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MATH39522

CONTINGENCIES 2 - COURSEWORK

Deadline 11.00am 15 March 2024

Answer ALL THREE questions

The total number of marks in the paper is 20.

Please note the following before you get started.

• You may submit handwritten answers or typed - either is fine. Please though ensure that you check your submission is readable and the correct document is submitted before the deadline.

• Please also submit a copy of your R code. This means that you will need to submit the two files within the same attempt - don’t press submit until both are loaded. If you do submit your answers, press submit, then submit your computer code it may overwrite the original submission. Please follow the instructions in the submission box on Blackboard.

• Recall that the work you hand in must be yours and yours alone.

• You can email me if you have any questions about the coursework.

• Context: In this coursework we calculate the premium for a policy with a term of 20 years that provides sickness and death benefits. The amount and type of benefit is varied across the three questions - the first question considers level lump sum benefits, the second question considers lump sum benefits that increase with time and the final question looks at the provision of an income while in the sick state, rather than a lump sum.

1. Consider an insurance policy for a healthy life aged 40 exact which pays lump benefits immediately on becoming sick for the first time or immediately on death. The policy has a term of 20 years.

The lump sum benefit on death is £20,000 and on becoming sick for the first time is £10,000.

Throughout this coursework note that there are three states, healthy (H), sick (S) and dead (D) and use the following assumptions:

• Mortality is AM92 (ultimate).

• The constant force of transition from the healthy state to the sick state between the ages of x and x + 1, where x is an integer, is calculated as follows:

 = 0.02 + 0.002x

(a) Calculate the one year dependent probabilities of death and sickness for a healthy life between the ages of 40 and 60, i.e. for the term of the policy.   [5 marks]

(b) Hence calculate the expected present value of the benefits provided by the policy.   [4 marks]

(c) Assume a premium P is paid annually in advance for the term of the policy, while the life is in the healthy state. Calculate P.   [3 marks]

[Total 12 marks]

2. Now consider the same policy except the benefit payments are different as follows:

• The death benefit is initially £20,000, but this amount is increased at an annual compound rate of interest of 2.5% applied continuously.

• The amount of the lump sum sickness benefit is increased continuously from the start of the policy at age 40 exact though to the point at which the life becomes sick for the first time and the lump sum payment is made. The increasing amount of benefit is calculated as £1,000 × t, where t is the length of time in years and part years between the start of the policy and the point when the life becomes sick for the first time.

Based on these revised benefits, calculate the revised premium for the policy, assuming again that the premium P is paid annually in advance for the term of the policy, while the life is in the healthy state.

[Total 3 marks]

3. Consider the policy in question 1 again. Now the policy is changed so that rather than receiving a lump sum on becoming sick for the first time, a continuous annual payment of £10,000 is made from becoming sick until the life either recovers or dies. For the purposes of this question there is no upper limit placed on the period of payment, so long as the life remains sick.

Consider a distinct model for people who are initially in the sick state. For that model, assume that constant forces of transition apply as follows:

•  = 0.20 for all ages x on transition from sick to healthy,

•  = 0.02 for all ages x on transition from sick to dead.

and that the interest rate remains at 4%.

(a) Calculate the expected present value of the sickness benefit for a life that has just become sick.   [3 marks]

(b) Apply this result to calculate the premium for the revised policy for the healthy life aged 40 exact. Assume again that the premium P is paid annually in advance for the term of the policy, while the life is in the healthy state.   [2 marks]

[Total 5 marks]