# CT4 Models Points To Remember

Going through the whole study material even if you are trying to mug up the summary of each chapter could be a tedious job. If all you need is a quick revision of CT4 Models then you are in the right place. This article will also serve you as an Interview Guide for CT4.

Before heading on, I would like to share my personal experience with statistics based actuarial science papers. Guys, never memorize the mathematics used in the paper, your main focus should be to understand the working and formulation of statistical models because the only aim of this paper is to provide the grounding in stochastic processes and survival models along with their application.

So, let’s dive into the pool of important and conceptual questions that make up CT4!

Q: What is a Stochastic Process?
Ans: A stochastic process is a time-dependent random phenomenon, i.e. a collection of time-dependent random variables {Xt : t ϵ J}, one for each time t in the time set J.
Whereas the state space S (The set of possible values of Xt) and time domain J could be discrete or continuous.

Q: What is stationarity? Explain when a stochastic process is said to be:

1. Strictly stationary
2. Weakly stationary

Ans:  In a stochastic process:

1. Strictly Stationary
A process is said to be strictly stationary if all the statistical properties of a process remain constant throughout. Statistically, if the joint distribution of any set of values {X1 , X2 ,….., Xn} is same as the joint distribution of {X1+k , X2+k ,….., Xn+k} where k is the lag between the two sets.
2. Weakly Stationary
A process is said to be weakly stationary if the first two moments do not vary with time, i.e. E(Xt) and V(Xt) are constant and Cov(Xt , Xs) depends only on lag t-s.

Q: What is an increment and when it is said to be independent?
Ans: An increment of a stochastic process is the amount by which its value changes over a period of time.
A stochastic process is said to have independent increment if for all t and u>0, the increment Xt+u – Xt is independent of values of the process up to time t. For example- Random Walk.

Q: Define Markov property?
Ans: If the probabilities for the future values of the process are dependent only on the latest available value, then the process is said to possess Markov property.
And, the stochastic process which possesses a Markov property is called a Markov process.

Q: Explain the White noise process?
A
ns: A discrete-time stochastic process consisting of independent or uncorrelated random variables is known as white noise process.

Q: What is a Markov Chain?
Ans: A Markov process with a discrete state space and discrete time set is known as the Markov chain.

Q: Write Chapman-Kolmogorov equation?

Ans: pij (m,n) = ∑kϵS   pik(m,l) pkj(l,n)

Q: What is the full form of NCD Model? Why is it important?
Ans: NCD Model- No Claim Discount Model
It is run by motor insurance companies. Under this, the company offers discounts on the premium paid by the policyholders. It is an example of a Markov Chain.

For instance, let’s say the levels of discount are: {0%, 25%, 40%}
Where the policyholder’s status is subjected to the following rules:

• All new policyholders start at 0% level.
• If no claim is made, the policyholder moves a level up or remain at the highest level.
• If the claim is made, the policyholder moves a level down or remain at the lowest level.
• The probability of making a claim is p.

This model is designed to discourage the false claims.

Q: Explain random walk?
Ans: The Random walk is an example of a Markov chain. The increments of a random walk are IID (Independent and identically distributed). We can also say, the values move up or down by completely random amounts at each time step.

Additionally, the simple random walk has unit step-sizes of +1 and -1.
In a symmetrical random walk, the probability of moving up or down are equal i.e. ½.

Q: When does the Markov chain is said to be irreducible?
Ans: A Markov chain is said to be irreducible if any state j can be reached from any state i in any number of steps.

Q: Explain the Poisson process in detail?
Ans: Let N(t) be an increasing, integer-valued process starting at 0. Let λ>0. Then N(t) is a Poisson process if any of the four conditions hold:

• N(t) has stationary, independent increments and for each t, N(t) has a Poisson distribution with parameter λt.
• N(t) is a Markov jump process with independent increments and transition probabilities over a short time period h given by:

P[ N(t+h)- N(t)  = 1] = λh + o(h)

P[ N(t+h)- N(t)  = 0] = 1 – λh + o(h)

P[ N(t+h)- N(t)  ≠ 0,1] = o(h)

• The holding times To , T1 ,… of N(t) are independent exponential random variables with parameter λ and To+1+2+……+(n-1) = n.
• N(t) is a Markov jump process with independent increments and transition rates given by:

Q: What is the Markov jump process?
Ans: A Markov jump process is a stochastic process with a continuous time set and discrete state space that satisfies the Markov property.

Q: Differentiate between time-homogeneous and time-inhomogeneous Markov jump processes?
Ans: A Markov jump process is said to be time-homogeneous if the transition probabilities P(X= j/ Xs = i) depends only on the length of the time interval, t – s.

A Markov jump process is said to be time-inhomogeneous if the transition probabilities P(X= j/ Xs = i) does not depend only on the length of the time interval, t – s but, also on the states i and j at time s and t.

Q: Define the term holding time?
Ans: The holding times are inter-event times. In other words, the time spent in a particular state between transitions. For the process given, the ith holding time Ti-1 is the time spent in state i-1 before the transition to state i.

Q: What are Markov jump chains?
Ans: The jump chain of a Markov jump process is the sequence of states that the process enters. The time spent in each state is ignored. The jump chain is a Markov chain in its own right.

Q: How can we model mortality?
Ans: We can model mortality by assuming that future life is a continuous random variable taking values between 0 and some limiting age ω. From this starting point, we calculate probabilities of survival (tpx) and death (tqx) for an individually aged x over a period of t years.

Q: What is censoring and explain its types?

Ans: Censoring in an experiment occurs when there is an incomplete information about a study, participant, duration of lifetime etc.

Types of censoring are:

• RIGHT CENSORING: When the censoring mechanism which cut shorts the observation in progress, then the data is right censored.
• LEFT CENSORING: It is the censoring mechanism which prevent us from knowing when the entry into the state we wish to observe took place.
• INTERVAL CENSORING: If the information received only allow us to say that an event of interest fell within some interval of time, then the data is said to have interval censoring. Both right and left censoring can be seen as a special case of interval censoring.
• TYPE 1 CENSORING: If the censoring times are known in advance then the mechanism is called type one censoring.
• TYPE 2 CENSORING: If the experiment is continued until a pre-decided no. of deaths (or decrement under consideration) have occurred then type 2 censoring is said to be present.
• RANDOM CENSORING: Random censoring arises when the individual leaves the experiment due to some reason other than death(or decrement under consideration) and the time of leaving is not known in advance.
• NON- INFORMATIVE CENSORING: If the censoring mechanism gives no information about the future lifetime of an individual after the experiment is over, then non-informative censoring is present.
• INFORMATIVE CENSORING: If the censoring mechanism gives the information about the future lifetime of an individual after the experiment is over, then informative censoring is present.

Q: What are the two non-parametric based models used to evaluate survival functions? Also underline their assumptions?

Ans: The two non-parametric models used to evaluate survival functions are:

1. KAPLAN-MEIER MODEL ASSUMPTIONS:
– Data must be homogeneous.
– Observations should be independent.
– Data possesses non-informative censoring.
2. NELSON-AALEN METHODASSUMPTIONS:
– Data must be homogeneous.
– Observations should be independent.
– Data possesses non-informative censoring.

Q: What is a covariate?
Ans:
A covariate is any quantity recorded in respect of each life, such as age, sex, type of treatment etc.

Q: How do fully parametric model work?
Ans: Fully parametric model assume a lifetime distribution based on a statistical distribution whose parameters are then estimated.

Q: How do proportional hazard model work?
Ans: In the proportional hazard model the hazard function for ith life, λi (t;zi) may be written as:

λi (t;zi) = λo (t) g(zi)

The baseline hazard λo (t) is a function only of the duration t and g(zi) is a function only of the covariate vector zi.
The hazards of different lives are independent of the baseline hazard and are in the same proportion at all times. This proportion depends on the values of the covariates recorded for each life.

Q: Explain the Cox proportional hazard model. Why is it a proportional hazard model?
Ans: It is a semi-parametric proportional hazard model under which the force of mortality for an individual is given by:

λ(t;zi) = λo(t) exp(βziT)

The force of mortality is proportional to the baseline hazard λo(t).
The Cox model is a proportional hazard model because the hazards of different lives are independent of the baseline hazard and are in the same proportion at all times. This proportion depends on the values of the covariates recorded for each life and the values of the regression parameter β.

Q: What do we assume in the binomial and Poisson model?
Ans: BINOMIAL MODEL– We assume each life as an independent Bernoulli trial with probability qx of dying during the year of age.
We record our observations only as deaths and survivals and lose any information about the actual timing of the deaths.
POISSON MODEL- We assume that the force of mortality is constant between integer ages and the number of deaths has a Poisson distribution with µExc, where µ is the parameter of Poisson distribution and Exc is central exposed to risk.

Q: Compare the two-state, binomial and Poisson models?
Ans: The maximum likelihood estimators of the underlying rate or force of mortality under each model are consistent and unbiased.

The two-state model and Poisson model can easily be extended to allow for multiple decrements, whereas the binomial cannot.

All models perform well when transition intensities are low.

Q: State principle of correspondence?
Ans: It states that the death data and the exposure to risk must be defined consistently, ie the numerator (dx) and denominator (Exc) must correspond.

Q: What is graduation?
Ans: The process of using statistical techniques to improve the estimates provided by the crude rates is called graduation.

Q: What are the aims of graduation?
Ans: The aims of graduation are:

• To produce a smooth set of rates that are suitable for a particular purpose.
• To remove random sampling errors.
• To use the information available from adjacent ages to improve the reliability of the estimates.

Q: What are the three methods of graduation?
Ans: The three methods of graduation are:

• GRADUATION BY PARAMETRIC FORMULA: We assume that mortality can be modeled using a mathematical formula.
• GRADUATION BY REFERENCE TO STANDARD TABLE: We assume that there exists a simple relationship between the observed mortality and an appropriate standard table.
• GRAPHICAL GRADUATION: We draw a curve by hand on a graph of the crude estimates.

Q: What are duplicate policies? How do they affect the investigation? Mention a way to overcome this problem?
Ans: Duplicate policies refer to those similar extra policies taken by the single policyholder. Duplicate policies distort the results of an investigation.

Allowance can be made for the increase in the variance of the number of claims observed due to the existence of duplicate policies.

So, that’s all from my side. The above questions summaries the whole CT4 paper. I hope it helps you! For further help, you can send your queries here.