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
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
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
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
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
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
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
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
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
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
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
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
Initial expenses:
Renewal expenses:
Interest:
6% per annum
￡ 250 plus 50% of the gross annual premium
￡ 75 per annum, inflating at 1.92308% per
annum, at the start of the second and subsequent
policy years and 2.5% of the second and
Claims expense:
subsequent monthly premiums
￡300 inflating at 1.92308% 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
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
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
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(K65<k)≤0.05. So P(K65<k):
3
p65=0.9527; so k=4. The premium for the policy if K65=4 is...?
Lecture Notes 10
15
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
If:
Prospective vs Retrospective
●
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
The End*
*until the second half of this subject next year...
Lecture Notes 10
18