2. Consider two independent risks X1 and X2 distributed as...

2. Consider two independent risks $X_1$ and $X_2$ distributed as follows: $P(X_1 = 90) = 1/3$, $P(X_1 = 150) = 2/3$ and $P(X_2 = 120) = 1/2$, $P(X_2 = 150) = 1/2$. Assume the insurer has wealth 400 and calculates the premium using the Principle of Zero Utility with utility function $U(x) = x - .001 x^2$. (a) Calculate $\pi_{X_1}$, $\pi_{X_2}$, $\pi_{X_1 + X_2}$ (b) Does the Additivity Property hold?

Transcript

00:01 Hello students today we will discuss about this question in this question we are given we need to consider a decision we facing an initial valve that is w is equal to 20 and on certain income that can take the following values that is a1 is to minus 10 with probability of a 1 is equal to 0 .2 a 2 is equal to 0 with the probability of a 2 is equal to 0 .3 a that is equal to 10 with probability of a 3 is equal to 0 .3 and a 4 is equal to 20 with the probability of a 4 is equal to 0 .2.

00:41 Now the decision maker has the risk performance that will be represented by the unity function that is u of w plus a is equal to minus e raise to minus 0 .1 multiplied by w plus a.

00:57 So here in the part a we need to find the risk premium that is equal to question mark and certainary certainty equivalent.

01:10 So here first of all, the expected monetary value that is emv that can be given as that is equals to summation of ai, probability of ai, where ai is the income earned with the probability of a .i.

01:30 So that is equals to we can 0 .2 multiplied by minus 10 plus 0 multiplied by 0 .3 plus 10 multiplied by 0 .3 plus 20 multiplied by 0 .2 is equals to 5.

01:46 Now the utility function for the initial wealth w and the written a that will be u of w plus a is equal to minus e raised to minus 1 multiplied by w plus a plus a.

02:00 The expected utility that can be calculated, eu of w plus a, that is equals to summation of u of w plus ai, probability of ai.

02:16 So substituting the values of w is equal to 20 and a1, a2, and the probability of a1 and a2.

02:25 So that is e of u 20, 20 minus 10, that is equal.

02:31 To u of 20 minus 10 plus 0 .2 plus u of 20 plus 0 sorry here we can write u w plus a 20 plus 0 0 plus 0 plus u of 20 plus 10 multiplied by 0 .3 plus u of 20 plus 20 plus 20 multiplied by 0 .2 so that for here we will get minus 0 .132 now the certainary equivalent of the uncertain income is that the certain income c, so that will be provided the utility as the uncertain income...

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