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Why is the maximum premium that specific formula vs the minimum premium?

  • Thread starter Johnson Adeleke
  • Start date
J

Johnson Adeleke

Member
E[U(a-X)] = U(a-P) vs E[U(a+(Q-Y))] = U(a).
Why is this the maximum premium formula? I suspect it is some sort of semantics because I see that the maximum premium formula contains a loss random variable: X, in it whereas the minimum premium contains a fixed potential loss: Y. Even then, if I try to explain this to myself I notice that there is a difference in 'initial state of wealth'; for instance, in the minimum premium formula the utility begins at initial wealth: U(a) vs the maximum premium, where it is deducted: U(a-P).
 
Hi Johnson

In both formulae, X and Y are random variables representing losses:

- X is the loss faced by the individual seeking insurance
- Y is the loss faced by the insurance company

Although the example on Page 32 of Chapter 2 makes it look like Y is a fixed amount of £500, it is actually a random variable as:

Y = £500 with probability 0.5
Y = £0 with probability 0

In both formulae, P and Q represent premiums:

- Q is the maximum premium that the individual seeking insurance is prepared to pay
- P is the minimum premium that the insurer is prepared to charge

Taking a = £10000 say, we have, for the individual:

E[U(10000 - X)] = U(10000 - P)

which is saying that an individual is indifferent between a certain level of wealth of £10000 less a premium, and a level of wealth of £10000 together with an unknown loss X.

Taking a = £10000 say, we have, for the insurance company:

E[U(10000 + Q - Y)] = U(10000)

which is saying that the insurer is indifferent between a certain level of wealth of £10000, and a level of wealth of £10000 together with an unknown loss Y and a premium Q.

You are right to query that P and Q are being treated differently, appearing on different sides of the equation. Is there perhaps an argument that the insurer would only take on the loss with the premium, ie they have to go together, you wouldn't have one without the other? For example, the insurer:

- either has £10000
- or has £10000, insures the loss and receives the premium

where as the individual

- either has £10000 less the premium
- or has £10000 and faces the loss

Interesting query
Anna
 
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