Remember..Prospective Reserves

Transcription

1 Remember..Prospective Reserves Notation: t V x Net Premium Prospective reserve at t for a whole life assurance convention: if we are working at an integer duration, the 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

2 Reserves - recursively It is possible to calculate reserves recursively...provided some conditions are met: reserves at different times must be calculated on same basis and the basis must be the same as that used for the net premium Lecture Notes 8 2

3 Reserves - recursively Lecture Notes 8 3

4 Reserves - recursively Or, only in terms of qx 's: Bad taste actuarial joke of the day (you have been warned!) Why did the goth want to become an actuary? Lecture Notes 8 4

5 Mortality Profit He wanted to get paid to predict death and destruction....in that vein, to 'mortality profit'... We start out with a policy where certain premiums are 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

6 Mortality Profit If we have sold a large number of pure endowments and 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 people die/die sooner than we expected...? If we have sold a large number of term assurances...? If we have sold a large number of endowment assurances...? Lecture Notes 8 6

7 Mortality Profit Of course we will also make either a profit or a loss depending on whether our expenses are higher/lower than we thought, our returns on assets are different to what we thought, 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 we have least control over... Lecture Notes 8 7

8 Death Strain at Risk The maximum 'loss' the life office will make if the person dies in the next year is the death strain at risk The death strain (DS) is We have already 'saved' at the end of the coming year t + 1 V x +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

9 Expected Death Strain For a single policy the expected amount of death strain is the expected death strain (EDS) which is simply the Death Strain multiplied by the probability of death Lecture Notes 8 9

10 Actual Death Strain For a single policy the actual amount of death strain is the actual death strain (ADS) which is simply the 'realised value' of the Death Strain Lecture Notes 8 10

11 ...Mortality Profit Mortality profit = Expected Death Strain Actual Death Strain Lecture Notes 8 11

12 ...For a portfolio of properties Total DSAR = (S-( t+1 V+R)) Total EDS = q x+ t (S-( t+1 V+R)) =q x+ t (Total DSAR) Total ADS = ADS for each policy = (S-( t+1 V+R)) where sum is over deaths only Mortality Profit = Total EDS Total ADS Lecture Notes 8 12

13 ...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) Lecture Notes 8 13

14 Thiele's Differential Equation Remember it is possible to calculate reserves recursively (...provided some conditions are met) Rearranging a bit: Lecture Notes 8 14

15 Thiele's Differential Equation If we move to continuous time, instead of a 'difference' we will get a differential equation, and we call it Thiele's Differential Equation. There are different versions of the equation for different policy types. One is given in your yellow books pg 37, which policy is this for? Lecture Notes 8 15

16 Deriving Thiele's Equation: Write the equation for the prospective (usually) reserve at time t and derive w.r.t. t Generally it is easier to differentiate survival 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 want Lecture Notes 8 16