Question: Given the yearly benefit of q = 12 x 10 000, the interest rate assumption of 4,2 % and mortality assumptions; the guaranteed amount F, can now be calculated through equation (25) (the last...

Question:
Given the yearly benefit of q = 12 x 10 000, the interest rate assumption of 4,2 % and mortality assumptions;


the guaranteed amount F, can now be calculated through equation (25) (the last equation below).

So I need you to calculate F for me if possible.
Remarks:
In the text I have written x-hat(v). this is for the x symbol that have a v on top of it.
You might not need this information but z is 65 and t₀
is 40…

3.2 Stochastic Cash Flow

The cash flow describes the amount and the time for in- and out payments that occurred or will occur in the future. It will be regarded as continuous since that is the most common way in practice. A cash flow is stochastic if it for any time depends on an uncertain, stochastic outcome ?. For instance, you are promised 1 000 SEK in ten years from now given that you are alive at that point. The sample space for this event would be O = {dead, alive} and the payment will only occur if ?= alive. Here focus lie on a cash flow that is a function of a stochastic variable Y = Y(?), ? ?O, on the form


Where ƒ are a deterministic function that is most commonly used in insurance models. A cash flow in longevity insurance depends not only on age but also on a stochastic life span. If we let x be a deterministic cash flow we can get a stochastic cash flow x-bar through

(9)
The indicator function is to be regarded as;

(10)
or in other words, as a function on O. In de following the outcome ? will be left out from the definition. The discounted value of the amount
x-bar(u)du
at age t is given by

(11)
Where the rate r_t
now is allowed to be stochastic. Summarizing all contributions to present value for all future ages
t
gives us; (12)
The expected value given that the individual are alive at time s (ie. given that T>s);

(13)
The interest rate is supposed to be deterministic (ie. the function is per definition constant throughout the sample space O).

3.3 Contract

The typical deterministic cash flow (9) given above can be divided into two parts: x-hat that stand for the period where the insurance are paid and x-hat(v) that is the payoff period. Stating the limit between the two as z and t₀
as the age where the contract is established gives,

(14)
Looking at a pure endowment insurance with regular payments to time z, if regarded as a continuous problem the agreement can be represented as the cash flow presented in equation (9), where

(15)
Here
p
is the premium paid by the insured each year up to age z, and q is the insured’s payoff each year after reaching age z (i.e x-hat(v)(t) = p and x-hat(t) = q).
Repayment protection means that if the insured will die before the given age, T


(16)
where F is the guaranteed amount. Further the deposit stream x-hat(v) have to meet the relation

(17)
The guaranteed amount is thereby equal to the accumulated premiums paid and the amount that are necessary to cover the benefit.
As mentioned before the theoretical, actuarial, price of longevity insurance is determined according to the equivalence principal. For the cash flow x-bar this means that it has to be construted so that the expected value of the present value at the time when the contracts is made is zero. Matemathical this means that

(18)
is valid for t₀, were t₀
is the age of the insured when the contract is made. Looking at the contract presented on equation (9), x has according to (13), satisfy

(19)
Since we have divided x into to x-hat and x-hat(v) , assuming constant rate r we get

(20)
This equation is the pricing relation assumed to be used when pricing longevity insurance of pure endowment typ.
In this type of insurance where the benefit side x-hat = q is considered fixed, the premium has according to the equivalence principal (20) to be determined so that x-hat(v) = p satisfies

(21)
at time t₀. From this we can easily solve p as:

(22)
______________
applies to a contract with repayment benefits. According to the equivalence principal the following must apply;

(23)
which gives,

(24)
Finally the present value of the payment stream x-hat(v) at the time z, be the same as the guaranteed amount F;

(25)