |  2,678 |  3,581 |  3,485 |  843 |  4,121 | 3,683 | 2,473 | 483
|  (25.3%) |  (33.8%) |  (32.9%) |  (8.0%) |  (38.3%) | (34.2%) | (23.0%) | (4.5%)
41a45 |  2,633 |  4,133 |  4,522 |  938 |  4,153 | 4,134 | 3,080 | 642
|  (21.5%) |  (33.8%) |  (37.0%) |  (7.7%) |  (34.6%) | (34.4%) | (25.6%) | (5.3%)
46a50 |  2,198 |  3,765 |  5,194 |  1,135 |  4,201 | 4,703 | 3,892 | 871
|  (17.9%) |  (30.6%) |  (42.3%) |  (9.2%) |  (30.7%) | (34.4%) | (28.5%) | (6.4%)
51a55 |  1,354 |  2,545 |  4,258 |  1,062 |  3,265 | 4,117 | 4,166 | 922
|  (14.7%) |  (27.6%) |  (46.2%) |  (11.5%) |  (26.2%) | (33.0%) | (33.4%) | (7.4%)
56a60 |  818 |  1,313 |  2,591 |  707 |  1,527 | 1,877 | 2,315 | 714
|  (15.1%) |  (24.2%) |  (47.7%) |  (13.0%) |  (23.7%) | (29.2%) | (36.0%) | (11.1%)
61a65 |  309 |  442 |  917 |  316 |  524 | 646 | 796 | 274
|  (15.6%) |  (22.3%) |  (46.2%) |  (15.9%) |  (23.4%) | (28.8%) | (35.5%) | (12.2%)

Table 4. Percentage of expenses in each expense level by sex and age range.
The expenses of all years are aggregated, so these percentages refer to the
percentage of expenses, rather than the percentage of individuals, as each
individual has five expenses, one for each year.

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