Utilizing the utility function in actuarial calculations
The model of expected utility can explain the very existence of the institution of insurance. In this model, the policyholder is a person not prone to risk and making reasonable decisions. The decisionmaking mechanism in the presence of uncertainty consists in comparing not expected payments arising as a result of decisions, but expected utility of these payments.
The decision maker connects a certain amount of u (w), with the size of his capital w (from welfare, welfare  welfare) usually without even realizing it; function and (x) is called utility function (utility function).
The insured is ready to pay an insurance premium, greater than the mathematical expectation of an insurance loss. The utility theory explains this phenomenon. A potential insured is going to conclude an insurance contract for a certain object with a random amount of damage X when an insured event occurs. The probability of damage is known  p and the initial capital of the insured w.
The policyholder must make a choice from two options (Table 2.4):
 conclude an insurance contract and pay a gross premium for it;
 do not pay the BP and go for the risk of losing the amount of X.
Table 2.4
Policyholder options
Insured Event 
Having an insurance contract 

is 
No 

No 
w  BP 
w 
is 
w  PSU 
wX 
For the insured is better, when its capital w grows. However, the sense of utility of capital is not directly proportional to the value of X. If the policyholder loses a significant portion of the capital, it may be catastrophic for him. Therefore, the insured must carefully weigh the outcomes, in which his capital will fall to a very low level. The undesirability of events leading to large damage is well explained by the low value of the utility function. On the other hand, growing capital is usually perceived as a positive event, but the richer the insured, the less weight is given to his desire to increase capital by a certain fixed amount Aw.
This pattern is explained by and (x)  the utility function, estimating how far different outcomes are for the decision maker.
If the policyholder concludes a full insurance contract, the utility function will look like:
If the policyholder does not enter into an insurance contract, the utility function will be:
If , the policyholder decides whether to conclude an insurance contract, otherwise refuses.
The insurer and the insurer have each its own utility function  and respectively. From the insurer's point of view, if it is necessary to choose one of two losses X x or X , then the individual compares the mathematical expectations and and selects the loss with the greatest expected utility.
The maximum premium that he is willing to pay for insurance loss X, is defined as the decision regarding the BPP1ax of the utility equilibrium equation from the insurer's point of view:
(2.27)
This is the zeroutility bonus (see the 11th principle in paragraph 2.3).
At the equilibrium point, the customer does not care whether he has an insurance contract. Insurance is beneficial for the client if the left side of the equation exceeds the right one.
Fig. 2.11. View of a typical utility function of a nonrisk insured
The insurer has its own utility function U and its capital W, and it can similarly determine the minimum premium of BPM1  П, which the policyholder must make. The equation of utility equilibrium from the position of the insurer:
(2.28)
If the insurer's BPS is larger than BTP1, P of the insurer, then both parties, having chosen a premium between BP and BPM, increase their utility by concluding an insurance contract.
The utility function is the growing function of capital (see Figure 2.11), it should have the following properties.
1. The function and (x) should be nondecreasing:
The first derivative of the utility function is nonnegative. A larger value of capital usually corresponds to a larger value of the utility function, so the utility function must be nondecreasing.
2. The rate of increase and (x) should decrease:
The second derivative of the utility function and (x ) is less than or equal to zero. The welfare gain gives a smaller (in relative expression) increase in utility and, than a reduction in the level of welfare by the same amount (see Figure 2.11). Persons with this utility function are called persons "with diminishing marginal utility."
3. The function is invariant with respect to the linear transformation: and (x) and au (x ) + b equivalents, since the sums are equally ranked.
The decision maker makes a choice between the options containing the uncertainty. From the standpoint of comparing the random losses X y and X 2, the utility function u, (x) is equivalent to the function au 2 (x) + B, and & gt; 0.
Jensen's inequality (Jensen) reads:
If v (.r) is a convex down function, and X is a random variable, then the mathematical expectation:
equality is achieved then, and only if v (.r) is linear on the set where the random variable X, is concentrated or when (X) = 0 (i.e., X = const).
It follows from the Jensen inequality that for the convex upward of the utility function and (x)
Therefore, persons with diminishing marginal utility (and <(x) <0>) are rightly called risk averse , but they prefer a deterministic (nonrandom ) payment of the BP to a random payment X.
The decision made by the insured will depend not only on p, X and BP, but also on the choice of the utility function.
The following utility utility classes are most commonly used:
1) quadratic function:
2) the logarithmic function:
3) exponential function:
4) power function:
The utility function is applicable to the description of rational behavior in simple situations, it is difficult to justify its appearance in certain cases. In practice, the utility function is usually determined by expert judgments, since it is impossible to find a utility function that would satisfactorily describe all the objects of insurance. In general, the question of the choice of the utility function is the most complex and uncertain in utility theory, as noted in many actuarial sources.
You can find out the simplified definition of the premium that the policyholder is willing to pay by asking him what kind of premium the BP is willing to pay to avoid a loss of the size X, that can occur with probability p, i.e. to estimate BPA for the risk X for a given function u (.r).
You can more accurately estimate the BPA based on the risk aversion coefficient of the utility function:
(2.29)
The higher the coefficient of risk aversion, the greater the premium the insured is willing to pay for insurance of the same risk. The maximum premium that the policyholder will be willing to pay is:
(2.30)
(2.31)
EXAMPLE 2.9
The insurer uses the exponential utility function U (x) = ae ~ aA with the parameter a> 0 and has capital W.
a) Find the riskaversion factor for a given utility function.
b) What is the minimum premium of BPU, n for which the insurer agrees to accept the risk of X?
Solution
a) The coefficient of risk aversion for a given utility function is found by formula (2.29):
Thus, for the exponential utility function, the coefficient of risk aversion is equal to the parameter of the utility function, is constant and does not depend on the initial capital W.
b) The minimum premium for which the insurer can accept such a risk, guided by this utility function, is determined from equation (2.28):
Using the properties of mathematical expectation, we get:
Thus, the minimum premium is:
where  the generating function of random moments
of the value of X, the argument of which is a (see Section 3.2.2).
We thus obtained nothing more than an "indicative award"  see paragraph 2.3 of this chapter and the 7th indicative principle of calculating the premium. The indicative premium does not depend on the current capital of the insurer W, and it agrees with what we received in the point a) of the decision  the risk aversion coefficient of the exponential utility function is a constant.
Using the utility function and (x) when choosing the best insurance contract (or optout of insurance in general) is as follows.
Recall that w  the initial capital of the insured (its level of welfare); BP  payment for insurance policy; X  The damage occurred during the validity of the term of the contract (a random variable having in the general case a definite law of distribution of damage with density/(. r)).
Then the principle of equivalence in the case of full protection is expressed by the equality of the values of the utility function for insurance (the left side of the equation) and the rejection of insurance (the right side of the equation) (since losses are a random variable, we can estimate only the mean expectation of the utility function) (see (2.27)):
The insured benefits insurance, if
If we have a partial and partial protection contract (see paragraph 1.7 and table 2.2), the policyholder can also pay part of the losses X. In this case, it is necessary to find the mathematical expectation of the utility function from the balance of his capital:
Thus, having calculated the utility function for various variants of insurance contracts and refusal of insurance in general, it is possible to find out what behavior will be most beneficial to the insured from the point of view of utility theory.
The mathematical expectation of the utility function and (x) for all variants is:
(2.32)
where f (x) is the damage distribution density in the interval from a to b (if the damage is limited, otherwise b = + °°); x & gt; rest is the remainder of its capital but after a year.
First of all, in solving problems, the boundary conditions and (x) are defined:
(the insured was not insured, there were no losses at all);
(the insured was not insured, the losses are maximal).
All values of the utility function must be within the limits:
Disclaimer of insurance corresponds to the utility function
When using partial damage compensation contracts, the utility function is calculated from the rest of the capital w OCT after paying the insurance premium and the loss left on the insurer's own deduction.
Values of the utility function for various options of insurance contracts are determined in accordance with the terms of the contracts  gross premiums and the share of risk liability.
The maximum of the values found utility function and will indicate the most beneficial for the insured contract.
EXAMPLE 2.10
The potential insured has capital w = 500 cu. and uses the utility function and (x) = Inx to evaluate its choice. Possible damage to the object of insurance available to it is distributed evenly in the interval (0; 400) cu. What is more advantageous for him from the point of view of the utility function:
a) refusal of insurance;
b) contract with full protection with an insurance premium of 200 cu.
c) protection contract with a conditional franchise 1 = 50 cu with a premium of 140 cu;
d) contract with an unconditional deductible Z. = 60 cu with a premium of 120 cu;
e) contract of partial protection on the following conditions: full compensation of damage up to 200 cu. and compensation for half the damage after 200 cu. with an insurance premium of 100 USD
Solution
It is necessary to calculate the utility function for various variants of insurance contracts and to refuse insurance in general and thus to find out what behavior will be most beneficial to the insured from the point of view of utility theory.
Define the boundary conditions:
(there will be no losses at all);
(losses are maximum).
All values of the utility function in this task should be in the range
The mathematical expectation of the utility function u (w) is (2.32):
where /(.r) is the damage distribution density in the interval from a to b; daost is the remainder of the capital after payment of the insurance premium under the contract and the losses remaining on the own deduction of the insured.
By the condition of the problem, the damage is distributed uniformly on the interval [0; 400], its probability density function according to the uniform distribution law:
And then the mathematical expectation of the utility function in this task:
a) Waiver of insurance means the payment of any lost loss from its capital, then the mathematical expectation of the utility function:
In the conditions of our task:
Values of the utility function for various options of insurance contracts are determined in accordance with the terms of the contracts  gross premiums and the share of risk liability.
b) In an insurance contract with full protection with an insurance premium of 200 cu. the policyholder will reduce his capital to a gross premium of $ 200, but he will be paid all the losses that have occurred x.
Capital by the end of the period will be: and the value of the utility function:
c) In an insurance contract with a conditional franchise L = 50 у.е. and a premium of 140 cu capital in any case will be reduced by a gross premium of $ 140, and when small losses do not exceed the deductible, by the amount of these losses. If the damage exceeds the deductible, it will be fully reimbursed:
Get the value of the utility function:
d) In an insurance contract with an unconditional franchise L = 60 cu and a premium of 120 cu. capital in any case will decrease by a gross premium of 120 cu. and with the occurrence of small losses not exceeding the deductible by the amount of these losses. If the damage exceeds the deductible, it will be reimbursed after deducting the deductible:
Get the value of the utility function:
e) In the insurance contract with partial protection for full compensation of damage up to 200 cu. and compensation for half the damage after 200 cu. with an insurance premium of 100 cu. the capital in any case will decrease by a gross premium of $ 100, and upon the onset of small losses, exceeding 200 cu, they will be fully compensated. If the damage exceeds 200 cu, it will be reimbursed by half:
Get the value of the utility function:
Now let's compare the results and choose the conditions with the maximum utility function:
Boundary conditions 
The value of the utility function 

There are no losses 

6,215 
Losses max 

4,605 
Contract of insurance 

Optout of insurance 

5,617 
Full Protection 

5,704 
Conditional deductible 

5,877 
Unconditional deductible 

5,782 
Partial Protection 

5,753 
As you can see, a small advantage of the contract with a conditional franchise. The most disadvantageous of those considered is the full protection contract  apparently due to a too high premium. However, any of the proposed contracts are more beneficial for the insured than a waiver of insurance in terms of the utility function.
Answer: according to the theory of utility function, it is more profitable for the insured to make out any of the types of insurance contracts offered by the insurer than to refuse insurance in general. However, the most beneficial is a contract with a conditional franchise.thematic pictures
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