usufruct

i called a number to activate my new credit card. the woman was quite chatty, though with arrière pensée. i was staying on the phone waiting for her to say "your account is activated," but she kept telling me about great new features of the card that i might have been ignorant of.

she mentioned a program whose name sounded like CREDIT PROTECT PLUS, but then moved onto another matter without giving any details. looking at my credit history, she wondered whether i would like to transfer some of the balance of a student loan that i held towards my new card, with its low introductory APR. the student loan allows me to forbear or defer with little hassle, so it is good that it stays where it is.

she mentioned CREDIT PROTECT PLUS again, something of which i would enjoy a free trial, which i would be enrolled in shortly, but then went to the next offer. i could use the card to add some money to my checking account, so that i'd "have a little extra spending money." unless this "spending money" was for drugs or to pay my rent, it wouldn't make sense to do that when i could simply charge expenditures to my new card with its attractive introductory APR.

finally, we come to CREDIT PROTECT PLUS. she said that i would be receiving all of the details in the mail shortly and that my free trial would start right away. if i canceled after 30 days then i would not be charged. the program is a type of insurance: if you go through some important life event, such as a college graduation, marriage, birth of a child, death in the family, hurricane, the policy will activate and it will pay your minimum balance for you for up to six months. the cost of the premium would be $0.98 per billing cycle for every $100 of balance on my account.

a 0.98% interest rate compounded monthly is equivalent to an annualized rate of 12.4%. so if i accept this service, then my APR will effectively be 14.7% during the introductory period and 30.7% afterwards. no shit.


...


now let's calculate the usurer's utility for a customer who agrees to this. we will imagine two types of customers

1) those who pay only the minimum required payment each month, which is equal to the new interest accrued plus 1% of the balance.
2) those who pay their balance in full each month.

the usurer's profit (the interest paid) for a month where the policy is not claimed, (let's say a 30 day month), and b is the balance, is

P(b) = 0.0231b for customer type (1) (to wit, $2.31 per $100 of balance)
P(b) = 0.0098b for customer type (2) (to wit, $0.98 per $100 of balance)

in a month where the policy is claimed, the usurer's profit is

P(b) = -0.0002b for both types of customers (to wit, -$0.02 per $100 of balance)

now we introduce the probability that a given customer successfully makes a claim on the policy, p(1) and p(2) for customer type (1) and (2), respectively.

the expected value of the usurer's profit for a given month is

P(b,p) = b[0.0231(1-p(1)) - 0.0002p(1)) for customer type (1)
P(b,p) = b[0.0098(1-p(2)) - 0.0002p(2)) for customer type (2)

now let's determine the necessary value of p such that the usurer's monthly profit exceeds what it would have been in the event that the customer had not elected to receive the policy,

(to wit,

P'(b) = 0.0133b for customer type (1)
P'(b) = 0 for customer type (2) )

i.e.

P(b,p) > P'(b)

the solutions are

p(1) < 0.42
p(2) < 0.98

this means that the program is of positive utility to the usurer if customers of type (1) are less than 42% likely, in a given month, to successfully claim against the policy, and if customers of type (2) are less than 98% likely to do the same.