Question:

a. If f(x) denotes the PDF of the Lognormal(,2) distribution function prove that ∫???(?)????=exp(??+12?2?2)[Φ(Uk)−Φ(Lk)]
where (z) is the cumulative distribution function of the standard normal distribution. Write down the values for ?? and ??. [12 marks]
b. If ?=6 and ?=1 find (i)∫?(?)??∞500 [6 marks] (ii)∫??(?)?? [6 marks]5000
c. An insurer took out an unlimited individual excess of loss cover with retention limit £500. If the claims are assumed to follow a Lognormal distribution with ?=6 and ?=1, find the mean of the claim amount paid by the insurer. [6 marks]
[Total 30 marks]
2. The number of claims each year from a portfolio of insurance policies over n years were X1,X2,…,Xn. The insurer assumes that the annual number of claims have a Binomial distribution with index m and unknown parameter p. The prior information indicates p follows the Beta distribution ?(?)=Γ(?+?)Γ(?)Γ(?)??−1(1−?)?−1
(i) Write down the likelihood for p. [6 marks]
(ii) Find posterior distribution of the parameter p. [11 marks]
(iii) Find the Bayesian estimate of p under quadratic loss. [5 marks]
(iv)Show the Bayesian estimate can be written in the form of a credibility estimate. [8 marks]
[Total 30 marks]
3. An insurance company is reviewing claim amounts arising from a particular portfolio of insurance policies over the past 3 years. The data is split into four different regions of England.

Answer:

Answer1

Part a

 

Here  denotes the PDF of the lognormal () distribution function –

Proof –

Let  has the lognormal distribution with the parameter (), then for

By this we came out with the function –

 

By recalling that if   has the normal distribution with mean  and the standard deviation, then   has the moment generating function given by –

 

Now considering the value of the upper and the lower region of the distribution

 

In the above equation  are the cumulative distribution function of upper and the lower  limits for the normal distribution.

 

For the values of the  & .

Here for the above values we have to consider the cumulative limits of the distribution-

 

 

 

 

Part b

 

• = ?

If   and

 

As the value of, so it is treated as rational number-

Putting, we have

 

= 665

 

• = ?

If   and

 

As the value of, so it is treated as rational number-

Putting, we have

 

= 166285.40

 

Part c

An insurer took the unlimited individual excess of loss cover with the retention limit .

Given:

The basic mean of the claim amount paid by the

 

Where with the probability statistics the value of k = 1.5

+ 500

= 502.310

Answer2

Given –

The number of the claims each year from a portfolio of insurance policies over n years were-

With the annual claims of binomial distribution with the index m and the unknown parameter p follows the Beta distribution as-

 

Part (i)

 

The likelihood for the p can be calculated as-

 

Figure 1: Basic graph for the Beta Distribution

From the above graph, the for the binomial distribution we can take the value of

 

=

=

Now considering the likelihood for  prior probability p =0.75

 

= 1.125

By this the likelihood is found as 3.

 

Part (ii)

 

The posterior distribution of the parameter p can be calculated as-

Posterior distribution = likelihood x prior probability

=

= 0.843

Part (iii)

 

The Bayesian estimate of p under quadratic loss can be estimated as-

 

=

=

Now considering the likelihood for prior probability p =0.5

 

= 0.2

 

Part (IV)

 

The Bayesian estimate can be written in the form of a credibility estimate as-

Here the credibility estimate can be calculated as

Z = 1- Bayesian estimate

= 1- 0.2

= 0.8

 

Answer 3

 

 

The claim amounts for an insurance company arising from a particular portfolio of insurance policies over the past 3 years.

Part (i)

 

The pure premium for all four regions in 2015 by using Empirical Bayes Credibility Theory model 1 (EBCT model 1) can be calculated as-

Figure 2: Table for four different data is split into regions of England

The pure premium (EBCT model 1) is calculated by using the

 Pure premium (EBCT model 1) for South region =

=

= 2775

 

Pure premium (EBCT model 1) for East region =

=

= 2250

 

Pure premium (EBCT model 1) for Midlands region =

=

= 1650

Pure premium (EBCT model 1) for North region =

=

= 2250

 

Part (ii)

 

 

The total number of policies sold for this particular portfolio of insurance policies for the given below portfolio as-

Figure 3: : Table for four different data is split into regions of England

Total policies sold for South region =

=

= 25.5

 

Total policies sold for East region =

=

= 13

Total policies sold for Midlands region =

=

= 16.5

Total policies sold for North region

=

= 22.5

 

References

 

Sclater, N. (2017). Learning Analytics Explained. Routledge.

Laryea, P. N. A. (2016). KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY (Doctoral dissertation, INSTITUTE OF DISTANCE LEARNING, DEPARTMENT OF MATHEMATICS, KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY).

Dickson, D. C. (2013). Risk Modelling in General Insurance, Gray Roger J., Pitts Susan M., Cambridge University Press, 2012, 393 pp.(hardback). ISBN: 9780521863940. Annals of Actuarial Science, 7(02), 345-346.

Hull, P. (2016). Estimating Hospital Quality with Quasi-experimental Data.

Davis, A. M. (2014). Measuring student satisfaction in online math courses.