Category Archives: Math and Statistics

Extra Credit Opportunity for Finance 4335 (due prior to the beginning of class on Thursday, September 7)

I have decided to offer the following extra credit opportunity for Finance 4335. You can earn extra credit by building an Excel spreadsheet which replicates the Standard Normal Distribution Function table for positive z values ranging from 0.00 to 3.09 (in .01 increments).  Helpful hint – you can obtain cumulative probabilities for all 310 z values (comprising 31 rows and 10 columns) by using the NORMSDIST function that is built into Excel.

This extra credit assignment must be emailed as a file attachment to risk@garven.com prior to the beginning of class on Thursday, September 7; I will not accept this assignment in any other way. I will use your grade on this assignment to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest grade).

The Birthday Paradox: an interesting probability problem involving “statistically independent” events

Following up on my previous blog posting entitled “Statistical Independence,” consider the so-called “Birthday Paradox”. The Birthday Paradox pertains to the probability that in a set of randomly chosen people, some pair of them will have the same birthday. Counter-intuitively, in a group of 23 randomly chosen people, there is slightly more than a 50% probability that some pair of them will both have been born on the same day.

To compute the probability that two people in a group of n people have the same birthday, we disregard variations in the distribution, such as leap years, twins, seasonal or weekday variations, and assume that the 365 possible birthdays are equally likely.[1] Thus, we assume that birth dates are statistically independent events. Consequently, the probability of two randomly chosen people not sharing the same birthday is 364/365. According to the combinatorial equation, the number of unique pairs in a group of n people is n!/2!(n-2)! = n(n-1)/2. Assuming a uniform distribution (i.e., that all dates are equally probable), this means that the probability that no pair in a group of n people shares the same birthday is equal to p(n) = (364/365)^[n(n-1)/2]. The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different. Therefore, its probability is p’(n) = 1 – (364/365)^[n(n-1)/2].

Given the assumptions listed in the previous paragraph, suppose that we are interested in determining how many randomly chosen people are needed in order for there to be a 50% probability that at least two persons share the same birthday. In other words, we are interested in finding the value of n which causes p(n) to equal 0.50. Therefore, 0.50 = (364/365)^[n(n-1)/2]; taking natural logs of both sides and rearranging, we obtain (ln 0.50)/(ln 364/365) = n(n-1)/2. Solving for n, we obtain 505.304 = n(n -1); therefore, n is approximately equal to 23.[2]

The following graph illustrates how the probability that a pair of people share the same birthday varies as the number of people in the sample increases:

New Picture (1)

[1] It is worthwhile noting that real-life birthday distributions are not uniform since not all dates are equally likely. For example, in the northern hemisphere, many children are born in the summer, especially during the months of August and September. In the United States, many children are conceived around the holidays of Christmas and New Year’s Day. Also, because hospitals rarely schedule C-sections and induced labor on the weekend, more Americans are born on Mondays and Tuesdays than on weekends; where many of the people share a birth year (e.g., a class in a school), this creates a tendency toward particular dates. Both of these factors tend to increase the chance of identical birth dates, since a denser subset has more possible pairs (in the extreme case when everyone was born on three days of the week, there would obviously be many identical birthdays!).

[2]Note that since 33 students are enrolled in Finance 4335 this semester, this implies that the probability that two Finance 4335 students share the same birthday is roughly p’(33) = 1 – (364/365)^[33(32)/2] = 76.5%.

Statistical Independence

During yesterday’s Finance 4335 class meeting, I introduced the concept of statistical independence. During tomorrow’s class meeting, 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 2017 is 1%, whereas my probability is 2%. Then the likelihood that we both get into auto accidents during 2017 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

Hello Class,

Overall, well done on the problem set. The majority of you have a very firm grasp of the subject. For those of you that struggled, I would once again suggest studying how to take derivatives and the implications of derivatives very carefully should you wish to do well in the course. Additionally, even if you know the material well, be sure to carefully check your work for any intermediate calculation errors, or for incorrectly reading the question and hence not appropriately answering the question.

Best Wishes,

Alexander

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

The next two class meetings will be devoted to covering various topics in probability and statistics that are important for Finance 4335. On Tuesday, August 29, class will begin with a quiz on the assigned readings (“The New Religion of Risk Management” and “Normal and standard normal distribution“). Furthermore, Problem Set 1 will be due at the beginning of class that day.

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 Tuesday, September 26). Specifically, we will delve into 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 our attention upon variance as our risk measure. Most basic finance models (e.g., portfolio theory and the capital asset pricing model, or CAPM) 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 to which 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 us with a very 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, we’ll move on to other topics including demand for insurance, asymmetric information, portfolio theory, capital market theory, option pricing theory, and corporate risk management.

Calculus and Probability & Statistics recommendations…

Since many of the topics covered in Finance 4335 require a basic knowledge and comfort level with differential calculus and probability & statistics, the first class meeting (August 22) will include a mathematics tutorial, and the second class meeting (August 24) will cover probability & statistics. I know of no better online resource for brushing up on (or learning for the first time) these topics than the Khan Academy.

So here are my suggestions for Khan Academy videos which cover these topics (unless otherwise noted, all sections included in the links which follow are recommended):

Finally, if your algebra is a bit rusty, I would also recommend checking out the Khan Academy’s review of algebra.