How would actuarial science refer to the idea that the premium should increase as number of insured

Augustus Acevedo

Augustus Acevedo

Answered question

2022-07-09

How would actuarial science refer to the idea that the premium should increase as number of insured entities decreases.
In other words, what is the technical term for the intuition that I would charge a higher premium per car to insure just one car than to insure 1000 cars, if such intuition is even justified.
As a sketch of my thinking on this, I have defined risk-adjusted premium as the total premium to cover losses divided by the standard deviation of the total loss on the portfolio of policies. I believe both the numerator and the denominator can be deduced from the law of large numbers.

Answer & Explanation

Keely Fernandez

Keely Fernandez

Beginner2022-07-10Added 14 answers

Say each car pays a premium of q and has accident probability p over some horizon. All accidents cause a full loss of the car value (=1). All cars are identical. Accident rates are independent. With n cars insured, let X i be the loss on car i = 1 , 2 , . . . n.You want to charge enough in premium so that your losses are covered with high probability. Say you want to choose q so that P ( n q i = 1 n X i > 0 ) = 0.99. The term i = 1 n X i is a binomial random variable with prob p and n trials, so for this purpose we can replace with a normal random variable with mean n p and standard deviation n p ( 1 p ) . A little algebra shows that the premium q needs to be set to at least p + 2.33 p ( 1 p ) / n . So there you see that the premium you need to charge (per car) goes down with n.

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