2. (38 pts) The probability distribution for the Ludo Corporation's pre-tax income is given by $100 with prob. 1/3 $70 with prob. 1/3 $20 with prob. 1/3 There is a progressive tax code such that the tax rate is 10% for the first $40 of income and 40% for all income above $40. (a) (3 pts) Calculate the Ludo Corporation's expected pre-tax income. (b) (6 pts) Calculate the Ludo Corporation's expected after-tax income. (c) (20 pts) An insurance company offers the Ludo Corporation the option to purchase one of the two following products: "Small Loss Insurance": Will pay up to $30 in indemnity to cover all losses in pre-tax income which reduce pre-tax income from $100. (So if Ludo Corporation's pre-tax income is $70 or $20, this policy would pay an indemnity of $30.) "Large Loss Insurance": Will pay up to $80 in indemnity to cover all losses in pre-tax income which reduce pre-tax income from $100. (So if Ludo Corporation's pre-tax income is $70, this policy would pay an indemnity of $30. If Ludo Corporation's pre-tax income is $20, this policy would pay an indemnity of $80.) i. (4 pts) What is the actuarially fair premium price Ps (in dollars) for Small Loss Insurance? ii. (4 pts) What is the actuarially fair premium price PL (in dollars) for Large Loss Insurance? iii. (12 pts) Suppose that Small Loss Insurance is available at the actuarially fair premium price Ps which you found in part (i), but that Large Loss Insurance is only available at a loaded premium price of PL +3, where PL is the actuarially fair price you found in part (ii). Given these prices, which of the insurance policies (if any) should Ludo Corporation purchase in order to maximize its expected after-tax income? (Assume that taxes are collected based on income levels after all premium and indemnity payments have been made.) Show your work and explain your reasoning.