Tutorial 1

Question 1

1. Define a stochastic process, including the definition of the state space and time set.

Answer 1

A stochastic process is a collection of random variables indexed by time. The process is denoted {𝑋𝑡: 𝑡 ∈ 𝐽}, where the values in the set 𝐽 are called the time set of the process. The set of values that the random variables 𝑋𝑡 are capable of taking is called the state space, 𝑆, of the process

Question 2

2. Explain the difference between a stochastic and deterministic process.

Answer 2

A deterministic model does not contain any random component and the output is
determined once the fixed set of inputs and the relationships between them have been defined.
A stochastic model allows for the input components to be random in nature. The output is therefore also random in nature, and several independent runs are required for each set of inputs so that statistical theory can be used to help in the study of the results.

Question 3

3. Give an example of a stochastic process with:
   a) A discrete state space and time set;
   b) A discrete state space and continuous time set;
   c) A continuous state space and discrete time set; and
   d) A continuous state space and time set

Answer 3

a) Students on actuarial degree: State space is the four years of the degree, 𝑆 = {1,2,3,4}, time set is discrete as students can only move years in September. Time set is 𝐽 = {0,1,2, … }.
b) Marital status: An insurance company classifies policy holders as single, married, divorced, widowed or dead. Marital status can change at any time. The state space is 𝑆 = {𝑆, 𝑀, 𝐷𝑖, 𝑊, 𝐷𝑒}, and the time set is 𝐽 = 𝑅+.
c) Deaths per annum: If we want to model the number of deaths in the UK per annum, we could use a continuous state space due to the large numbers, 𝑆 = 𝑅+, as we are measuring annually, the time set is discrete – 𝐽 = {0,1,2, … }.
d) Births: If we want to model the number of births in the UK, we could use a continuous state space due to the large numbers, 𝑆 = 𝑅+. As births can occur at any time the time set would be continuous with 𝐽 = 𝑅+.

Question 4

4. Consider the following random walk: A stock price process begins at time 𝑡 = 0 with 𝑋0 =
5. The price can move up or down by 1p each day with probability 𝑝 and 1 − 𝑝
   respectively.
   Derive the following probabilities:

𝑃(𝑋4 = 100, 𝑋7 = 103|𝑋0 = 100)
𝑃(𝑋2 = 98, 𝑋3 = 97|𝑋0 = 100)
𝑃(𝑋2 = 100, 𝑋6 = 102|𝑋0 = 100)
𝑃(𝑋2 = 102, 𝑋5 = 101|𝑋0 = 100)
𝑃(𝑋2 = 100, 𝑋6 = 100|𝑋0 = 100)
𝑃(𝑋4 = 100, 𝑋8 = 100|𝑋0 = 100)

Answer 4

𝑃(𝑋4 = 100, 𝑋7 = 103|𝑋0 = 100)
The sample paths and probabilities are as follows:
100, 101, 102, 101, 100, 101, 102, 103 𝑝5(1 − 𝑝)2
100, 101, 100, 101, 100, 101, 102, 103 𝑝5(1 − 𝑝)2
100, 101, 100, 99, 100, 101, 102, 103 𝑝5(1 − 𝑝)2
100, 99, 100, 101, 100, 101, 102, 103 𝑝5(1 − 𝑝)2
100, 99, 100, 99, 100, 101, 102, 103 𝑝5(1 − 𝑝)2
100, 99, 98, 99, 100, 101, 102, 103 𝑝5(1 − 𝑝)2
This gives a total probability of 6𝑝5(1 − 𝑝)2.
We could also have derived this as follows:
There are six ways of achieving 𝑋4 = 100 given 𝑋0 = 100, each with probability 𝑝2(1 − 𝑝)2.
There is just one way of achieving 𝑋7 = 103 given 𝑋4 = 100, which has probability 𝑝3.
Combining these, we have probability 6𝑝2(1 − 𝑝)2 × 𝑝3 = 6𝑝5(1 − 𝑝)2.

Similarly…
𝑃(𝑋2 = 98, 𝑋3 = 97|𝑋0 = 100) = (1 − 𝑝)3
𝑃(𝑋2 = 100, 𝑋6 = 102|𝑋0 = 100) = 2𝑝(1 − 𝑝) × 4𝑝3(1 − 𝑝) = 8𝑝4(1 − 𝑝)2
𝑃(𝑋2 = 102, 𝑋5 = 101|𝑋0 = 100) = 𝑝2 × 3𝑝(1 − 𝑝)2 = 3𝑝3(1 − 𝑝)2
𝑃(𝑋2 = 100, 𝑋6 = 100|𝑋0 = 100) = 2𝑝(1 − 𝑝) × 6𝑝2(1 − 𝑝)2 = 12𝑝3(1 − 𝑝)3
𝑃(𝑋4 = 100, 𝑋8 = 100|𝑋0 = 100) = 6𝑝2(1 − 𝑝)2 × 6𝑝2(1 − 𝑝)2 = 36𝑝4(1 − 𝑝)4

Question 5

5.
a) State the 5 key elements of the actuarial control cycle.
b) Describe how the actuarial control cycle may be used by an undergraduate student in their university life.

Answer 5

a) Specify the problem
Develop a solution
Monitor experience
Consider the general economic and commercial environment
Act with professionalism and integrity
b) One example may be:
Specify the problem:
A student realises he/she is spending too much money, and will be in breach of their
overdraft limit before the end of the academic year.
Develop a solution:
The student decides to:
1) Get a part time job
2) Approach the bank to discuss the feasibility of increasing the overdraft limit
3) Set up a week by week budget of planned income and expenditure to ensure the overdraft limit is not breached in future
Monitor experience:
The student monitors actual income and expenditure on a weekly basis, and
compares this to the budget.
He/she revises the budget week by week to reflect savings/overspending.
Have regard to the general economic and commercial environment:
Regard is paid to the overdraft charges that may be incurred, and the interest that
may be charged. This includes potential changes to interest rates that may be
charged, given the current economic climate.
Have regard to professionalism and integrity:
Regard is also paid to the desire of the student to avoid borrowing money from
parents and friends.
Other well described examples are acceptable.

Question 6

Exam standard question (CT4 September 2016 Q7)
6.
a) List EIGHT factors which should be considered when deciding whether a model is suitable for a particular purpose. [4]
A colleague has been asked to present a model which might be used to determine the number of new schools required throughout a country over the next 40 years. He forgot all about it until the last minute when he was reading an article in a newspaper about immigration and education which provided some figures to back up the article. Your colleague has the following suggestion for a model:
* Start with the number of children in the education system over the last twenty years (as provided by the country’s central statistical office). Project these forward using a straight line approach.
* Use the number of immigrants predicted to arrive in each of the next five years as given in the newspaper article. Apply to this an estimate of “number and age of children for each immigrant” also provided by the newspaper. Project this forward also using a straight line approach.
* Add the two together to get the total number of children in the education system for the next 40 years.
b) Assess whether this model is suitable with regards to SIX of the factors which you listed in your answer to part (i). [6]

Answer 6

a) Bookwork:
* The objectives of the modelling exercise
* The validity of the model for the purpose to which it is to be put
* The validity of the data to be used
* The validity of the assumptions used
* The possible errors associated with the model OR the fact that the parameters used are not a perfect representation of the real world situation being modelled
* The impact of correlations between the random variables (or input variables) that
“drive” the model
* The extent of correlations between the various results produced from the model
* The current relevance of models written and used in the past
* The credibility of the data input
* The credibility of the results output
* The dangers of spurious accuracy
* The costs of buying or constructing, and of running the model
* Ease of use and availability of suitable staff to use it
* The risk of the model being used incorrectly or with wrong inputs
* The ease with which the model and its results can be communicated
* Compliance with the relevant regulations
* The existence of clear documentation
b) Application
The objectives of the modelling exercise
The validity of the model for the purpose to which it is to be put
The model is not hugely valid as it does not address the number of schools directly, for example by dividing the number of pupils by average school size or considering when existing schools may become obsolete, the presence of competition etc.
The validity of the data to be used
The Central Statistical Office data will be fine, but that gained from the newspaper will be of limited validity as estimates of future migrants arriving may be heavily skewed by the political bias of the newspaper. Estimates of birth rates and migration rates are generally valid data for this exercise.
The validity of the assumptions used
Straight line projection is dubious over 40 years, especially on immigration numbers.
The possible errors associated with the model OR the fact that the parameters used are not a perfect representation of the real world situation being modelled
The total number of school children in 40 years’ time is very susceptible to errors in the parameters, for example the difference between straight line projection following a baby boom will give a rapidly increasing number, whereas if the baby boom is over, the numbers may decline.
The impact of correlations between the random variables that “drive” the model.
It is quite likely that the estimate of new arrivals and the children per household of new arrivals will be biased in the same direction, i.e. both overstated or understated.
The extent of correlations between the various results produced from the model.
If you overestimate the number of children in the education system in, say 5 years’ time, you will most likely overestimate the number of children in, say, 30 years’ time as these latter will be the next generation, the children of those in the system in 5 years’ time.
The current relevance of models written and used in the past
The government/local authorities should have models which are still relevant even if the need parameters adjusting.
The credibility of the data input
The data from the newspaper may be of doubtful credibility. It would be worth
examining them in the light of past trends to see whether they fall within the range of past data.
The credibility of the results output
This model will give a very crude answer which is pretty difficult to have much faith in.
Again, it will be worth examining the output in the light of recently past trends to see whether they mark a break with the past.
The dangers of spurious accuracy
There is no point calculating the number of children to many significant figures when the assumptions are so approximate and the size of individual schools so variable.
The cost of buying or constructing, and of running the model
An advantage of the model is that it is very inexpensive.
Ease of use and availability of suitable staff to use it
The model is very easy to use.
The risk of the model being used incorrectly or with wrong inputs
This is low, as the model is so simple.
The ease with which the model and its results can be communicated
Another advantage of the model is that it is very simple to communicate.
Compliance with the relevant regulations
Regulations are unlikely to be applicable in this case. However changes in legislation
concerning immigration might be an issue.
The existence of clear documentation
It should be easy to produce clear documentation.

Question 7

FIN 2012 – Section 1 Tutorial – In Class Question
FIN2012 May 2020 Q1
A senior actuary asks you to make a small adjustment to an input parameter in an existing model. A colleague had done the original work on the model but is on holiday and so you have been asked to run the update in their place. The senior actuary isn’t anticipating a big change in the results.
When you run the model with the updated parameter, the change in the output is much greater than expected. Suggest some reasons why this could be the case. [6]

Answer 7

Sample solution
* Error in input parameter (previous or updated run)
* Error in other parameter (previous or updated run)
* Model is stochastic, and so more than one run should be done, and the
distribution of results analysed
* Data has been updated since previous run
* Senior actuary’s estimate of change was inaccurate, and the change in output
from the model is actually correct
* Perhaps due to correlation with other parameters
* You aren’t experienced with the model and run it incorrectly
Credit for any sensible point, up to 1.5 marks per point. No credit for pure
regurgitation of bookwork