Anualidades vidas múltiples

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Contingent & Reversionary
•Contingent benefits are payable to a specified person, only if another another person specified person is alive or dead – so they are payable contingent upon

•Reversionary benefits are payable to a specified person, from the moment of death of another specified person

•Usually, we talk of a contingent assurance (so payable on the death of one person, depending onwhether another person is alive or dead); and a reversionary annuity which pays a pension to one person starting from the death of another person (usually a spouse).
Lecture Notes 2

1

Contingent & Reversionary
•Commonly reversionary annuities are found in the pensions world – a pensioner will have a pension of (say) £10,000 pa, and upon his/her life. death, his/her spouse will receive(say) £5,000 pa for the rest of their

•Note here that the reversionary benefit requires that the pensioner is no reversionary benefit payable.

dies before the spouse. If the spouse dies before the pensioner, there

Lecture Notes 2

2

Contingent & Reversionary
•The probability that x dies before y (i.e. first) sometime within the next t years is Note here that we don't care whether ysurvives t years or not, just that x dies both before y and within the next t years

•The probability that x dies after y (i.e. second) but still within the next t years is

Note here that because x has to die after y and within t years; both x and y must be dead in t years' time.
Lecture Notes 2 3

Contingent & Reversionary
•The positioning of the number over the subscripts shows whichlife which deaths must occur in order for that payment to be made. we are concerned about (timing wise). The value shows the order in

•In the above, we care about whether x has died first. So we are take into account whether or not y is alive as well.

essentially worried about the timing of x's death, but we do need to

Lecture Notes 2

4

what can we say about

Lecture Notes 2

5 Contingent & Reversionary
•The probability that x dies before y (i.e. first) sometime within the next t years is

Lecture Notes 2

6

Contingent & Reversionary

Lecture Notes 2

7

Contingent & Reversionary
•The probability that x dies after y sometime within the next t years is

Lecture Notes 2

8

Contingent & Reversionary
•We can also consider more complicatedconditions, for instance that x must die more than m years after y. There is no standard notation for this type of condition. Our technique is to express as an integral and then simplify.
•Example:

•Two lives, (x) and (y), are assumed to be independent with respect to mortality and are each dies more than 10 years after (y). [4]

assumed to be subject to a constant force of mortality of 0.01.Calculate the probability that (x) S 2002 Professional paper, (100 total, 3 hours)

Hence or otherwise calculate the expected present value of £10,000 payable immediately on the death of (x) provided (x) dies more than 10 years after (y), using δ = 5%.
Lecture Notes 2

9

Contingent Assurances
•Let Z denote the present value of a contingent assurance, which pays £1 immediately on the death of(x) if (y) is then alive (note (y) must outlive (x) for this benefit to be payable). The benefit may or may not be payable to (y) – eg if (x) is a parent and (y) a four year old child....

Lecture Notes 2

10

Contingent Assurances

Lecture Notes 2

11

Contingent Assurances
•If we consider a contingent assurance payable at the end of the year of death; we find

•Here we suddenlydevelop the problem of x and y both dying in the of the year. It is more common to work with benefits payable immediately on death.
Lecture Notes 2

same year. The benefit will still be paid, even if y dies before the end

12

Contingent Assurances
•As in Contingencies I,

and similarly for most other types of assurances. •Normally we evaluate the integral, using numerical techniques...
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