Math 210:
Theory of Interest
(“Financial Mathematics I”)
Professor Rick Gorvett
374 Altgeld Hall
University of Illinois
at Urbana-Champaign
Fall 2015
Syllabus
• Office Hours: 3-4 pm Tuesdays, 3-4 pm
Wednesdays, or by appointment
• Textbook: Mathematical Interest Theory by
Vaaler and Daniel (2009, 2nd Edition)
• Exam dates: 3 exams, per syllabus
• Grades: Exams, homeworks, other
assignments
Me
• Director of the UIUC Actuarial Science Program
• MBA (University of Chicago)
• Ph.D. in Finance (UIUC)
• FCAS: Fellow of the CAS
• ASA: Associate of the SOA
(I wish)
• CERA: Chartered Enterprise Risk Analyst
• Actuarial corporate / consulting experience
Class Objectives
• Understand the mathematical foundations of
finance
• Learn Exam 2 / FM material
• Appreciate this material in a broad, crossdisciplinary framework
Motivation for the
Theory of Interest
• Borrowing and investing
– There is a “price” associated with such
transactions
– I borrow $1 from you for one day; how
much do you want in return?
– What about borrowing for 1 year? 5 years?
10 years?
– What about borrowing $10? $100? $1,000?
$10,000?
Motivation for the
Theory of Interest (cont.)
Time Horizon
1 yr
$
Amount
5 yrs
10 yrs…
1
10
100
1,000
10,000
Increasing “Charge”
Definitions of “Interest”
• “Interest may be defined as the compensation
that a borrower of capital pays to a lender of
capital for its use.”
– Kellison, The Theory of Interest, p. 1
• Interest is “the time value of money”
– Broverman, Mathematics of Investment and Credit, p. 1
• “Cost of using money.”
– Dictionary of Finance and Investment Terms, p. 203
Historical / Cultural Context
• “And if your brother becomes poor…. you
shall not lend him your money at interest.”
– Old Testament, Leviticus 25:35-37
• “You shall not lend upon interest to your
brother…. To a foreigner you may lend
upon interest, but to your brother you shall
not lend upon interest.”
– Old Testament, Deuteronomy 23: 19-20
Historical / Cultural Context
• “The most hated sort (of wealth-getting), and
with the greatest reason, is usury, which
makes a gain out of money itself, and not
from the natural object of it. For money was
intended to be used in exchange, but not to
increase at interest.”
– Aristotle, Politics Book 1, Ch. 10
• “Usury”: lending money at an unreasonably
high interest rate
Historical / Cultural Context
• “(A group)…. systematically increased its
wealth by usury in defiance of a law passed
by Caesar the Dictator…. First, the Twelve
Tables prohibited anyone from exacting
more than 10% when, previously, the rate
had depended on the caprice of the wealthy.
Subsequently,…. interest was reduced to
half that amount, and finally compound
interest was wholly forbidden.”
– Tacitus, The Annals Book 6, Ch. 16
Historical / Cultural Context
• “…. you may avoid usury by a simple shift
of the intention. ‘It would be downright
usury,’ says he, ‘to take interest from the
borrower, if we should exact it as due in
point of justice; but if only exacted as due
in point of gratitude, it is not usury….’ ”
– Pascal, Provincial Letter VIII
Historical / Cultural Context
• “The lowest ordinary rate of interest
must…. be something more than sufficient
to compensate the occasional losses to
which lending, even with tolerable
prudence, is exposed.”
– Adam Smith, The Wealth of Nations
Historical / Cultural Context
While studying, especially for actuarial exams,
keep in mind a quote:
• “Do you think that you shall enter the
Garden of Bliss without such trials as came
to those who passed away before you?”
– Joseph Campbell, The Power of Myth – quote
from the Quran (Koran),
Data per FRED, St. Louis FRB, for 3-Month T-Bills, Secondary Market
In This Class…
• We tend to refer to the (one) interest rate
– In reality, there are many interest rates
– Varying across several dimensions
• We tend to assume interest rates do not
change
– In reality, interest rates are stochastic
Typical Interest Theory
Problem
• Four quantities
• If you know three, you can determine the
fourth
2
1) Initial value
2) Interest rate
3) Time period
4) Final value
1
4
3
Next Time…
• Begin discussing interest rates
• Accumulation and amount functions
• Simple versus compound interest
• Effective versus nominal interest