Actuarial Joke of the Day

Transcription

1 Actuarial Joke of the Day Two people are flying in a hot air balloon and realize they are lost. They see a man on the ground, so they navigate the balloon to where they can speak to him. They yell to him, "Can you help us - we're lost." The man on the ground replies, "You're in a hot air balloon, about two hundred feet off the ground." One of the people in the balloon replies to the man on the ground, "You must be an actuary. You gave us information that is accurate, but completely useless." The actuary on the ground yells to the people in the balloon, "you must be in marketing." They yell back, "yes, how did you know?" The actuary says," well, you're in the same situation you were in before you talked to me, but now it's my fault." Lecture Notes 10 1

2 Gross Premiums A gross premium is a premium which allows for expenses and also for future expected levels of bonus in a withprofits policy. Lecture Notes 10 2

3 Expense types There are two main types of expense: Overhead expenses those that in the short term do not vary with the amount of business written (so a 'fixed cost') Direct expenses those that vary with the amount of business written (so a 'variable cost') In the long term some overhead expenses will become direct expenses Lecture Notes 10 3

4 Expense types Overhead expenses: printer cost, office rental, actuary's salary, administrator's salary (?), regulatory license Direct expenses: postage for that policy, printing for that policy, commission paid for the policy, administrator's salary (?), performance bonuses, per policy regulatory fee Lecture Notes 10 4

5 How do we pass the expenses on? Overhead expenses are usually added to the premium using an 'average overhead expense per policy' approach so all policies share equally in these Direct expenses are usually allocated to each policy depending on how much we think they will be for that policy. We call this according to their drivers. Drivers are the factors that affect the expense level for each type of expense. Some types of expense will be per policy, other drivers include the sum insured, type of policy, or premium payable Lecture Notes 10 5

6 Expense Investigation An expense investigation is when a life company estimates its expenses (both direct and overhead) and decides how to split them between various types of policies. The company wouldn't split the expenses down to an individual policy, but would work out the expenses to be allocated to certain groups of policies (eg, 30 yr term assurances). Lecture Notes 10 6

7 When are the expenses payable? It makes sense for overhead expenses to be shared among policies each year, every year they are charged as renewal expenses (eg: the company will still need an office 20 years from now if the policy is still going) Direct expenses are due either at the start (initial expenses) or arise regularly through the term of the policy (renewal expenses) or arise at the end of the policy (terminal expenses) Lecture Notes 10 7

8 The nature of the expenses? Most expenses will be linked to inflation some will be linked to earnings It is important that the future increase in expenses is allowed for so that the life company doesn't make a loss on the policy Lecture Notes 10 8

9 Gross Future Loss rand. variable For a whole life non-profit assurance, the gross future loss random variable, assuming initial expenses of I, level annual expenses of e and terminal expenses of f is: where G denotes the gross premium pa If using the equivalence principle, the gross premium G will be calculated so that the above expression is zero. Lecture Notes 10 9

10 Gross Future Loss rand. variable For a whole life with-profit assurance, the gross future loss random variable will need to allow for future bonuses as well if bonuses are at a compound annual rate of b, vesting annually in arrears: Note that for all these types of calculations you need to take care regarding the timing of bonuses and also of whether or not I includes e for the first year Lecture Notes 10 10

11 Gross Future Loss rand. variable Just as for net premiums, benefits &/or premiums (&/or expenses and bonuses now!) can be payable annually, mthly, in advance, in arrear, immediately on death, at the end of the year (or some other time period) of death, deferred, can increase at a compound rate, a simple rate... Lecture Notes 10 11

12 A life insurance company issues a 35-year non profit endowment assurance policy to a life aged 30 exact. Level premiums are payable monthly in advance throughout the term of the policy. The sum assured of 75,000 is payable at maturity or at the end of year of death of the life insured, if earlier. (i) Show that the monthly premium is approximately 74. Basis: Mortality:AM92 Select Interest: 6% per annum Initial expenses: 250 plus 50% of the gross annual premium Renewal expenses: 75 per annum, inflating at % per annum, at the start of the second and subsequent policy years and 2.5% of the second and subsequent monthly premiums Claims expense: 300 inflating at % per annum Inflation: For renewal and claim expenses, the amounts quoted are at outset, and the increases due to inflation start immediately. [7] (ii) The insurance company calculates a surrender values equal to the gross retrospective policy value, assuming the same basis as in (i) above. Calculate the surrender value at the end of the 30th policy year immediately before the premium then due. [7] Note you don't yet know what a surrender value is, but the question tells you it's equal to the gross retrospective policy value (i.e. the gross retrospective premium reserve) Lecture Notes 10 12

13 Gross Premiums without the equivalence principle The equivalence principle sets the expected future loss to zero so it's a 'break even' level of premium But remember life companies want profit We can price so that our gross premium solves, for instance: Lecture Notes 10 13

14 Gross Premiums without the equivalence principle Alternately, we could price so that the probability that the loss is positive is less than z% This is sometimes called a percentile premium Lecture Notes 10 14

15 Example: a 65 year old takes out a whole life assurance policy, payable at the end of the year of death, with sum assured 10,000. Premiums are payable annually in advance. Find the smallest premium such that the probability of a loss on the policy is less than 5%. Assume AM92 & 4% interest Solution: The expected future loss r.v. is: So, if we know K65 = k, we could set a premium, P (k), so that our future loss is zero. The smaller k, the higher P (k). So we want to find the largest k such that P(K 65 <k) So P(K 65 <k): p =0.9527; so k=4. The premium for the policy if K =4 is...? Lecture Notes 10 15

16 Gross Premium Reserves The premium used in a gross premium reserve is the gross premium actually being charged Gross premium reserves always allow explicitly for expenses, but the basis used may be specifically for reserving and not for pricing (working out the premium) There is no set rule for how future bonuses should be allowed for in gross premium reserves (so read the question or state your assumptions clearly in an exam) Lecture Notes 10 16

17 Prospective vs Retrospective If: the retrospective & prospective reserves are calculated on the same basis (interest rate, mortality and expenses), and the basis is the same as the basis used to calculate the original gross premium, and the original gross premium was determined using the equivalence principle then the retrospective & prospective gross reserves will be equal Lecture Notes 10 17

18 The End* *until the second half of this subject next year... Lecture Notes 10 18