CHAPTER ONE

INTRODUCTION

Background of the Study

Insurance is the purchase of security or the assured, anxious to protect oneself against a risk. It is a form of risk management strategy used to hedge against risk of contingent loss. Vehicle usage has embedded morbidity and mortality risk plus loss through theft and fire in several countries (Maddala G.S., 2005). In view of this Motor vehicle insurance was developed to take care of the possible loss that may arise from use of motor vehicles (Amoo G. K, 2002; Onafalujo, et al, 2011). This take the form of equitable transfer of the risk of a loss, from the vehicle owner to the insurer in exchange for a premium, and can be thought of as a guaranteed for devastating loss. Motor insurance, which is also known as Automobile insurance, is gradually becoming the most common form of insurance. Motor insurance was introduced to protect motorist from potentially enormous financial loss from operating a vehicle. Therefore, policy makers require motorist to purchase motor insurance cover to protect innocent third parties as well as the art fault motorist from liability (Outreville, J.F 1990, Aeron-Thomas, A. 2002).

In Ghana, there are primarily two types of motor insurance; namely, third party insurance and comprehensive motor insurance. Third party insurance covers only the cost of repairing the vehicle following an accident while comprehensive motor insurance policy cover the risk of the cost of repairing the vehicle following an accident, the cost of a new vehicle if it is stolen or damaged beyond economic repairs as well as legal liability claim against the driver or owner of the vehicle following the vehicle causing damage to a third party (Esho, N.,et al., 2004).

Comprehensive car insurance is not an obligation in most countries of the world. However, if a car owner want to guard against financial jeopardy, comprehensive auto insurance is the best because it covers compensation for car accidents and other kinds of misfortune. If damage is made to another party, harm to the person or damage to a property, this car insurance policy will safeguard policy holder in that kind of situation. It also covers the policy holderâs own vehicle and medical expenses. However, this policy has the highest premiums compared to the third party auto insurance policy. Moreover the coverage for the damages to policy holderâs car and the other party’s vehicle, comprehensive insurance covers damages that consequences to other

non-car crash incidents.

In Ghana, insurance is an accepted thing to do, as everyone must protect themselves against risks in life. The needs of insurance in Ghana for example cuts across travel insurance, fire insurance, motor insurance, marine insurance etc. Though quite competitive, the governing laws of insurance have also increased with time. For example, in Ghana, Insurance Act 58 stipulates that all car owners insure their vehicles against unintended risks and damages. Despite the many posing questions asked on insurance, Ghanaians are gradually accepting the fact of paying premiums to protect themselves and their loved ones in case of loss and damage.

According to USAID (2006) report, although insurance has become a widely used financial service since first introduced in ancient Greece, its benefits have yet to reach much of the developing world. Evaluate the potential to develop insurance schemes, annuity markets and insurance products for low-income populations; it is believe some countries like Ghana may have great potential to benefit from the development of specific lines of insurance, given adequate infrastructure and conditions. Outreville (1990) posits that the share of total insurance premium generated in developing countries remain at low figure even though these countries contain more than 75 percent of the world population.

The risk involved in allocating insurance funds has boosted the need for statistical methods and principles. Insurance funds may be invested in several asset categories, which include bonds, equities and others with the aim of increasing investment real returns to meet claim payment demands and other financial obligations. The puzzle of percentage allocation of insurance fund into investment portfolios is subject to insurance liabilities of which part is systematic and the other stochastic. Insurance liability relating to stochastic constraints, which is of much concern to this research work forms comparatively huge part of the total liability of insurance companies. It is solely linked to claim demands and uncertainties relating to it by which the policy contract mandates the insurer to forfeit. Insurance companies employ statistical methods of estimation to acquire concrete information about these uncertain liabilities with which decisions pertaining to assets allocation, expected monthly claims and payment targets as well as future insurance pricing can be ascertained. This is known in insurance administration as the expected claim liability estimation problem.

The concern of an insurer when establishing a base premium is to ensure that the premium is sufficiently large to fulfill its financial obligations. Experience rating systems in general and credibility theory in particular constitute an efficient step to determine a fair premium. Experience rating system takes into account the past individual experience of an insured when establishing the insuredâs premium.

Statement of the problem

Claim estimation is an important component of automobile insurance companies in the world and Ghana is no exception. However, expected claim liability estimation is a general problem for all insurance companies in Ghana. Although most of them try to use various statistical methods of estimation to get uncertainty liabilities with decisions pertaining to assets allocation, expected monthly claims, payment targets and future insurance pricing. The concern of an insurer when establishing a base premium is to ensure that the premium sufficiently large to fulfill its financial obligations.

Another challenge for the actuaries then is to employ a proper statistical technique to analyze insurance data. Claims and risks have long been estimated using a pure algorithmic technique or a simple stochastic technique (WÃ¼thrich & Merz, 2008). These methods result in poor estimations. Huang, Zhao and Tang (2009) consider a risk model in which the claim number process is treated as a Poisson model and the individual claim size is assumed to be a fuzzy random variable. JÃ¸rgensen and Souza (1994) suggested a Poisson sum of Gamma random variables called Tweedie to estimate insurance risk. According to Smyth and JÃ¸rgensen (2002), there is also another problem in that the proposed Tweedie model does not permit the separate estimation of probability and claim size.

Experience rating systems in general and credibility theory in particular constitute an efficient step to determine a fair premium. It takes into account the past individual experience of an insured when establishing the insuredâs premium. For claim actuaries, claim modeling is very crucial since a good understanding and interpretation of loss distribution is the backbone of all the decisions made in insurance industry regarding premium loading, expected profits, reserves necessary to ensure profitability and the impact of re-insurance (Boland, 2006).

General insurance companies are unable to find an appropriate statistical distribution for their large volumes of claims severity and test how well this statistical distribution fits their claims data. Hence, the need to use the Bu Ìhlmann credibility model which easily computes a certainty estimates for future claim expectation.

1.3 Objectives of the Study

The main objective of the study will be to model the number of auto insurance claims occurring under motor accidents using Bu Ìhlmanns credibility model. The specific objectives of the study would include the following:

To develop Bu Ìhlmanns credibility model, predicting auto insurance claim counts

Use the model to calculate credibility weighted estimates for the expected number of claims per year for each policy holder.

Use the weighted estimates to estimate the number of claims in 2015

1.4 Justification of the Study

Insurance is the most important parts of welfare system. Insurance provides financial protections against unplanned losses that are unforeseen and unexpected and facilitates a redistribution of social benefits in society. The basic idea of insurance is that individuals transfer their risk to an insurance company, and for this they pay premiums. In the light of the contributions of the insurance companies in the countryâs developmental agenda, this study would serve as a source of information on insurance claim liabilities for most of the insurance companies in Ghana .Furthermore, the findings of the study would also enlighten policy holders to understanding some issues on insurance claims.

The study will help the experts in the motor insurance companies in their informed decisions by appreciating the challenges on the field and also advised the underwriters from this position of knowledge. Finally, the study is expected to add knowledge to the existing insurance claim liabilities procedures since they need information on claim counts to estimate insurance premiums and would also serves as a basis for future research.

1.5 Methodology

The data for the study would be obtained from the motor department of the Quality Insurance Company. The data will be on insured vehicle specifying; the type, weight of the vehicle, age of entry, sex and claim counts of the policy holder . Other literature materials on estimation of auto insurance claims were collected from National Insurance Commission.

In this study Bu Ìhlmanns credibility model would be employed as the main methodology to analyze the data.

1.6 Scope of the Study

The study will cover only the Quality Insurance Company and its insurance claim liabilities due to the limited time period available within the academic calendar to conduct this study. But the findings from the study can be generalised to insurance companies in the country.

1.7 Organization of the Study

The study would be organized into five chapters. Chapter one will provide introduction which include, background of the study, statement of the problem, objective of the study, justification of this study, methodology of the study and the scope of the study on auto insurance claim liabilities. In chapter two, we shall present a review of some related literatures on insurance companies of Ghana and auto insurance claim liabilities. Methodology which would contain a development Bu Ìhlmanns credibility model would be done in chapter three of the study. The model implementation and discussion of results obtained would be detailed in Chapter four. Finally, the conclusions, and recommendations of the study will be presented in chapter five.

2.0 Introduction

This chapter will review theories on mortality and discuss extensively the factors affecting mortality and deliberate on various literature on mortality in the Ghanaian environment.

2.1 Mortality theories

The study of mortality has always been a natural subject for actuaries. Benjamin Gompertz (1825) first introduced his theories which was then said to be based on âphilosophical principlesâ

As some prefer to describe his theories as scientific and biological principles. Barnett (1968)

William Makeham (1860) made some additions to Gompertz theories which is an age independent component, to the exponential growth.

Other theories also followed up which served as bases for various models such as Heligman and Pollard laws, Heligman and Pollard, (1980). The Gompertz-Makeham function described by Forfar et al. (1988) simplifies the original models proposed by Gompertz and Makeham. Renshaw (1991) and Renshaw et al. (1997), aslo used generalized linear and non-linear models for adjusting the Gompertz and Makeham models. DebÂ´on et al (2005).

Most of these models has itâs fundamentals on two component parts of mortality, one being the rate of senescent deaths, and respective rates of different types of anticipated deaths.

2.2 Factors affecting mortality

There are various factors which play important roles in determining the mortality of groups of people. These factors are intermingled and very difficult to disassociate one from the other. Yet it is very necessary for the government to identify specific mortality risk factors in order for measures to be put in place to curtail the effects. Aside age and sex, on a broader note, standard of living can be used to describe the major factor influencing mortality, and this includes various factors such as occupation, nutrition, housing, climate, education and genetics.

Some critics do argue that death rate is peculiar to each individual as the various elements contributing to quality of life varies from one person to another. This seems quite plausible as there are some information which may be privy to the individual only and computing mortality based on these known facts may be more accurate. In spite of its merits this approach may be very time wasting when dealing with a large population, where grouping lives into homogeneous groups will be faster in producing results.

2.2.1 Education and mortality

Various studies have provided evidence that higher educational levels reduce mortality Sorlie et al. (1995) as cited in Whitehouse and Zaidi (2008) , Showed that in the US 60% of working aged men with 16 years of education are 60 % less likely to die than those with 12 years of education and those with 5-8 years of education are 35% more likely to die. Huisman et al. (2004) , Elo and Preston (1996), had similar finding though their percentages do vary. Attanasio and Emmerson (2001) on the other Hand argue based on their findings that education affects morbidity but not mortality. Whitehouse and Zaidi (2008) argue that education is related to mortality mainly because it increases income. And higher income implies lower mortality.

On the other hand the relationship between mortality of women and educational level is proven to be quite weak. Huisman et al. (2004). Preston and Elo (1995) also find that the death rates inequalities due to educational levels for men has widened whereas this has contracted for working â”age women and has also widened for pensioners than people of working age.

2.2.2 Income and mortality

The effect of income on mortality has been widely researched as majority argue that higher income reduces mortality. For pensioners the major source of income is the monthly pension received.

Pension is an amount promised by an organization to persons on retirement until their death. In order to be able to have an ideal or fair amount of pensions to be paid, the mortality experience should be estimated in order to have a fair idea of life expectancy. However life expectancy has exceeded previous estimates and this poses a major challenge to insurance companies. This implies that mortality in recent times has decreased dramatically as people tend to live longer.

Can this be attributed to the fact that people are receiving higher income hence lower mortality? or are there other factors playing a major role in the mortality experience of recent times. Well, in spite of the widely accepted relationship between income and mortality there are some studies which have different views.

According to Synder and Evans (2006) lower pension results in low mortality in spite of the positive relationship established between health and mortality. It is generally assumed that the higher pensions should avail the pensioner to better health facilities and healthier lifestyles implying a longer healthier life. There are other views that generally not because of higher income or higher socio economic background that improves mortality but rather implementation of national health insurance which helps provide better health facility for economically developing countries. In contrast to these findings Longue and Black (2004) established that higher pensions results in lower mortality which tends to follow majority of other research Pappas et al. (1993), Schalick et al. (2000). In longue and blacks study however the pension amount was dependent on the level of disability and pensions were computed based on the average pension received between the period 1874 and 1890.

2.2.3 Occupation and mortality

The job people do greatly affects their mortality. Some jobs are highly risky in terms of been prone to accidents whereas other exposes workers to toxic chemicals. Some jobs are also not physically engaging leading to poor health. Occupations also reflects the educational level and level of income, therefore it is expected to have a strong relationship with mortality.

Researchers have tried to establish the relationship between occupation and mortality and a strong link has been established by many. Sorlie et al. (1995) and Rogers et al. (2000) show that higher occupations with higher income results in lower death rates.

2.3 Pensioners

Pensioners on the other hand are a category of people who have retired from active service either due to old age or invalidity.

The population of pensioners have are in three groups, classified according to the reason for pension, they are as follows;

Pensioners who retired at the normal retirement age and are now receiving a pension. In Ghana the normal retirement age is 60 years.

There are pensioners who retired under the ill-health retirement rules of the scheme and are now receiving a pension benefit. This is termed invalidity pension.

And lastly pensioners who retired under the early retirement rules of the scheme and are now receiving a pension benefit, in Ghana the early retirement category fall between ages 55-59 years.

But for the purpose of this research, the focus will be on people who have retired due to old age and early retirement only. In this regard it is prudent to imply that the factors facing mortality of pensioners will be different from the factors influencing mortality of the youth or people in active service.

2.3.1 Pensioner Mortality; the Ghana experience

Mortality rates, in recent times has been decreasing globally, hence emphasizing the critical importance of investigation into mortality forecasting and factors influencing this phenomenon. Many researchers have made projections based on falling death and birth rates with most countries having estimates of older persons dominating the population than younger persons in the next few years.

According to Johnson et al (2005) Japan has 31.6% of its population being 60 years and above followed by Europe with 22% and Africa having a population which is ageing progressively faster with 10% of its population aged 60 years and above by 2050.

Ogwumike and Aboderin (2005) made an assessment that the population of Africans who are 60 years and above will increase four fold from 45.7 million to 182.6 million in 2050. The aged in the African population will concentrate more in the sub-Sahara specifically Ghana which will have the highest rise in old age population.

Naturally as the human body ages beyond certain limits it becomes weaker, productivity reduces drastically and become more prone to diseases and consequently death. So by the laws of nature, as we grow older our mortality becomes higher. However in the past few years the average life expectancy has been increasing as according to Bizz 101 (2015) ,in 1980 life expectancy in Ghana was 51 for men and 53 for women and as at 2012 life expectance has increased to 62 for men and 64 for women. This brings into light the fact that, the average life expectancy has increased by 11 years over the period of 32 years. But however, as this is worth celebrating for the human race in general, insurance and pension industries may deem this development as draining their profit away since they will keep paying life annuities and pensions as long as the individual is alive. These finding instigated the argument that the legal retirement age should be increased in order for the country to benefit from its experienced older population.

The increasing life expectancy over the years implies that in the near future the population of older ages will keep increasing as stated by Ogwumike and Aboderin (2005) they gave estimates that in 2005, 5.4% of the Ghanaian population were 60 years and over and projected 7.6% and 14.6% of the population to be 60 and over in 2025 and 2050 respectively.

Pensioners in Ghana however are still in the bracket of high mortality as compared to other countries which has life expectancy way above 62 and 64, and there are various reasons for this.

The major threat facing mortality of pensioners in Ghana is poverty which affects the physical emotional and mental wellbeing. In Ghana and many other African countries, policies to reduce poverty are mainly focused on children, mothers and the youth. Hence the quality of life after pension becomes the sole responsibility of the pensioner with reliance on the meager pension salary and family support which is not even guaranteed. Recently there has been some efforts by the ministry of gender to help senior citizens by providing free medical care for people over 65 years.

According to the 2010 Ghana population and housing census older people in Ghana are in the following age brackets 60- 74 (young-old), 75- 84 years (old-old) and 85+ years (very old). These categories of older persons are mostly pensioners, both in the formal and informal sectors of the economy.

The vast majority of the older persons are in the informal sector with minority in the formal sector. Most pension schemes however are focused on the small minority of public and other formal sector workers leaving larger majority with no income security after pension.

In Ghana, aside pension coverage for formal sectors there are provisions for pension coverage for the informal sector which are both managed by The Social Security and National Insurance Trust. This may have contributed to Ghana having a rather increasing ageing population as compared to other African counties.

However it is an established fact that older pensioners have lower income than their younger counterparts and even worst off when compared with younger working age people. In Johnson and Stears (1995) findings from UK data 80 year olds were 20% worse off than 65 year olds and in US, Radner (1987) finds that 65-69 year olds had incomes 30% higher on average than 80-84 year olds. Therefore generally poorer populations in most countries constitute majority of pensioners. Having low income has its implications with mortality which isnât quite favorable.

Research on the Ghanaian mortality experience is broadly focused on child and maternal mortality. With very few focus on mortality of pensioners.

From the Ghana Statistical Service 2010 census data age specific death rates were computed from ages less than one to 80 years plus, to compare mortality at different ages and sexes. However in all age groups female death rates are lower than male death rates. In age groups 50-54 years and above, females have lower death rates than males in urban areas than rural areas. Form 75-79 females in rural areas have higher death rates than those in the urban areas. On a broader note however there isnât much difference in death rates between ages 15-54, but ages 55 and above have quite large differences in death rates for both males and females.

Source: 2010 Population and Housing Census;

National Analytical Report (2013)

Fig 2.1 Age patterns of mortality by sex (2010)

2.4 Approaches to mortality estimates

Mortality forecasting has been a major research area in recent times due to increasing longevity patterns worldwide. Predicting mortality experience is very complex and a challenging task and many disciplines have interest in this area hence different models and approaches have been developed. We have the casual models or non-demographic models and the actuarial and demographic models. Keilman (2003), further explains that Demographic or actuarial models are typically collective models which describe how the mortality of groups of individuals evolves over time. It typically uses historical data to estimate changes over time and usually, mortality risks are primarily broken down by demographic variables such as age and sex. Non-demographic or causal models emphasize causal factors in mortality such as health, socio-economic status, environmental conditions and access to health care, Hudson (2006). That is, the focus here is the factors that influence the life chances of the individual.

According to Booth and Tickle (2008) mortality modeling are categorized in three broad approaches, which are expectation, explanation and extrapolative approaches. Generally the expectation approach is subjective and conservative, explanation approach is restricted to certain causes of death with known determinants and the extrapolative approach makes use of statistical methods of estimation, its emphasis is on age patterns and trends over time.

In most cases of mortality forecasting some aspects of each approach is applied, hence exact distinction between the three approaches is not clearly spelt out.

2.4.1 Expectation approach

This approach is purely based on expert opinion, most statistical agencies and actuaries were known to have heavily relied on this approach in the past, but currently are adopting a more complicated extrapolative approach. Continuous Mortality Investigation Bureau (2002, 2004, 2005, 2006, 2007) as cited in Booth and Tickle (2008).

Targeting of life expectancy is one method that was mostly used, here a value is assumed for a future date and a specified path is determined Olshansky (1988).

Targeting has also been conventionally applied to age-specific mortality reduction factors. The Continuous Mortality Investigation Bureau (CMIB) of the UK Institute of Actuaries and Faculty of Actuaries has projected the proportion of lives of exact age x who die before attaining exact age x+1,q_x, from year 0 by multiplying by the t -year reduction factor RF(x,t)

RF(x,t)= a(x)+[1-a(x)].ã[1-f_n (x)]ã^(t/n)

a(x) and f_n (x) represent respectively the ultimate reduction factor and the proportion of the total decline ( 1-a(x)) assumed to occur in n years. The approach embodies an exponential decline in mortality over time to the asymptotic value, a(x)and uses expert opinion to set targets a(x) and f_n (x).Booth and Tickle (2008).

The merits of the expectation approach is that it includes epidemiological and demographic factors, and other relevant information in a qualitative way. This approach is however prone to bias and subjectivity, expert opinion is also subject to âassumption dragâ. Where expectation trial behind actual experience instead of the other way round.

2.4.2 Explanation approach

Explanatory methods of mortality forecasting are based on structural or causal epidemiological models of certain causes of death involving disease processes and known risk factors. Boot and Tickle (2008).

This approach needs an in-depth understanding of medical conditions and socio economic situation that leads to death. The understanding of causes of death and mortality is yet to be fully developed. Therefore the use of this approach solely is rare. Most often a hybrid of extrapolative, expectation and explanation approach is used.

Most of the models used in this approach are regression based and therefore are within the generalized linear model framework Tabeau et al (2001). The difference between this and regression based extrapolative model is that it incorporates explanatory variables or risk factors, which are either lagged or forecast.

Boot and tickle further state that the main advantage of the explanatory approach is that feedback mechanisms and limiting factors can be taken into account.

2.4.3 Extrapolative approach

As noted earlier this approach makes use of historical data and the assumption here is that past trends will continue into the future. This assumption cannot always hold as some excerptions do occur that will alter trends. Situation such as the outbreak of EBOLA, AIDS or other epidemics or natural disasters which can claim the lives of large numbers of the youth. This approach is used for long term forecasting hence large data samples is most appropriate.

In earlier years mortality forecasting was quite simple and it involved some degree of subjective judgment Pollard (1987). Until quite recently, more complicated approaches were developed of which the Lee Carter model has been predominant Stoeldraijer et. al (2013). This method summarizes age specific mortality and period for a single population and how mortality evolves over time. Lee and Carter 1992, Stoeldraijer et. al (2013).

The strengths of the extrapolative approach hence the Lee Carter model includes its robustness in age specific mortality with linear trends Boot et al (2006). The Lee Carter model can also be seen as a special case of log â”linear model for contingency tables Bishop et al (1975), King (1989).

2.5 Some models for mortality estimates

Ronald D. Lee and Lawrence Carter introduced the Lee Carter (LC) model in 1992 using US data to forecast age-specific mortality from 1990 to 2065.(Lee, Carter 1992) this method is most widely used for forecasting age-specific mortality today.

Girosi and Gary (2007) show that this model is a special case of a considerably simpler, and most often unbiased, random walk with drift model.

The first step of the Lee carter model consists of modeling mortality rates using logm_xt in terms of vectors a and b along the age dimension and T along the time dimension such that

ãlog mã_xt= a_x+b_(x ) T_t

Where a_x , b_(x ) and ã Tã_t are parameters to be estimated .The constraints here are such that the bâs sum up to one and the T’s sum up to zero.

Some extensions on the Lee Carter model includes the Generalized Lee Carter model by Renshaw and Haberman (2006). Here a cohort effect is added to the original model giving us;

logm(x,t)=a_x+b_x k_t+c_x y_(k-t)

This model is also exposed to identification problem hence the following constraints are imposed, the kâs sum up to zero, the b_xâs sum up to one, the c_xâs sum up to one and y_(k-t)âs sum up to zero. Cairns et al 2009

Currie (2006) introduces the Age Period Cohort (APC) model

logm(t,x)=a_x+(1/n_a ) k_t+(1/n_a ) y_(t-x)

Where n_a is the number of ages in the data set. The APC model has its origins in medical statistics. The constraints imposed are such that k_t sums up to zero and y_(t-x) also sums up to zero.

According to Cairns et al (2009), Renshaw and Haberman (RH), Lee Carter and Currie all have the underlying assumptions that qualitatively ,age, period and cohort effect are different in nature and hence the need for them to be modelled in different ways.

The continuous mortality investigation bureau (CMIB) employed the extrapolating formulae derived from past mortality data to make predictions for future mortality improvements. Cairns (2000) identified three uncertainties involved in using this formulae.

Model uncertainty- the appropriate statistical model to be adopted may be unclear

Parameter uncertainty- uncertainty in the parameters to be used in the model

Stochastic uncertainty- the exact outcome is partly dependent on random fluctuations, hence the outcome of these models are deemed stochastic.

2.6 calculating crude rates

There two models used depending on whether the interest lies in the continuous rates, that is force of mortality or discrete rates, that is probability of death.

2.6.1. Poisson model

This model estimates the force of mortality Î¼_(x+1/2) at time x+1/2 the distribution for the number of death Dx at age x.

Dx~Poisson(E_(x ) Î¼_(x+1/2))

With expectation E_(x ) Î¼_(x+1/2) , Î¼_(x+1/2) is the expected number of deaths for age (x,x+1) and E_(x ) is the number at risk for age x. The force of mortality here is computed by u=d/(E_X^C ) where

E_x^c is the central exposed to risk given by E_x^(c )= E_x-1/2 d

2.6.2. Binomial model

The binomial model is an important probability model used where there are two possible outcomes in a discrete data set. In probability of death estimates the major two outcomes are either an observed life dies or stays alive.

This model is used to estimate the probability of death q_x . The number of death d has a distribution of D ~ Binomial (N,q_x). Where N is the number of lives observed .The estimate of probability of death under this model is given by

q_x=d/N

with an unbiased mean of q_x and variance q_x (1-q_x)/N

The assumptions underlying this model are that;

uniform distribution of deaths: this implies an increasing force of mortality

tqx = t.qx (0â¤tâ¤1)

The Balducci assumption: this implies a decreasing force of mortality tween integer ages

1-tqx+t = (1-t) qx ( 0â¤ t â¤1)

constant force of mortality

tqx =1- e-ut ( 0â¤ t â¤1)

2.7 Graduation

Graduation is defined by Haberman and Renshaw (1996), as the methods by which crude probabilities are fitted to provide estimate that are a smooth function of age.

A desirable graduation should have features such as smoothness, adherence to data and should be suitable for the purpose.

Smoothness and adherence to data do conflict with each other. When graduated rates show little adherence to data, yet achieves a smooth curve, this will result in over graduation and the vice versa, where there is more adherence to data but inadequate smoothness is achieved, this will result in undergraduation.

The purpose for which the work is done serves as a guideline to determine certain expectations.

For example in the pension or annuity companies it will be detrimental to overestimate death as it will imply that pension or annuity cost estimations will be based on overestimated death, and Payments may be made for longer periods than expected.

2.7.1 Methods of Graduation

Graduation methods are classified in two, we have the parametric methods and nonparametric methods, and this depends on whether they adjust the data to a function or directly achieves smoothness. DebÂ´on et al 2005.

However the two main approaches to graduation includes graduation by parametric formula, and graduation nonparametric formula.

Graduation by Parametric formula

The main widely used models are the Gompertz(1825) and Makeham(1860) models. These are best fitted with older age groups. The Gompertz assumes force of mortality grows exponentially with age whereas Makeham adds a constant (which could be accidental death which has no relation with age) to the exponential growth of mortality (which is inclined to senescence death.

Over time as mortality among the young and middle age increased it became diï¬cult to use Makeham formula to obtain desirable graduation, DebÂ´on et al pro cit. This led to the introduction of new models known as the Heligman and Pollard laws (Heligman and Pollard, 1980).

The Gompertz and the Makeham models has the following formulae;

Gompertz (1825) Î¼_(x )=Bc^X

Makeham (1860) Î¼_(x )=A+Bc^x

Where Î¼_x is the force of mortality and A, B and c are parameters to be estimated.

Graduation by non-parametric formula

By this methods graduated rates are estimated by smoothing crude rates calculated from the original data. This is best used when some information on data used is unavailable, since the age-dependent function assumption is not a requirement as it is in parametric methods.

According to Planchet (2013), there are two approaches to smooth crude rates under this methods.

The Exogenous Approaches

This is where actual mortality experience is compared with a standard table. This becomes a reference table where which is adjusted to the experience of a given data

The Endogenous Approaches

this involves analyzing information on the crude rates and using that to obtain a smooth set of rates which can be used to make realistic projections. It is best used for large volume of data and can produce biased results when a small data sample is used.

According to Deboân et al(2006) some of the well-known methods includes the kernel smoothing and splines smoothing

Kernel smoothing

This was initially developed to estimate density functions

Ï_x=(k((x-x_i )/b))/(â’_(i=1)^râ’ãk((x-x_i)/b)ã)

Where b is the bandwidth and k is the kernel function.

Splines smoothening

The objective here is to find the estimator q_x by minimizing

w=â’_(i=1)^nâ’ããw_j (q_j-q_j)ã^2+Î´â’^(n-m)â’ã(â^m q_j)ã^2 ã

The smoothening parameter is defined as

âq_j=q_j-q(j-1)

â^m is the difference operator applied m times.

2.8 Desirable properties of theoretical models

There are certain criteria that are expected to be met when modeling, and mortality rates modeling is no exception. These criterion are in other words desirable properties which are expected from a good model. Cairns further explains some of these properties;

Parsimony in this context implies a model with fewer parameters. This is more preferable than a model with numerous parameters. Though more parameters indicates more individual homogenous groups, it makes the model complex and prone to errors and oversights.

Transparency also means that the model should be clear and uncomplicated to avoid inappropriate use. No part of the output or process should be obscured whatsoever.

Thirdly, whether the model has the ability to generate sample paths for underlying and unobservable death rates and whether cohort effects are allowed for, should they be present. Finally the ability of a model to produce a nontrivial correlation structure between the year-on-year changes in mortality rates at different ages.

He further states that none of the three models explained have a clear cut distinctions as to whether they possess the property of parsimony and transparency but clearly all the above models have the ability to generate sample paths. With the exception of LC, the other two models, that is RH and Currie do incorporate cohort effects. The nontrivial correlation structure is obviously not present in all three models.

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