1.
(a) You are an actuary returning to your office after lunch in a restaurant. In the restaurant you bumped into a client who asked you for some advice and, on reflection, you are rather uncomfortable about what you said to them. First they asked you to recommend a suitable piece of valuation software. You suggested a programme called “EasyVal” but now you remember there is a cheaper and better programme which has just become available. Unfortunately it has a long complicated name which you always have difficulty remembering. Next they asked you to recommend a suitable inflation assumption for the forthcoming year. They said they thought 3% was a sensible figure, but would like your opinion. You remembered that the research team in your firm had recently published a central estimate of 2.5%. Not wanting to disagree too much with your client, you proposed 2.75% but now you are worried that it was a higher figure than you’re comfortable with.
(i) What mental biases might have influenced you in these two discussions. [2 marks]
(ii) Discuss how you might try to avoid similar mistakes in the future. [4 marks]
(b) It’s now 12 months later and you have just seen the most recent annual inflation rate which was 2.5%. Relieved that 2.5% is exactly the figure you suggested to your client in the restaurant, you send them a quick email pointing this out. What mental bias have you fallen victim to, and how can you try to avoid it in the future? [2 marks]
(c) You work for an insurance company that has hired a psychologist to help them design their marketing material for travel insurance policies. The company sells policies under two brands: a “value” brand, which is a low cost simple product, and a “premium” brand with enhanced coverage, higher maximum claims limits and some other benefits. Both brands provide the same expected profit margin because the value of the benefits is in the same ratio as the premiums, although the value brand has a significantly lower cost. The psychologist explains that there are two ways of framing the results of buying any insurance policy. One focuses on the cost of the policy, while the other focuses on the benefits provided by a good policy. For each of the two brands:
(i) Explain which type of framing is likely to be appropriate. [4 marks]
(ii) Discuss the features that might be given prominence in the marketing material. [4 marks]
(iii) Suggest a possible scenario for a television advert for each brand. [4 marks]
Total: 20 marks
2.
A car hire customer has reserved a car online and paid in advance. When they pick up the car they are offered the chance to purchase an insurance policy which will cover the excess payment of £1000 for which they are liable if there is significant damage to the car. The policy costs £10 and, in making the decision, the customer uses Prospect Theory with a logarithmic value function given by: v(x) = ln(1+x/10) for x ≥ 0 Where x represents the gain or loss on this transaction.
(a) Show that this value function represents risk aversion in the domain of gains. [2 marks]
(b) Explain why −λln(1+|x|/10) is a suitable functional form for the value function in the domain of losses (i.e. for x < 0), and find the range of appropriate values for λ. [4 marks]
(c) The customer estimates the probability of incurring the £1000 charge is p. Calculate the range of values of p for which the customer will purchase the insurance. [6 marks]
(d) Explain how, under Prospect Theory, the customer’s estimate of p might relate to the actual probability if: 1. they think there is a very low probability of damage; and 2. they think there is a high probability of damage. [2 marks]
(e) The car hire company changes the way it sells insurance policies. The policy is automatically included when the car is reserved and the premium added to the advance payment. When the customer collects the car they are given the option of declining the policy and receiving a refund of the premium. Without doing any calculations, explain how this might alter the number of policies sold to customers who use the decision-making framework described above. [6 marks] Total: 20 marks
3.
The effective annual rate of interest iuni of a particular investment (Investment A) can take one of the two values 2% or 3% with equal probability.
(a) If a single investment of 1 is made at time zero, state a condition for the expected return after n years to equal 1.025n . [2 marks] An alternative investment (Investment B) has an effective annual rate of interest ilog such that the variable (1+ilog) is lognormally distributed with parameters µ and σ 2 , as defined in the Formulae and Tables for Actuarial Examinations. The mean and variance of (1+ilog) are equal to the mean and variance of (1+iuni).
(b) Show that: µ = 0.02468, and σ 2 = 0.00002380. [7 marks]
(c) Calculate the mean and variance of the accumulated amount after 10 years of an investment of 1 at time zero in Investment A. [5 marks]
(d) State and explain the mean and variance of the accumulated amount after 10 years of an investment of 1 at time zero in Investment B. [2 marks]
(e) Calculate the probability that the accumulated return on the Investment B is greater than 1.3. [4 marks]
(f) Explain how you could calculate without simulation the probability that the accumulated return on the first investment is greater than 1.3 (do not do any calculations). [4 marks]
Total: 24 marks
4.
An insurance company estimates that individual claim amounts on a particular group of policies are independent and each follows a gamma distribution with parameters α = 0.0048 and λ = 0.012, where claim amounts are measured in multiples of £‘000s. Claims arise as a Poisson process with annual rate κ = 100.
(a) Calculate the mean and standard deviation of the annual aggregate claim amount from these policies. [5 marks]
(b) The company assigns initial surplus capital of 100 to this business and charges premiums with a loading of 40%, payable at the start of the year. Calculate the probability of ruin at the end of the first year by assuming the aggregate annual claim amount follows a Normal distribution. You should ignore expenses. [4 marks]
(c) A new manager worries that the Normal approximation underestimates the probability of ruin and suggests that a Pareto distribution might be a better model. They have calculated that a Pareto distribution with parameters αpar = 3.8293 and λpar = 113.17 has the same mean and standard deviation as the Normal distribution used in Part (a). Explain the manager’s suggestion and recalculate the probability of ruin under the new model. [6 marks]
Total: 15 marks
5.
The table below shows incremental claims paid and premium income on a portfolio of insurance policies:
(a) Assuming an ultimate loss ratio of 94%, calculate the Bornhuetter-Ferguson estimate of outstanding claims. [14 marks]
(b) State the assumptions underlying this estimate. [4 marks]
(c) Discuss the main difference in approach between the Bornhuetter-Ferguson method and the basic chain-ladder method. [3 marks]
Total: 21 marks
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