ECON 143 HU Finance Corporations and Society Problem Set

BusGen143/Econ 143/Public Policy 143/Sustain 143/Political Science 127A/International Policy Studies 227
Finance, Corporations and Society
Fall 2023
Professor Anat Admati
Problem Set #1
Due: Sunday, October 8, 2023, 11:59 pm
Note: You can discuss the problems with classmates but you must submit your individual
write-up of the solutions. Please submit your solutions on Canvas.
1. The Importance of Using Spreadsheets Correctly
You want to start Australia’s answer to Silicon Valley’s Tech Giants, an Uber competitor
called Kangaroo. Kangaroo’s proprietary technology™ interfaces with Elon Musk’s
Neuralink to allow customers to get a ride to anywhere they are thinking about within a 15
miles radius. You predict that this technology will cost you AUD (Australian dollar) 5 billion
to develop in immediate investment and a further AUD 1 billion in exactly one year. Starting
a year later (in two years from now), you confidently predict that you will have a cash flow
of AUD 500 million and that over the subsequent 5 years, cash flows will grow by AUD 500
million each year. You assume that there is no risk associated with this opportunity.
Year Cash Flows (AUD billions)
0
-5
1
-1
2
0.5
3
1
4
1.5
5
2
6
2.5
a) Calculate the Net Present Value of Kangaroo (in AUD) if you discount the future at
1.2% a year (compounded annually), using only a calculator and paper and
pen/pencil. Is this a worthwhile (meaning a positive NPV) investment under these
assumptions? What would be your answer if discounting at 5% a year? At 10% a year?
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b) We live in the 21st century, and we should use 21st century tools. In either a Microsoft
Excel Spreadsheet or on a Google Sheet, type the following formula into an empty
Cell:
= NPV(0.1, -5,-1,0.5,1,1.5,2,2.5)
Is there anything strange about your answer? What’s going on? Hint: Try using the
formula
= -5 + NPV(0.1,-1,0.5,1,1.5,2,2.5)
2. Feeling TIPSy
A bond is a type of debt security where the issuer (or seller) of the bond promises to make
fixed payments to the buyer of the bond at pre-determined future dates. Bonds are often issued
as “coupon bonds,” which make periodic payments called coupons and a final payment that
also includes a so-called principal payment at a final maturity date. A zero-coupon bond is a
bond that promises only one final payment and no intermediate payments. U.S. Treasury bills
(also known as T-bills) are zero-coupon bonds issued by the U.S. Treasury that mature (i.e.,
make their only and final payment) in one year or less.
The Consumer Price Index (CPI) is a measure of annual inflation computed by the Bureau of
Labor Statistics (BLS). An index value of 100 corresponds to no inflation, and inflation is
often stated in terms of increases in CPI.
a) Suppose you buy a 1-year treasury bill at the start of year 0. This involves cash flows
of $-1,000 at date 0 (you pay $1,000 today) and $1,050 at date 1, a year later (you
receive $1,050 in one year). That is, the nominal interest rate you receive is 5% per
year, compounded annually. Suppose also that the CPI increases from 100 at date 0 to
104 at date 1 (i.e., annual inflation was 4%). What is the real rate of return you earn
by investing in this treasury?
Treasury Inflation Protected Securities (TIPS) are a special kind of bond that can be used to
hedge (i.e., protect) against inflation risk. The way a TIPS works is as follows. First, there is
a contracted “real” rate of return 𝑟 that the issuer of the security (the US Treasury
Department), commits to paying. Second, there is an “inflation compensation” component.
The inflation compensation component is the percent change in a specific price index that the
bond terms specify, typically the Consumer Price Index (CPI) computed by the Bureau of
Labor Statistics (BLS). The bond payments are adjusted proportionally to the CPI, so the total
nominal return on a 1-year TIPS is (1 + 𝑟)(1 + 𝜋), or, in other words, investing $1 today in
a 1-year TIPS pays off $(1 + 𝑟)(1 + 𝜋) in a year, where 𝜋 is the inflation rate.
b) Suppose you buy today a 1-year (zero coupon) TIPS which costs $1,000 and promises
you $1,030 plus inflation compensation linked to the CPI. Suppose today’s CPI is 100
and in one year it increases to 104 (i.e., the annual inflation rate is 4%). What is the
real interest rate you get on the TIPS? What is the nominal interest rate?
The CPI is computed as the average price change for products within a basket determined by
the BLS. In other words, it demonstrates how much the cost of consumption has changed for
a “representative” consumer whose purchases are similar to the basket used by the BLS.
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c) Suppose your consumption basket is substantially different from the one used to
compute the CPI. For example, suppose that you live in a rural area and have to drive
more than the average American.1 If the cost of your consumption basket increases
more than the CPI-based inflation (because gasoline prices increased relatively more
than other prices and you spend more on gasoline than the representative consumer
reflected in the CPI index), is the real return you get on the TIPS described in Part (b)
higher than, lower than, or equal to the one you found in part b? Explain your answer.
3. Credit Scores for Sale
“Credit-builder” loans are small short-term loans that lenders promote as helping customers
to establish credit histories and also save for the future. Customers typically receive a
relatively small loan — say, $1,000 — and agree to have the money set aside in a special
savings account. The money is locked up while the borrower pays off the loan in quarterly
installments, typically over a year or two. Once the loan is fully repaid, the savings account
is “unlocked” and the borrower is given access to the money, plus any interest earned. The
loan payments are reported to the major credit bureaus, helping to establish a credit history
that can then enable the borrower to qualify for more traditional loans and credit cards.
Typically, customers pay an upfront fee as well as paying interest on the loan. The savings
account or certificate of deposit is held at a bank that’s insured by the Federal Deposit
Insurance Corporation (FDIC), earning minimal interest. If the loan is not paid in full, the
borrower receives a refund of what was paid up to that point.
Credit 4 Kids offers Allison a $1,000 credit builder loan with an APR of 12% to be paid over
one year in equal quarterly payments. The savings account in which the funds will be invested,
according to the terms of the transaction, is fully insured by the FDIC and has an APR of
1.2%, compounded quarterly. The deposit will be locked for one year with no possibility to
withdraw the funds. Assume that such savings accounts are available to everyone. Hint: The
present value of receiving $C each quarter for N years (starting one quarter from now) if you
are earning APR of r compounded quarterly (that is, a quarterly interest rate of 𝑟/4) is
1
]
𝑟 4𝑁
(1 + )
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a) If Allison takes the credit builder loan and makes all her quarterly payments for a year,
she will receive $Y in one year when her savings account is unlocked. Recall that the
savings account earns 1.2% APR compounded quarterly. What is $Y?
𝑃𝑉 =
𝐶
× [1 −
𝑟 ⁄4
b) Suppose the quarterly payment Allison will need to make if she takes the credit builder
loan described above (i.e., $1,000 for one year with 12% APR paid in equal quarterly
installment) is $X. Write a condition that X must satisfy and explain. Then calculate
the value of X.
See “The Missing Data in the Inflation Debate,” Austan Goolsbee, New York Times, December 30, 2021, Link
(Library link)
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c) Assume Allison has a paying job as well as $1,500 in savings that she wants to
maintain for emergencies. Allison is considering an alternative way to build her credit.
Specifically, she was just able to obtain, and will soon activate, a Simplicity credit
card. The card has no annual fee and zero APR for the first year. Allison does not have
a credit history, and she therefore has a small credit limit of $1,000 on the Simplicity
card.
Assume that the payment history for the Simplicity card is reported to credit bureaus
and can help improve Allison’s credit score, and that the credit score update depends
on the amount of the loan and on the payment history (specifically, whether full
payments were made on time), but not on the interest rate on the loan. Assume that
Allison is trying to decide whether she should take the credit builder loan from Credit
4 Kids described above or borrow on the Simplicity card (but she will not do both).
Also assume Allison will not default if she takes either loan. Can you tell which option
Allison should choose and why? Explain.
d) You and some of your friends consider starting a company that you tentatively call
Credit 4 Trees to compete with lenders such as Credit 4 Kids in making credit builder
loans, targeting students at a prestigious private university in North-West California
with a strong affinity for pine trees. You assess that if you have at least 7,000
customers, their fees will cover the ongoing expenses in managing the accounts. You
are now trying to raise the initial funding to start Credit 4 Trees. Assuming you will
always be able to maintain at least 7,600 customers, is there a risk that Credit 4 Trees
will incur losses on the loans it makes to customers? If so, describe a scenario in which
the company will lose. If not, explain why.
4. IID Genomics
Suppose you have $2M and you want to invest it in personal genomics testing companies with
a focus on inherited cancer gene screening. Each company needs $2M to develop their
respective testing kit. With 30% probability, the testing kit is successful and you get $10M;
with probability 70% it is ruined and you get $0. Furthermore, the success of one company
doesn’t depend on success of the other companies.
Option A: Invest in 1 company.
Option B: Borrow an additional $2M and invest in two companies in total. Assume the
interest rate is zero and the bank will lend to you any amount you want. If the final return
is negative, you still must pay (sell your house, borrow from friends and family or any
other means), and assume unless stated otherwise that you will be able to pay.
a) Suppose that you take Option B. What is the probability that both companies are
unsuccessful? What is the probability that both companies are successful? Statistical
hint: for independent events X and Y, the probability of both events occurring is the
multiplication of the probabilities of each event: Prob (X and Y) = Prob(X)×Prob(Y).
b) What is your expected total cash flow from investing in option A? What is your
expected total cash flow if you follow option B?
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c) Suppose you borrowed $2M and took option B. Describe the scenario(s) related to
your investments such that you don’t have enough money to pay the bank from the
investment returns. What is(are) the probability(probabilities) of that(those)
outcome(s)?
d) Now suppose you can borrow $4M and invest in 3 independent companies in total.
What is the expected total cash flow? Describe the event(s) in which you don’t have
enough money to pay the bank from the investment returns alone. What is (are) the
probability (probabilities) of that (those) event(s)?
e) Finally, suppose that if the investment cash flow is lower than the amount you owe to
the bank, then you default and return only remaining cash flows from the investment
(if there are any). Essentially, the lender has no recourse to your other assets. Before
giving you the loan, the bank realizes the loan is risky (there is a chance it will be
repaid less than what it lends). Suppose the bank’s required return for risky loans is
3% (i.e., the bank wants to achieve an expected return of 3%). Follow a similar
approach to the one we took in class to figure out what interest rate the bank will
demand from you when you apply for a loan of $2M and when you apply for a $4M
loan. Is the interest rate different? Explain your intuition. Hint: view the investment
from the bank’s perspective and solve for the cash flows they need to have given the
probabilities of the scenarios in which they are paid given that there is a 70%
probability of total ruin and 30% of success for each investment and the investments
are independent.
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