A few commenters have
said that they like the recondite stuff, so I did quite enjoy, earlier today,
throwing out a blog post talking about the CFTC and the CDS market and hazard
rates without bothering to spell everything out. Maybe I went too far –
but it just so happens that
Kling Bryan Caplan has a blog entry today which allows me to spell out some of the assumptions
I was making. Here he is on bankers:
Borrowers rarely default on their loans. Nevertheless, differences in default
rates have huge effect on rates of return. Suppose, for example, that two
lenders charge 3% interest, but one has a default rate of 1% and the other
has a default rate of 2%. The first lender has twice the rate of return of
the second. After all, when the first guy lends out $100, he gets back .99*$103=$101.97,
while the second only gets back .98*$103=$100.94.
Caplan’s rhetorical point is right, but his mathematics is wrong, unless the
recovery rate on the loans is zero. When a borrower defaults, the bank
doesn’t get back nothing. Usually the borrower has already repaid some principal
and interest, and nearly always the bank can sell the loan to a collections
agency for more cash still. Let’s say the lender with a 2% default rate has
a 50% recovery rate – then his total return is .98*103+.02*50=101.94.
See, I’ve doubled his return at a stroke!
Caplan goes on to talk about education in similar terms:
People often enroll for
a year of school, attend for a while, and then give up. And sometimes they
attend for a full year, but get failing grades. Either way, they spend most
or all of the cost of a year of education (including foregone earnings), with
little or no benefit. It’s analogous to defaulting on a loan – you spend the
resources, but don’t get the return.
The omission matters. If there is a 7% return to successfully completed education,
but a 3.5% default rate, the expected rate of return is 3.5% – half as big
as the naive estimate.
Once again, Caplan is assuming a recovery rate – the value of a non-completed
transaction – of zero, which is ridiculous. Obviously an extra year’s
education has some value. Let’s say that the 7% return to a successfully-completed
four-year degree is comprised of 1% for each year completed, plus an extra 3%
for the piece of paper you get at the end. Then if 3.5% of students "default"
– drop out after one year, say – then the expected rate of return
is 0.965*1.07+0.035*1.01=1.0679, for a 6.8% return.
It’s important not to write off anything completely – either loans or