Week 2 Insurance Theory
1. Suppose you think that poorly educated families are less able to smooth consumption
in the absence of unemployment insurance than are well-educated families. How would
you empirically test this supposition? What types of data do you want to use?
(1) Low-middle-high education and unemployment insurance. Through regression analysis
(2) Consumption smoothing; unemployment insurance; pensions, an income you take from
your current self and transfer it to your future-self (saving to smooth consumption)
(3) Select families high education and low education in sample, to minimize the variation of
education levels within family.
2. Your utility function U = ln(2C) where C is the amount of consumption you have in a
given period. Your income is 40,000 euro per year and there is a 2% chance that you
will be involved in a catastrophic accident that will cost you 30,000 euro next year.
a. What is your expected utility without insurance?
Expected utility: E(U) = Ū = p1U(y1) + p2U(y2)
P1 = 0.02 P2 = 0.98
Y1 = ln(2*10.000) Y2 = ln(2*40.000)
E(U) = Ū = 0.02 * ln(2C) + 0.98 * ln(2C)
E(U) = Ū = 0.02 * ln(2*10.000) + 0.98 * ln(2*40.000)
E(U) = Ū = 11,26
Its the weighted average of 2 possible points of utility depending on the likelihood of the 2
occasion.
b. Calculate an actually fair insurance premium. What would your expected utility be
were you to purchase the actuarially fair insurance premium?
Actuarial premium = p1 (Y2 – Y1)
0.02 * (40.000-10.000)
0.02 * (30.000)
= 600
Actuarial fair insurance premium (AFIP): It just covers the costs of a bad event, not making a
profit. Look at the probability of that bad event and the costs of that event.
You lost 30K at the probability 0.02% that you will lose income. He is willing to pay €600 as
a premium for a certain income. You have to deduct the 600 form the 40.000 since that’s the
premium, but in return you have the certainty of having an income equal to 40.000-600 =
39.400€ (total income – premium). <U = log (2* 39.400) = 11.27 > What you see its
beneficial for him to pay insurance.
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,c. What is the most that you would be willing to pay for insurance, given your utility
function?
ln(2*40.000-p) = 11.26
2(40.000-p) = e11.26
40.000-p = (e11.26)/2)
P = 40.000 – ((e11.26/2)
P = 40.000 – 38.906 = 1.094
So utility without insurance should be equal to with insurance.
11.26205603 = log (2C)
C = (exp (11,26)/2) = 38,906 <You do the inverse of LN, so you get e11.26
40.000 – 38.906 = 1.094 < What you pay at maximum to get insured.
3. Chimnesia has two equal-sized groups of people: smokers and nonsmokers. Both
types of people have utility U = ln(C), where C is the amount of consumption people
have in any period. So long as they are healthy, individuals will consume their entire
income of 15,000 euro. If they need medical attention (and have no medical insurance),
they will have to spend 10,000 euro to get healthy again, leaving them with only 5,000 to
consume. Smokers have a 12% chance of requiring major medical attention, while
nonsmokers have a 2% chance.
Insurance companies in Chimnesia can sell two types of policies. The “low deductible”
(L-) policy covers all medical costs above 3,000 euro, while the “high-deductible” (H-
)policy only covers medical costs above 8,000 euro.
a. What is the actuarially fair premium for each type of policy and each group?
P1 = 12% Y1 = 15000
P2 = 2% Y2 = 5000
If either of them gets sick they both got to pay 10.000. The policy insures them both, so you
got to reduce the amount of 10.000 with the policy. There are 2 types or risks with 2 types of
insurances. This is from the perspective of the insurer. The costs are 10.000, but he covers
3000 or 8000. So the difference is 7000 or 2000.
L-policy (covers costs >3000€)
10.000 – 3000 = 7000€
Smokers: 12% * 7000 = 840€
Non-smokers: 2% * 7000 = 140€
H-policy (covers costs >8000€)
10.000 – 8000 = 2000€
Smokers: 12% * 2000 = 240€
Non-smokers: 2% * 2000 = 40€
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, b. If insurance companies can tell who is a smoker and who is a nonsmoker and charge
the actuarially fair premiums for each policy and group, show that both groups will
purchase the L-policy.
In the first part of the formula the smoker or non-smoker uses the insurance so he has to pay
to premiums for either the H-policy or L-policy. In the second part of the formula you put the
risk that nothing happens and you just pay the premium. The formula which gives the highest
U (utility) will be picked. You need to check what gives the highest utility for them to pick a
policy.
U(y1) = ln(C-Mc-p)
U(y2) = ln(C-p)
P1 = 0.12
P2 = 0.88
Utility = p1 * ln(C-Mc-p) + p2 * ln(C-p)
Usmokers(L) = 0,12 * ln(15.000-3000-840) + 0,88 * ln(15.000-840) = 9.530
Usmokers(H) = 0,12 * ln(15.000-8000-240) + 0,88 * ln(15.000-240) = 9.506
Usmokers(N) = 0,12 * ln(15.000-10.000) + 0,88 * ln(15.000) = 9.483
In this option the smokers choose policy L because it gives the highest utility 9.530 > 9.506 >
9.483. Less insurance = less utility.
Unonsmokers(L) = 0,02* ln(15.000-3000-140) + 0,98 * ln(15.000-140) = 9.602
Unonsmokers(H) = 0,02 * ln(15.000-8000-40) + 0,98* ln(15.000-40) = 9.598
Unonsmokers(N) = 0.02 * ln(15.000-10.000) + 0.98 * ln(15.000) = 9.594
In this option the smokers choose policy L because it gives the highest utility 9.602 > 9.598>
9.594
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1. Suppose you think that poorly educated families are less able to smooth consumption
in the absence of unemployment insurance than are well-educated families. How would
you empirically test this supposition? What types of data do you want to use?
(1) Low-middle-high education and unemployment insurance. Through regression analysis
(2) Consumption smoothing; unemployment insurance; pensions, an income you take from
your current self and transfer it to your future-self (saving to smooth consumption)
(3) Select families high education and low education in sample, to minimize the variation of
education levels within family.
2. Your utility function U = ln(2C) where C is the amount of consumption you have in a
given period. Your income is 40,000 euro per year and there is a 2% chance that you
will be involved in a catastrophic accident that will cost you 30,000 euro next year.
a. What is your expected utility without insurance?
Expected utility: E(U) = Ū = p1U(y1) + p2U(y2)
P1 = 0.02 P2 = 0.98
Y1 = ln(2*10.000) Y2 = ln(2*40.000)
E(U) = Ū = 0.02 * ln(2C) + 0.98 * ln(2C)
E(U) = Ū = 0.02 * ln(2*10.000) + 0.98 * ln(2*40.000)
E(U) = Ū = 11,26
Its the weighted average of 2 possible points of utility depending on the likelihood of the 2
occasion.
b. Calculate an actually fair insurance premium. What would your expected utility be
were you to purchase the actuarially fair insurance premium?
Actuarial premium = p1 (Y2 – Y1)
0.02 * (40.000-10.000)
0.02 * (30.000)
= 600
Actuarial fair insurance premium (AFIP): It just covers the costs of a bad event, not making a
profit. Look at the probability of that bad event and the costs of that event.
You lost 30K at the probability 0.02% that you will lose income. He is willing to pay €600 as
a premium for a certain income. You have to deduct the 600 form the 40.000 since that’s the
premium, but in return you have the certainty of having an income equal to 40.000-600 =
39.400€ (total income – premium). <U = log (2* 39.400) = 11.27 > What you see its
beneficial for him to pay insurance.
1
,c. What is the most that you would be willing to pay for insurance, given your utility
function?
ln(2*40.000-p) = 11.26
2(40.000-p) = e11.26
40.000-p = (e11.26)/2)
P = 40.000 – ((e11.26/2)
P = 40.000 – 38.906 = 1.094
So utility without insurance should be equal to with insurance.
11.26205603 = log (2C)
C = (exp (11,26)/2) = 38,906 <You do the inverse of LN, so you get e11.26
40.000 – 38.906 = 1.094 < What you pay at maximum to get insured.
3. Chimnesia has two equal-sized groups of people: smokers and nonsmokers. Both
types of people have utility U = ln(C), where C is the amount of consumption people
have in any period. So long as they are healthy, individuals will consume their entire
income of 15,000 euro. If they need medical attention (and have no medical insurance),
they will have to spend 10,000 euro to get healthy again, leaving them with only 5,000 to
consume. Smokers have a 12% chance of requiring major medical attention, while
nonsmokers have a 2% chance.
Insurance companies in Chimnesia can sell two types of policies. The “low deductible”
(L-) policy covers all medical costs above 3,000 euro, while the “high-deductible” (H-
)policy only covers medical costs above 8,000 euro.
a. What is the actuarially fair premium for each type of policy and each group?
P1 = 12% Y1 = 15000
P2 = 2% Y2 = 5000
If either of them gets sick they both got to pay 10.000. The policy insures them both, so you
got to reduce the amount of 10.000 with the policy. There are 2 types or risks with 2 types of
insurances. This is from the perspective of the insurer. The costs are 10.000, but he covers
3000 or 8000. So the difference is 7000 or 2000.
L-policy (covers costs >3000€)
10.000 – 3000 = 7000€
Smokers: 12% * 7000 = 840€
Non-smokers: 2% * 7000 = 140€
H-policy (covers costs >8000€)
10.000 – 8000 = 2000€
Smokers: 12% * 2000 = 240€
Non-smokers: 2% * 2000 = 40€
2
, b. If insurance companies can tell who is a smoker and who is a nonsmoker and charge
the actuarially fair premiums for each policy and group, show that both groups will
purchase the L-policy.
In the first part of the formula the smoker or non-smoker uses the insurance so he has to pay
to premiums for either the H-policy or L-policy. In the second part of the formula you put the
risk that nothing happens and you just pay the premium. The formula which gives the highest
U (utility) will be picked. You need to check what gives the highest utility for them to pick a
policy.
U(y1) = ln(C-Mc-p)
U(y2) = ln(C-p)
P1 = 0.12
P2 = 0.88
Utility = p1 * ln(C-Mc-p) + p2 * ln(C-p)
Usmokers(L) = 0,12 * ln(15.000-3000-840) + 0,88 * ln(15.000-840) = 9.530
Usmokers(H) = 0,12 * ln(15.000-8000-240) + 0,88 * ln(15.000-240) = 9.506
Usmokers(N) = 0,12 * ln(15.000-10.000) + 0,88 * ln(15.000) = 9.483
In this option the smokers choose policy L because it gives the highest utility 9.530 > 9.506 >
9.483. Less insurance = less utility.
Unonsmokers(L) = 0,02* ln(15.000-3000-140) + 0,98 * ln(15.000-140) = 9.602
Unonsmokers(H) = 0,02 * ln(15.000-8000-40) + 0,98* ln(15.000-40) = 9.598
Unonsmokers(N) = 0.02 * ln(15.000-10.000) + 0.98 * ln(15.000) = 9.594
In this option the smokers choose policy L because it gives the highest utility 9.602 > 9.598>
9.594
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