Here is something for you to work on during Spring Break. You do NOT have to turn anything in! :-)

1. Watch the video  https://www.youtube.com/watch?v=cpwSGsb-rTs   and enjoy the connections to M362K.

2. I mentioned a kind of a banking business in class, which gives us the opportunity to put into
practice many of the ideas we have used so far. Let me review the situation, then let you match
this to the concepts we have been developing.

The business is called Lending Club, and it matches investors (who want to lend money to make a
profit) with borrowers (who are willing to pay interest to borrow money). The borrowers request a loan
(of some multiple of $25, not more than $35,000), and multiple borrowers offer to lend them $25 portions
of the total. (Any lender can offer multiple shares to any borrower.) So for example an investor
can invest $5000 this way by offering $25 loans to each of 200 borrowers.

In practice, the borrowers can request either 36- or 60-month terms, and pay back their loans monthly,
but for this model problem let us assume the borrowers repay their loans all at once, 36 months after
they borrow the money in the first place.

In a typical case, the borrow would repay each lender $40 for each $25 borrowed in the first place.
The $15 difference is the interest that the borrower pays the lender for the privilege of borrowing
the money for three years. This works out to about a 17.0% annual interest rate  (  (40/25)^(1/3) - 1 ),
a bit less than credit-card interest rates.

Sadly for the investors, not all borrowers will repay. Some will declare bankruptcy, others will
skip town or otherwise make collection impossible. In practice, the defaulting borrowers stop
making payments only after a few monthly payments have been made successfully, but in this simplified
model there is only one payment to make and so either it is made or it isn't.


a. Suppose you have made loans of $25 each to 200 borrowers, as above. Now assume that 10% of these
loans are not repaid. What is the investor's annual rate of return? 

b. Suppose you are told that there is a 10% chance that any individual borrower will default, but
that you cannot know with any greater certainty which borrower will default and which won't. If you make
200 loans of $25 each, how many defaulters would you expect? What do you expect your annual rate
of return to be? ((Collections/Investment)^(1/3) - 1 ) ? 

c. What is the probability that you will not have any loans default? What is
the probability they will all default? If you had one more default than you 
expected, would you be surprised? How many "extra" defaults would make you say,
"Wow, it seems remarkable that I should have that many people default"?
What is the probability that you have actually lost money after 3 years?

d. Now repeat problems a,b,c under the assumption that you instead lent $50 each to 100 people.
What changes, and why? What does this tell you about how to invest?

e. If each of my borrowers paid the full $40 as desired, I'd end up with $8000. Suppose I believe
it to be fair that I end up with $8000 as my compensation for risking my money. How much should
each person be asked to repay me (after 3 years), taking into account that (probably) 10% of
them will not repay me at all, so the remainder have to make up the difference?


Some borrowers will object that they are being asked to atone for the sins of others, even
though they are "sure" they themselves will not default. They will point to their excellent
credit history and so on. For this reason, Lending Club grades each borrower into one of many
categories, some of which are more or less likely to default. There is no way to be
absolutely sure that a borrower will repay, but we can isolate an "A" group of borrowers
for which we expect only 2.5% to default. Likewise there are groups B,C,D,E with default
rates of 5%, 10%, 15%, and 17.5%.

f. Suppose I make $25 loans to 40 people in each group. If each borrower is asked to repay $40,
what is my expected return from each group? How much should tbe members of each group be asked
to repay so that I can expect each group to repay me $1600? (Obviously less for the "A"
borrowers and more for the "E" borrowers.) For each of the groups, what is the probability
that I will lose money on the loans to that group? If you were the investor, how would
this affect your investment strategy?

g. If I ask my "A" group to pay me $40 per person, I might get $1600. Probably one or more
will default and I'll get less. But complete this sentence: "With at least 95% probability,
I will have only ___ or fewer defaults". Use that to complete this sentence:
"If I ask my A group people to repay me $___, then with at least 95% probability I
will collect at least $1200". What is the expected return from the A group when they
are asked to repay this amount?

h. Repeat exercise  g  with each of the other four groups. That is, set the payback
amounts for each group not to have the same EXPECTED revenue from each group, but
rather to have each group "almost surely" return you $1200. (That way you will
almost surely be making at least a little profit.) Which group then provides
the best expected return?


Feel free to comment about what this says about your investment strategies, fairness
to investors, or your how the probability calculations correlate with your intuition!


As an aside: please don't mess up your finances to the extent you have to repay a
loan at interest rates of 20%/yr or even higher! Your misery will become some investor's
easy cash...