Remember..Prospective Reserves
Notation:
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Vx
t
Net Premium Prospective reserve at t for a
whole life assurance
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convention: if we are working at an integer duration, the
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reserve is calculated just before any premium due at that
date and just after any benefit payable in arrear at that
date
Lecture Notes 8
1
Reserves - recursively
It is possible to calculate reserves recursively
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...provided some conditions are met:
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reserves at different times must be calculated on
same basis
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and the basis must be the same as that used for the
net premium
Lecture Notes 8
2
Reserves - recursively
Lecture Notes 8
3
Reserves - recursively
Or, only in terms of qx's:
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Bad taste actuarial joke of the day (you have been warned!)
Why did the goth want to become an actuary?
Lecture Notes 8
4
Mortality Profit
He wanted to get paid to predict death and destruction.
...in that vein, to 'mortality profit'....
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We start out with a policy where certain premiums are
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paid to the life company, and the life company will pay
out certain benefits depending on the survival or death of
the policyholder.
Lecture Notes 8
5
Mortality Profit
If we have sold a large number of pure endowments and
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more people die than we expected (or die sooner than we
expected) when we worked out the premiums – will we
make a profit or a loss?
If we have sold a large number of annuities and more
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people die/die sooner than we expected...?
If we have sold a large number of term assurances ...?
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If we have sold a large number of endowment
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assurances...?
Lecture Notes 8
6
Mortality Profit
Of course we will also make either a profit or a loss
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depending on whether
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our expenses are higher/lower than we thought,
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our returns on assets are different to what we thought,
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our policyholders surrender their policies or lapse
(more on those later)
But for now we'll just focus on mortality as it's the thing
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we have least control over...
Lecture Notes 8
7
Death Strain at Risk
The maximum 'loss' the life office will make if the person
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dies in the next year is the death strain at risk
The death strain (DS) is
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We have already 'saved' – at the end of the coming year
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– t + 1Vx +R to cover the expected future payments from
the policy; so the 'cost' is the payment made on death less
the amount set aside to cover payments under the policy
Lecture Notes 8
8
Expected Death Strain
For a single policy the expected amount of death strain is
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the expected death strain (EDS) which is simply the Death
Strain multiplied by the probability of death
Lecture Notes 8
9
Actual Death Strain
For a single policy the actual amount of death strain is
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the actual death strain (ADS) which is simply the 'realised
value' of the Death Strain
Lecture Notes 8
10
...Mortality Profit
Mortality profit =
Expected Death Strain – Actual Death Strain
Lecture Notes 8
11
...For a portfolio of properties
Total DSAR = ∑ (S-(t+1V+R))
Total EDS = ∑ qx+ t(S-(t+1V+R))
=qx+ t(Total DSAR)
Total ADS = ∑ADS for each policy
=∑(S-(t+1V+R)) where sum is over deaths only
Mortality Profit = Total EDS – Total ADS
Lecture Notes 8
12
...Example
A life insurance company offers special endowment contracts that mature at age
65. Premiums are payable annually in advance on 1 January each year. The sum
assured payable at the end of year of death during the term is one half of the sum
assured that will be paid if the policyholder survives until maturity. Details of these
contracts in force on 31 December 2007 are: Exact age 60; Total sums assured
payable on maturity £12,250,000; Total annual premiums: £440,000
The claims in 2008 were on policies with total sums assured of £200,000 and
annual premiums of £7,000. Calculate the mortality profit or loss in 2008 given
that the company calculates reserves for these contracts using the gross prospective
method using AM92 Ultimate mortality, 4% pa interest and nil expenses.
Previous Actuarial Profession paper – 9 marks (total marks 100, three hour paper)
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Lecture Notes 8
13
Thiele's Differential Equation
Remember it is possible to calculate reserves recursively
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(...provided some conditions are met)
Rearranging a bit:
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Lecture Notes 8
14
Thiele's Differential Equation
If we move to continuous time, instead of a 'difference'
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we will get a differential equation, and we call it Thiele's
Differential Equation.
There are different versions of the equation for different
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policy types. One is given in your yellow books pg 37,
which policy is this for?
Lecture Notes 8
15
Deriving Thiele's Equation:
Write the equation for the prospective (usually) reserve
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at time t and derive w.r.t. t
Generally it is easier to differentiate survival
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probabilities (why?) so rewrite q's and associated life
functions as far as possible in terms of p's and
annuities/pure endowments
rearrange the resulting formula to get to the form we
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want
Lecture Notes 8
16