Statistical Independence

During today’s Finance 4335 class meeting, I introduced the concept of statistical independence. On Thursday, much of our class discussion will focus on the implications of statistical independence for probability distributions such as the binomial and normal distributions which we will rely upon throughout the semester.

Whenever risks are statistically independent of each other, this implies that they are uncorrelated; i.e., random variations in one variable are not meaningfully related to random variations in another. For example, auto accident risks are largely uncorrelated random variables; just because I happen to get into a car accident, this does not make it any more likely that you will suffer a similar fate (that is unless we happen to run into each other!). Another example of statistical independence is a sequence of coin tosses. Just because a coin toss comes up “heads,” this does not make it any more likely that subsequent coin tosses will also come up “heads.”

Computationally, the joint probability that we both get into car accidents or heads comes up on two consecutive tosses of a coin is equal to the product of the two event probabilities. Suppose your probability of getting into an auto accident during the coming year is 1%, whereas my probability is 2%. Then the likelihood that we both get into auto accidents during the coming year is .01 x .02 = .0002, or .02% (1/50th of 1 percent). Similarly, when tossing a “fair” coin, the probability of observing two “heads” in a row is .5 x .5 = 25%. The probability rule which emerges from these examples can be generalized as follows:

Suppose Xi and Xj are uncorrelated random variables with probabilities pi and pj respectively. Then the joint probability that both Xi and Xj occur is equal to pipj.

Problem Set 1 Hint…

Problem Set 1 is due at the beginning of class tomorrow. Here is a hint for solving the 4th question on this problem set.

The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing the total cost of operating a hospital; a sure-fire way to minimize total cost would be to not even have a hospital in the first place. Indeed, if you were to differentiate the total cost function given by C = 4,700,000 + 0.00013X2 with respect to X, this is what the math would tell you.

In part “a” of the 4th question, you are asked to “derive” a formula for the relationship between cost per patient-day and the number of patient days; in other words, what you are interested in determining is what is the most cost-efficient way to scale a hospital facility such that the cost per patient-day is minimized. Once you obtain that equation, then you’ll be able to answer the question concerning optimal hospital size.

Gamma Iota Sigma Interest Meeting: Tuesday, August 28 at 5:30 pm in Foster 226

(Note from Dr. Garven: I am posting this announcement on behalf of Gamma Iota Sigma, which is a professional business fraternity for which I am a faculty advisor.)

You’re invited to an interest meeting for Gamma Iota Sigma (GIS), the international risk management and insurance (RMI) business fraternity.  The purpose of GIS is to promote, encourage, and sustain student interest in insurance, risk management, and actuarial science as professions and to facilitate interaction with the business community.

Members of GIS have the opportunity to:

– meet executives from insurance and risk management companies;
– learn about internship and job opportunities in insurance and other risk management related companies;
– attend and network at industry conferences in places like Chicago, Atlanta, Phoenix and elsewhere;
– meet and enjoy fun and fellowship with students of like interests.

The first meeting of GIS for the 2018-2019 academic year is Tuesday, August 28 at 5:30 pm in Foster 226. Anyone who wants to learn more about risk management and insurance is welcome, regardless of major.

Apple Is a Hedge Fund That Makes Phones

This is a fascinating article in today’s Wall Street Journal about how Apple is, for all intents and purposes, a highly levered hedge fund, thanks to its wholly owned Braeburn Capital subsidiary which accounts for 70% of the book value of Apple’s assets.

Quoting from this article,

“Similar shadow hedge funds abound within S&P 500 industrial companies. Most disclose less information than Apple about their activities… in 2012 these corporations managed a combined portfolio of $1.6 trillion of nonoperating financial assets. Of this amount, almost 40% is held in risky financial assets, such as corporate bonds, mortgage-backed securities, auction-rate securities and equities.”

The (gated) Journal of Finance article upon which this WSJ op-ed is based is available at

Big companies need to disclose more about their investments.

A much more “rigorous” way to calculate 1+1 = 2

One of my Baylor faculty colleagues pointed out an entertaining and somewhat whimsical parody on the use of math in applied economics and finance which first appeared in the Nov.-Dec. 1970 issue of The Journal of Political Economy, entitled “A First Lesson in Econometrics” (at least I found it entertaining :-)).  Anyway, check it out!

File Attachment: JPEMathParody.pdf (30 KB)

Plans for next week’s Finance 4335 class meetings, along with a preview of future topics

Next week in Finance 4335 will be devoted to tutorials on probability and statistics. These tools are critically important in order to evaluate risk and develop appropriate risk management strategies for individuals and firms alike. Next Tuesday’s class meeting will be devoted to introducing discrete and continuous probability distributions, calculating parameters such as expected value, variance, standard deviation, covariance and correlation, and applying these concepts to measuring expected returns and risks for portfolios consisting of risky assets. Next Thursday will provide a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions are showcased.

While I have your attention, let me briefly explain what the main “theme” will initially be in Finance 4335 (up to the first midterm exam, which is scheduled for Thursday, September 27). Starting on Tuesday, September 4, we will begin our discussion of decision theory. Decision theory addresses decision making under risk and uncertainty, and not surprisingly, risk management lies at the very heart of decision theory. Initially, we’ll focus attention on variance as our risk measure. Most basic finance models (e.g., portfolio theory, the capital asset pricing model (CAPM), and option pricing theory) implicitly or explicitly assume that risk = variance. We’ll learn that while this is not necessarily an unreasonable assumption, circumstances can arise where it is not an appropriate assumption. Furthermore, since individuals and firms are typically exposed to multiple sources of risk, we need to take into consideration the portfolio effects of risk. To the extent that risks are not perfectly positively correlated, this implies that risks often “manage” themselves by canceling each other out. Thus the risk of a portfolio is typically less than the sum of the individual risks which comprise the portfolio.

The decision theory provides a particularly useful framework for thinking about concepts such as risk aversion and risk tolerance. The calculus comes in handy by providing an analytic framework for determining how much risk to retain and how much risk to transfer to others. Such decisions occur regularly in daily life, encompassing practical problems such as deciding how to allocate assets in a 401-K or IRA account, determining the extent to which one insures health, life, and property risks, whether to work for a startup or an established business and so forth. There’s also quite a bit of ambiguity when we make decisions without complete information, but this course will at least help you think critically about costs, benefits, and trade-offs related to decision-making whenever you encounter risk and uncertainty.

After the first midterm, the remainder of the semester will be devoted to various other risk management topics, including the demand for insurance, asymmetric information, portfolio theory, capital market theory, option pricing theory, and corporate risk management.

Lagrangian Multipliers

There is a section in the assigned “Optimization” reading tomorrow on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9) may be skipped without loss of continuity.  The primary purpose of this chapter is to re-acquaint students with basic calculus and how to use the calculus to solve so-called optimization problems.  Since the course only requires solving unconstrained optimization problems, there’s no need for Lagrangian multipliers.

Besides reading the articles entitled “Optimization” and “How long does it take to double (triple/quadruple/n-tuple) your money?” in preparation for tomorrow’s meeting of Finance 4335, make sure that you fill out and email the student information form as a file attachment to prior to the beginning of tomorrow’s class.  As I explained during yesterday’s class meeting, this assignment counts as a problem set, and your grade is 100 if you turn this assignment in on time (i.e., sometime prior to tomorrow’s class meeting) and 0 otherwise.

CFA® Information Session on Monday, August 27th @ 4:30-5:00 pm in FOS 143/144

CFA® Information Session

When: Monday, August 27th @ 4:30-5:00 pm

Where: FOS 143/144

Do you want to be an investor or a financial professional?

Do you want to challenge yourself within the field of investment and finance?

Do you want to differentiate yourself in the job market?

The Chartered Financial Analyst (CFA) program is a globally recognized standard for measuring the competence and integrity of financial analysts, and a valued credential by investment firms, banks, and financial institutions around the world.   If you have unanswered questions about the CFA program and want to find out if it is right for you, come discover its advantages from industry professionals with the CFA designation. Topics will include:

  • Who should pursue the CFA designation
  • The CFA charter and your career
  • Who employs CFA charterholders
  • International recognition
  • Requirements for taking the exams and receiving the charter
  • Exam topics and preparation
  • Scholarships

Stop by, enjoy some food, and learn about the CFA designation.  This session is open to all students. If you have questions, please contact

Finance 4335