Question

lizabeth, the manager of the medical test firm Theranos, worries aboutthe firm being sued for botched results from blood tests. If it isn’t sued, thefirm expects to earn profit of $120, but if it is successfully sued, its profit will beonly $10. Elizabeth believes that the probability of a successful lawsuit is 20%.If fair insurance is available and Elizabeth is risk averse, how much insurancewill she buy? (Hint: Assume that Elizabeth starts with a wealth ofwand shebuys insurance coveringx≤(120−10) = 110 of her loss. Write down herexpected utility as a function ofxand then take a derivative with respect toxto find the optimal insurance. If you want more hints, have a look at the secondinsurance problem solved in the lecture notes.)

Answer 1

The value of variance is '0' at full insurance while expected profit therefore the individual will buy insurance $100

Explanation:

Given that:

Elizabeth, the manager of the medical test firm Theranos, worries about the firm being sued for botched results from blood tests.

Earn Profit -$120

Successfully sued Profit -$10

Probability -20% -0.80

so that above data we shown in below table

Profit Probability

120 0.08

10 0.08

so that expected profit & variance under this situation are

expected profit =(120) (0.08) +(10)(0.80)

= 96+8

=104

expected profit =104

Variance = (120-104)2 (0.08)+(10-104)2 (0.08)

=(256)(0.08)+(8836)(0.08)

= 9116.8

Variance = 9116.8

Further it is given that the individual is risk averse and is offered fair insurance since the probability loss is 20% so the fair insurance simples a premium of 0.08 for each dollar insurance coverage

so it is required to be determine how much insurance individual will buy

A fair insurance implies that the amount of insurance (coverage )bought has no impact on the expected values of individuals expected profit, but a higher insurance implies a lower variance

A lower variance is desired by a risk average individuals

Therefore in this a risk average individual will purchase full insurance because a full insurance would mean that the variance expected profiles will be minimized without any reduction in expected profit.

The individual faces a potential loss of 100 hence a full insurance would be an insurance of 100

since the probability loss is 20% the fair insurance implies a premium of 0.20 for each dollar of insurance

premium of 22 = (0.02*100) for full insurance

so individual buys full insurance , than she faces the following below table

Profit probability

104(120-22) 0.98

104(10-22+100 0.20

The expected profit variance

expected profit (104)(0.98)+ (104) (0.20)

= 101. 92 +20.8

=104

Variance (104-104 )2 (0.98)+(104)4 (0.20)

=(0)2 (0.98)+(104-104)2 (0.20)

(0)2 (0.98) +(0)2 (0.20)

= 0+0

= 0

As shown above the value of variance is '0' at full insurance while expected profit therefore the individual will buy insurance $100