HPA 420 IN-CLASS EXERCISE
Answer Guide for Pricing Practice
February 15, 2000
In your local health insurance market, 75 people out of 100 people are low risks with a 10% chance of being hospitalized in a year. The rest are high risks, with a 20% chance of being hospitalized. The average cost of a hospitalization, in either risk group, is $20,000.
(1)What is the actuarial value of a health plan that covers all hospital expenditures after a $1000 deductible?
(A)For a low-risk buyer?
(B)For a high-risk buyer?
There are two ways to think about solving this problem.
(a)Make use of the knowledge that actuarial value is average claims per policy. Calculate total claims and then divide by the number of people to get the average. Here’s the process for low risk buyers…
Total number of hospital admissions: 10 percent of 75 people = 7.5 admissions
Claims per hospitalization: $20,000 received by the hospital, less the deductible paid by the insured = $19,000
Total claims: 7.5 admissions * $19,000 = $142,500
Average claims: $142,500 divided by 75 = $1900
Notice the following shortcut in doing this arithmetic
7.5 * $19,000$19,000
7510
because 7.5/75 is just the admission rate (1/10) that you started with.
(b)Just plug into the formula that I gave you:
Actuarial value = (average utilization)* (provider payment – copay)
For low-risks, the relevant values are
Average utilization:.10
Provider payment: $20,000
Copay: The $1000 deductible that the insured pays.
Actuarial value = (.10)* ($20,000 - $1000) = (.10)*($19,000) = $1900
(2)If the loading fee is 5%, and you are selling an experience-rated hospital plan, what is the premium for
(A)Low risks?
(B)High risks?
Since this is an experience rated plan, you will base the premiums on different actuarial values (average claims) for low and high risks, based on the knowledge that their claims will differ. To calculate the premiums, you just need to “load up” the actuarial values.
Low risks:Premium = (actuarial value) * (1 + load) = ($1900)*(1.05) = $1995
High risks:Premium = (actuarial value) * (1 + load) = ($3800)*(1.05) = $3990
(3)If the loading fee is 5% and you are selling a community-rated plan, what is the premium for
(A)Low risks?
(B)High risks?
Here you have to realize that a community-rated premium is based on the actuarial value (average claims) for everyone in the community. Since everyone’s premium is based on the actuarial value for the community, everyone pays the same premium.
Here, there are three ways to think about solving this problem.
(a)Calculate total claims for the entire community. Then divide by the number of people/policies to get average claims (aka, “actuarial value”). Then add the load.
Total claims for community = Total claims for low risks + total claims for high risks
75 low risks * (.10) * ($19,000) + 25 high risks * (.20) * ($19,000)
$142,500 + $95,000= $237,500
Divide total claims by 100 (number of people):
$237,500 divided by 100 = $2375 (which is ave. claims or actuarial value for the
community.)
Then add the load to get the premium:
Premium = (actuarial value) * (1 + load) = ($2375) * (1.05) = $2493.75
(b)Realize that the average actuarial value for the community will be the average actuarial value over low and high risks.
75 low risks * $1900 + 25 high risks * 3800
100
Then add the load to get the premium:
Premium = (actuarial value) * (1 + load) = ($2375) * (1.05) = $2493.75
(c)Since the cost of a hospitalization is the same for low and high risks, you can also plug into the usual formula for actuarial value—after you calculate the average utilization rate for low and high risks. Note: this would not work if the cost of a hospitalization differed between the two groups—because there would be two different values of “average payment to provider” to plug into the formula below.
75 low risks *.10 + 25 high risks*.2= .125
100
Actuarial value = (.125)* ($20,000 - $1000) = (.125)*($19,000) = $2375
Premium = (actuarial value)*(1 + load) = $23758*(1.05) = $2493.75